THEORETICAL MODELING OF ULTRAFAST OPTICAL - FIELD INDUCED PHOTOELECTRON EMISSION FROM BIASED METAL SURFACE S By Yi Luo A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Electrical Engineering Doctor of Philosophy 2021 ABSTRACT THEORETICAL MODELING OF ULTRAFAST OPTICAL - FIELD INDUCED PHOTOELECTRON EMISSION FROM BIASED METAL SURFACES By Yi Luo Laser - induced electron emission from nanostructures offers a platform to coherently control electron dynamics in ultrashort spatiotemporal scales, making it important to both fundamental research and a broad range of applications , such as to ultrafast elec tron microscopy, diffraction, attosecond electronics , strong - field nano - optics , tabletop particle accelerators, free electron laser s , and novel nanoscale vacuum devices. T his thesis analytically studies nonlinear ultrafast photoelectron emission from biased metal surface s, b y solving the time - dependent Schrödinger equation exactly . Our study provides better understanding of the ultrafast control of electrons and offer s useful guidance for the future design of ultra fast nanoelectronics. First, we present an analytical model for photoemission driven by two - color laser fields. We study the electron energy spectra and emission current modulation under various laser intensities, frequencies, and relative phase between the two lasers. We find strong modulation for both the energy spectra and emission current (with a modulation dept h up to 99%) due to the interference effect of the two - color lasers. Using the same input parameter, our theoretical prediction for the photoemission current modulation depth (93.9%) is almost identical to the experimental measurement (94%). Next , to inve stigate the role of dc field, we construct an analytical model for two - color laser induced photoemission from dc biased metal surface s . We systematically examine the combined effects of a dc electric field and two - color laser fields. We find the strong mod ulation in two - color photoemission persists even with a strong dc electric field. In addition, the dc field opens up more tunneling emission channels and thus increases the total emission current . Application of our model to time - resolved photoelectron spe ctroscopy is also demonstrated, showing the dynamics of the n - photon excited states depends strongly on the applied dc field. We then propose to utilize two lasers of the same frequency to achieve the interference modulation of photoemission by their relative phase. This is motivated by the eas ier access to single - frequency laser pairs than two - color lasers in experiment s. W e find a strong current modulation (> 90%) can be achieved with a moderate ratio of the laser fields (< 0.4) even under a strong d c bias. Our study demonstrates the capability of measuring the time - resolved photoelectron energy spectra using single - frequency laser pairs. W e further extend our exact analytic model to photoelectron emission induced by few - cycle laser pulses. The single formulation is valid from photon - driven electron emission in low intensity optical fields to field - driven emission in high intensity optical fields , and is valid for arbitrary pulse length from sub - cycle to CW excitation, and for arbitrary pulse repetition rate. We find the emitted charge per pulse oscillatorily increases with pulse repetition rate, due to varying coherent interaction of neighboring laser pulses. For a well - separated single pulse, our results recover the experimentally obser ved vanishing carrier - envelope phase sensitivity in the optical - field regime. We also find that applying a large dc field to the photoemitter is able to greatly enhance the photoemission current and in the meantime substantially shorten the current pulse . Finally, we construct analytical models for nonlinear photoelectron emission in a nanoscale metal - vacuum - metal gap . Our results reveal the energy redistribution of photoelectrons across the two interfaces between the gap and the metals. Additionally, we f ind that decreasing the gap distance tends to extend the multiphoton regime to higher laser intensity. The effect of dc bias is also studied in detail. Copyright by YI LUO 2021 v ACKNOWLEDGMENTS First, I would like to express my appreciation to my advisor, Prof. Peng Zhang. During my four - year Ph . D . study at Michigan State University, Prof. Zhang provided me with continuous encouragement and support. His expertise and generous guidance helped me achieve my professional and personal growth during this journey. Secondly, I wish to thank my thesis committee member s : Prof. Qi H ua Fan, Prof. Chong - Y u Ruan and Prof. John Albretch , and I also wish to thank Prof. John Verboncoeur, Dr. John Luginsland and Prof. Sergey Baryshev. Their insightful comments make my research work greatly improve d . Thirdly, I would like to express my thank to my colleagues : Dr. Yangyang Fu, Dr. Deqi Wen, Dr. Patrick Wong, Dr. Sneha Banerjee, Dr. Asif Iqbal, Janez Krek and Yang Zhou, and friends who I met , for their help, support, and contribution. I also would like to acknowledge the support from the funding agencies that made this work possible: the Air Force Office of Scientific Research ( YIP Grant No. FA9550 - 18 - 1 - 0061 ) , and the Office of Naval Research ( YIP Grant No. N00014 - 20 - 1 - 2681 ) . Finally, and most importantly, I would like to thank my parents for always loving, understanding, and encourag ing me in pursuit of my dream and career aspiration. vi T ABLE OF C ONTENTS LIST OF TABLES ................................ ................................ ................................ ..................... viii LIST OF FIGURES ................................ ................................ ................................ ..................... ix CHAPTER 1 INTRODUCTION ................................ ................................ ................................ 1 1.1 Background ................................ ................................ ................................ ........................... 1 1.2 Photoelectron Emission Mechanisms ................................ ................................ .................... 2 1.2.1 Multiphoton Over - Barrier Emission ................................ ................................ ............... 2 1.2.2 Tunneling Emission ................................ ................................ ................................ ........ 3 1.2.3 Keldysh Parameter ................................ ................................ ................................ .......... 7 1.3 Theoretical Models for Photoemission from Metal Surfaces ................................ ................ 8 1.3.1 Three - Step Model ................................ ................................ ................................ ........... 8 1.3.2 Fowler - Nordheim Equation ................................ ................................ .......................... 10 1.3.3 Quantum Analytical Model ................................ ................................ .......................... 11 1.4 Organization of This Thesis ................................ ................................ ................................ 14 CHAPTER 2 TWO - COLOR LASER INDUCED PHOTOEMISSION ................................ 16 2.1 Introduction ................................ ................................ ................................ ......................... 16 2.2 Photoemission Without DC Bias ................................ ................................ ......................... 17 2.2.1 Analytical Model ................................ ................................ ................................ .......... 17 2.2.2 Results and Discussion ................................ ................................ ................................ . 21 2.2.3 Summary on Photoemission without DC Bias ................................ ............................. 28 2.3 Photoemission with DC bias ................................ ................................ ............................... 29 2.3.1 Analytical model ................................ ................................ ................................ .......... 29 2.3.2 Results and Discussion ................................ ................................ ................................ . 33 2.3.3 A pp lication to Time - Resolved Photoelectron Spectroscopy ................................ ........ 44 2.3.4 Summary on Photoemission with DC Bias ................................ ................................ .. 47 2.5 Conclusion ................................ ................................ ................................ ........................... 48 CHAPTER 3 PHOTEMISSION MODULATION BY TWO LASERS OF THE SAME - FREQUENCY ................................ ................................ ................................ ............................ 50 3.1 Introduction ................................ ................................ ................................ ......................... 50 3.2 Analytical model ................................ ................................ ................................ ................. 51 3.3 Results and Discussion ................................ ................................ ................................ ........ 53 3.4 Conclusion ................................ ................................ ................................ ........................... 58 CHAPTER 4 FEW - CYCLE LASER PULSES INDUCED PHOTOEMISSION ................ 59 4.1 Introduction ................................ ................................ ................................ ......................... 59 4.2 Analytical Formulation ................................ ................................ ................................ ....... 60 4.3 Results and Discussion ................................ ................................ ................................ ........ 65 4.4 Conclusion ................................ ................................ ................................ ........................... 72 vii CHAPTER 5 PHOTOELECTRON EMISSION IN A NANOSCALE GAP ....................... 74 5.1 Introduction ................................ ................................ ................................ ......................... 74 5.2 Photoelectron Transport without DC Bias ................................ ................................ .......... 75 5.2.1 Analytical Model ................................ ................................ ................................ .......... 75 5.2.2 Results and Discussion ................................ ................................ ................................ . 79 5.2.3 Summary on Photoelectron Transport without DC Bias ................................ .............. 84 5.3 Photoelectron Transport with DC Bias ................................ ................................ ............... 85 5.3.1 Analytical model ................................ ................................ ................................ .......... 85 5.3.2 Resu lts and Discussion ................................ ................................ ................................ . 92 5.3.3 Summary on Photoelectron Transport with DC Bias ................................ ................... 98 CHAPTER 6 SUMMARY AND SUGGESTED FUTURE WORK ................................ ...... 99 6.1 Summary ................................ ................................ ................................ ............................. 99 6.2 Suggested future work ................................ ................................ ................................ ....... 101 APPENDI CES ................................ ................................ ................................ ........................... 102 A PPENDIX A: Two - color laser induced photoemission without dc field ............................. 103 A PPENDIX B: Two - color laser induced photoemission with dc field ................................ ... 103 A PPENDIX C: Few - cycle laser pulses induced photoemission ................................ ............. 104 A PPENDIX D: Photoelectron transport in a nanoscale gap without dc bias .......................... 104 A PPENDIX E: Photoelectron transport in a nanoscale gap with dc bias ............................... 105 BIBLIOGRAPHY ................................ ................................ ................................ ..................... 107 viii L IST OF TABLE S Table 1: List of achieved strong local dc fields (after field enhancement) of sharp tips before breakdown for eight materials . ... 47 ix LIST OF FIGURES Figure 1.1: Multiphoton over - barrier emission. Electron inside the metal is excited to a continuum state by absorbing enough photon energy and then escapes from the metal surface. W and are the work function and Fermi energy of metal, respectively. .. 3 Figure 1.2: Log - scale plot of photoelectron yield from a sharp gold tip as a function of laser power with 800 V dc bias (red curve) and without dc bias (blue curve) [9]. 3 Figure 1.3: Optical field emission. The potential barrier near the metal surface greatly oscillates with time under the illumination of strong laser field, enabling the electron tunneling emission. W and 4 Figure 1.4: Log - scale plot of photoelectron yield from sharp gold nanotip as a function of laser energy. The decreasing slope with the increasing incident energy indicates the transition of the dominant emission process from multiphoton over - barrier emission to optical field emission [11]. .. 5 Figure 1.5: Photon - assisted tunneling emission. The tunneling potential barrier near the metal s urface is formed under the strong dc field. Electron tunneling emission is possible even with a weak laser field. W and are the work function and Fermi energy of metal, respectively. .. 6 Figure 1.6: Experimentally measured photoelectron energy sp ectra from tungsten nanotip with strong dc field [46]. Figure 1.7: Comparison between the experimentally measured QE under low dc electric field (black points) and calculated QE under low (red solid line) and high (blue dashed line) dc electric field [72]. 0 Figure 1.8: (a) Fowler - Nordheim plots of field emission (blue squares) and dc - assisted optical tunneling (red circles) [8]. (b) Fowler - Nordheim fit to the experimental measurements (bright orange dashed line) [19]. .. . 1 Figure 1.9: (a) Photoelectron energ y spectra with increasing laser field. Left three plots show the experimental measurements [54]. analytical model [6]. (b) Photoemission current as a function of applied dc field. Left plot is the experimental result [9]. .. 1 3 Figure 2.1: Energy diagram for electron emission through a wiggling potential barrier induced by two - color laser fields across the metal - vacuum interface at x = 0. Electrons with initial energy of x are excited to emit through n - photon absorption, with a transmitted energy of , with being an integer. The fundamental and the harmonic laser fields are and , respectively . and are the Fermi energy and work function of the metal, respectively. ... 18 Figure 2.2: Photoelectron energy spectra, calculated from Equation (19). (a) - (e) Energy spectra under different combinations of two - color laser fields (at frequency ) and (at frequency 2 ), for the special case of . (f) - (j) Energy spectra for various phase differences . The unit of laser fields and is V/nm in all figures. 3 Figure 2.3: Normalized total time - averaged emission current density for the phase differences = 0 and . (a) - (b) total time - averaged current density as a function of the second - harmonic laser field , under various fundame ntal laser fields . (c) - (d) as a function of , under various . The laser intensity is related to the laser electric field as I [W/cm 2 ] = 1.33 × 10 11 × ( F 1 [V/nm]) 2 . The dotted lines represent the scale . 4 Figure 2.4: Current modulation depth. (a) Normalized total time - averaged emission current density as a function of the phase difference , under different . (b) Magnification of the bottom area of (a). (c) Semi - log plot o f in (a). is fixed at 1.6 V/nm in (a) - (c). (d) Electron energy spectra of (point A) and (point B) for = 0.1375 in (c). (e) Current modulation depth as a function of the field ratio for different 0.5, 1.6, and 10 V/nm. 26 Figure 2.5: Total time - dependent emission current density for the phase differences = 0 and . (a) - (b) Total time - dependent emission current density as a function of the space and time t . (c) - (d) Total emission current density at = 100 as a function of time t . Dotted lines in (c) and (d) are for the total time - dependent laser field + . The fundamental laser field = 1.6 V/nm. The second harmoni c ( laser field = 0.22 V/nm (experimental laser parameters in Reference [57]). When = 0, the normalized time - averaged emission current density = 5.23 ; when = , = 7.31 . 27 Figure 2.6: Effects of the harmonic order. The emission current modulation depth , the maximum and minimum time - averaged current density, and as a function of harmonic order . The fundamental laser field and the harmonic laser field are 1.6 V/nm and 0.22 V/nm, respectively (intensity ratio of 2%). 28 Figure 2.7: Energy diagram for photoemission under two - color laser fields and a dc bias. Electrons with initial energy are emitted from the dc biased me tal - vacuum interface at x = 0, with the transmitted energy of , due to the n - photon contribution [multiphoton absorption ( n > 0), tunneling ( n = 0), and multiphoton emission ( n < 0)], where n is an integer. The fundamental and harmonic laser fields are and , respectively. The dc electric field is . The photon energy of the fundamental (harmonic) laser is ( ). and are the Fermi energy and work function of the metal, respectively. 29 xi Figure 2.8: Photoelectron energy spectra under different in - phase (i.e., ) laser fields (at frequency ) and (at frequency ) and dc fields . In ( a) - (c) is fixed as 1 V/nm, and in (d) - (f) is fixed as 10 V/nm. The - photon process (that is the horizontal axis) is given with respect to the fundamental laser frequency, which measures the energy of the emitted electrons. The units of dc field and laser fields and are V/nm in all figures. 3 Figure 2.9: Normalized total time - averaged emission current density , for the phase difference between the two - color lasers (a) - (c) = 0, and (d) - (f) = , as a function of the fun damental laser field , under various combinations of the second - harmonic laser field and dc electric field . The laser intensity is related to the laser electric field as I 1,2 (W/cm 2 ) = 1.33 × 10 11 × ( F 1,2 (V/nm)) 2 . 35 Figure 2.10: Normalized time - averaged emission current density through the n th channel, for the phase difference between the two - color lasers (a) - (c) = 0, and (d) - (f) = , as a function of the fundam ental laser field , for various dc electric fields , when the second harmonic laser field = 5 V/nm. Dotted lines represent the normalized total emission current . 36 Figure 2.11: Normalized total time - averaged emission current density for the phase difference between the two - color lasers (a) - (c) = 0, and (d) - (f) = , as a function of the dc electric field , for different fundamental laser fields and se cond - harmonic laser fields . Intertwined curves in (d) - (f) indicate the strong interference effect of the two lasers. 38 Figure 2.12: Normalized time - averaged emission current density through the n th channel for the phase difference (a) - (c) = 0, and (d) - (f) = , as a function of the dc electric field , for various second - harmonic laser fields , when the fundamental laser field is = 7 V/nm. Dotted lines represent the normalized total emission current . . 39 Figure 2.13: Emission current modulation depth. (a) Normalized total time - averaged emission current density as a function of the phase difference , for different dc electric fields , with - laser - field and 2 - laser - field fixed at 1.6 V/nm and 0.22 V/nm respectively (experimental laser parameters in Reference [57]). (b) Energy spectra of the emission current at different for the case of V/nm in (a). A, B, C, and D denote the cases of 0, , , and in (a), respectively. (c) Current modulation depth in (a) as a function of the dc field . The unit of dc field is V/nm in all figures. 1 Figure 2.14: Normalized total time - averaged emission current density as a function of harmonic order , for the phase difference (a) = 0, and (b) . The fundamental - laser field = 1.6 V/nm, and the harmonic - laser field = 0.22 V/nm. 2 Figure 2.15: Effects of the image - charge - induced barri er lowering on the total emission current for various , , , and . The solid (dotted) lines represent the cases with (without) the xii image charge effect, calculated using effective work function (work function ). The gray dashed lines show the scale . 3 Figure 2.16: Time - resolved photoelectron energy spectra for the tungsten nanotip. (a) Comparison between the experimentally measured electron counts from Figure 3 in Reference [57] (see solid lines) and fitting results (see dotted lines). (b) Normalized total time - averaged emission current density as a function of the phase difference between the two - color lasers , for different , with fixed 1.6 V/nm. Blue and r ed lines denote the experimentally observed emission electron current and the sine fit from Fig ure 2(b) of Reference [57], respectively. (c),(d) Energy spectra for various dc fields when (c) = 0, and (d) = . (e),(f) Photoelectron spectra at dif ferent phase delays for the dc field (e) = 0.01 V/nm and (f) = 0.09 V/nm. (g),(h) Projection of the spectra in (e) and (f) on the plane, respectively. Except (b), the fundamental laser (1560 nm) field = 1.8 V/nm and the second - harmonic la ser field = 0.3 V/nm for all other figures (experimental laser parameters in Reference 45 Figure 3.1: Time - resolved photoelectron energy spectra. (a),(b) Energy spectra as a function of the phase difference between the two lasers , for dc field (a) = 0 and (b) = 0.8 V/nm. (c),(d) Projections of the spectra in (a),(b) on the - plane respectively. (e),(f) Projections of the spectra in (a),(b) on the n - plane respectively. Here, the laser fields = 1.8 V/nm and = 0.3 V/nm (experimental parameters in Reference 3 Figure 3.2: Photoemission current modulation. (a) Normalized total time - averaged emission current density < w > a s a function of phase difference for different , when the dc field = 0. (b) Semilog plot of < w > in (a). (c) Current modulation depth (solid lines) as a function of the laser field ratio for different dc fields . Dotted (dashed) l ines in (c) are for the maximum (minimum) emission current density at = 0 ( = ). Here, the laser field is fixed as 1.8 V/nm. . 55 Figure 3.3: (a) Emission current modulation depth (solid lines) as a function of the dc field with and without the image - charge - induced potential barrier lowering (or the Schottky effect), for laser fields = 1.8 V/nm and = 0.3 V/nm. The case without Schottky effect is calculated by replacing W eff with the nominal work function of metal W in Equation (27). (b) Modulation depth (solid lines) as a function of for different laser field ratios , with the effective work function W eff . is fixed at 1.8 V/nm in (b). In (a),(b), the dotted (dashed) lines are for the maximum (minimum) emission current density at = 0 ( = ). 56 Figure 3.4: Normalized total time - averaged emission current density as a function o f the phase difference , for various (a) cathode materials and (b) incident wavelengths. In (a), the laser wavelength 800 nm ( = 1.55 eV). The nominal work function of different materials is W Ag = 4.26 eV [107], W w = 4.31 eV [57][60], W Mo = 4.6 eV [107], W Cu = 4.65 eV [107], and W Au = 5.1 eV [6][107]. In (b), the metal is tungsten. Here, the dc field is 0.8 V/nm and the laser fields and are fixed at 1.8 and 0.3 V/nm, respectively. 57 xiii Figure 4.1: (a) Sketch o f photoelectron emission from a biased emitter under the illumination of a laser pulse train with a time period T . (b) A single laser pulse with carrier - envelope phase (CEP) and full width at half maximum (FWHM) of the field envelope . The red curve and black dotted lines denote the time evolution of laser electric field and laser pulse envelope, respectively. 61 Figure 4.2: Effects of time separation between adjacent laser pulses on photoelectron ener gy spectra and total emission charge density Q. (a) Laser electric field for different . From top to bottom, = 13, 29, 160, and 276 fs, corresponding to 5, 11, 60, and 100, respectively. (b) Energy spectra for different . denotes the la ser photon order (with single photon energy 1.55 eV). (c) Q as a function of . The inset shows the magnification of (c) between 9 and 30, where A, B, C and D denote 9, 11, 13 and 15 respectively. (d) Photoelectron energy spect ra near the maximum at A, B, C and D in the inset of (c). 65 Figure 4.3: CEP modulation in energy spectra with different pulse duration . (a) Laser electric field for different when CEP = 0 and . (b) Energy spectra as a function of for different . (c) Extracted energy spectra of = 0 and from (b). (d) Linear plot of energy spectrum for = 4.4 fs in (c). (e) Normalized current modulation magnitude = ( - ave )/ ave as a fun ction of for different . Here, ave = ( max + min )/2 denotes the averaged value of with respect to . (f) Current modulation depth =( max - min )/( max + min ) as a function of . 67 Figure 4.4: CEP sensitivity of total emission charge density Q under different laser fields . (a),(b) Difference between Q and its averaged value Q ave as a function of for different with pulse duration (a) = 4.7 fs and (b) = 8.8 fs. For a given , Q ave = (Q max + Q min )/2. Dashed lines indicate the shift of the phase for the CEP modulation. (c) Difference between the maximum and minimum values of charge Q max - Q min in the curves of (a) and (b), as a funct ion for different . Points A, B and C denote 7, 9, and 10 V/nm, respectively. (d) Photoemission charge phase Q as a function of for different . Q is obtained by using B cos( + Q) to fit the curves in (a) and (b), with B = Q max - Q min . 68 Figure 4.5: Time - dependent emission current density w(t) at the surface ( , with surface oscillatory current excluded ) as a function of time t for pulse duration = 4.7 fs at CEP when Q max (top row) or Q min (bottom row) occurs, under different laser fields at (a),(b) V/nm; (c),(d) V/nm; and (e),(f) V/nm. The values of laser field correspond to case A, B and C in Figure 4.4(c), respectively. The value of CEP in each panel corresponds to the occurrence of Q max (top row) or Q min (bottom row) i n Figure 4.4(a). The blue lines are for emission current density, red lines for laser field, and black dotted lines for laser pulse envelope. The optical half cycles of the laser field in (a), (b), (e), (f) are numbered as 0, 1, 2 and the center cycle with the highest peak. Only positive optical half cycles are shown. 69 Figure 4.6: Total time - dependent emission current density w( , t) under the dc field V/m and V/m. (a),(b) w( , t) including surface oscillation currents as a function of the xiv space and time t . Solid white lines show the corresponding cl assical trajectories. Dotted white lines show the positive half cycles of the laser electric field. (c),(d) Emission current density w(t) at = 50 and 100, as a function of time t . The time - dependent current in all figures is normalized in terms of the t ime - averaged emission current . Here, the laser pulse duration = 8.8 fs and the peak laser field V/nm. When V/m, = 2.5 ; When V/m, = 2.1 . 1 Figure 5.1: Energy diagram for photoelectron emission in a nanoscale metal - vacuum - metal junction under a single - frequency laser field. Electrons with the initial energy are emitted from the surface at x = 0, with an energy of , due to n - photon contribution. Here, by symmetry, electron emission from the surface at x = d can be modeled in the same way (but with an opposite sign of instantaneous laser field). 76 Figure 5.2: Normalized time - averaged photoemission current density und er various gap sizes and laser fields. (a) Total emission current density as a function of gap distance d for different laser fields . Dashed lines denote the emission current density from a single surface when the metal on the right - hand side in Fig ure 5. 1 is removed, which is obtained from Ref erence [6]. (b) Energy spectra for photoelectrons transmitted into the metal on the right - hand side for different d and . (c) Photoelectron energy spectra for electrons inside the vacuum gap and in the metal on the right - hand side under different for d = 2 nm. For the curves for photoelectrons inside the gap, white - filled diamond markers denote the absolute value of negative emission current density through the n th channel. 0 Figure 5.3: (a) Normalized total time - averaged emiss ion current density and (b) difference between total emission current and emission current from a single surface as a function of laser field for different gap distances d. The single surface case is obtained from Reference [6]. The dashed line in (a) denotes the scale of with n = 4. (c) Emission current density as a function of laser field for gap spacing d = 3, 5, and 11 nm. Here, laser field regimes are labeled with n = 4 and n = 5 (cf. the areas filled with different colors), which means the dominant emission process in this field regime is four - or five - photon absorption. 8 2 Figure 5.4: Total time - dependent current density as a function of time t and space x , under various laser fields and gap distances d . Here, the time - dependent current density is normalized in terms of the time - averaged current density . In all figures, the units of and d are V/nm and nm, respectively. The dott ed lines show the position of . 3 Figure 5.5: Normalized total time - averaged emission current density as a function of gap distance d for various (a) laser wavelengths and (b) metal materials. Photoelectron energy spectra for different (c ) laser wavelengths and (d) metals, for d = 2 nm. In (a) and (c), the metal is assumed to be gold. In (b) and (d), the incident wavelength is 800 nm. The work function of different materials is W Ag = 4.26 eV [107], W W = 4.31 eV [106], W Mo = 4.6 eV [107], W Cu = 4.65 eV [107], and W Au = 5.1 eV [6][107]. The laser field 4 xv Figure 5.6: (a) Schematic of metal - vacuum - metal nanogap with a dc bias V under the illumination of laser field. d is the gap distance . (b) Energy diagram for photoelectron emission from left metal - vacuum interface of the gap in (a). Electrons with the initial energy would see a potential barrier subjected to a positive dc electric field = (> 0) and laser field . (c) Energy diagram for photoelectron emission from right metal - vacuum interface of the gap in (a). Electrons would see a potential barrier with a negative dc electric field = (< 0) and laser field with of opposite sign of tha t in (b) at any time instant for a given laser field. Figure 5.7: Photoelectron energy spectra for dc field (a) = 1 and 3 V/nm and (c) = - 1 and - 3 V/nm. Emission mechanisms when (b) = 1 and 3 V/nm and (d) = - 1 and - 3 V/nm. Here, laser field = 1 V/nm and gap distance d = 5 nm. Figure 5.8: Normalized total time - averaged emission current density as a function of laser field for various dc fields . The gap distance d is fixed at 5 nm. The black dashe d lines display the scale . Here, n = 2.8, 1.9, 6.5 and 12.4 when 1, 3, - 1, and - 3 V/nm is consistent with the observed orders of domination multiphoton emission channel in Figures 5.7(a) and 5.7(c). 4 Figure 5.9: Normalized total time - averaged emission current density as a function of gap distance d for different dc fields and laser fields . Dashed lines denote the emission current density from single surface, which is obtained from Reference [6]. Figure 5.10: (a),(b) Normalized total time - averaged emission current density as a function of dc field for different laser fields . (c) The dependence of net emission current density on the applied dc bias for di fferent laser fields . Here, gap distance d is fixed at 5 nm. Figure 5.11: Time - dependent emission electron density from left metal surface of the nanogap in Figure 5.6(a) as a function of time t and space x under various combinations of dc and laser fields. Solid white lines show the corresponding classical trajectories [76]. Dotted white lines show the laser electric field. Here, gap distance d is fixed at 5 nm. The units of dc field and laser field are V/nm in all figures. Figure 5.12: Time - dependent emission electron density from right metal surface of the nanogap in Figure 5.6(a) as a function of time t and space x under various combinations of dc and laser fields. Here, gap distance d is fixed at 5 n m. The units of dc field and laser field are V/nm in all figures. 1 CHAPTER 1 INTRODUCTION 1.1 Background Ultrafast science concerns the study of electronic dynamics and motion in ultrashort timescale with the aid of ultrafast lasers. This field has been widely explored in atomic and molecular systems. The main observation include s above - threshold ionization [1] [2] and high - order harmonic generation [3] . In r ecent decade, a new research direction has emer ged in the ultrafast science field, which is the study of laser - induced electron emission from solid nanostructures [4] [6] . Utilizing the solid - state nanostructure s [7] [23] , especially those made of metals , enables the nanoscopic confinement of optical field s and the resulting l arge field enhancement factor on the nanosurface. The former provides the possibility for the control of ultrafast electron emission on the nanometer scale, which is fundamentally important to the development of high - resolution electron microscopy [24] [27] , highly coherent electron source s [28] [30] and novel nano - vacuum electronic devices [31] [36] ; the latter enables the access to strong - field optics with low laser intensity , which can reduce the requirement for the laser experimental system and avoid thermal damage on the structure when illuminated by strong laser field s [4][37] . P hotoemission is also important to the development of vacuum electronics, high power electromagnetic sources and amplifiers, and high current cathodes [ 38] [45] . The initial work on ultrafast laser - induced elect ron emission from nanostructure is reported by Hommelhoff and his colleagues [7] . They demonstrated the nonlin earity of ultrafast photoe lectron e mission from a tungsten nanotip driven by low - power femtosecond laser. A variety of photoemission properties from metallic nanostructures were subsequently revealed, including the transition from multiphoton emission to strong optical - field emission [11] , dc - assisted tunneling 2 emission [6][8][46] , surface - plasmon boosted emission [20][37][47] [49] , dense - arrays enhancement effect [50][51] , dependence of emission distribution on optical orientation [52][53] , carrie r - envelope - phase (CEP) sensitivity [13][54] [56] , modulation effect of two - color lasers [57] [62] , and rectification effect of metal - vacuum - metal nanogap [63] [66] . 1.2 Photoelectron Emission Mech anisms Photoemission mechanisms in general depend on the local optical field intensity. This section summarizes the photoemission process es from metal surface in different field intensity regimes . 1.2.1 Multiphoton Over - Barrier Emission Under the illumination of a weak laser field, the main photoemission process is m ultiphoton over - barrier emission , where the electron inside the metal is excited to a continuum state by absorbing a threshold number of photons or more photons and then escape s from the metal surface (see Fig ure 1.1 ). The photoemission yield follows a power law in the incident laser intensity, and the exponent denotes the threshold numbe r of photons needed to overcome the potential barrier . Figure 1.2 displays the experimentally measured multiphoton emission current from a sharp gold tip as a function of the incident laser power [9] . F or zero dc bias (see the blue line in Figure 1.2 ), the fourth - order power dependence indicates the electron in side the tip needs to absorb at least four photons for the emission , which is consistent with the ratio of the work function of gold ( 5 eV) over incident single photon energy of 1.5 eV (for 828 nm laser ) , W / 3.3. 3 Figure 1.1 : Multiphoton over - barrier emission . E lectron inside the metal is excited to a continuum state by absorbing enough photon energy and then escape s from the metal surface . W and are the work function and Fermi energy of metal, respectively. Figure 1. 2: Log - scale plot of photoelectron yield from a sharp gold tip as a function of laser power with 800 V dc bias (red curve) and without dc bias (blue curve) [9] . 1.2.2 Tunneling Emission Optical field emission O ptical field emission occurs in the strong laser field regime , where the potential barrier near the metal surface greatly oscillates with time , enabling the electron tunneling into the vacuum with less photon absorption than multiphoton over - barrier emission (see Fig ure 1.3 ). O ptical field 4 emission only occurs during the positive half laser cycl e s , as shown in Fig ure 1.3. Bormann et al , [11] first ly reported the optical field emission from nanostructure , and their main experimental observation is displayed in Fig ure 1.4 . With increasing laser energy, the slope of photoemission current decreases, indicating the transition of dominant emission from multiphoton over - barrier emission to optical field emission. Figure 1. 3 : O ptical field emission . T he potential barrier near the metal surface greatly oscillates with time under the illumination of strong laser field , enabl ing the electron tunneling emission . W and are the work function and Fermi energy of metal, respectively. 5 Figure 1. 4: Log - scale plot of photoelectron yield from sharp gold nanotip as a function of laser energy. The decreasing slope with the increasing incident energy indicates the transition of the dominant emission process from multiphoton over - barrier emission to optical field emission [11] . Photon - assisted tunneling emission For the sharp metal lic tip, a strong dc field can be easily obtained at the apex due to strong field enhancement near the tip , inducing a narrow barrier near the metal sur face. This makes the electron tunneling emission possible, even in the weak laser field regime (see Figure 1.5) , which is referred as photon - assisted tunneling emission (or dc - assisted optical tunneling) . Figure 1.6 displays the experimentally measured photoelectron energy spectra from a tungsten nanotip with strong dc field , where the photon - assisted tunneling emission is the main emission process [46] . 6 Figure 1. 5: Photon - assisted tunneling emission . T he tunneling potential barrier near the metal surface is formed under the strong dc field. Electron tunneling emission is possible even with a weak laser field. W and are the work function and Fermi energy of metal, respectively. Figure 1. 6: Experimental ly measured photoelectron energy spectra from tungsten nanotip with strong dc field [46] . 7 In addition, due to the image charge effect (or Schottky effect ) induced by the strong dc field, a significant reduction of potential barrier appears at the surface (see Figure 1.5) , which is given by , where is the elementary charge, is the local dc field, and is the free space permittivity . The decreased barrier height can greatly increase the photoelectron emiss i on yield. 1.2.3 Keldysh Parameter Keldysh parameter is used to define the limit between multiphoton over - barrier emission and optical field em ission [67] . It is given by , where is the work function of metal and is the ponderomotive energy which describes the time - averaged kinetic energy of an electron with charge - e and mass in an oscillating electric field with the angular frequency and field amplitude F , For (weak optical field), the dominant emission mechanism is multiphoton over - barrier emission. For (strong optical field), the optical field emission dominates . When the Keldysh parameter is close to 1, the contribution from multiphoton over - barrier and optical field emission coexists . 8 1.3 Theoretical Model s for Photoemission from Metal Surface s A variety of theoretical approaches have been developed to describe and understand the underlying photoelectron emission mechanisms, such as Fow l er - Dubridge model [68] [71] , three - step model [72] [74] , p erturbative theory [11][75][76] , Floquet method [76][77] , Fowler - Nordheim tunneling approximation [8][19][55] , and directly solving the time - dependent Schrödinger equation (TDSE) [6][7][13][76][78] [80] . In this section, we introduce the commonly used three - step model, Fowler - Nordheim equation , and quantum analytical model based on the TDSE. 1.3.1 Three - Step Model Three - step model considers photoelectron emission as three sequentially independent processes: (1) E lectrons inside the metal are excited to higher energy s t ates by absorbing the incident photon; (2) Excited electrons migrate to the metal surface , where electron - electron scattering effect is included ; (3) Electrons with the energy larger than the potential barrier energy escape from the metal surface . The photoemission quantum efficiency (QE), defined as the ratio of the number of emi ssion electrons over that of incident photons, is expressed in terms of the probabilities of these three steps [72][73] , (4) where is the metal surface reflectivity as a function of optical frequency , is the Fermi - Dirac function, describing the distribution of electron energy states inside the metal, is the Fermi energy of metal, is the effective work function 9 in cluding the Schottky effect, is the probability an electron reaches the metal surface without electron - electron scattering, is the angle between the electron velocity and the surface normal, is the azimuthal angle , , where is the maximum escape angle for electron s with the total energy . At low temperature ( ), the Fermi - Dirac function can be approximated by Heaviside step function . When the photo n energy is close to the effective work function of the metal, will be near ly normal to the metal surface. Thus, the angle dependence of can be ignored. With these assumptions, Equation (4) can be simplified to [72][73] , As shown in Figure 1.7, the QE calculated from Eq uation (5) exhibits good agreement with the experimental measurements for copper surface s . However, this model is constructed by the classical treatment, thus it only works in the multiphoton over - barrier emission regime instead of the strong optical field regime with quantum mechanical tunneling [7 4] . 10 Figure 1. 7: Comparison between the experimentally measured QE under low dc electric field (black points) and calculated QE under low (red solid line) and high (blue dashed line) dc electric field [72] . 1.3.2 Fowler - Nordheim Equation Fowler - Nordheim equation describes the field emission where electron s tunnel through a narrow potential barrier due to a strong static electric field. The formula is given by [8][81][82] , w here is the field emission current density, e is the elementary charge, F is the local dc electric field, is the Planck constant, is the effective work function, for field emission , m e is the electron mass, and with . Fowler - Nordheim equation is also frequently used to calculate the photoemission rate j for the dc - assisted optical tunneling or strong optical field emission by directly replacing the electric field 11 F in Eq uation ( 6 ) with the sum of applied dc field and time - dependent laser electric field [8][83] Figure 1.8 shows that the experimentally measured field emission, dc - assisted optical tunneling emission and optical field emission can be well described by the Fowler - Nordheim scaling. Nevertheless, Fowler - Nordheim equation is only valid in the strong optical field regime instead of the multiphoton over - barrier emission regime . Figure 1.8: (a) Fowler - Nordheim plots of field emission (blue squares) and dc - assisted optical tunneling (red circles) [8] . (b) Fowler - Nordheim fit to the experimental measurements (bright orange dashed line) [19] . 1.3.3 Quantum Analytical Model Solving the TSDE is a quantum approach to describe the photoelectron emission, where the interaction between the electrons inside the metal is ignored. In 2016, Zhang and Lau [6] developed (a) ( b ) 12 a n quantum analytical model for the photoemission due to a combination of a dc field and a laser field , by exactly solving the TDSE , where is the reduced Plank constant, m e is the electron mass, and is the time - dependent potential energy being 0 inside the metal ( ) and in the vacuum ( ) respectively, with being the Fermi energy of the metal, the effective work function including the Schottky effect and the elementary charge. Here, both external electric fields a re assumed to be perpendicular to the flat metal surface. Based on the triangular potential barrier, the exact solution of electron wavefunction inside the metal and in the vacuum are obtained [6] , where and are the e lectron wave number, and are the Airy functions of the first kind and second kind respectively, , the drift kinetic energy , and the ponderomotive energies . The transmission coefficient can be obtained 13 from the boundary conditions that both the electron wave function and its derivative are continuous at x = 0 (see Referen ce [6] ) . U sing the probability current density, the time - averaged normalized emission current density, defined as the time - averaged ratio of the transmitted probability current density over the incident probability current density, , can be obtained as, where denotes the normalized emission current density through the th channel with emitted electron energy due to the n - photon contribution. Figure 1.9: (a) Photoelectron energy spectra with increasing laser field. Left three plots show the experimental measurements [54] analytical model [6] . (b) Photoemission current as a function of applied dc field. Left plot is the experimental result [9] . Right plot is results [6] . As shown in Figure 1.9 , the calculation from the quantum model recovers the experimental ly measur ed trends on the energy spectra for the transition from multiphoton to optical field emission 14 and the voltage and laser power dependence of photoelectron yield. Th ese good agreement with the experimental results display the validity of Schrödinger - based analytical mod el in both multiphoton over - barrier regime and optical field regime. O ur theoretical model in this thesis is also derived from the TDSE. 1.4 Organization of This Thesis In this thesis, we develop analytical quantum models to study ultrafast optical - field induced photoelectron emission from biased metal surfaces , by solving TDSE exactly . We consider two - color laser induced photoelectron emission with and with out dc bias, interference modulation of photoemission using two lasers of the same frequency, nonlinear ultrafast photoemission from a dc - biased surface triggered by few - cycle laser pulses , and laser induced photoelectron transport in nanogaps. Chapter 2 pres ents analytical model s for nonlinear ultrafast photoelectron emission from metal surface induced by two - color laser fields with out and with dc bias, by exactly solving the TDSE. The photoelectron energy spectra, emission current density and current modulat ion under various combinations of laser intensities , frequencies, dc fields, and phase differences of the two - color lasers are analyzed. The application of our model to the time - resolved photoelectron spectroscopy of one dimensional (1D) system is exemplified. Chapter 3 explores the modulation to photoemission current and dynamics of multiphoton excited states using t wo lasers of the same frequency. The effects of different laser fields, wavelengths, cathode materials, and dc bias are analyzed in detail . T he capability of measuring the time - resolved photoelectron energy spectra using single - frequency laser pairs is dem onstrated. Chapter 4 presents an analytical model for nonlinear ultrafast photoemission from a dc - biased surface triggered by few - cycle laser pulse s , by exactly solving the TDSE. O ur exact model is valid 15 for arbitrary pulse length from sub - cycle to CW excitation, and for arbitrary pulse repetition rate. The photoelectron energy spectra, emission current and emission charge density with different combinations of laser pulse repetition s , duration s , laser int ensit ies , CEP and dc fields are explored , showing good agreement with the experimental observations. This work offers clear insights to the photoelectron energy distribution and spatiotemporal dynamics of electron emission with different ultrashort pulse s and dc fields. Chapter 5 presents analytical model s for ultrafast photo electron emission in a nanoscale metal - vacuum - metal gap driven by a single - frequency laser field . W e study the dependence of photoelectron spectra and emission current on gap distance, laser intensit y , wavelength, and meta l material s . This work may provide useful guidance for the future design of ultrafast optoelectronic devices, such as photodetector s . Chapter 6 gives a summary and an outlook to future work s . 16 CHAPTER 2 TWO - COLOR LASER INDUCED PHOTOEMISSION 2.1 Introduction Two - color laser induced photoelectron emission from nanostructure is reported by Förster and his colleagues [57] in 2016. They found a substantial emission current modulation of 94% for tungsten nanotip s via the control of the relative phase between a strong fundamental laser and a weak second - harmonic laser , due to the interference effect between quantum emission path ways. This provides a new platform for coherently controlling electron dynamics in ultrashort spatiotemporal scales by the phase difference between the two - color lasers. By optimizing the employed laser and dc electric fields, Paschen et al [59] reported a nearly perfect two - color emission current modulation of up to 97.5% for tungsten nanotip in 2017 . O ther aspects of two - color photoemission from metallic nanostructures are also studied , including laser polarization dependence [58] , interac tion of two - color lasers with free electron beams [84] and plasmon - assisted emission [85] . Despite these recent studies on two - color photoemission from metallic nanostructure , the correlation between laser fields , applied dc bias and various underlying emission processes is still not well understood . The parametric dependence of the photo electron emission needs substantial further study. In this chapter, we present quantum analytical model s for nonlinear ultrafast photoelectron emission from metal surface induced by two - color laser fields with out and with dc bias, by exactly solving the TDSE [61][62] . Our model s are valid for arbitrary laser intens ities , harmonic orders, phase differences between the two lasers, dc bias and metal work function and Fermi level. V arious emission processes, including multiphoton over - barrier emission, dc - assisted tunneling emission and optical field emission , are all included in the single formulation. We comprehensively analyze 17 the photoel ectron emission properties , including energ y spectra, emission current density, and current modulation, under various combinations of laser intensities and frequencies, dc fields, and relative phase of the two - color lasers. We study the effects of image charge induced by the dc field on the emissio n current, which gives an examination on the sensitivity of photoemission to the shape of potential barrier. The application of our analytical model to the time - resolved photoelectron spectroscopy of one dimensional (1D) system is also demonstrated . The ma terial of this chapter is based on our published papers References [61] and [62] , and is presented with permission from the copyright holders . 2.2 Photoemission Without DC Bias 2.2.1 Analytical Model Our one - dimensional (1D) model (see Figure 2.1) considers electron s with initial energy are excited to the higher energy state by absorbing photon energy and then get emitted from the metal - vacuum interface at x = 0 , under the illumination of two - color laser fields, and , where and are the magnitudes of the laser fields, is the fundamental laser frequency, is a positive integer, and is the relative phase. We assume both laser fields are perpendicular to the metal surface, and cut off abruptly at the surface . The sudden screening of external fields is justified [6] , because the laser penetration depth (i.e. , skin depth) is typically much smaller than t he laser wavelength (e.g. , for the gold, the skin depth of 800 nm l aser wavelength is around 4 nm). 18 Figure 2.1 : Energy diagram for electron emission through a wiggling potential barrier induced by two - color laser fields across the metal - vacuum interface at x = 0. Electrons with initial energy of are excited to emit through n - photon absorption, with a transmitted energy of , with being an integer. The fundamental and the harmonic laser fields are and , respectively. and are the Fermi energy and work function of the metal, respectively. A time - varying potential barrier would be created at the metal - vacuum interface x = 0 , where , and are the Fermi energy and work function of the metal respectively, and is the elementary charge. To make the analytical treatment possible, image charge effects are not included in Equation (12). However, our previous work [6] demonstrated a very good approximation to include the image charge potential in our model, by simply replacing the work function W with the effe ctive work function due to Schottky barrier lowering. The electron wave function is solved from the TDSE , where is the reduced Plank constant, m e is the electron mass, and is the potential energy given in Equation (1 2 ). 19 An exact solution to Equation (13) for is obtained [61] (see A ppendix A for the method ), w here , , , , is the transmission coefficient, the drift kinetic energy , the ponderomotive energies , and , and is the electron initial energy . Because of the time periodicity, Equation (14 ) re presents the superposition of transmitted electron plane waves with energies , due to multiphoton absorption ( n > 0), tunneling ( n = 0), an d multiphoton emission ( n < 0) [6][76] . For x < 0, the solution to Equation (13 ) is [61] , which denotes the superposition of an incident wave and a set of reflected waves , where , , a nd is the reflection coefficient. I t has been verified that most of the reflected current is thr ough the initial energy level ( n = 0) [6] . . By matching the solutions in Equations (14) and (15) from the boundary conditions that both and are continuous at x = 0 , and taking Fourier transform, we obtain, in nondimensional quantities [6] , , , , , , , , , , , the following equation, 20 where is the Dirac delta function , and , and are given by, with . Since Equation ( 17 ) is deriv ed from the conditions that electron wave function and its first derivative are continuous at the metal - vacuum inter face ( x = 0 ) , and i n Equation ( 17b ) denote the phase factor of the wave function in the th state and of its spatial derivative at = 0, respectively. and are the th Fourier coefficients of and the product of and , resp ectively . The transmission coefficient (and therefore the reflection coefficient ) is obtained from Equation ( 16 ). The emission current density is then calculated from the probability curren t density , where is obtained from Eq uation (14). 21 T he normalized emission current density , defined as the ratio of the transmitted probability current density over the incident probability current density , , is found in nondimensional form as , w here , and . The normalized time - averaged emission current density is found to be , where represents the emission current density through the th channel, with emitted electrons of energy due to the n - photon contribution . 2.2.2 Results and Discussion In this chapter, u nless mentioned otherwise, the default values for the calculation are as follows: the wavelength of the fundamental laser field is 800 nm ( = 1.55 eV ), the harmonic laser field is with the frequency of (i.e. ) , the metal is assumed to be g old [ 6 ][ 11 ][5 6 ] [61] [6 2 ] [ 76 ] , with Fermi energy =5.33 eV and the work function W =5.1 eV , and s ince most of the emission electrons from sources are located near the Fermi level [6][61][62][76][86][87] , we choose the electron initial energy for simplicity. First, in order to understand the detailed underlying emission processes , the photoelectron energy spectra , u nder different two - color l aser fields (at frequency ) and (at second harmonic 2 ) , for various phase differences between two laser fields are displayed in F igure 22 2 .2. It can be seen that t he dominant emission process is the four - photon absorption ( ) for the fundamental laser (or two - photon absorption for the second - harmonic laser), where electrons at the Fermi level need to absorb at least four photons to overcome th e potential barrier ( ( see Fig ure 2. 1). The tunneling emission channels ( ) is closed . W hen the two laser fields are in phase ( = 0), the photoelectron emission spectrum becomes broader and the total emission current density i ncreases when either or increases, since more channels open up for electron emission. When is small (see Fig ure 2 .2( a) ) , the emission spectrum is very close to that driven by the second harmonic laser alone, indicating dominates the emission process. As increases (from Fig ure 2 .2( a ) to 2.2( e )) , the emission spectrum gradually transits to that driven by alone, indicating the laser field dominating the emission process changes from to . During the transition process, the competition between and fo r dominating the electron emission process causes the dip in Figure 2.2(c). In Fig ures 2 .2(d) and 2.2( e ) , the dip shifts to larger n as increases , due to the channel closing effect [6][76] . When either or , the results recover those of single frequency laser induced photoemission [6][76] . Figure s 2 .2( f ) - ( j ) show that the emission spectra can be greatly modified as changes, due to the interference effect between two lasers. For example, when changes from to , the emission process with the highest probability shifts from the four - photon to five - photon absorption. Figure 2.3 shows the normalized total time - averaged emission current density under various combinations of and , for the phase difference = 0 and . In Figures 2.3(a) and 2.3(b), when is small ( / 10), is insensitive to , because the fundamental laser dominates the emission process. As increases, the current density gradual ly approaches the scale with n = 2 ( see Figures 2.3(a) and 2.3(b) ) , indicating two - photon absorption for the second - 23 harmonic laser (or four - photon with respect to the fundamental laser) is the main emission process. The gradual change of the slo pe of is due to the opening of higher emission channels, as seen in Figure 2.2. When ( see Figure 2.3 (b) ) , a series of new dips appear in the curves as compared to those when ( see Figure 2.3 (a) ) , indicating strong interference effects between the two lasers. The interference effect is also reflected in that the total current density with = 1 V/nm changes from being larger than with = 0 to being smaller ( see the green and dark blue l ines in Figures 2.3(a) and 2.3(b) ) . The sharp drops of at 13 V/nm in Figures 2.3 (a) and 2.3(b) are due to the channel closing effect [6][76] , which is accurately predicted by taking , giving = 12.4 V/nm. Similar behaviors of as a function of are observed in Figures 2.3 (c) and 2.3(d). Figure 2. 2: Photoelectron energy spectra, calculated from Equation (19). (a) - (e) Energy spectra under different combinations of two - color laser fields (at frequency ) and (at frequency 2 ), for the special case of . (f) - (j) Energy spectra for various phase differences . The unit of laser fields and is V/nm in all figures. 24 Figure 2.3 : Normalized total time - averaged emission current density for the phase differences = 0 and . (a) - (b) total time - averaged current density as a function of the second - harmonic laser field , under various fundamental laser field s . (c) - (d) as a function of , under various . The laser intensity is related to the laser electric field as I [W/cm 2 ] = 1.33 × 10 11 × ( F 1 [V/nm]) 2 . The dotted lines represent the scale . The total time - averaged emission current density as a function of is shown in Fig ures 2.4(a) - 2.4(c), for various laser field s with fixed 1.6 V/nm. The total emission current density oscillates as a sinusoidal function of , showing striking resemblance to the experimentally measured emission current (see Figure 2(b) in Reference [57] ) . A s decreases, 25 the maximum and minimum of both decrease, but the corresponding for the maximum and minimum remain almost unchanged. The modulation depth, defined as , reaches a maximum value of approximately 99% when = 0.1375 (or intensity ratio of 2%). For tungsten and the fundamental laser wavel ength of 1560 nm as in Reference [57] , we obtain the modulation depth of 95.5% and of 93.9%, when setting the work function in Equation (12) to be 4.3 eV and 3.6 eV (effective work function with Schottky effect), respectively. The latter is almost identical to the experimentally measured modu l ation depth of 94% in Reference [57] . Despite the excellent agreement between the theoretical predictions and experimental results , we should stress that our model assumed one - dimensional flat metal surface, whereas the experiment used nanometer scale sharp emitt er [57] . The sharpness of the emitter may introduce varying field enhancement and Schottky lowering factor along the emission surface, nonuniform off - tip electron emission [53] , and even quantized energy levels inside the emitter [88] . In addition, our model neglects the image charge potential, laser pulse shape, laser penetration depth , incident electron energy distribution inside the meal, and surface effects (e.g. , local surface roughness, grain boundaries, and different crystal plane terminations). As further decreases, drops. When reache s 0, becomes a constant, with zero as expected, as shown in Figure 2.4(c). Figure 2.4(d) compares the electron energy spectra at the peak and valley of the current modulation for = 0.1375, where the dominant emission process shifts from four - p hoton to five - photon absorption. Figure 2.4(e) summarizes the modulation depth as a function of , for different strength s of the fundamental laser field . As the laser field increases, the location of the peak modulation depth shifts to larger , since a larger laser field is needed to balance the increase of for achieving the same modulation depth. 26 Figure 2.4 : Current modulation depth. (a) N ormalized total time - averaged emission current density as a function of the phase difference , under different . ( b) Magnification of the bottom area of (a). (c) Semi - log plot of in ( a ). is fixed at 1.6 V/nm in (a) - (c). (d) Electron energy spectra of (point A) and (point B) for = 0.1375 in (c). (e) Current modulation depth as a function of the field ratio for different 0.5, 1.6, and 10 V /nm . Figure 2. 5 shows the time - dependent electron emission current density as a function of the space and time t , for laser field V/nm and laser field V/nm (experiment al laser parameters in Reference [57] ). When is greater than 20 (beyond the strong surface current oscillation region), the emission current keeps the same temporal profile with only a phase shift as increases ( see Fig ures 2. 5(a) and 2.5 (b) ) , which is primarily due to the drift and acceleration motion of electrons under the influence of laser fields . As the phase difference varies from 0 to , becomes significantly smaller, due to the interference effect of two lasers, which also causes the total time - averaged emission current density to decrease from 5.23 to 7.31 . Fig ures 2. 5(c) and 2.5 (d) show the total emission current density at = 100 as a function of time t . It is shown that and the total laser field have a 27 clear phase shift , which means the peak value of time - dependent total emission current density do es not occur at the peak value of the total incident laser field . As the phase difference changes, the temporal profile of emission current density for a fixed also has a phase shift due to the interference effect between the two lasers . The full width at half maximum ( FWHM ) of the modulation of the ultrafast current pulses in Fi g ure 2. 5 is approximately 0.62 fs, which is significantly shorter than the period of the fundamental laser period of 2.67 fs. Figure 2.5: Total time - dependent emission current density for the phase differences = 0 and . (a) - (b) T otal time - dependent emission current density as a function of the space and time t . (c) - (d) T otal emission current density at = 100 as a function of time t . Dotted lines in (c) and (d) are for the total time - dependent laser field + . The fundamental laser field = 1.6 V/nm. The second harmonic ( laser field = 0.22 V/nm (experiment al laser parameters in Reference [57] ). When = 0, the normalized time - averaged emission current density = 5.23 ; when = , = 7.31 . 28 T he effects of harmonic number on the emission current modulation are shown in Fig ure 2. 6. As increases, modulation depth decreases, due to the reduced interference between the two - color lasers. Note that superimposing the fourth harmonic laser ( = 4) on the fundamental laser leads to the largest and . This is in agreement with the prediction [6] that the maximum emission current occurs whe n the single photon energy (that is the fourth harmonic photon here) roughly equals the potential barrier, . Figure 2.6: Effects of the harmonic order. The emission current modulation depth , the m aximum and minimum time - averaged current density, and as a function of harmonic order . The fundamental laser field and the harmonic laser field are 1.6 V/nm and 0.22 V/nm, respectively (intensity ratio of 2%). 2.2.3 Summary on Photoemission without DC Bias In this section, an analytical model for ultrafast electron emission from a metal surface due to two - color lasers is constructed, by solving the TDSE exactly. Our model demonstrates great tunability on the photoelectron spectr a , emission current, and current modulation, via the control of the phase delay, relative intensity, and harmonic order of the two - color lasers . We identify the condition for the maximum emission current modulation depth (99%) by superimposing a weak harmonic laser 29 on a fundamental laser. Using the same input parameters, our theoretical prediction for the photoemission current modulation depth (9 3.9%) is almost identical to the experimental results (94%). S uch two - color induced photoemission may inspire new route towards the design of future ultrafast nanoelectronics. 2.3 Photoemission with DC bias 2.3.1 Analytical model T he addition of dc bias to the metal makes the potential barrier near the metal - vacuum interface at x = 0 narrower , c ompared to the case without dc bias (see Figure 2.7). The time - dependent potential barrier near the interface reads [62] , Figure 2.7: Energy diagram for photoemission und er two - color laser fields and a dc bias. Electrons with initial energy are emitted from the dc biased metal - vacuum interface at x = 0, with the transmitted energy of , due to the n - photon contribution [multiphoton absorption ( n > 0), tunneling ( n = 0), and multiphoton emission ( n < 0)], where n is an integer. The fundamental and harmonic laser fields are and , respectively. The dc electric field is . The photon energy of the fundamental (harmonic) laser is ( ). and are the Fermi energy and work function of the metal, respectively. 30 where is the applied dc electric field which is assumed to be perpendicular to the flat metal surface . Other parameters have the same definition as that in Equation (12). By solving the TDSE subjected to the potential energy given in Equation (20), the exact solution for is found to be [62] (see Appendix B for the method), where , , , , , the drift kinetic energy , the ponderomotive energies , and , and are the Airy functions of the first kind and second kind respectively, showing an outgoing wave traveling to the + x direction (see Figure 2.7) [6][62][82][86] , repr esents the transmission coefficient, and is the initial energy of the electron. It is easy to find that Equation (21 ) is periodic with the time period of , therefore Equation (21 ) is readily to be recast into a Fourier series, which denotes the superposition of transmitted traveling electron waves with energies . These ladder eigenenergies are made possible by multiphoton absorption ( n > 0), tunneling ( n = 0), and multiphoto n emission ( n < 0) [6][61][62][76] . The exact solution of electron wavefunction for x < 0 is , 31 which denotes the superposition of an incident plane wave and a set of reflected waves with reflection coefficient and energies , where , and . Applying the boundary conditions that both and are continuous at x = 0, Fourier transform yields, in nondimensional quantities [6][61][62] , , , , , , , , , , , , the following equations, where , , and are given by, w ith , , , , , and . and in Equations 32 (24c ) and (24d) represent the phase factor of the n th - state wave function and of its spatial derivative at = 0 respectively. and are the th Fourier coefficients of and respectively . The transmission coefficient (and therefore the reflection coefficient ) is calculated from Equation (23 ). The normalized emission current density is defined as the ratio of the transmitted probability current density over the incident probability current density, , where the probability current density is . Thus, the normalized instantaneous emission current density is found as, where , and is defined in Equation (24f) . The normalized time - averaged emission current density is obtained as , 33 2.3.2 Results and Discussion Figure 2.8: Photoelectron energy spectra under different in - phase (i.e. , ) laser fields (at frequency ) and (at frequency ) and dc fields . In (a) - (c) is fixed as 1 V/nm, and in (d) - (f) is fixed as 10 V/nm. The - p hoton process (that is the horizontal axis) is given with respect to the fundamental laser frequency, which measures the energy of the emitted electrons. The unit s of dc field and laser fields and are V/nm in all figures. T he photoelectron energy spectra for different combinations of in - phase ( two - color laser fields (at frequency ) and (at frequency ) and dc fields are shown in Fig ure 2. 8. The results are calculated from Equation (26 ), except for the dc f ield cases ( Fig ure s 2 .8 (a) and 2 .8 (d)), which are obtained from Equation (19) . W hen the dc field is turned off (see Figures 2 .8 (a) and 2 .8 (d)), the dominant emission process is the four - photon absorption ( n = 4) for the fundamental laser, indicating the electron at the Fermi level needs to absorb at least four photons to overcome the potential barrier (see Figure 2.7 ) . This is consistent with the ratio of the work function over the fundamental laser photo n energy, 3.29. Applying a strong dc field 34 to the metal is able to open the tunneling emission channels below the over - barrier emission threshold ( n < 4), as shown in Figures 2 .8 (b) - 2 .8 (c) and 2 .8 (e) - 2 .8 (f). This is because the dc field could sufficiently narrow the potential barrier at the metal - vacuum interface ( x = 0) (see Figure 2.7 ), enabling the dc - assisted tunneling emission process for . As increases from Figures 2 .8 (b) to 2 .8 (c) and 2 .8 (e) to 2 .8 (f), the potential barrier becomes narrower, increasing the probability of electron emission through the tunneling channels, and the emission channel with the highest probability shifts towards the direct tunneling process ( n = 0), which is consistent with the observation in Refere nce [6] . For a given dc field , as either of laser fields ( or ) increases, the energy spectra become broader, because more emission channels are open up and contribute to photoemission. In the meantime, the dominant emission process shifts to the channel with larger n , which is due to the fact that electrons have to absorb sufficient number of photons to overcome the increasing ponderomotive energies and with increasing laser fields strength , exhibiting the transition from the multiphoton regime to optical - strong - field regime. These observations are consistent with previous experimentally and theoretically obtained energy spectra [6][12][14][54][61] . Since is fixed at 1 V/nm in Figures 2 .8 (a) - 2 .8 (c) , whereas is fixed at a larger value of 10 V/nm in Figures 2 .8 (d) - 2 .8(f), the spectra in Figures 2 .8 (d) - 2 .8 (f) are gener ally broader than those in Figures 2 .8 (a) - 2 .8 (c). In general, when the dc field or the laser field or becomes much stronger than the other two , the total current emission is dominated by th is largest field. Figure 2.9 shows the normalized total time - averaged emission current density as a function of the fundamental laser field , for different second harmonic laser fields and dc fields , when and . When the second harmonic field increases, becomes less sensitive to , since gradually dominates the emission process. For single - frequency laser - induced electron emission [6] , it is confirm ed that, in 35 the multiphoton regime, the slope of the curve of versus follows the scale ; this indicates that the dominant emission process is the n - photon process . This scale is not strictly valid for the two - color photoemission here; howe ver, the change of the slope of the curves could still manifest the shift of the main - photon emission process. For instance, a s the dc field increases from Figures 2.9(a) to 2.9(c) for and from 2.9(d) to 2.9 (f) for , the slope of for a given decreases , since the dominant emission process shifts to the lower emission channels. Figure 2.9: Normalized total time - averaged emission current density , for the phase difference between the two - color lasers (a) - (c) = 0, and (d) - (f) = , as a function of the fundamental laser field , under various combinations of the second - harmonic laser field and dc electric field . The laser intensity is related to the laser electric field as I 1,2 (W/cm 2 ) = 1.33 × 10 11 × ( F 1,2 (V/nm)) 2 . 36 Figure 2.10: Normalized time - averaged emission current density through the n th channel, for the phase difference between the two - color lasers (a) - (c) = 0, and (d) - (f) = , as a function of the fundamental laser field , for various dc electric fields , when the second harmonic laser field = 5 V/nm. Dotted lines rep resent the normalized total emission current . Th e above trend is also reflected in Figure 2.10, which shows the normalized time - averaged emission current density through the n th channel as a function of the fundamental laser field , for fixed = 5 V/nm. For both cases of and , when increases from 1 to 4 V/nm, the dominant emission channel shifts from n = 3 to n = 2 in general (see Figures 2.10(a), 2.10(b), 2.10(d), and 2.10 (e)). When r eaches 8 V/nm, the dominant emission process transits from the two - photon absorption ( n = 2) for 7 V/nm to single - photon absorption ( n = 1) for > 7 V/nm (see Figures 2.10(c) and 2.10 (f)). It is clear that the direct tunneling ( n = 0) is almost ind ependent of the laser field but very sensitive to the dc field . 37 When the phase difference changes from 0 to , due to the interference effect between the two lasers, new dips appear in the curves of , which can cause change s of the dominant emission process when increase s. For example, in Figure 2.10 (e), the dip in the curve of n = 2 at around 5.5 V/nm changes the dominant emission to the process instead of the process otherwise observed. The dips are also reflected in the total emission current (see Figures 2.9(d) and 2.9 (e )), which is consistent with our previous observation of two - color laser induced emission without a dc bias (see Figure 2.3). As the dc field becomes larger, these new dips gradual ly disappear , as shown in Fig ures 2.9(f) and 2.10 (f) , because the interference effect of the two lasers is masked by the strong dc field. The total emission current density as a function of the dc field for different laser fields and is shown in Figure 2.11 . When the phase difference of the two lasers = 0, the total emission current density increases as either of the laser fields ( or ) increases. When the dc field becomes larger, becomes less sensitive to the laser fields, since the Fowler - Nordheim - like fi eld emission [81] due to the dc electric field becomes more important than the over - barrier photo emission. The curves in Figure 2.11(a) resemble the experimentally measured trends of the voltage - and power - dependent electron flux (see Figure 2 in Reference [9] ) . As shown in Figures 2.11( d ) - 2.11( f ), w hen , due to the interference effect of the two lasers , the curves are intertwined , indicating strong nonlinear dependence of the emission current on the laser fields . Fo r large ( 7 V/nm) and small V/nm) in Figure 2.11(d) , remains almost the same as that with in Figure 2.11(a) , since the interference effect is suppressed by the dc field. 38 Figure 2.11: Normalized total time - averaged emission current density for the phase difference between the two - color lasers (a) - (c) = 0, and (d) - (f) = , as a function of the dc electric field , for different fundamental laser fields and second - harmonic laser fields . Intertwined curves in (d) - (f) indicate the strong interference effect of the two lasers. Figure 2.12 shows the emission current density as a function of the dc field for the case of = 7 V/nm. It is clear tha t the dominant multiphoton emission process shifts to smaller as increases. As increases, these shifts would occur at larger dc field . For example, when , the shifts of three - photon emission to two - photon emission occur at 3.5, 4, a nd 4.5 V/nm when = 1, 5, and 10 V/nm in Figures 2.12(a) - 2.12(c), respectively. The shifts of the dominant emission process also depend strongly on the phase difference . F or = 5 V/nm, a new dip appears in the curve of n = 2 when as compared to the case of , leading to 39 the change of the dominant emission channel (i.e. , two - photon process in Figure 2.12 (b) vs single - photon process in Figure 2.12 (e) at 7.5 V/nm). Figure 2.12: Normalized time - averaged emission current den sity through the n th channel for the phase difference (a) - (c) = 0, and (d) - (f) = , as a function of the dc electric field , for various second - harmonic laser fields , when the fundamental laser field is = 7 V/nm. Dotted lines represent the normalized total emission current . The combined effects of the dc field and the interference between two - color lasers on the energy spectra and total emission current are shown in Figure 2.13; this reveals the strong effects of the dc bias on the photoemission current modulation depth . Figure 2.13(a) shows t he effects of phase difference of the two - color lasers on the total emission current density , under different dc fields . Here , the - l aser - field and the 2 - laser - field are fixed as 1.6 V/nm and 0.22 V /nm respectively (intensity ratio of 2%) . It is clear that oscillates as a function of with a period of 2 , which show s a close resemblance to the experimental observation (see Figure 2(b) in 40 Reference [57] ). As the dc field increases, also increases . T he maximum ( minimum ) values of occur around ( , when the two - color lasers are in phase (180 out of phase). Figure 2.13 (b) shows the photoelectron energy spectra of at different in a single period for the case of V/nm in Figure 2.13(a) . When = 0 (A), (B), and (D) , the electron emission probability through the dominant ch annel ( n = 3) driven by two - color lasers is larger than that driven by the strong fundamental laser field alone . However, when (C) , the emission through n = 3 driven by the two - color lasers becomes smaller than that driven by alone, due to the strong interference effect. The emission current driven by the two - color lasers is always larger than that driven by the weak second harmonic laser field alone , r egardless of . These observations are in excellent agreement with the experimentally measured electron spectra ( see Fig ure 3 in Reference [57] ). Figure 2.13 (c) summarizes the modulation depth in Figure 2.13 (a), defined as , as a function of the dc field . When is zero , the modulation depth is as high as 99% [61] . As increases, decreases because the interference effect is gradually suppressed by . When = 8 V/nm, drops to approximately 2.98%, showing a strong dependence of current modulation on the dc bias. It is important to note that even when the dc bias reaches 3 V/nm (significantly larger than the laser fields V/nm and V/nm, corresponding to a ratio of ), a current modulation 70% can still be achieved. This suggests a practical way to maintain a strong current modulation , while increasing the total emission current by orders of magnitude, by simply adding a strong dc bias for two - color laser induced electron emission. 41 Figur e 2.13: Emission current modulation depth. (a) Normalized total time - averaged emission current density as a function of the phase difference , for different dc electric fields , with - laser - field and 2 - laser - field fixed at 1.6 V/nm and 0.22 V/nm respectively (exper imental laser parameters in Reference [57] ). ( b) Energy spectra of the emission current at different for the case of V/nm in (a). A, B, C, and D denote the cases of 0, , , and in (a), respectively. (c) Current modulation depth in (a) as a function of the dc field . The unit of dc field is V/nm in all figures. Since photoelectron emission paths ( or channels) depend strongly on the incident laser frequencies, as well as the interferences between them, superimposing different order of harmonic lasers on the fundamental laser can lead to different photoemission current s . Figure 2.14 shows the effects of the harmonic order on the total emission current density induced by the two - color lasers of frequency and under various dc fields . When the dc field = 0, the maximum value of occurs when = 4 . T his is because the maximum emission current happens when the single - photon energy (of the fourth - harmonic laser in this case) roughly equals the potential barrier ( ) [6][61] . By comparing Figure 2.14 with Figure 2.8 , it is found that the harmonic order where the maximum emission current occurs coincides with the channel 42 number of the dominant - photon process (with respect to the fundamental frequency ), for a given combination of , , and . As observed in Figure 2.14, as increases, the value of for the maximum shifts to a smaller number . This is consistent with the observation in Figure 2.8 that a larger dc field changes the dominant - photon process to a smaller . When 7 V/nm, the electron emission becomes almost independent of the frequency ( ) of harmonic laser , since th e Fowler - Nordheim - like field emission dominates the emission process. When changes from 0 to , for small ( 4 V/nm) and ( 4) , the emission current density has a distinct reduction due to the interference effect of the two lasers . However, for larg e ( 7 V/nm), the emission current is almost independent of , for all harmonic orders of the second laser . Figure 2.14: Normalized total time - averaged emission current density as a function of harmonic order , for the phase difference (a) = 0, and (b) . The fundamental - laser field = 1.6 V/nm, and the harmonic - laser field = 0.22 V/nm . Our calculations so far are based on the sharp triangular potential profile (see Figure 2.7 ), which does not include the image charge effects (or Schottky effect) due to the applied dc field . Our earlier work [6] demonstrated that the effects of image - charge - induced Schottky barrier lowering on photoemission can be accurately approximated in our model , by simply replacing the work function W in Equation (20 ) with the effective work function , 43 Figure 2.15: Effects of the image - charge - induced barrier lowering on the total emission current for various , , , and . The solid (dotted) lines represent the cases with (without) the image charge effect, calculated using effective work function (work function ). The gray dashed lines show the scale . where is the free space permittivity. A comparison between the total emission current density with and without the image - charge - induced bar rier lowering is shown in Figure 2.15 . Due to the reduction of potential barrier ( < W ), the emission current increases when considering the image charge effect. A larger dc field increases the emission current more significantly ( V/nm in Figures 2.15(a), 2.15(b) vs V/nm in 2.15(c), 2.15(d)), since a smaller effective barrier is created. As increases, the difference between the emission current with and with W becomes smaller. The increase of the emission current due to the inclusion of the image effect is rel ative insensitive to the phase delay of the two - color lasers. It is also 44 important to note that with the inclusion of , the slope of decreases , as observed from the value of n in the scale , which indicates that the number of photons involved in the dominant emission process decreases, be cause of the deduction of the potential barrier near the metal surface. 2.3.3 A pp lication to Time - Resolved Photoelectron Spectroscopy Photoelectron spectroscopy is one of the most popular techniques to study the composition and electronic states of solid surfaces by analyzing the energy spectra [89], [90] . Particularly, the time - resolved photoemission spectroscopy enables the measurement of short lifetime of the intermediate states , such as the image - potential states on metal surface, via control of the time delay between the pump and probe photons [91] [93] . In this part, we demonstrate the application of our analytical model to describe the dynamics of di fferent n - photon excited states in time and energy. As shown in Figure 2.16 (a), our 1D model is able to provide excellent fitting to the measur ed photoelectron spectra in Reference [57] fo r the tungsten nanotip, by using a dc field of 0.01 V/nm and an effective work function of eV. Furthermore, the current modulation profile (both magnitude and shape) obtained from our 1D model in section 2.2 [61] agrees very well with the experimentally observed sinusoidal variation with a period of for the relative phase delay , as shown in Figure 2.16 (b). Notably, other models, including simple tunneling rate model and 1D time - dependent density functional theory (TDDFT), fail to describe the experimental resu lts of the sinusoidal pro file (see supplementary material of Reference [57] ). 45 Figure 2.16: Time - resolved photoelectron energy spectra for the tungsten nanotip. (a) Comparison between the experimentally measured electro n counts from Figure 3 in Reference [57] (see solid lines) and fitting results (see dotted lines). (b) Normalized total time - averaged emission current density as a function of the phase difference between the two - color lasers , for different , with fixed 1.6 V/nm. Blue and red lines denote the experimentally observe d emission electron current and the sin e fit from Fig ure 2(b) of Reference [57] , r espectively . (c),(d) Energy spectra for various dc fields when (c) = 0, and (d) = . (e),(f) Photoelectron spectra at different phase delays for the dc field (e) = 0.01 V/nm and (f) = 0.09 V/nm. (g),(h) Projection of the spectra in (e ) and (f) on the plane, respectively. Except (b), the fundamental laser (1560 nm) field = 1.8 V/nm and the second - harmonic laser field = 0.3 V/nm for all other figures (experimen tal laser parameters in Reference [57] ). The photoelectron energy spectra from the tungsten nanotip under various dc fields are shown in Figures 2.16 (c) and 2.16 (d), for and , respectively. In the calculation, for each dc field, the effective work function is approximated by de termining the peak value in the surface barrier profile under dc bias [94][95] , , where the second term is the axial potential profile near a parabolic tip of radius of curvature r with d being a constant (= 83 nm to fit the spectra in Figure 2.16(a)) [95] , and the third term is the image charge potential of a spherical surface, with being the Schottky constant [94] . It is important to note that the 46 photoelectron spectra are very sensitive to the applied dc field , as shown in Figures 2. 16 (c) and 2.16 (d). The shift of the dominant emission process to a smaller with larger dc field agrees with the trend in Figure 2.8 . More importantly, the emission current density is increased by more than three orders of magnitude as is gradual ly increased from 0.01 to 0.09 V/nm, which could strongly facilitate the experimental detection of photoemission. When the relative time delay changes from - to , the variations of the spectra during one period for 0.01 and 0.09 V/nm are shown in Figures 2.16 (e) and 2.16 (f), respectively. To clearly observe the dynamics of different excited states in time, Figures 2.16 (g) and 2.16 (h) show the projection of the energy spectra in Figures 2.16 (e) and 2.16 (f) on the plane respectively. When the dc field is small , with = 0.01 V/nm (see Figure 2.16 (g)), all the n - photo orders of the spectra are modulated in the same way as a function of the relative phase delay , in agreement with the results in Reference [57] . The risi ng ten dency of the points along the phase difference from to 0 indicates the population of the n - photon excited intermediate states induced by lasers, while the decreasing signal from 0 to implies the decay of the excited states. When is increased to 0.09 V/nm (see Figure 2.16( h)), it is interesting to find that due to the effect of the dc fie ld, various n - photon excited states behave differently with respect to time delay . For instance, the one - photon tunneling state is almost invariable as changes from to , but the two - photon state decreases significantly at . This is in contrast to the two same - frequency induced photoemission, where the dynamics of multiphoton excited states remains same under different dc bias (see Figure 3.1 ). In addition, for a small dc field, the value of n for the dominant excitation state remai ns unchanged over the relative phase delay ( see Figure 2.16 (g)), which means the energy of the n - photon excited intermediate state is independent of the time delay 47 [91] . Ho wever, when the dc field is larger, n for the dominant excitation state changes with the relative phase delay ( see Figure 2.16 (h)). For electron emitters under a dc bias, it is important to prevent breakdown and premature failure of the emitter tips. Table I lists the local dc fields (aft er field enhancement) of sharp tips that have already been achieved in experiments before breakdown for eight materials. It is known that nanostructures survive large fields better for short pulse durations. Thus, l ocal dc field up to 10 V/nm or larger val ue at sharp tips may be realized in experiments via either laboratory - scale setup based on pu lsed capacitor discharge [9][96] , or powerful THz pulses [97] . . Table 1: List of achieved strong local dc fields (after field enhancement) of sharp tips before breakdown for eight materials . Achieved local dc field (V/nm) Au 8.8 [9] W 9.64 [96] Cu 10.35 [96] Mo Pt/Ir Carbon fiber Carbon nanotube 8.09 [96] 16 [97] 10.46 [41] 14 [98] CNT fiber 9.85 [42] [44] 2.3.4 Summary on Photoemission with DC Bias In this section, we construct an exact analytical model for photo electron emission from a dc biased metal surface induced by two - color laser fields, by solving the time - dependent Schrödinger equation. Our calculations reveal underlying various emission process, including multiphoton over - barrier emission, dc - assisted tunneling emission and optical field emission , for different dc 48 and laser fields, and recover the trend in the experimentally measured energy spectra and voltage - an d power - dependent electron flux . Besides the properties of the two - color lasers, including relative phase, intensity and frequency, our model shows the addition of a dc field to the metal surface can provide great tunability of the photo emission energy spectra and current modulation depth for two - color laser - induced photoemission. Furthermore, the dc bias can increase the emission current by orders of mag nitude. This increase of the current emission is due to the combined effects of potential barrier narrowing and barrier lowering. Our results suggest a practical way to maintain a strong current modulation while increasing the total emission current by orders of magnitude in two - color laser induced electron emission, by simply adding a strong dc bias and a weak harmonic laser. This work will enable applications requiring both high current level and strong current modulation, such as miniaturized particle accelerators, photoelectron microscopy, and ultrafast electron sources. Moreover, being verified against the experimentally measured time - resolved photoelectron energy spectra, the results from our model are expected to guide future experiments on time - re solved photoemission spectroscopy. 2.5 Conclusion In this chapter, we present quantum analytical solution s for highly nonlinear ultrafast photoelectron emission from metal surface s driven by two - color laser fields with and with out a dc bias, by exactly solving the TDSE . We systematically study the photoelectron energy spectra, emission current density, and current modulation under various combinations of laser intensities and frequencies, dc bias , and phase differences of the two - color lasers . Our model show s great tunability on the photoelectron spectr a , emission current, and current modulation depth, via the control of the phase delay, relative intensity, harmonic order of the two - color l asers and dc fields . The results are in good agreement with experimental measurements on the two - color photoelectron 49 energy spectra and current modulation from a sharp nanotip. Our results suggest a practical way to maintain a strong current modulation in the meantime to increas e the total photo emission current by orders of magnitude, by simply adding a strong dc bias and a weak harmonic laser. Application of our model to time - resolved photoelectron spectroscopy is also exemplified, showing the dynamics of the n - photon two - color excited electronic states depends strongly on the applied dc field. Our study may inspire new routes towards many appli cations requiring both high photoemission current and strong current modulation, such as tabletop particle accelerator, X - ray sources and time - resolved photoelectron microscopy . 50 CHAPTER 3 PHOTOEMISSION MODULATION BY TWO LASERS OF THE SAME - FREQUENCY 3.1 Introduction Although two - color laser induced electron emission from nanoemitters provides an attractive platform for modulating photoelectron emission by the relative phase difference between the two - color lasers and shows promises for the potential a pplication of time - resolved photoelectron spectroscopy [62][99] , the two - color laser system typically relies on the generation of higher order harmonics of a fundamental laser [57] [60][85] , which, because of its stringent requirements on the experimental setup and its relative low efficiency, greatly limits the accessibility o f the two - color laser system. For hig her intensity lasers, harmonic generation becomes increasingly complex and difficult to control [100][101] . In this Cha pter , we propose to utilize two lasers of the same frequency to modulate the photoelectron emission by their relative phase delay. This is motivated by the simple experimental implementation of single - frequency laser pairs , e.g. via a beam splitter with various coating materials to control the reflection and transmission of incident light [102] [105] . The tw o same - frequency lasers may be tuned to have virtually arbitrary ratio of intensities (in contrast to a small harmonic - to - fundamental intensity ratio in the two - color laser system [57] [60][85] ) , thus providing a much larger parameter space to assess the interference effect of the two lasers and the induced photoelectron emission . Using the quantum mechanical model in Reference [6] , we study the photoemission modulation properties for a dc - biased metal cathode illum inated by two laser fields with the same frequency. We investigate the modulation of photoemission current and the dynamics of multiphoton excited states for different laser fields, wavelengths, cathode materials, and dc fields. Our study demonstrates the capability of measuring the time - resolved photoelectron 51 energy spectra using single - frequency laser pairs. The material of this chapter is based on our published paper in Reference [106] and is presented with permission from the copyright holder. 3.2 Analytical model Under the action of two same - frequency laser fields and and a dc electric field , the time - dependent potential barrier near the surface of the cathode reads [6][61][62][74][106] , where is the Fermi energy of the metal cathode, is the effective work function with Schottky effect [6] , with being the nominal work function, is the elementary charge, is the free space permittivity , is the distance away from the cathode surface ( , and is the magnitude of the total laser field due to the two laser fields and , . (28) From Equation (28), it is clear that the magnitude of the total laser field depends strongly on the phase delay of the two lasers , which is expected to provide similar current modulation to that in the two - color laser setup [61][62] . The resultant phase , the effect of which becomes important for photoemission only in very short laser pulses when carrier - envelope phase matters. For laser pulses long er than about 10 cycles, it can be well approximated by continuous - wave excitation for photoemission [6] . Thus, in the calculation of this chapter , we ignore the effects of the absolute phase and set without loss of generality [106] . Bas ed on the quantum analytical theory of photoemission in Ref erences [6] [74] , the time - averaged normalized emission current density, defined as the time - averaged ratio of the transmitted 52 probability current density over the incident probability current density, , can be obtained as, where denotes the normalized emission current density through the th channel with emitted electron energy due to the n - photon contribution, is the reduced Plank constant, m is the electron mass and represents the transmission coeffi cient of electron wave functions , which is calculated from , where is the Dirac delta function, and are integers, and are the Fourier coefficients , with and , where , , and with . Here, and are the Airy functions of the first kind and second kind respectively , is the ponderomotive energy , and a prime denotes derivative with respect to the argument. For the special case of zero dc field , the time - averaged normalized emissio n current density becomes [6][74] , 53 where is still calculated from E q uation ( 30 ) with and unchanged, but with and . 3.3 Results and Discussion Figure 3 .1: Time - resolved photoelectron energy spectra. (a),( b) Energy spectra as a function of the phase difference between the two lasers , for dc field (a) = 0 and (b) = 0.8 V/nm. (c),(d) Projections of the spectra in (a),(b) on the - plane respectively. (e),(f) Projections of the spectra in (a),( b) on the n - plane respectively. Here, the laser fields = 1.8 V/nm and = 0.3 V/nm (experimental parameters in Reference [57] ). In Figure 3.1 , we plot the calculated photoelectron energy spectra as a function of the phase difference between the two lasers for different dc fields . The wavelength of both lasers is 800 nm ( 1.55 eV ). The metal is assumed to be tungsten [7][13][57] , with a Fermi energy 7 eV and a work function W = 4.31 eV. S ince most of the electrons emitted from sources are 54 located near the Fermi level [6][76][86][87] , we choose the electron initial energy for simplicity. Note with laser fields = 1.8 V/nm and = 0.3 V/nm for the special case of , the total normalized emission current density in Figure 3.1 is and 8.71×10 - 5 , for the DC field and 0.8 V/nm respectively. Using free - electron theory of metal [74] , we find the corresponding emission current density is 5.74×10 2 A/cm 2 and 6.75×10 4 A/cm 2 , resp ectively. When the dc field is turned off, the dominant emission process is three - photon absorption ( n = 3) (see Figures 3.1(a) and 3.1 (e)). This is consistent with the ratio of the work function of tungsten over the photon energy, . By changing the phase difference between the two lasers, the electron emissio n varies sinusoidally (see Figures 3.1(a) and 3.1 (c)). When applying a large dc field to the cathode, the tunneling emission channels ( n 2) are opened up, as shown in Figu res 3.1(b) and 3.1 (f). This is because the dc field adequately narrows the surface potential barrier, in addition to the Schottky - effect - induced barrier lowering, enabling the tunneling emission process. In the case of V/nm, the dominant emission p rocess is shifted to two - photon absorption. From Figures 3.1(c) and 3.1 (d), it is found that the multiphoton excited states ( n 3) vary with respect to the phase delay sinusoidally in the same way, with the maximum at and the minimum at , for both values of dc bias . This is in contrast to the two - color laser induced photoemission, where the dynamics of multiphoton excited states changes under different dc bias (see Figure 2.16(g) and 2.16(h) ). The one - photon ( n = 1) absorption and direct tunneling ( n = 0) process are almost independent of for the case of V/nm , as shown in Figures 3.1(d) and 3.1 (f). 55 Figure 3.2 : Photoemission current modulation. (a) Normalized total time - averaged emission current density < w > as a function of phase difference for different , when the dc field = 0. (b) Semilog plot of < w > in (a). (c) Current modulation depth (solid lines) as a function of the laser field ratio for different dc fields . Dotted (dashed) lines in (c) are for the maximum (minimum) emission current density at = 0 ( = ). Here, the laser field is fixed as 1.8 V/nm. The sinusoidal modulation in the total emission current density is shown in Figures 3.2 (a) and 3.2 (b), for the case of V/nm. When the laser field ratio increases , the maximum emission current at = 0 increases , while the minimum emission current at = decreases, due to the more profound interference of the two lasers . Figure 3.2 (c) shows the modulation depth, , as a function of laser field ratio under different dc fields . For a given , increases as increases, and it reaches the maximum value of 100% when = . It is important to note that, in order to reach a large modul ation depth 90%), only a small laser field ratio is needed even with a strong dc 56 field, e.g. , when = 1 V/nm. The dependence of on the dc field (see Figure 3.2 (c)) is not monotonic and wi ll be examined further in Figure 3.3 below. Figure 3.3 : (a) Emission current modulation depth (solid lines) as a function of the dc field with and without the image - charge - induced potential barrier lowering (or the Schottky effect), for laser fields = 1.8 V/nm and = 0.3 V/nm. The case without Schottky effect is calculated by replacing W eff with the nominal work function of metal W i n Equation (27 ). (b) Modulation depth (solid lines) as a function of for different laser field ratios , with the effective work function W eff . is fixed at 1.8 V/nm in (b). In (a),(b), the dotted (dashed) lines are for the maximum (minimum) emission current density at = 0 ( = ). As discussed before, b esides making the surface potential barrier narrower, the dc bias induces a reduction of the barrier height via the image charge effect (or the Schottky effect), which strongly influences the photoemission processes [6][62] . In Figure 3.3 (a), we compare the emission current modulation depth as a function of the dc field with and without the Schottky effect. When Schottky effect is not considered, gradua lly decreases with . It is clear that the Schottky effect greatly alters the dependence of modulation depth on the dc field (see solid lines in Figure 3.3 (a)). The change of originates from the change of the maximum (minimum) values of emission current with the Schottky effect, as shown by dotted (dashed) lines in Figure 3.3 (a). As varies, the effective potential barrier changes, which induces an increase (decrease) in 57 the emission current when the ratio becomes close r to (further away from) an integer, where resonant - pho ton absorption occurs (see Figures 6 and 7 of Reference [6] ). This resonant emission process causes the nonlinear behavior of as a function of dc field . Figure 3.3 (b) shows the modulation depth as a function of the dc field for different laser field ratios with fixed 1.8 V/nm . As gradually approaches the maximum value of 1 for the full range of dc field from 0 to 1 V/nm. This is consistent with the observation in Figure 3.2 (c). Note that when is increased from 0 to 1 V/nm, the total emission current density can be increased by orders of magnitude. Figure 3.4: Normalized total time - averaged emi ssion current density as a function of the phase difference , for various (a) cathode materials and (b) incident wavelengths. In (a), the laser wavelength 800 nm ( = 1.55 eV). The nominal work function of different materials is W Ag = 4.26 eV [107] , W w = 4.31 eV [57][60] , W Mo = 4.6 eV [107] , W Cu = 4.65 eV [107] , and W Au = 5.1 eV [6][107] . In (b), the metal is tungsten. Here, the dc field is 0.8 V/nm and the laser fields and are fixed at 1.8 and 0.3 V/nm, respectively. We also examine the photoemission current modulation depth for cathode materials with different work functions in Figure 3.4 (a) and for various incident laser wavelengths in Figure 3.4 (b). We fix the dc field = 0.8 V/nm and laser fields = 1.8 V/nm and = 0.3 V/nm. Under the same illumination condition, the electron emission current depends strongly on the work 58 function ; however, the modulation depth varies only slightly. This is because is predominantly determined by the ratio of the laser field strengths. Figure 3.4 (b) shows the effect of laser wav elength on both emission current and modulation depth for a tungsten cathode. The nonlinear dependence may also be attributed to the change of the ratio near resonant - photon processes [6] . 3.4 Conclusion In this chapter, we propose to utilize two lasers of the same frequency to modulate the photoelectron emission by their phase delay. Compared to the two - color laser configuration, single - frequency laser pairs can be more easily implemented in experiments since they relax the requirement of higher order harmonic generation, which becomes increasingly difficult in the high laser intensity regimes. The intensity ratio of the single - frequency laser pairs can be tuned over a much wider range than the two - color laser system. Using t he quantum model, we find a strong current modulation (> 90%) can be achieved with a moderate ratio of the laser field s (< 0.4) even under strong dc bias. The nonlinear effects of dc field, cathode materials, and laser wavelength on both the emission curre nt level and modulation depth are also examined. The strong dependence of photoelectron energy spectra on the phase delay of the two lasers demonstrates a promising potential for the application of time - resolved photoelectron spectroscopy using single - freq uency laser pairs. 59 CHAPTER 4 FEW - CYCLE LASER PULSE S INDUCED PHOTOEMISSION 4.1 Introduction U ltrashort pulsed laser induced photoelectron emission from nanostructure enables the control of electron motion on sub - optical - cycle time scale , by - envelope phase ( CEP ) [13][19][54] . This may pave the way towards the subfemtosecond and subnanometer probing of electron motion in so lid - state systems and the generation and measurement of attosecond electron pulses . [6][61][62][74] [13][76][78] Fowler - Nordheim equation based mode ls are commonly used to calculate the ultrashort pulsed photoemission rate [8][19][55] but it is only valid in the strong optical field regime (see section 1.3). To explicitly reveal the interplay of various emission processes under different regimes and to systematically characterize the parametric scalings of photoemission characteristics, an exact quantum theory under ultrashort pulsed condition is highly desira ble. In this chapter, we 60 [13][54][55] The material of this chapter is based on our published paper in Referen ce [56] and is presented with permission from the copyright holder. 4.2 Analytical Formulation a n optical electric field ( see Figure 4.1) of a Gaussian laser pulse train with a time period T = 2 L of the form , , , with , ( 32 ) [108] Equation 61 [6][56][61][62][74][106] Fig ure 4.1 : (a) Sketch of photoelectron emission from a biased emitter under the illumination of a laser pulse train with a time period T . (b) A single laser pulse with carrier - envelope phase (CEP) and full width at half maximum (FWHM) of the field envelope . The red curve and black dotted lines denote the time evolution of laser electric field and laser pulse envelope, respectively. [6][62] Equation By solving the TDSE Equation 62 [6][81][86] [6][56][61][62] 63 qõ qõ Equation qõ qõ 64 qõ qõ (see Appendix C for the method ). 65 4.3 Results and Discussion [6][11][76] [6][76][86][87] Figure 4 . 2 : Effects of time separation between adjacent laser pulses on photoelectron energy spectra and total emission charge density Q. (a) Laser electric field for different . From top to bottom, = 13, 29, 160, and 276 fs, corresponding to 5, 11, 60, and 100, respectively. (b) Energy spectra for different . denotes the laser photon order (with single photon energy 1.55 eV). (c) Q as a function of . The inset shows the magnification of (c) between 9 and 30, where A, B, C and D denote 9, 11, 13 and 15 respectively. (d) Photoelectron energy spectra near the maximum at A, B, C and D in the inset of (c). 66 [13][54] [6] 67 Figure 4 . 3 : CEP modulation in energy spectra with different pulse duration . (a) Laser electric field for different when CEP = 0 and . (b) Energy spectra as a function of for different . (c) Extracted energy spectra of = 0 and from ( b). (d) Linear plot of energy spectrum for = 4.4 fs in (c). (e) Normalized current modulation magnitude = ( - ave )/ ave as a fun ction of for different . Here, ave = ( max + min )/2 denotes the averaged value of with respect to . (f) Current modulation depth =( max - min )/( max + min ) as a function of . 68 [55] [55] Figure 4 . 4 : CEP sensitivity of total emission charge density Q under different laser fields . (a),(b) Difference between Q and its averaged value Q ave as a function of for different with pulse duration (a) = 4.7 fs and (b) = 8.8 fs. For a given , Q ave = (Q max + Q min )/2. Dashed lines indicate the shift of the phase for the CEP modulation. (c) Difference between the maximum and minimum values of charge Q max - Q min in the curves of (a) and (b), as a function for different . Points A, B and C denote 7, 9, and 10 V /nm, respectively. (d) Photoemission charge phase Q as a function of for different . Q is obtained by using B cos( + Q) to fit the curves in (a) and (b), with B = Q max - Q min . 69 To uncover the physical origin of the vanishing CEP sensitivity behavior and the CEP phase shift in the photoemission charge, we plot the time - dependent electron emission current density w ( t ) at the surface ( ) as a function of time, under different laser fields and CEP for = 4.7 fs, as shown in Fig ure 4.5 . The laser field strengths of 7, 9, and 10 V/nm used in Fig ure 4.5 correspond to cases A, B and C in Fig ure 4 .4 (c), respectively . By observing th ese time - dependent current pulses, it is clear that electron emission starts at the beginning of each positive half cycle in a given laser field pulse. When = 7 V/nm (case A before the dip in CEP sensitivity in Fig ure 4 .4 (c)), even - numbered positive op tical half cycles (Fig ure 4. 5(a)) drive more photoelectron emission than odd - numbered positive optical half cycles (Fig ure 4. 5(b)). However, as the laser 70 field is increased to 10 V/nm (case C after the dip in CEP sensitivity in Fig ure 4. 4(c)), odd - numb ered positive half cycles trigger more electron emission than even - numbered cycles (cf. Fi gures . 4. 5(e) and 4. 5(f)). This indicates in the strong field regime, there exists a competition between even and odd positive half - cycle contributions to photoelectr on emission, and thus a phase shift in as shown in Fig ure 4. 4(d) , with varying CEP. At = 9 V/nm (case B at the dip in CEP sensitivity in Fig ure 4. 4(c)), Q max - Q min becomes minimal, where Q max and Q min occur at and , respectively. The competition between electron emission from neighboring positive optical half cycles also leads to the dips in CEP sensitivity and phase shifts at = 5 V/nm and 9 V/nm for = 8.8 fs in Fig ures 4. 4(c) and 4. 4(d). It is important to note that, for clarity, we plot in Fig ure 4. 5 only the emitted current density that eventually escapes from the surface, whereas the local strong oscillatory current density near the surface typically associated with photoemission (e.g. see Fi g ures 4. 6(a) and 4. 6(b) below, and also References [6][61][62][76][109][110] ) is filtered out. This is possible in our exact analytical calculation using Eq uation ( 39 ), by excluding the high n - order (and l - order) terms, which is verified to give the strong oscillatory surface currents only. This is also consistent with previous study that the high energy regime in the photoelectron spectra is due to surface oscillations and rescattering (cf. Fig ure 4 in Ref erence [109] ). It is also noteworthy that, though electron emission starts at the beginning of every positive optical half cycles in the laser pulse, there is typically a time delay between the peak of the positive optical half cycle and the peak of the emission current pulse, as seen in Fig ure 4. 5. Furthermore, a stronger positive optical half cycle does not necessarily lead to a higher current pulse emission, which, however, depends strongly on the emission from neighboring half cycles in a laser pulse. These observatio ns indicate that further examination is needed on the validity of the widely used 71 Fowler - Nordheim rate equations, in which current emission follows closely the optical positive half cycles, to study the CEP sensitive, time - dependent strong - field photoemiss ion [55] . Figure 4.6 : Total time - dependent emission current density w( , t) under the dc field V/m and V/m. (a),(b) w( , t) including surface oscillation currents as a function of the space and time t . Solid white lines show the corresponding classical tr ajectories. Dotted white lines show the positive half cycles of the laser electric field. (c),(d) Emission current density w(t) at = 50 and 100, as a function of time t . The time - dependent current in all figures is normalized in terms of the time - averag ed emission current . Here, the laser pulse duration = 8.8 fs and the peak laser field V/nm. When V/m, = 2.5 ; When V/m, = 2.1 . 72 [76] [110] 4.4 Conclusion 73 74 CHAPTER 5 PHOTOELECTRON EMISSION IN A NANOSCALE GAP 5.1 Introduction Due to the promise for potential applications to ultrafast and highly sensitive photodetection in the room temperature, laser - driven electron emission in the nanometer - scale two - tip junctions has drawn strong recent interests [17][20][23][63] [66][111] . Rybka et al. [17] reported laser - induced sub - femtosecond photoelectron tunneling in a nanoscale metal - vacuum - metal gap. Higuchi et al. [63] explored the rectification effect of dc - biased two - metal - nanotip junction in ultrafast multiphoton photoemission. Ludwig et al. [23] presented the strong dependence of dynamics of nanoscale electron transport between two metal tips on the temporal profile of driving laser pulses. Turchetti e t al. [66] studied the impact of dc bias on photoemission from metal surfaces surrounding a nano - vacuum gap. Typically, numerical solutions of the time - dependent density function theory [23 ][ 64 ][ 65 ][ 112 ][ 113] and Schrödinger equation [66 ][ 114] are implemented to study the photoemission properties in nanoscale gaps, but the underlying physics for the interplays between electron emission process, laser field, gap size and materials is not always transparent, especially when transitioning among dif ferent emission regimes. In this chapter , by exactly solving the TDSE , we present analytical model s for nonlinear ultrafast electron emission and dynamics in a nanoscale metal - vacuum - metal junction without and with dc bias driven by a single - frequency las er field. Using the analytical formulation, we investigate the photoelectron transport with various gap distances, laser intensities, wavelengths, dc bias and metal materials. Our results provide clear insights to the energy distribution of emitted photoel ectron and spatiotemporal emission dynamics inside the metal - vacuum - metal junction. Part 75 of the material of this chapter is submitted to Optics Letters and another journal article is also planned . 5.2 Photoelectron T ransport without DC Bias 5.2.1 Analytical Mo del Our one - dimensional (1D) model (see Fig ure 5. 1) considers electrons with initial energy emitted from the surface at x = 0, under the action of laser field , where is the amplitude of the laser field and is the angular frequency . T he laser field is assumed to be perpendicular to the flat emitter surface, and cut s o ff abruptly at the surface [6][66] , thus the time - dependent potential energy in the entire regime reads [6][61][62][74][106] , (43) where and are the Fermi energy and work function of the left metal respectively , and is the elementary charge. By solving the TDSE Equation the electron wave function for is found to be , which denotes the superposition of an incident plane wave with initial energy and a set of reflected plane waves with reflection coefficient and energies , wher e the wavenumber and . For (in the gap), the exact solution of electron wave function is (see Appendix D for the method), 76 are emitted from the surface at , due to n (45) which shows the superposition of a set of electron waves travelling towards + x direction with coefficient and towards - x direction with coefficient inside the gap , where the drift kinetic energy , and the ponderomotive energ y . For x d , an exact solution of electron wave function is easily obtained , which represents the superposition of transmitted electron plane waves with energies , due to multiphoton absorption ( n >0), direct tunneling ( n =0) and multiphoton emission ( n < 0) [6][76] , where the wavenumber and is the transmission coefficient. 77 By imposing the boundary conditions that both the electron wave function and its derivative are continuous at x = 0 and x = d , and taking Fourier transform, we obtain, in nondimensional quantities [6][61][62] , , , , , , , , , , the following equations, where , , , , , , , , and are given by, 78 with . The coefficients , , and (and therefore ) is then calculated from Equations (47), (48) and (49). The normalized transmitted current density is defined as the ratio of the transmitted probability current density over the incident probability current density, , where the probability current density . Thus, the normalized instantaneous current density inside the gap (0 < x < d ) is, where , , , and . The corresponding time - averaged emission current density is obtained from the numerical integration of Equation (52) over time, In the metal on the right - hand side ( x > d), the normalized instantaneous transmitted current density is found as, 79 where . The time - averaged transmitted current density is, where represents the time - averaged transmitted current density through - photon process, with transmitted e lectrons of energy [6 ,76 ] . Due to cur rent continuity, the time - averaged current density obtained from Equation (52) and Equation (54) are equal, which has been verified in our calculations. 5 . 2.2 Results and Discussion Using the analytical solution presented above, we analyze the photoe lectron e mission properties under different combinations of gap distances and laser fields . Unless mentioned otherwise, the default value of the laser wavelength is 800 nm ( eV ) , the metals on both sides of the gap are assumed to be gold [17][20][65][ 111] , with Fermi energy = 5.53 eV and work function W = 5.1 eV , and the photoemission current is calculated from Equation ( 54 ). Since most of the electrons are emitted with initial energies near the Fermi level [6][76][86][87] , we choose the electron initial energy for simplicity. Figure 5. 2(a) show s the dependence of total time - averaged transmitted current density on the gap distance d under different laser fields . When the laser field is off (i.e. , ), the current is contributed only by direct tunneling, which rapidly decreases as gap distance increases. After applying a laser field, the current decreases initially as increases , closely following the scaling for the case of , where direct t unneling domina tes. As increases further , for a given laser field, the current oscillates around a constant value ( cf. the dashed 80 lines) , which is found to be the photoemission current from a single metal surface ( i.e., when the metal on the right - ha nd side in Figure 5.1 is removed). The oscillation behavior is attributed to the interference of electron waves inside the gap due to reflections from the metal - vacuum interfaces, for various gap distances d . Here, we ignore the effects of image charge and space charge, thus the oscillation amplitude of remains almost unchanged with increasing d . This oscillation behavior is similar to that found in field emission from dielectric coated surfaces [11 5 ][11 6 ] . The quantum interference of electron waves is also demonstrated experimentally in Ref erence [ 1 3], where the distinct peaks in energy spectra arise from the interference of electron waves re - scattering at the emitter tip. Figure 5. 2(b) shows the energy spectra for photoelectrons transmitted into the right - side metal for different gap distances d and la ser fields . It can be seen that for a smaller l as er field ( = 1 V/nm), as d de creases, the dominant emission shifts from four - photon [6] 81 over - barrier emission ( n = 4 , cf. the ratio of metal work function over single photon energy W / 3.29) to tunneling emission ( n < 4). As laser field increases ( = 4 V/nm and 8 V/nm), this shift of the dominant emission process becomes less prominent, because the potential barrier inside the gap becomes less sensitive to the gap distance under strong laser fields. Figure 5. 2(c) compares the energy spectra for photoelectrons inside the gap and in the right - side metal for d = 2 nm. It is found that a lthough the total emission current is equal in these two regions, the energy distribution of photoelectron s is quite different. In particular , the time - averaged current densities for all n - photon channels are positive in the right - side metal , while some of them are negative in side the gap (see the open diamond markers in Fig ure 5. 2(c)). Negative value of means electrons excited through those n - p hoton processes are reflected backwards inside the gap. Additionally , n - photon processes with contribute more significantly for transmitted electrons in the right - side metal than those inside the gap, which becomes more pronounced for larger laser int ensity. In Fig ure 5.3 (a), we plot the total time - averaged emission current density as a function of laser field with various gap distances d . For the vacuum gap with d 1 nm, the slope of increases with , indicating the dominant emission process shifts to higher order n - photon absorption. This is consistent with the results shown in Fig ure 5.2 ( b ). For the cases with larger gap distances, the slope of becomes insensitive to the gap distance and follows that of photoemission current from a single metal surface. The scale approaches with n = 4, indicating four - photon absorption dominates the emission process. Figure 5.3 (b) displays the difference between the total emission current in a nanogap and emission current from a single surface , where the difference becomes more pronounced in the larger laser intensity regime. Besides, it is interesting to find that the location of channel - closing - induced drop of 82 Figure 5.3: (a) Normalized total time - averaged emission current density and (b) difference between total emission current and emission current from a single surface as a function of laser field for different gap distances d. The single surface case is obtained from Ref erence [ 6 ]. The dashed line in (a) denotes the scale of with n = 4. (c) Emission current density as a function of laser field for gap spacing d = 3, 5, and 11 nm. Here, laser field regimes are labeled with n = 4 and n = 5 (cf. the areas filled with different colors), which means the dominant emission process in this field regime is four - or five - photon absorption. emission current density (i .e., the location of transition between the dominant four - and five - photon absorption in Fig ure 5.3 (c) , determined by observing the shift of the peak of the emitted electron energy spectra ) shifts to larger laser field for smaller gap distance d . This indicates that decreasing the gap distance (before entering the direct tunneling regime) can extend the multiphoton regime to higher laser intensity. This may be explained by the fact that the shape of the potential barrier becomes less sensitive to the l aser field strength for a smaller gap distance, thus allowing the dominant n - photon process to remain over a larger range of laser fields (or laser intensities). Figure 5.4 shows the time - dependent current density as a function of space and time t for different combinations of laser field and gap distance d . It is seen that, besides the surface oscillation current near the metal - vacuum interface at , some electrons are back reflected from the vacuum - metal interface at into the vacuum gap approximately at the beginning of 83 and d are V/nm and nm, respectively. The dotted lines sh ow the position of . second half cycle of the laser fields (i.e., . This is shown by the change of from red to dark blue around in Figures 5.4(e), 5.4(f), 5.4(h) and 5.4(i), where the red region denotes positive current density propagates in the + x direction and the dark blue region in - x direction. As the gap distance d increases, more interference patterns of current density inside the gap are formed. The full width at half maximum (FWHM) of the emission current pulse is about 0.63 fs, which is greatly shorter than laser period of 2.67 fs. We examine the total emission current density as a function of gap distance d for different incident wavelengths in Fig ure 5.5 (a) and for metals with various work functions in Fig ure 5.5 ( b ). It is found that the oscillation amplitude of increases when the laser photon energy ( , with be ing the laser wavelength) becomes closer to the metal work function , indicating 84 stronger interference of electron waves inside the gap when . Figures 5.5 ( c ) and 5.5 ( d ) show the photoelectron energy spectra for different laser wavelengths in Figu re 5.5(a) and for different metals in Figure 5.5(b) with d = 2 nm, respectively. The shift of the dominant emission to larger n - photon process is due to the increasing ratio of . [107] [106] [107] [107] , [6][107] is fixed as 4 V/nm for all the cases. 5. 2.3 Summary on Photoelectron Transport without DC Bias In this section , we present an analytical solution for photoelectron emission and transport in a nanoscale metal - vacuum - metal junction driven by a single - frequency laser field, by exactly solving the time - dependent Schrödinger equation. The analytical model is valid for arbitrary gap distance, laser intensity, wavelength and metal work function and Fermi level. Our calculation exhibits the transition from direct tunneling to multiphoton induced electron emission and the 85 oscillatory dependence of photoemission current on t he gap distance in the multiphoton regime. Our results demonstrate the energy redistribution of emitted p hotoelectrons across the two interfaces of the nanogap. We also find that decreasing the gap distance (but before transiting into the direct tunneling regime) can ex tend the multiphoton regime to h igher laser intensity . T he nonlinear effects of laser wavelength and materials on the gap - size dependence are examined. 5.3 Photoelectron Transport with DC Bias 5.3.1 Analytical model With the external applied dc voltage V shown in Figure 5.6(a) , the symmetry of the metal - vacuum - metal system is broken, which means under the same illumination condition, the left and right metal surfaces of the nanogap in Fig ure 5.6 (a) have different photoemission properties . Therefore , we analytically model photoele ctron emission from the left and right metal surfaces, respectively. Fig ure 5.6: (a) Schematic of metal - vacuum - metal nanogap with a dc bias V under the illumination of laser field. d is the gap distance. (b) Energy diagram for photoelectron e mission from left metal - vacuum inter face of the gap in (a) . Electrons with the initial energy would see a potential barrier subjected to a positive dc electric field = ( > 0) and laser field . (c) Energy diagram for photoelectron e mission from right metal - vacuum inter face of the gap in (a) . Electrons would see a potential barrier with a negative dc electric field = ( < 0) and laser field with of opposite sign of that in (b) at a ny time instant for a given laser field. For the photoe mission from the left metal - vacuum inter face of the nanogap in Fig ure 5.6 (a), electrons with the initial energy would see a potential barrier subjected to a positive dc electric 86 field = ( > 0) and laser field , as shown in Figure 5.6(b) . Thus, t he time - dependent potential energy in the whole regime reads as [6][61][62][74][106] , (5 5 ) where and are the Fermi energy and work function of the left - side metal in Figure 5.6(a) respectively , and is the magnitude of the applied dc bias . Other parameters have the same definition as that in Equation (43). By solving the TDSE Equation the electron wave function for is, which denotes the superposition of an incident plane wave with initial energy and a set of reflected plane waves with reflection coefficient and energies , wher e the wavenumber and . For (in the gap), the exact solution of electron wave function is found to be (see Appendix E for the method), which represents the superposition of a set of transmitted and reflected electron waves inside the gap, where , and and are the coefficients. For x d , an exact solution of electron wave function is, 87 which shows the superposition of transmitted electron plane waves with energies , due to multiphoton absorption ( n >0), direct tunneling ( n =0) and multiphoton emission ( n <0), where the wavenumber and is the transmission coefficient. By applying the boundary conditions that both the electron wave function and its derivative are continuous at x = 0 and x = d , and taking Fourier transform, we obtain, in nondimensional quantities [6][61][62] , , , , , , , , , , , the following equations, where , , , , , , , , and are given by, 88 w ith , , and . The coefficients , , and (and therefore ) is then calculated from Equations ( 59 ), ( 6 0 ) and ( 6 1 ) . The normalized transmitted current density is defined as the ratio of the transmitted probability current density over the incident probability current density, , where the probability current density 89 . Thus, the normalized instantaneous transmitted current density i n the metal on the right - hand side of Figure 5.6(a) ( i.e., x > d) is found to be , where . The time - averaged transmitted current density is, where represents the time - averaged transmitted current density through - photon process, with transmitted electrons of energy [6] . For the photoe mission from right metal - vacuum inter face of the gap in Figure 5.6(a) , electrons would see a potential barrier subjected to a negative dc electric field < 0) and laser field , as shown in Fig ure 5.6 (c). Thus, t he time - dependent potential barrier in Fig ure 5.6 (c) is [6][61][62][74][106] , (6 5 ) where and are the Fermi energy and work function of the right metal in Figure 5.6(a) respectively , and is the magnitude of the applied dc bias . Other parameters have the same definition as that in Equation (43) , with of opposite sign (i.e., 180 degree out of phase) of that in Figure 5.6(b) at any time instant for a given laser field. S olving the TDSE Equation the electron wave function for , 90 where the wavenumber and , and is the reflection coefficient. For (in the gap), the exact solution of electron wave function is found to be (see Appendix E for the method), where , and and are the coefficients. For x d , an exact solution of electron wave function is, where the wavenumber and is the transmission coefficient. B y applying the boundary conditions that both the electron wave function and its derivative are continuous at x = 0 and x = d , and taking Fourier transform, we obtain the following equations, 91 where , , , , , , , , and are given by, 92 with , , and . The coefficients , , and (and therefore ) is then calculated from Equations ( 69 ), ( 7 0 ) and ( 7 1 ). The normalized instantaneous transmitted current density i n the metal on the left - hand side of Figure 5.6(a), defined as the ratio of the transmitted probability current density over the incident probability current density, , is obtained as where . The time - averaged transmitted current density is, 5.3.2 Results and Discussion In the calculation of this section, positive dc field ( > 0) and negative dc field ( < 0) cases denote the electron emission from left metal surface and right metal surface of the vacuum nanogap with the external dc voltage V ( = ) shown in Fig ure 5.6 (a), respectively. Unless mentioned otherwise, the default value of the laser wavelength is 800 nm ( eV ) , the metals on both sides of the gap are assumed to be gold [1 7][20][65][111] , with Fermi energy = 5.53 eV and work function W = 5.1 eV , and the photoemission current is calculated from Equations (6 4 ) and (7 4 ). Since most of the electrons are emitted with initial energies near the Fermi level [6][76][86][87] , we choose the electron initial energy for simplicity. In Fig ure 5.7 , we plot the photoelectron energy spectra under different applied dc bias with fixed gap distance d = 5 nm . Increasing the dc field from 1 to 3 V/nm increases the left - to - right photoelectron transmission current by about two orders of magnitude and shifts the corresponding 93 dominant electron emission process from three - photon absorption to two - photon absorption, as shown in Fig ure 5.7 (a). This is becaus e under a larger dc field, the potential barrier near the left metal - vacuum interface becomes narrower, enabling the less photon transition process (cf. Fig ure 5.7 (b)). Nevertheless, with the larger dc field , the right - to - left photoelectron current dramati cally decreases and the dominant emission shifts to the higher order multiphoton absorption (see Fig ure 5.7 (c)). This can be explained by that under a stronger dc bias V , electrons from the right - side metal sur face of the gap need to absorb more photons to overcome the potential barrier with increased height in the gap for the emission , as shown in Fig ure 5. 7 (d) . T hese observed changes are also well reflected in Fig ures 5.8 (a) and 5.8 (b) which show the total time - averaged tran smission current density from left (Fig ure 5.8 (a)) and right (Fig ure 5.8 (b)) metal surface of the nano gap of F ig ure 5.6 (a) as a function of laser field with different applied dc bias . Here, t he increasing (decreasing) slope of the curve of with dc field manifests the shift of main emission process to the larger (smaller) n - photon absorption. Also, the slop of versus follows the power - law scaling of photoemission , indicating the dominant n - photon emission process . The value of n is consistent with the observation in Fig ures 5.7 (a) and 5.7 (c) ( cf. the cases with = 1 and 3 V/nm)). 94 Fig ure 5.7: Photoelectron energy spectra for dc field (a) = 1 and 3 V/nm and (c) = - 1 and - 3 V/nm . Emission mechanisms when (b) = 1 and 3 V/nm and (d) = - 1 and - 3 V/nm. Here, laser field = 1 V/nm and gap distance d = 5 nm. Figure 5.8: Normalized total time - averaged emission current density as a function of laser field for various dc fields . The gap distance d is fixed at 5 nm. The black dashed lines display the scale . Here, n = 2.8, 1.9, 6.5 and 12.4 when 1, 3, - 1, and - 3 V/nm is consistent with the observed orders of domination multiphoton emission channel in Figures 5.7(a) and 5.7(c). In F ig ure 5.9 , we plot the total time - averaged emission current density as a function of gap distance d under various dc fields and laser fields . Without the dc field ( = 0), the emission current from left or right metal surface would continuously oscillate around the current from 95 a si ngle surface (cf. the dashed line s ) as d increases, which is due to the interference of electron plane waves inside the gap, and the oscillation amplitude remains unchanged, due to the exclusion of image charge and space charge effects in our calculation . After applying a strong dc field , it is found that the oscill ation behavior in the photocurrent is gradually suppressed with the increasing d , and the left - to - right emission current eventually approaches that from single metal surface. Besides , our calculation shows with a very narrow gap ( d < 0.5 nm), the emission current from the left and right surfaces has the same order of magnitude, regardless of applied laser intensity and dc bias . This is because the gap - dependent direct tunneling emission dominates the transmission . As the gap distance d increases, c ompared to the emission current from left metal surface, the current from right surface is more greatly suppressed . This manifests that varying the gap distance is able to greatly tune the dc - induced rectification on the photoelectron emission in a nanogap . Our calculation displays the gap distance of larger than 1 nm is enough to achieve full rectification. Fig ure 5.9 : Normalized total time - averaged emission current density as a function of gap distance d for different dc fields and laser fields . Dashed lines denote the emission current density from single surface, which is obtained from Ref erence [ 6 ]. Figures 5 .10 (a) and 5 .10 (b) show the total emission current density as a function of dc field under different laser fields . The calculated exponentially increasing and decreasing trend with dc field exhibits good coincidence with the numerical simulation ( see Fig ure 3 in R ef erence [ 66 ]). F igure 5 .10 (c) displays the net emission current density, defined as the difference 96 between the left - to - right and the right - to - left emission current, as a fu nction of dc field for laser field = 0.4, 0.8 and 1 V/nm . It can be seen that as dc bias approaches 0, the net emission current also approaches the minimum value of 0. When the dc bias increases, the net current exponentially increases, indicating it is gradually dictated by the left - to - right photocurrent. This uncovers the rectification effe ct of external dc bias on the photoemission in a nanoscale gap [ 17 ]. In Fig ures 5.11 and 5.12 , we plot the spatiotemporal evolution of emitted electron density from left and right metal surfaces respectively, under different combinations of dc and laser el ectric fields. Here, the gap distance d is fixed at 5 nm. As clearly seen in Fig ure 5.11 (a), with the external dc field = 1 V/nm and laser field =0.1 V/nm, parts of electrons emitted from left metal surface are reflected back and forth inside the gap, which is in line with the numerical simulation results in Fig ure 2(a) of Reference [66] . Also, o ur calculation shows increasing the laser field c auses Fig ure 5.10 : (a),(b) Normalized total time - averaged emission current density as a function of dc field for different laser fields . (c) The dependence of net emission current density on the applied dc bias for different laser fields . Here, gap distance d is fixed at 5 nm. 97 more oscillatory emission features within the gap (cf. Fig ures 5.11 (a) - 5.11 (c)), which is due to the stronger quiver motion of emitted electrons under strong laser electric fields. When adding a large dc field = 5 V/nm ( see Fig ures 5.11 (d) - 5.11 (f)), dc field - like electron emission pattern dominates the whole regime, and due to the strong acceleration, electrons enter the r ight - side metal with higher velocity (cf. the slope of classical trajectories). On the other hand, for the photoelectron emission from right surface ( see Fig ure 5.12 ), the addition of 1 V/nm dc field confines most of electrons inside the vacuum gap , and o n ly when the laser field is increased up to 8 V/nm could a small part of electrons escape from the gap into the left metal ( cf. Fig ure 5.12 ( c )). Similar trend with the increasing is observed in Fig ures 5.12 (d) - 5.12 (f), except that most electrons are constrained in the strong surface oscillation regime when applied dc field is 5 V/nm. Fig ure 5.11: Time - dependent emission electron density from left metal surface of the nanogap in Fig ure 5.6 (a) as a function of time t and space x under various combinations of dc and laser fields. Solid white lines show the corresponding classical trajectories [ 76 ] . Dotted white lines show the laser electric field. Here, gap distance d is fixed at 5 nm. The units of dc field and laser field are V/nm in all figures. 98 Fig ure 5.12: Time - dependent emission electron density from right metal surface of the nanogap in Figure 5.6(a) as a function of time t and space x under various combinations of dc and laser fields. Here, gap distance d is f ixed at 5 nm. The units of dc field and laser field are V/nm in all figures. 5.3.3 Summary on Photoelectron Transport with DC Bias In this section , by exactly solving the TDSE , we present analytical model s for photoelectron emission fr om left - and right - side surfaces of a dc - biased nanoscale metal - vacuum - metal gap driven by a single - frequency laser field . Our results reveal the underlying photoemission process, time - averaged emission current and spatiotemporal dynamics of photoelectrons from both sides of the nano gap under different combinations of dc bias, laser fields and gap distances. Our calculation shows the addition of a large dc field can greatl y reduce the interference effect induced oscillation in the total emission current , and demonstrate s that in addition to the applied dc bias , changing the gap distance is also able to achieve strong rectification to the photoelectron emission in a dc - biased nano - vacuum gap. Our results may be helpful for the future design of ultrafast optoelectronic devices, such as photodetectors. 99 CHAPTER 6 SUMMARY AND SUGGESTED FUTURE WORK 6.1 Summary In this thesis , we develop quantum analytical model s to study nonlinear ultrafast optical - field induced photoelectron emission from biased metal surface s, by exactly solving the TDSE. We consider two - color laser induced photoelectron emission with and without dc bias, interference modulation of photoemission using two lasers of the same frequency, nonlinear ultrafast photoemission from a dc - biased surface triggered by few - cycle laser pulses, and laser induced photoelectron transport in nanogaps. Our analytical solution is valid for arbitrary laser parameters, including laser frequency, intensity, relative phase between two lasers, pulse duration, repetition rate , carrier - envelope phase , applied dc fields, gap distances , metal work function and Fermi level. Various emission processes, such as multiphoton over - barrier emission, dc - assisted optical tunneling emission and dc or optical field emission, are all included in our simple formulation. We provide comprehensive analysis of the photoelectron emission properties under different comb inations of laser parameters and dc fields. Under the illumination of two - color laser fields, our results show strong tunability on the photoelectron spectra, emission current, and current modulation, via the control of the phase delay, relative intensity , harmonic order of the two - color lasers, and dc bias, exhibiting good agreement with the experimental measurements . Application of our model to time - resolved photoelectron spectroscopy is demonstrated. Our study also suggests a practical way to maintain a strong current modulation, in the meantime, greatly increase the total emission current in two - color laser - induced electron emission, by simply adding a strong dc bias and a weak harmonic laser. 100 For the two - same frequency lasers induced photoelectron emission, we find strong interference modulation on electron emission can be achieved with low threshold value of the laser field ratio even with a strong dc field. Our study demonstrates the capability of using interference modulation by single - frequency laser pairs for practical measurements of time - resolved photoelectron energy spectra . With few - cycle laser pulses, we identify the new signature of coherent interaction of adjacent laser pulses on photoemission , that is , the emitted charge per pulse oscillatorily changes as the laser pulse separation increases . For a well - separated single pulse, our calculations recover the experimentally measured features of sinusoidal CEP modulation to photoelectron emission and vanishing CEP sensitivity with a phase shift in strong optical - field regime. Moreover, we find adding a large dc field is able to greatly enhance the photoelectron current and shorten the current pulse. For the photoelectron emission in a metal - vacuum - metal nanogap, our calculation reveals the underlying photoemission processes, including direct tunneling, dc - assisted optical tunneling and over - barrier emission, and the transition between them, under different combinations of gap distance increases, dc bias and laser fields. For the zero dc field, our results show the oscillatory dependence of photoemission current on the gap distance in the multiphoton regime and the energy redistribution of photoelectrons across the two interfaces between the gap and the metals. We also find that decreasing the gap distance (before entering the direct tunneling regime) tend s to ex tend the multiphoton regime to h igher laser intensity . With the addition of large dc bias, the interference induced oscillation in photo current from metal - vacuum interface of the gap is found to be significantly reduced with the increasing gap spacing. Additionally, our calculation demonstrate s 101 that besides the applied dc bias, changing the gap distance is also able to achieve great rectification to the photoelectron emission in a dc - biased nano - vacuum gap. 6.2 Suggested future work As the works in this thesis are analytically solving the TDSE exactly, it is important to compare our solutions with those of perturbative treatments widely used in th e literature, and the inverse LEED and LEED wave functions used for scattering problems [67][ 118 ] - [120] . It is also important to consider the effect of space charge in the electron emission process [ 34 ] [36] . Suggested future work would also include the theoretical modeling of ultrashort pulsed laser induced photoelectron transport in nano - vacuum gap s and the rectification effects in nanogaps formed with dissimilar materials. It would also be interesting to study the effects surface states and materials ( e.g. , semiconductor and two - dimensional materials) by considering the energy dependent electron supply function inside the material and work function variations along the emission surface in the future . Ultimately, it is envisioned to build a hybrid model using our exact analytical solution s for simulating electron emission in practical geometries, such as sharp metal tips or cathodes with surface roughness, where effects such as the electron emission a ngle and space charge can be incorporated. The time - dependent field distribution near the emitter may be first calculated using a Maxwell solver. Next, our exact model can be applied along the surface of the emitter to give the instantaneous photo emission current. The emitted electrons can then be loaded into particle - in - cell pusher to account for the detailed space charge effects and electron dynamics. Once such a tool becomes available, it would find immense applications in various areas, such as solid - st ate physics, strong fields, ultrafast sciences, vacuum electronics, and accelerators and beams. 102 APPENDI CES 103 EXACT SLOTUION OF ELECTRON WAVE FUNCTION Following Truscott [6][11 7 ] , the time - dependent potential energy for x (see Appendix A, B, and C) or (see Appendix D and E ) can be written as . Thus, the TDSE can be transformed to the coordinate system , t , where , the displacement , and , by assuming that , with , and being a constant . Then, w e have, , ( A1 ) with . By separation of variab les, in Eq uation (A1) can be easily solved. From , we obtain exact solution of electron wave function. A PPENDIX A: Two - color laser induced photoemission without dc field Based on the method above, we have the potential energy , with and in the vacuum ( x ) , and . (A2) From , we obtain Eq uation ( 14 ) with . A PPENDIX B: Two - color laser induced photoemission with dc field We have the potential energy , with , and in the vacuum ( x ) , and 104 , (A3) where is the solution of the equation , where [81][86] . From , we obtain Eq uation ( 21 ) with . APPENDIX C: Few - cycle laser pulses induced photoemission We have the potential energy , with and (see Equations (33) and (34)), and , (A4) where is the solution of the equation , where [ 81][86] . From , we obtain Equation (35 ) with and . For the special case of dc field and carrier - envelope phase , with n being an integer, the solution of is revised by merely displacing in Equation (A4) with exp[ i . A PPENDIX D: Photoelectron transport in a nanoscale gap without dc bias We have the potential energy , with and , and 105 . (A5) denotes the electron wave travelling towards + x electron wave travelling towards x direction. Due to the reflection of electron waves at metal - vacuum surfaces of x =0 and d (see Figure 5.1 ), the electron wave function in side the vacuum gap ( ) should be the superposition of wave functions towards + x direction and x direction. Then, from , we obtain E quation (45) with . A PPENDIX E: Photoelectron transport in a nanoscale gap with dc bias For the photoemission from left metal - vacuum inter face of the gap in Figure 5.6(a), we have the potential energy , with where and , and (A6) where is the solution of the equation , where [81][86] . Here, denotes the electron wave travelling towards + x + wave travelling towards x direction. Due to the reflection of electron waves at metal - vacuum surfaces of x =0 and d (see Figure 5.6(a) ), the electron wave function inside the vacuum gap ( ) should be the superposition of wave functions towards + x direction and x dire ction. 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