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Ana‘s? ......u..:)...\...fir.\.>.wflmu ...: IL .. u. I... x179... 7...; This is to certify that the thesis entitled A Rayleigh-Brillouin Spectrophotometer presented by Stuart J. Gaumer has been accepted towards fulfillment of the requirements for Ph .D . degree in Chemistry émé 6 (gm/’7‘“) / Major professor U Date February 201 1973 0-7639 ABSTRACT A RAYLEIGH-BRILLOUIN SPECTROPHOTOMETER BY Stuart J. Gaumer A Rayleigh—Brillouin spectrophotometer was designed, constructed, and characterized. This instrument consists of: (l) a vibration free optical table, (2) a variable scattering angle mechanism, (3) a temperature controlled sample cell system, (4) lasers, (5) a piezo-electrically scanned Fabry-Perot interferometer and, (6) an analog electronic detection system augmented by a photon-counting- digital-data collection system. Furthermore, in order to provide versatility, the optics may easily be changed to accommodate different types of scattering experiments. The versatility, accuracy and precision of this apparatus were characterized by performing some classical Rayleigh light scattering experiments, measurement of depolarization ratios and their temperature and wavelength dependence, and some more modern experiments such as the eval- uation e — O — .— R 2n _. _J ”m (2-13) wherein we have introduced the definitions: IO and ki are the intensity and wave vector of the plane-polarized incident beam (a time averaged intensity), R is the distance from the scattering volume element to the 18 observation point, ¢ is the angle between R and the electric field vector of the incident wave, k E n(kO—ki) is the change in wave vector in the media of refractive index n and is given by k = [5| = 2nki sin (6/2) (2—14) for this quasi-elastic scattering at the observation angle 0, p(k,t) is the spatial Fourier transform Idgé p(£,t)ei}i°£ and >, mathematically equivalent to g;>" is the Fourier associated, time dependent density-density correlation function. Many of the mathematical properties and physical interpretations of this double ensemble average are elucidated by Van Hove, Komarov and Fisher. In summary, let it be said that 'b is a conditional ensemble average. It is the average of the values which o*(k,t) can have at time t after it had the initial value o*(k,o). After obtaining this ensemble average, one calculates the final ensemble average of all the possible weighted products of this with the initial values p(k,o). Linearized Hydrodynamic Liquid Theory According to the work of Mountain (22—27), Nichols and Carome (28), this conditional ensemble average may be 19 obtained from an equation of motion originating from a linearized hydrodynamic theory of the liquid. The pertinent equations are (l) the continuity equation: Sp —— + p0 V'X.= O (2‘15) 3t (2) the longitudinal part of Navier-Stokes equation modified by the addition of time—dependent viscosity terms (for the justification of this see (27): 3v _ _ _ i . mpo 3; - VP1 + (3 ns+nv) VIV K) 1 N t +2 [ndt-t') vtv-yt'ndt' (2-16) j: and (3) the energy transport equation: as 2 moOTO ;;-= K V T (2-17) In these equations: 0 = 00 + p1(£,t) (2-18) T = TO + Tl(£,t) (2—19) p0 and T0 are space-time number averages, p1, T1 and p1 are the excess number density, temperature and hydrostatic 20 pressure respectively, S is the entropy per unit mass, m is the molecular mass, K is the thermal conductivity, Us ”V are the frequency independent shear and volume (intrinsic) viscosities respectively, N is the number of frequency- dependent viscosity terms nj(t) and Z(£,t) is the irrota- tional part of the local velocity field whose average is zero. To solve these phenomenological equations, Mountain et a1. invoked the Laplace time-frequency transform: 00 0(11,S)= f e‘Sto<1_<_,t)dt (2-20) 0 and obtained the solution: k? a sin ¢ 1 _ 2 _ 1(0)) _ IO R <|p(l_<_,o)| >o(k,w) (2 21) wherein1 F(k,iw) O(k,u0 == 2 Re ———-———— (2-22) G(k,iw) bi k2 F(k,s) = 82 + S ak2 + bk2 + E + abk'+ 1 (l+sri) ab.k" 1 2 + 202 k (y—l) (2-23) 1 (1+sTi) Y + 1Inherent in this derivation is Mountain's (22) conversion of the memory factors of equation (2-16) to a form which utilizes Ti, the time constant associated withrk. 21 bik2 G(k,s) = s3 + 52 ak2 + bk2 + Z —_—————— (2—24) 1 (1+sr ) ab k'+ av2 k1+ + s vék2 + abk“ + E + 1 (1+sr.) y a E K/oOCV _ (2-25) b:(4 +) 22 ' 3 ns nv /mpo ( - 6) bi : ni/mpO (2-27) The expression for the structure factor o(k,w) can be converted into a form more convenient for discussing the spectra, namely partial fractions. To do this, one first converts the ratio of the F(k,s) and G(k,s) functions into a ratio of polynomials, FN(k,s) and GN(k,s), by multiplying both the numerator and denominator by the product of the (1+STi) terms. Then one (a) calculates the roots of the dispersion equation, GN(s) = O, (b) constructs a factored form of GN(s) and (c) splits o(k,w) into partial fractions using these factors as the denominators. For a liquid with a single relaxation process (N=1, for example carbon tetrachloride), these equations take on the forms: G1(s) = T1(s+PB-imb)(s+PB+imB)(s+TbT)(s+FCI) (2—28) 5 IET IbI o(k,w) = 2 ACT F:-:;;- + Cl (2-29) CT B 22 — TB TB +A +— BS Fg+(w+wB)2 FE+(w—wB)2 .J (2-29-- w+wB w—wB 2 cont.) +A — BA 2 2 2 _ 2 > FB+(w+wB) FB+(w wB) s The roots of equation (2-24), or Brillouin spectral parameters, (line widths and position) are related to the thermodynamic parameters of the liquid through some rather complex relationships. The purpose of experimental Brillouin spectroscopy is to determine Umespectralparameters directly. It is more convenient to compare experimental and theoretical values of thespectral parameters using numerial techniques rather than analytical expressions to calculate the roots of equation (2—24). Approximate expressions for these roots are available in the literature (22,23,26,28). Since Brillouin spectroscopy deals only with o(k,w), we can ignore absolute intensity measurements and fix the coordinate scale factor by requiring o(k,w) to integrate to unity. The other factors in the I(w) expression are related to classical light scattering experiments and are discussed extensively elsewhere. Fabry-Perot Interferometry To study the fine structure of a Brillouin spectrum, one needs a device of very high resolving power. Optical homo- or heterodyne techniques are of extremely high 23 resolving power but have a very limited spectral range; double grating and many interferometric techniques would easily scan the entire spectral range but have an insuffi- cient resolving power and present many other problems from a practical point of view. Fabry-Perot interferometry compromises these two extremes and presents the best method for resolving Brillouin spectra. The applications of Fabry-Perot interferometers have become extremely varied; these interferometers are used for Spectral resolution (dispersion) and as tuned resonant circuits for Optical generators, namely, cavities for lasers. The Fabry-Perot interferometers (first described by Fabry and Perot in 1897) originally consisted of a pair of plane—parallel mirrors. Later, Connes extended the concepts and technology to a pair of spherical mirrors. For laser cavities spherical mirrors are used primarily because it is easier to obtain higher finesse (resolving power) with them than with the plane—parallel mirrors from an experimental point of View. However, the spherical mirrors are not as versatile as the plane-parallel mirrors for Brillouin spectral analysis because their free spectral range is fixed. The theory (and application) of Fabry-Perot inter— ferometry has been discussed in the literature extensively (44-59). However, in order to conveniently explain the light scattering instrument later, the essential features will be presented here. 24 The physicalphenomenonresponsible for Fabry—Perot interferometry is the constructive or destructive recombina— tion of a multiple reflected beam. This is illustrated in Figure 2.1(a). The transmission of this multiple-beam interferometer is usually given by the analytical expression (an Airy function): It(9,>\) /T 2 l TIGIA)E = -—- (2—30) IO(0,A) l-R 4R 1 + sin26/2 (l—R)2 or TF 2 l TUBA) = — (2-31) n ' 2F 2 1 +'—— sin 6/2 n wherein: n/R H F E finesse = 2 for R——-rl.0 (l-R) (l-R) 4n 6 E scanning parameter = —— d cose A d E optical path separation of mirrors = n1 n E refractive index of media between reflecting surfaces 1 E physical separation of mirrors R = JR R2 E the geometric mean reflectivity of the two 1 mirrors 25 (a) IOO°/or Im+nfl' I"" I I / o Lo. o/ m“. 4 Ru~ \ \\ 7 III. \II \III . m I”" I\ I I / $25: >h_mzwbz. (b) Fabry—Perot Interferometry. Figure 2.1. 26 T = VTIT2 E the geometric mean transmission of the two mirrors 6 E angle of incidence In the derivation of this expression it was assumed that the mirrors were perfectly flat, perfectly parallel to one another, extended infinitely- perpendicular to their axis and do not absorb or scatter light. These assumptions are difficult to realize in practice. The imperfect nature of a practical instrument is conveniently dealt with by re— defining the finesse as: (2-32) WI hill—4 Each Fi is associated with one of the finesse degrading factors and is given by: .II/fi Reflectivity limited finesse, FR = (2-33) (l-R) Mirror-figure—of—merit limited finesse, A F = —— (2—34) f 2A A Mirror—tilt limited finesse, F = ——— (2-35) 2d¢ Aperture diffraction limited finesse, D2 F = (2-36) D 2m (for normally incident beam) 27 and Beam attenuation limited finesse, H F = — (2—37) L L wherein: L E (small) loss due to absorption and scattering in a single transit of the interferometer D E effective aperture diameter (at mirrors) A E mean wavelength o E mirror tilt angle in radians A E RMS mirror surface deformation Ff : m/2 if the mirrors are flat to A/m Historically, the Fabry-Perot spectra were recorded photographically; the various rings, illustrated in Figure 2.1(a), were photographed and the photograph was scanned with a densitometer across a diameter to obtain the spectra illustrated in Figure 2.1(b). However, in modern inter- ferometry, a pinhole and photomultiplier tube replace the photographic plate and the spectraare scanned by varying one of the variables (especially n or 1 but sometimes 6) in the scanning parameter 6 above. Our interferometer varies 1 by applying a voltage to a piezoelectric transducer which supports one of the mirrors. Instrumental parameters related to the spectra are: The spectral separation of two adjacent transmission maxima 28 is termed the free spectral range and determines the maximum frequency span which a spectrum may have (in other words, the maximum spectral range which can be observed) without overlap from adjacent transmission modes. The width (full width at half of maximum) of each of these transmission maxima, the apparent bandwidth observed for a perfectlly monochromatic incident beam, is termed the instrumental bandwidth. It determines the minimum resolvable Spectral increment. The ratio of the free spectral range to the bandwidth is equal to the experimental finesse. The acceptance angle is the maximum full-cone angle over which the incident beam may be collected without degradation of a specified finesse; it is given by A 1/2 A6 _<_ 2 —-— (2—38) Fd Expressions for these instrumental parameters and their dimensional units are summarized in Table 2.1. The effective parallelism and flatness of the mirrors may be greatly improved by selecting a high quality portion of the mirror's total aperture. However, in addition to reducing the beam intensity (hence, subsequent electronic signal to noise ratio), reducing the interferometer's aperture degrades the interferometer's effective finesse through walk-off and diffraction effects. An aperture diameter which maximizes the net finesse, the best 29 TABLE 2.1.--Instrumental Parameters for Fabry-Perot Interferometer. Wavenumber Wavelength Frequency Free spectral l A2 C range A0* = — AA* = __ Av‘k = _— 2d 2d 2d Bandwidth do 6A 6v Ao* AA* Av* Finesse, F F = F = F = do 6A 6v compromise between misalignment and diffraction or walk-off effects, is given by the expression: 1/3 Azd Dopt = (b (2‘39) when the mirrors are tilted at ¢ to each other. When a photoelectric method is used to record Spectra,a lens, focal length f focuses the beam onto a LI plane which contains a pinhole. The radii of the Fabry- Perot rings associated with different angles of incidence, d, of a monochromatic beam are given by Co = fL tan a (2-40) From this eXpression, we see that if the radius of the pinhole is too large, the instrument's finesse is obviously destroyed. However, the pinhole radius may 30 be increased to a point corresponding to the acceptance angle without degrading the instrument's linewidth. The pinhole may be made conveniently large (approximately 2 mm dia., assuming an acceptance angle of 1 milliradian and a focal length lens of 100 cm) by using long focal length collection lenses. Alignment and calibration will be explained later, in the chapter on the alignment of the entire instrumental system. Also, the inherent compromise between the inter- ferometer's free spectral range and bandpass, as well as serious stability problems will be discussed. Polarization Coherency Matrix With the invention of the laser, as well as many other improvements in modern optical technology, the matrix representations of optics are becoming more prevalent and provide great utility. The Jones calculus is especially useful to discuss the statistical properties of a propagating electromagnetic field and the action of optical devices on the beam relative to intensity, polarization and coherence. This formulism is extensively elucidated in the modern optics literature (44,60-63) and will not be given detailed consideration. However, for the reader's con- venience, the definitions, methods, etc. used in this work will be summarized. With the more classical vector representation of the monochromatic electromagnetic field, IF 31 g = (2—41) many physical properties of some propagating beams, for example an unpolarized (natural) or partially coherent beam, are difficult to discuss. To ease this difficulty we mathematically formulate the polarization coherency matrix (PCM) representation of the radiation field; namely xEx’:> Jxx ny d = (£err = } * * imply a time average. If we represent an instrument by the operator L, then the incoming E_and out- going E' fields are related by matrix multiplication: E' = L E (2-43) or, in terms of the PCM representation: d' = LuJ L+ (2-44) The PCM representation of some of the common (quasi- monochromatic) radiation fields and optical instruments are tabulated in (60). The total intensity of a field is given by: I = Tr J = J + J (2—45) xx yy — 32 The degree of polarization, P, is defined as the ratio of the intensity of the polarized part of the field to the total intensity of the field; namely Ipol P = (2-46) Itot and its definition in terms of the coherency matrix is: 4 det.J P = 1 r ——————— (2-47) (Tr J)2 The depolarization ratio is: o = = —— (2-48) Ivert Jxx The normalized cross—correlation function uxy is defined by J Xy u = (2-49) XY FER—y and gives the state of wave polarization. The utility of the PCM evolves from its measurabi4 lity. There are several sequences of measurements which will yield the PCM elements. The choice here was arbitrary; a rather extensive discussion of the chosen technique may be found in Born and Wolf (44). Basically, 33 this consists of placement of a retarder and polarizer into the beam and the measurement of the intensity for various orientations of them. If the retarder introduces a phase delay, 8, of the y component relative to the x component and the polarizer is at an angle 0 with respect to the x axis, then it may be shown that the transmitted intensity is given by: I(6,E)=J c0528 + J sinze + J e_l€cosesine + J elecosesine XX YY XY YX (2-50) In particular JXX = I(0,0) (2-51) = I n 2,0 2-52 yy (/ ) ( ) _ l n 3n is n n _ 3n n ny — ~2r 1(ZIO) 1(TIO) + EIIIZI-z') 104—"‘20 (2-53) _ l I I _ fl _ .1' I II _ All (2-54) if the different values of 8 correspond to different rotations of the polarizer but the two values for E, and O and 1V2, correspond to a quarter wave plate being in or out of the beam, respectively. 34 Convolution and Deconvolution Since the frequency distribution of the incident radiation and the transmission of the interferometer are not delta functions, the experimental spectrum is a broadened version of the true spectrum of scattered light. The experimental spectrum Ie is related to the true spectrum I by the convolution relationship t 00 Ie(w) = .[ f(w-w') It(w')dw' (2-55) —m wherein f(w-w') is the normalized instrumental profile which may be measured by directing a divergent laser beam into the detection Optics or approximated very well by the central peak of a pure liquid. w is the experimental frequency and w' is a dummy integration variable. However, a scanning interferometer has several transmission modes Spaced at a free spectral range (Af) apart. Thus, the observed spectra (Iobs) is really the overlap of several of the spectrum Ie’ namely: 00 = 2 Ie(w + n ZWAf) (2-56) n=-00 Iobs(w) or 00 IO (w) = E J. f(w+n2nAf-w')Ie(w')dw' (2—57) —00 35 Now, we recognize this as the sum of convolution integrals and apply the convolution theorem from Fourier analysis to obtain (D F[Iobs(w)]= 2 F[f(w+n2nAf)] F[1e(w)] (2-58) n:—oo Factoring out the "constant" factor F[Ie(w)]. recalling the shift theorem from Fourier analysis, that is F[f(w+n2nAf)] = F[f(w)] e”in2“AfC (2-59) and then regarding F[f(w)] as "constant" factor, we obtain FtlobSIw>J = FEIeIw>J F[f(w)] 2 e'in2"AfC nz—oo (2-60) or CK) FEIObS(w)] = F[Ie(w)] F[f(w)][l+2 X cos(n2nAf§)] (2-61) n=1 From this expression, if we measure IO S(w) and f(w), we may b calculate (deconvolute) the true spectrum: _ FEIobS(w)] I 1(w) = F (2-62) 00 F[f(w)][1+2 X cos(2nAf§)] 3 n=1 If the free spectral range and finesse of the interferometer are large enough, only the adjacent 36 interferometer modes contribute significantly to the measured spectrum and the above relation becomes: _ I F[Iobs(w)1 I(w) = F ' (2-63) I F[f(w)][l+2 cos(2nAf§)] Utilizing the numerical technique of Singleton (65), Cooley and Tukey, namely the fast Fourier transform (FFT), we may numerically evaluate these Fourier transforms (and inverse transform) easily and rapidly on a computer. We only need the data sets (experimental Spectra and instru— mental profile); we do not need to make any assumption relative to the analytical form of the instrumental profile or the observed spectrum. Furthermore, if the central point of either data set is really off center by we or mi, and if Iobs(w) and f(w) have the Fourier transforms Fe and FJ.- respectively, then, using the shift theorem: iwec FEIObS(w)] e F RE :._—_—____——: F[f(w)1 eiwic F i(we-wi)C (2-64) 1 Then, multiplying by the complex conjugate: RR* -_— .._____._.. (2'65) 37 or F lel IRI = (2-66) IR! 1 Thus, we see that a data set need not be centered with extreme accuracy. The Fourier transforms of off center data sets will be complex but if one takes the real part of this complex quantity, the data sets are inherently centered without knowing we and mi. CHAPTER III DATA ANALYSIS TECHNIQUES Introduction In order to study Brillouin spectroscopy conveniently, it was found useful to have available three computer pro- grams: (1) a program for the simulation of Brillouin spectra assuming various models; (2) a program to eliminate instrumental effects from an experimental curve and generate the true spectrum; (3)a program for the deduction or esti- mation of spectral parameters from an experimental data set after it was corrected for instrumental effects. The first and last of these programs will be discussed in following sections. Several programs were written for the purpose of eliminating particular experimental effects but since data collection procedures were not standardized, no single correction program was adopted. Experimental corrections include curve smoothing, baseline correction, frequency scaling, intensity scaling, shift to a more convenient data point spacing, deconvolution of the instrumental profile from the experimental curve, and elimination of the con- tributions from adjacent interferometer modes. These last two effects must be considered in the quantitative discussion of any spectrum. 38 39 Spectra Deconvolution The mathematical procedure to deconvolute the instrumental profile and eliminate the effects of adjacent interferometer modes is indicated in equation (2-63): (1) calculate the Fourier transform of the observed experimental spectrum,(2) calculate the Fourier transform of the instrumental profile, (3) divide this latter function into the first function, (4) divide by a trigonometric factor to eliminate adjacent interferometer mode effects (One now has the Fourier transform of the true spectrum.), and (5) take the inverse transform in order to obtain the true spectrum. Inverse Fourier transformation is equivalent to calculating a Fourier transform; thus, the important aspect of this deconvolution procedure is the numerical evaluation of a Fourier transform. These numerical tech— niques are discussed in the literature (64-69). The numerical technique adapted in the subroutine FFTSl is based on the Cooley—Tukey algorithm (65) for computing finite Fourier transforms. This technique is commonly referred to as fast Fourier transform. The calculation of a finite Fourier transform is advantageous over the calculation of a normal Fourier transform in that it involves numerical integration over a finite interval instead of an infinite interval. However, in order for this finite Fourier transform to be appropriate, the original function must lFortran listings of the computer programs and subroutines described in this work are available from the laboratory. 40 be aliased. Aliasing is the objective Of the subroutine ALIAS. The theory and details Of this technique are discussed by Cooley (64); briefly, aliasing involves the transformation of a general function into a periodic function. FFTS integrates over the aliased function's period. The details of applying ALIAS and FFTS depend upon the data collection techniques. Since these techniques have not yet been standardized, it is not conveniently possible to discuss in detail the application of these subroutines. In order to check the FFTS subroutine it was applied twice to several functions (gaussian, lorentzian, voigt and linear combinations Of these), and the result compared to the original function. The differences, representing errors in the calculations, were small enough in magnitude to be associated with the number Of significant digits carried internally by the computer. Brillouin Spectra Simulation A computer program designated as BRILSPEC was written to simulate a Brillouin spectrum assuming a particular model and a corresponding set Of parameters. In addition to the single relaxation model represented by equation (2-29), a double relaxation and a trilorentzian model were incorporated as Options in BRILSPEC. A double relaxation model is also defined in the chapter on theoreti- cal basis of Brillouin spectra (N=2); the trilorentzian 41 model is not. The trilorentzian model assumes that the Brillouin spectrum is composed Of three partially over- lapping lorentzian peaks, one central lorentzian peak and two additional lorentzian peaks shifted symmetrically from this central peak. The utilization of BRILSPEC first requires the choice Of a model and its characteristic parameters. The data input cards related to the possible models are illustrated in Tables 3.1, 3.2 and 3.3. For different models, only the third (and following) data card varies considerably. This card is formatted as 4F20.5 and the sequence of parameters is: amplitude, halfwidths, and shifts from central peak. By amplitude, we do not mean peak height; these amplitudes correspond to A in the following equation for a lorentzian peak: A B2+)<2 Table 3.3 is for a single relaxation model; the assumption Of other models corresponds to the insertion or omission Of corresponding Ai and Bi values in the sequence indicated in Table 3.3. If one chooses to have the computer plot data, some additional data input cards are required by the subroutine GRAPH. An explanation Of these data cards, and this subroutine, may be found in the instruction manual, "Subroutine Graph". by D. L. Knirk (70). 42 TABLE 3.l.--First Data Card for BRILSPEC. Parameter Column Format Code Definition and Comments JPTYP 5 I5 1 assumption of trilorenzian model 2 assumption Of single relaxation model 3 assumption Of double relaxation model blank exit, no more work TABLE 3.2.-—Second Data Card for BRILSPEC. Parameter Column Format Code Definition and Comments NP 1—5 15 number of data points from center out DW 6-20 F15.5 width of each data interval NKEY 25 15 0 calculate only half of spectra 1 calculate entire spectra NPLT 30 15 0 no curves 1 output curves TABLE 3.3.--Parameter Data Cards for BRISPEC. Parameter Column Format Definitions and Comments Al 1-20 F20.5 amplitude of left peak A2 21—40 " amplitude Of "primary" central peak A3 41—60 " amplitude of right peak A4 61-80 " amplitude of "secondary" central peak Bl 41-60 " half width Of left peak B2 61—80 " half width of "primary" central peak B3 1-20 " half width Of right peak B4 21—40 " half width of "secondary" central peak WL 41-60 " shift Of left peak WR 61—80 " shift Of right peak AL 1—20 " amplitude Of asymmetric component Of left peak AR 21-40 " amplitude Of asymmetric component of right peak Hp 44 Figures 3.1, 3.2 and 3.3 indicate the use Of this computer program in a qualitative sense. The different figures correspond to different choices of the parameters for the trilorentzian model. These calculations were initially undertaken to illustrate the relative effects Of relative peak heights, line width and line shifts on the overlap of the side peaks with the central peak. However, the most important application of this computer program is its conversion to a subroutine and subsequent incorporation into a computer program for the estimation of model parameters from an experimental curve. Brillouin Spectra Parameter Estimation A Brillouinspectrwmnwy be characterized by a set of parameters; the purpose of the computer program BSPEP is to deduce a set of "best estimates" for these model parameters from an experimentalspectrum. Ewmthermore, by comparing the results Obtained for a few different models, one may ascertain which model best describes the data. Figures 3.4 and 3.5 represent the initial and final states when a single relaxation model was assumed to describe the data. Figures 3.6 and 3.7 represent the initial and final states for the same data except a trilorentzian model was assumed. One may easily conclude that the single relaxation model describes the data better than the trilorentzian model. 45 I I I I ,1...” I - I »\ I I I I I I I O O O O O o, 0. o_ o 0. st '0 N "' O AllSNSlNI BAIIV'IBH OO'OI OO'B 00'9 00'17 OO'Z OO'O OO'Z' 00'9- 00'9- 00'8- OO'OI - IGHz) FREQUENCY Trilorentzian Model with Parameter Set A. Figure 3.1. ‘J” 46 OO'OI 00’. .m pom Hopoemumm Spas Homo: SMHNESOHOHMHB .m.m musmflm 318 >ozwacwmu 9 t .2 .0 ..a ..v r ..u nlv o o o o o o o G o 0 0 0 0 0 0 0 0 0 -.,.IHII.IIIII . j . _ _ _ a A III lLlJllLLlLiLllllLLlLLllllJlllLlllllllll 00.0 00.. 00.N 00.» 006 EAIIV'IBH ALISNBLNI _J' K . 47 'l OOCN 0013 0013 .0 wow Hmpmfimumm spas Home: COHNOGOHOHHHB 7 O O $100 >ozm30mm... .m.m mnsmwm 001M- oogu O a co. 3 ... V H A an oo.~| N l a; N his . 1 oomA 006. 48 OO'OI .Hmpoz coapmxmaom mamcflm How OHM w>HDO HOHBHGH d 008 009 "' OO 17 00'0 ANIOV .v.m musmflm >ozw30wmm 00'9' m._.HSO Hmcwm mu 9 ..v 3 0 O O 0 O O O O u a $10. >ozm30wm... ....n. z<.N...zmm0.:mh 442.... I... 1.4...Zw2Ewaxm I S O O .m.m musmflm 00°8- 00'OI- 00.0 0.0 0N0 0m.0 0¢.0 00.0 00.0 ON. 0 00.0 00.0 I 00.. BAIlV'IBH AlISNalNI I 50 q .I d -I It d d -I q q -I qI lllllllllllllllllUlillllllllllllllllli 0 0 0 0 0 0 0 0 0 0 0 c, a: «a e. '0. '0. v. '0. “E ‘- °. -' o 0 o 0 0 0 0 0 0 0 AllSNElNI BALLV'IBH OO'OI 00'8 00'9 00'? 00'?! OO'O OO'Z" OO'b" 009‘ 008" OO‘OI" (GHz) FREQUENCY Initial Curve Fit for Trilorentzian Model. Figure 3.6. 001M .Hopoz SMHNBSOHOHHHB HOM uflm O>Hso Hmcflm .h.m ougmflm firm. >ozw30mmn. 51 . 9 .9 .7 Z 0 7.. .7 F W m 0 0 0 0 .0 . . . . o o o o o m m m m m A . ..4q1..-Hfi\u\I-IIII 00.0 x \ \I x \ / 9.0 s I z 8.0 H 3 and W n. o¢.o A 3 00.0 mm . .... 000 N S 2.0 H. A n.<.rzw.2.mmn.xm I 0040 (00.. 52 BSPEP determines the parametric values which minimize the sum of the weighted squares of the residuals. The residuals are the differences between the experimental and theoretical intensities at a given frequency. The weighting factors are inversely prOportional to the variance of the Observation. Wentworth (71) and Wolberg (72) have discussed the definition, theory and application Of these statistical concepts. It should be emphasized that minimization Of this functional (sum of weighted residuals) implies an assumption Of the data having uncor— related, normally distributed errors and that the final curve fit is not biased by the residuals being coincidentally small. Several mathematical techniques for this minimization were investigated; the numerical technique incorporated into BSPEP may be identified as an iterative—linearized—least square technique. BSPEP uses computer time most efficiently because Of analytical evaluation Of derivatives, a natural evolution of statistics describing the solution, and a lack of versatility, generality incorporated into the other previously available programs. A program using the variable metric technique of Davidon (75) was not adopted. Even though it was efficient in terms Of the number of iterations required for convergence to the final solution, it still required more computer time for each iteration and for the evaluation of statistical parameters characterizing 53 the solution. A curve fitting program (74), in which derivatives are not calculated, and a program in which one uses differencing techniques to estimate the derivatives were not adopted either. The pertinent derivatives can be analytically evaluated fast enough that their increased accuracy reduces the total required computer time. BSPEP also differs from the other linearized least square type programs in that its normal equations are solved by diagonalizing, with Jacobian rotation, the coefficient matrix rather than using straight forward matrix inversion. This procedure allows one to incorporate diagnostic least square techniques (76) which yields information relative to the linear independence Of the parameters being evaluated. Since the coefficient matrix is symmetric, this method of solving these equations is among the fastest and most accurate methods. The numerics are also improved by setting the off diagonal elements of the diagonalized matrix identically to zero before calculating the reciprocals of the diagonal elements _and "rotating the matrix back". Not only does diagnostic least squares allow one to detect linear dependence (rigorous or coincidental for a particular data set), it allows one to identify the corresponding parameters. This mathematical technique also allows one to Obtain a solution in spite of linear dependence. Even though these solutions are not rigorously correct, they may have practical value (76). 54 These computer programs were compared using some actual experimental data and some calculated data sets. The calculated data sets utilize a single relaxation model and parameters taken from a published Brillouin Spectrum. for carbon tetrachloride (28). Randomly generated errors, with different RMS values were added onto the various data sets. The detailed theory Of the techniques used in BSPEP are discussed by Wentworth (71), Wolberg (72), Deming (73), and Curl (76) and will not be explained here. A flow chart illustrating this program appears in Figure 3.8. The first data card contains computer program codes and parameters; the details Of this data card are indicated in Table 3.4. The next card contains initial estimates of the Brillouin spectra parameters; the format for this card is 8F10.5. Then the cards containing the spectral data are read in format 8E10.5. Finally, control information for the subroutine GRAPH is read in if computer plotting is desired. These cards are discussed extensively by Knirk (70). The output of BSPEP is well labelled. The printed output for a given iteration contains: 1. experimental, theoretical and difference curves 2. weighted and unweighted rms errors 3. coefficient matrix for normal equations 55 I Input 1. program parameters 2. starting values for spectral parameters 3. Brillouin spectra intensities 4. plotting parameters assuming a Taylor series linearized version of "problem"— set up normal equations and apply least square subroutine to estimate better values for the spectral parameters calculate statistics characterizing curve fit and parameter values / \ test for convergence to acceptable solution A ‘ output information for this iteration A / 1 \ Output-final information Figure 3.8. Flow Chart for BSPEP. 56 TABLE 3.4.--Program Parameters for BSPEP. Parameter Column Format Code‘ Definition and Comments ND 1-5 IS number Of data points DW 6-20 F15.5 value of frequency interval SC 21-35 F15.5 intensity scale factor NP 36-40 15 number Of model parameters NPLT 41-45 I5 0 do not plot 1 plot IOIT 46—50 15 0 do not output iteration information 1 output curve fit statistics, etc. on each iteration EPS 51-60 F10.5 criterion for convergence IMAX 61-70 110 maximum number of iteration allowed TBETA 71-80 F10.3 statistical parameter for confidence interval Of estimated parameters 57 4. diagonalized coefficient matrix 5. Jacobi rotation matrix for the diagonalization 6. adjusted-diagonalized coefficient matrix 7. rerotated (inverse of original coefficient matrix) matrix 8. new values for the model parameters After the final iteration, in addition to the above if it was requested, the printed output contains: 1. variance-covariance matrix 2. final value for model parameters 3. unbiased estimate of parameters' standard error 4. correlation coefficient matrix 5. parameter confidence interval 6. experimental, theoretical and difference curves 7. weighted and unweighted rms errors In addition to this printed output, one may Obtain some plots from a Calcomp plotter. The exact nature of these plots (size, scaling, labelling, etc.), generated by the subroutine GRAPH is controlled by information read into the computer. If plots are requested, two plots are always Obtained. The first plot contains an experimental and a theoretical curve using the initial estimates of thespectralpwuaneters, see Figure 3.4. After the computer program has converged to a solution, a similar graph is generated. The final values for the spectral parameters are utilized in the theoretical curve, see Figure 3.5. 58 The convergence of BSPEP to a solution was studied as a function of the rms experimental error. This study has a practical application in that it indicates what type Of experimental precision and accuracy are necessary if one ultimately expects to deduce a liquid's parameters from its Brillouin spectra accurately. The theoretical data sets previously mentioned were used for this purpose. If the rms error was greater than 0.2% to 0.3%, BSPEP would not converge to numerically good solutions. This deficiency was also demonstrated by the evolution of linear dependence between the halfwidth and amplitude of the secondary (smallest contribution) central peak. The 0.2% convergence limit above can be increased if one applies the curve smoothing technique of Savitzky and Golay (77). A subroutine SMUTH was written for this purpose. CHAPTER IV INSTRUMENTAL SYSTEM General Design Aspects-Introduction A schematic of the Rayleigh—Brillouin spectrometer which was designed, constructed, OptimizedIand characterized is illustrated in Figure 4.1. This block diagram divides the spectrometer into its subsystems emphasizing the instru- ment's high versatility for both photometric and spectral experiments. Briefly, the instrument consists Of optical sources suitable for performing scattering experiments on matter in any state, devices for controlling the nature of a sample, Optical analyzers and detectors which characterize the scattered radiation. In order to provide a versatile, precision spectro- meter, the following components Or sybsystems were in- corporated: (1) A convenient vibration-free Optical table was constructed. (2) Stable, single frequency helium— neon lasers were investigated extensively; however, an argon ion laser (single mode, frequency stabilized) was finally adOpted. (3) The incident Optics subsystem consists Of a rotatable mirror which can be translated along an Optical rail. This subsystem controls the direction 59 60 .kuoaouosmoupoomm SHSOHHmeIanmHMOm .H.v musmwm ....ZD _ I C23 I Jomhzoo 022.2300 TI ...HIm0mm5m 255 ..H ..m IIIIIIIIIII mkwmzw 0003».E °°°°°oo °°°°° 00 A?" o the outside Of the copper cylinder. The cooling coil is bihelically wound and its ends extend through the tOp of the assembly for connection to an 75 MERCURY OR PLATINUM THERMOMETER TEMPERATURE ONT ROL PROBE EPOXY JACKET COPPER AMPLE CELL JACKET coou~c SCATTERING COILS Lunno HEATING HHRES ASBESTOS l80° VIEWING PORT COPPER BASE PLATE INSULATIN BASE PLATE CENTERING PIN ALUMINUM BAR ON ROTATING TABLE TOP Figure 4.3. Thermostatted Sample Cell. 76 external coolant supply. Asbestos tape was wound in between the tubing windings. Over the asbestos tape a nichrome wire heating element was wound bihelically, yielding a 13.5 Ohm heating element. The rest of the copper cylinder recession was filled with an asbestos paste (aqueous slurry) and dried. The nichrome wire was connected to binding posts on top of the housing. The copper cylinder also had a 3/8 inch high viewing port machined 180° around its circumference. A 7/8 inch thick epoxy resin shell was machined to fit over the copper cylinder and is bolted to the copper cylinder, on top, so that the entire jacketing system can conveniently be lifted on and off with the sample cell setting in place. The epoxy jacket has a corresponding 180 degree viewing port. Teflon thermometer connectors usually used with tapered glass joints were remachined, threaded and screwed into the epoxy jacket on top. One connector was located so that a thermometer would protrude into the center of the liquid sample and the other connector was located Off axis so that a thermometer would protrude into the copper cylinder. A hole through the epoxy shell into the copper cylinder provides a well for the controller's probe. Two other 4 mm diameter wells were drilled for the thermistor probes of the temperature monitor. One protrudes, Off axis, into the liquid sample and the other into the copper cylinder. 77 The insulating base plate is one inch thick and six inches in diameter. It is constructed Of transite. Only two types of sample cells were used. The first is a Brice-Phoenix, model T—101, standard square turbidity cell 30 X 30 X 60 mm. The second type Of cell is a standard cylindrical light scattering cell (89) with flat entrance and exit windows; its height was reduced to 7 cm so that it would fit inside the sample cell housing. These sample cells and the dimensions Of this sample cell system were chosen because: (1) They were immediately available. (2) They are easily replaced since they are standard light scattering cells. (3) They represent a reasonable compromise between large diameter cells for which optical imperfections Of the cell have less effect on the scattered light and a large amount Of sample is required to fill the cell. A thin film Of Wakefield (NO. 128) thermal conducting compound is inserted between the sample cell's base and the copper base plate upon which the cell rested. In addition to providing a good thermal connection to the COpper base plate, the thermal compound holds the sample cell in place while it is being centered and then while the copper cylinder-epoxy shell assembly is dropped down over the cell into position. The thermal conducting compound is also used to insure a good thermal connection between the cylindrical copper shell and the copper base plate. 78 The temperature of the sample is controlled by directly controlling the temperature Of the copper block, providing good thermal conductivity between the copper block and the sample cell, and allowing enough time for the sample to come to thermal equilibrium with the copper block. With this technique, the copper block provides a thermal mass larger than a sample by itself and enhances the ease of obtaining good thermal stability of the sample. Since the accuracy of the temperature controller's setting is only i O.5°C, the temperature Of the COpper block and sample were measured directly with a mercury thermometer unless very accurate measurements were desired. For the high precision case, an Electro Science Inc. model 300 PVB potentiometer/bridge and platinum resistance thermometer accessory was used. In order to facilitate knowing when a sample has reached thermal equilibrium, and to examine .the sample's thermal stability, a temperature monitoring unit was provided. When the temperature control parameters were appropriately set and the sample was allowed to reach equilibrium (less than 2 hours for temperatures up to 100° C), no thermal gradients were detectable by temperature measurements made vertically along the central axis or the Off center probe. The only temperature differences seen could be attributed to the i 0.001° C temperature instability Of the sample. 79 Even though this sample cell thermostat system was designed specifically for liquid samples, the instrument is not limited to experiments on liquids. Colleagues have successfully designed systems for handling over types of samples (87,88). Temperature Controller The temperature Of the copper block is controlled with a Yellow Springs Instrument Company model 72 proportional controller. A proportional controller was utilized to reduce the temperature cycling associated with on-off type control- lers. A model 409 temperature probe was incorporated and the control point, within the range 0 to 120° C, can be reset to within i 0.005° C. A variac transformer was placed between the controller's output and the nichrome wire heating element with a 2000 Ohm, 140 watt bleeder resistor across the variac's input to prevent fuse overloads. The variac was adjusted so that, at the control point, the conduction angle Of the controller's SCR's was close to 90° with a bandwidth setting of 0.1° C. This increased the controller's sensitivity and hence the temperature stability. The temperature stability was increased from the manufacturer's specification of i 0.005° C to i 0.001° C as measured with the temperature monitor described below. As the manufacturer indicates, the actual control point tempera- ture may deviate from the temperature setting by i 0.5° C. 80 To facilitate a more accurate temperature setting, a measurement Of the liquid's actual temperature versus the temperature setting was performed and the results are presented in Figure 4.4. Glycerin was used for this study and a platinum resistance thermometer, with an accuracy Of i 0.003° C, was used to measure the liquid's temperature. Temperature Monitor The block diagram in Figure 4.5 depicts the tempera- ture monitoring subsystem. Two thermistor type probes were used to provide good temperature sensitivity. Yellow Springs model 44201 thermilinear thermistor components were used because they also provide a linear response. The temperature monitoring subsystem has two associated electronic recording systems: One system has a fixed 0 to 100° C temperature range for a one volt full scale recorder; the other electronic system has five ranges from 0.0l° C to 100° C (on a l millivolt full scale recorder) and a baseline which may be changed from 0 to 100° C. This sensitive, zero shifted recording greatly enhances monitoring the sample temperature. A six pole—double throw switch is used to invert the probes relative to these two different recording systems. The f 15 volt power supply (for the Operational smplifiers) is a full wave rectifier, filter network with 15 volt zener diodes across its output for regulation. 81 .mmcflppom HOHHOHBSOO OHDBMHOQEOB How maOHpOouHOO .v.v mnsmflm ..oo. 02......wm wm:...I.n.n5w ”.950“. h..0> m. H , < muomoowm ] , -_ >I.n.n.3m mwkon. h...0> m >I.n.n.3w meson. ......0> N < m0.zom...0m..m m>0I....m> h... < _ " oom¢ _ 00mm " ..oa .l CON .«OQ 538% O OO» 9 \ OmwN .>OOO.N m..__ 0... A _ 00m .m wowconpomam HMOQHHHEHOSB .h.v mnsmflm _mhzmzonuzoo m>0I.I.m> 00. 6.» >E _. m . o. om . 000 ..0 Someone / 000m 5 8 5.0 oooom .uoqmafizmo .85. + 59.8% 20 mz 000 mlW .000 I X 00m. 00b 86 volts so that the input voltage to the cathode follower is zero volts with the temperature probe at zero degrees centigrade and still has a 10 mv/° C temperature coefficient. The input circuitry is floating above ground and is well shielded to reduce noise pick-up. The cathode follower (an Analog Devices 142B solid state Operational amplifier) is used to: (l) prevent loading of the voltage dividers, (2) to impendence match to any recording or measuring dEVise and (3) to change the floating,temperature sensitive signal to a groundable signal. The only significant differences between thermilinear electronic systems A and B are: (1) Because a Philbrick model P25A differential amplifier was used in an amplifying mode, the input signal was reversed in polarity to Obtain an increasing output for an increasing temperature. (2) The signal is amplified by 100 and, via an attenuator, this unit provides five temperature ranges, from 0.01° C to 100° C full scale, to a one millivolt recorder. (3) The voltage divider AB is quite different. The two 700 Ohm pots are set ("upper" one to about 617.47 Ohms and the 3 "lower" one to about 382.5 Ohms) so that the voltages VAF 5 5 and VA are 0.617 and 1.617 volts respectively. The 1K H Ohm pot is a 0.1% accurate, 0.1% linear Helipot with a digital (0.0 to 100.0) readout dial. The baseline Of the temperature recording may be changed from 0° C to 100° C and its temperature value can be read from the Helipot dial. 87 The temperature monitors were calibrated by setting: (1) the two volt power supply, (2) the three volt power supply, (3) Ein voltages and (4) zero reference voltages with a potentiometen after the trimmer pot etc. on the Operational amplifiers were appropriately adjusted. The accuracy of these systems is relatively unimportant Since they were used primarily to study temperature variations qualitatively. However, the temperature Of a constant temperature bath was measured simultaneously with this device and a platinum resistance thermometer and the differences were of a random nature with i 0.3° C rms deviations. This Observation is in accord with the manufacturer's specifica— tions Of f 0.15° C thermilinear accuracy and interchangeability Of probes, and a i 0.216° C linearity deviation for these model 44201 probes. Collection Optics and Beam Analyzer The collection optics collect some Of the light scattered by the sample and transform it into a form appro— priate for interferometric analysis; these Optics are illustrated in Figure 4.8. For the aperture diameters and positions adapted, the first aperture, Al' determines the acceptance angle Of the light beam into the interferometer. Aperture A2 determines the cone angle over which scattered light is collected. The lens L1 transforms the diverging scattered light beam into a nearly parallel beam incident normally onto the interferometer etalon. 88 .cfimue SOHBOONOQ accepmo m¢3k .m< mmszmw...‘ .13 36.3% 45:40 .NmOLmo .w ... 353m xom 232.23,. .nm< 82.9.6: 55.22.32. .~m< 6.. 532.23.: .54 whzmzomo‘ou whJ a fi _ fi _ 45.20500: q _ 4 _ Ta _ 0N.0N 00.n~ 0+.0N 00..-um 0N.¢N 00.¢N 0550 000.0 000.0 000.0 0h0.0 000.0 0N0.0 BUanBBdWEL '3. BWVOS "IVLNOZIHOH SNIOVBH M3809 "IVILN383JJICI 99 aluminum box AB2 is 41 inches long, 14 inches high, and 8-3/8 inches wide. Lens L2 collects the collimated beam exiting from the interferometer and focuses it onto the pinhole A forming a visible Fabry-Perot ring pattern on 3: the pinhole face or a screen which is Often placed there. Reference to equation (2—40) indicates that, in order to maximize pa for a given acceptance angle, we need to select the largest feasible fL. Hence, L2 is a 100 cm focal length, 7.1 cm diameter, achromatic lens; with an acceptance angle of 2 milliradians, the pinhole diameter may be increased to 4 mm without diminishing the effective finesse. The pinhole is necessary to block some of the secondary ring patterns due to the wedge shape Of the interferometer mirrors; it also reduces the effects of scattering by the interferometer mirrors. Aperature A5 was found to enhance these two improvements (87). Also, as Met (56) illustrates, when maximum transmission is not required (when scattering is excited with an argon ion laser), A greatly improves the finesse by reducing mirror 5 misalignment effects. A5 was reduced in diameter until it seriously limited the transmitted intensity. Typically, the diameter is about one centimeter. Lens L was held by an Ealing self-centering lens 2 holder and was coarsely adjusted in the vertical direction to approximately center the Fabry-Perot ring pattern onto the photomultiplier tube. The photomultiplier pinhole was 100 mounted in an Ealing centering microscope Objective holder so that it could be carefully centered onto the ring pattern. A5 is a Cenco variable aperture and is only mounted via its rod mount since, having a relatively large diameter, its alignment is not critical. Similar to CB CB is a 3-1/2 X 5-1/2 inch box 1' 2 camera bellows. It is cemented to metal plates, with large axial holes, and bolted to the aluminum box AB2 and the interferometer housing, reducing the entrance of extraneous light. Detection Electronics Introduction Even though it is broadened by the electron impacts along the dynode chain, the output signal Of a photomultiplier tube is a single pulse for each photon incident on the photocathode. (The photomultiplier tube used in this research yields a pulse of 16 nsec duration, 2 mv high across a 50 Ohm load resistor.) An output pulse may also arise from thermionic emission of an electron from one of the downstream dynodes; however it will yield a less intense pulse at the anode compared to a pulse of photon initiation. These pulses represent a background or noise signal. The traditional photometric electronics are characterized by a time constant long compared to the duration Of these output pulses; hence, the pulses are lOl averaged into a dc signal. As an alternative, the intensity of a photon beam may be measured by counting pulses over a' fixed period Of time with digital equipment. The advantages, disadvantages, and limitations of these direct current and photon counting techniques are being debated in the litera- ture (91-94). One Of the more important aspects of photon counting is the improvement in signal-to—noise ratio for low intensity light beams. In photon counting techniques, the noise pulses are of smaller peak height and may be discriminated, i.e. not counted. For these low intensity light beams, direct current measurements average the photon and noise pulses together and the photon component is not easily extracted. When the light beam intensity is increased, the photon counting techniques lose this signal- to-noise ratio advantage because: (1) The noise component becomes insignificant in the direct current measurements. (2) The signal pulses occur much more frequently and over- lap each other making it difficult to examine and count individual pulses. Anticipation of low light intensities for some Of the light scattering experiments of interest (especially Brillouin SpectrOSCOpy) indicated that photon counting should be incorporated into the instrumental system. However, photon counting was 29929.t° the direct current capabilities of this instrumental system primarily for convenience in collecting data digitally and ultimately automating this 102 instrumental system; the data is naturally of a digital form. Furthermore, it is anticipated that recent and future advances in photomultiplier tube technology and solid state electronics will greatly increase the advantages of photon counting relative to direct current measurements. Photomultiplier Tube, Housing, and Power Supply The photomultiplier tube is an EMI 9558B tube. It was chosen primarily for two reasons: First, it is Often recommended in the spectroscopy literature relative to photon counting as well as analog detection electronics. This literature indicates the tube represents a good compromise between eXpense and parametric values desired for photon counting. Also, this published data produces a basis for engineering the Optimal instrument for a given experiment, using either detection scheme. Second, the red response (S-20 type cathode), coupled with very low dark currents and a 2 X 106 gain, make this an extremely useful tube for use with He-Ne lasers without seriously compromising with sensitivities at other wavelengths down to the near ultra—violet wavelengths. The cathode sensitivity is 134 uamp/lumen and the dark currents are diagramatically presented in Figure 4.10. From these dark current measurements, it is evident that very little is gained by lowering the cathode temperature below -20° C, 103 a compromise with the manufacturer's specifications: "extensive investigation reveals that dry ice temperature (ca -60 degrees C) is optimum for best signal to back- ground ratio." The photomultiplier tube is excited by a power supply in the Lansing Research scanner. This voltage is variable between 0 and -l.8 Kv with i 0.02 per cent per hour drift. It is controlled by a ten-turn potentiometer with a clock- type, turns counting dial ensuring easy readability, resolution and resetability. With this continuously variable voltage supply, one may easily find the optimum voltage for a given eXperiment: a compromise between dark current and sensitivity. The photomultiplier tube was mounted inside a Products for Research, Inc. model TE-104 TS refrigerated chamber reducing thermionic emissions and corresponding dark currents. Under normal lab conditions, with a coolant water flow rate greater than 10 gph and less than 24° C, this refrigeration unit can maintain the photomultiplier tube's cathode at temperatures down to —20° C (control point continuously variable from —20° C to temperature of coOlant water) with a stability Of i 0.5° C. The chamber has a mu—metal shield to reduce external magnetic field effects. The tube socket is wired and then potted to prevent water condensation on the dynode resistor chain (220K resistors with a 1N3043B, 150 volt zener diode 104 DYNODE CHAIN V0 LTAGE 2000 VOLTS I500 VOLTS IOOO vous Io-II 700 vous 500 vous :2 4—0— . 0 I0"'2 300 vous <5” .3. *z' w -- 10"3 a: m D 0 3% < + IO''4 0 J 1 l J l l l I 1 1 l -25 -20 -l5 -IO -5 0 5 IO I5 20 25 TEMPERATURE °C. Figure 4.10. Photomultiplier Dark Currents. 105 between the cathode and first dynode) and the associated leak current problems. The anode is directly connected to its BNC connector without capacitive coupling, making this photomultiplier suitable for photon counting as well as analog detection. The refrigeration unit was bolted onto the end of box AB with the bolts extending inward so that a disc 2 could be used to cover the photomultiplier tube while alignment was pursued. The room lights were turned Off and laser beams blocked while the side of box AB2 was removed, allowing this disc to be bolted in place to protect the photomultiplier tube. This was quite inconvenient and the instrumental system was modified by inserting a sliding light mask mechanism between the refrigeration unit and box AB . This unit is functionally analogous to, and 2 utilizes a mask from a sheet film holder. P_icoam'meter The output current Of the photomultiplier is simultaneously fed into a Keithley model 417 picoammeter and the photon counting circuit. The picoammeter has 13 to 3 X 10—5 ampere, eighteen full scale ranges from 10— Well bracketing the range Of photomultiplier currents of interest. Normally the rise time is limited by external caPaCitance (cable capacitance) to 5 millisec and may be increased to 3 sec with a dampening control. The fast rise time allows oscilloscope display of Brillouin 106 spectra; slower response times permit "averaging out of noise" for strip chart recordings. Calibrated suppression currents up to 1000 times the full scale setting are available at the input. In addition to the supression of a steady background signal, this permits a full scale display of 0.1% variations of the total signal; thus, a weak Brillouin signal is extracta- ble from a relatively large dark current. The electrometer input tubeand solid state amplifier are completely contained within an "input head" unit which may be located remotely; when placed close to the photomultiplier tube, the cable capacitance limitation on the rise time is reduced as well as pick up noise. Recorders The output of the Keithley picoammeter is fed into a strip chart recorder, a digital voltmeter, and an oscillioscope individually or simultaneously. A Sargent model SRG solid state potentiometric strip chart recorder was utilized. The digital voltmeter is a Heath model EU‘805A Universal Digital Instrument Operating in its digital voltmeter mode and the oscilloscope is a Tektronix, type 564B, model 121N storage oscilloscope. The oscillOSCOpe's horizontal plug-in unit is a type 384 time base unit; the Vertical plug-in unit is a type 2A63 differential amplifier unit- A Spectra Physics model 420 Optical spectrum analyzer 4 5...... -.- 107 plug—in unit was also employed, particularly for the study of laser beams. The oscilloscope is more useful than the strip chart recorder for aligning the Optics for Brillouin spectroscopy and the digital voltmeter is advantageous for the collection of quantitative information in experi- ments other than Brillouin spectroscopy, especially for measuring the ratio Of two very different light intensities. Photon Counting A block diagram Of the adopted photon counting system is in Figure 4.11. Malmstadt and Enke (95) provide a good introduction to this subsystem and a bibliography of articles discussing circuitry details is available from Zatzi ck (96) . The principle limitation of digital equip- ment, that is, the frequency response which increases with increasing expense, is partially alleviated because the amplifier and discriminator units have higher frequency capabilities than the counter. The amplifier and discrimi- nator must deal with the signal and noise pulses whereas the counter only deals with the signal pulses, a significant difference at low light intensities. The photomultiplier output is fed into an EG&G model ANZOl/N quad amplifier module via a 50 Ohm load resistor. After amplification, the pulses are fed into the EG&G model T1310]. differential discriminator module. Both of these digital units have a 100 MHz frequency limit. The 108 mwomoomm amt—.2300 m0._.0+ , ql .l I I I J. l_ u _ mm5:¢_ . _ = n. w>2w_ “ _..l l I I IL :3. Eda. MEG oz< 112 ”:5 mo F! — x.un: mmzzw.. 0.00.. 114 -l.5 KV, ultimately preventing the AND gate to supply a SWEEP pulse signal to the interferometer scanner until the scanning voltage returns to zero and the PBl switch is used to restart the system. After a period of time, adjustable in the monostable multivibrator, the Q pulse being applied to the AND Gate returns to zero. Complementarily, a positive going pulse is generated at Q and is applied to the interferometer's PAUSE switch causing the piezoelectric element to stop scanning. The mirror remains stationary until the clock module generates another negative transition causing Q to go positive and the AND gate to apply a positive transition to the SWEEP switch. The clock module's output frequency is adjustable from 0.1 Hz to 100 Hz, controlling_the frequency of the sweep pulses applied to the interferometer scanner. The monostable multivibrator module may be adjusted so that the duration Of the Q (SWEEP) pulse (before 5 or PAUSE pulse is applied) is 10 nsec to 1 sec. The slope, set on the interferometer scanner, will control the distance which the mirror actually moves for a given Q pulse duration. If the monostable pulse duration is set to be longer than the input (clock) signal period, the monostable also functions as a frequency divider. This is useful in extending the sweep period beyond the one second limit of 115 the monostable pulse widths when the input (clock) signal period is longer than the monostable pulse duration. TO summarize, these two pulse width adjustments provide extreme versatility in choosing: (l) the number Of pauses (number Of data points) during an entire inter- ferometer sweep, (2) the time period of each pause (period Of time over which an intensity measurement may be made), and (3) coupled with the slope setting on the interferometer scanner, the distance the mirror moves between pauses. The voltage applied to the piezoelectric element (after attenuation) was recorded for many different settings Of the clock frequency, the monostable pulse widths, and the slope of the interferometer scanner. A portion Of one of these recordings is in Figure 4.14. This record cor— responds tO 10 seconds between SWEEP pulses; the sweep lasts for 4 seconds leaving a 6 second pause period. At these longer pause periods, the light intensity may be measured with a digital voltmeter across the picoammeter's recorder output instead of utilizing photon counting techniques. The only quantitative check performed on this digital control unit was a measure Of the reproducibility of the piezoelectric transducer voltages at the different pause points. The standard deviation Of these measurements (made with a DVM) is 0.03% or less, for a slow 3 volt/sec ramp setting and a clock frequency Of 10 sec. 116 u: (9 <1 F- .J C) :> .._' hd c. TIME —- Figure 4.4. Piezoelectric Transducer Voltage when Controlled by Digital Control Unit. 117 The digital control circuit discussed above cor- responds to the interferometer's single sweep mode; however, the circuit is easily extended to include a free run mode and to provide synchronization pulses for photon counting (or DVM) and data "print out". CHAPTER V ALIGNMENT AND CALIBRATION TECHNIQUES Intrdduction In this chapter, the alignment and calibration Of the instrumental system from initial assembly to a situation appropriate for Brillouin spectroscopy will be discussed. Brillouin spectroscopy is the most sensitive and illustra— tive of all these techniques and it is not necessary to repeat them in full for all types of experiments. The exceptions to this extensive routine, for experiments other than Brillouin spectrOSCOpy and repetitive Brillouin Spectroscopy experiments, will be self evident. Optical Table Leveling Three Of the axle jacks were lowered slightly, allowing the plane Of the table top to be determined by the other three. These latter three jacks were adjusted until the table tOp was level in both Of its horizontal directions as detected by a four foot carpenter level. Then, the levelness was checked by filling a piece Of 1/4 inch tygon tubing with water until the water level matched the table top. The tubing went down to the floor, along the floor and up near another corner of the table. When 118 119 the table was level, the water level in the tubing at both corners matched the table top. After leveling was accomplished with these three axle jacks, the other three axle jacks were adjusted to assume a load equal to the three leveling jacks. This equal load was accomplished by using a torque wrench to adjust the latter three axle jacks. Initial Assembly Before any of the detection Optics subsystems were placed onto the Optical table, two posts with pinholes 10 inches above the Optical table tOp were mounted at Opposite ends of the table. The laser, defining the detection Optics axis, was mounted and adjusted so that its beam was centered onto these two pinholes. This insured the Optical axis to be horizontal to the Optical table top and traveling in a direction (parallel to the Optical table's edge) which would allow one to conveniently bolt the sub- system assemblies into place. The aluminum boxes, with the Optical rails inside, were lifted into place and carefully positioned SO that when a pinhole (centered on a vertical post) was slid along the triangular rail it remained centered on the laser axis beam. This adjustment insures that translations along the rail will be the same as translations along the detection Optics axis, facilitating alignment of the individual components mounted on the rail. The aluminum boxes were then bolted into place. 120 The interferometer housing, resting on its I-beam pedestal, was set into place. The interferometer's dif— ferential screws and axis adjustors were set at their midpoint. Two adaptors were inserted into the interferometer housing's two-inch entrance and exit holes providing axial, 2 mm dia centering apertures. The interferometer's pedestal and legs were adjusted so that the Optical axis laser beam was centered onto these two apertures. Then the interferometer's position was checked, and readjusted so that when a mounted mirror was slid along the inter- ferometer's dovetail rail, the incident beam was always reflected back onto itself. This makes the dovetail rail parallel to the system's Optical axis and insures the interferometer will always be adjustable, regardless Of its mirror spacing. The I-beam pedestal was then bolted into place, and the interferometer's leg lock nuts were set. The rotating table assembly was lifted into place and adjusted (see the section on the variable scattering angle mechanism) so that its rotating axis intersected the Optical axis. TO facilitate this alignment, a vertical pin point adaptor was mounted onto the rotary table. Assuming a cylindrical sample cell, this insures that the collection Optics, when aligned appropriately, will always view the same scattering volume element,irrespective Of the table's rotatary position, (angle Of incidence or scattering angle). 121 Next, the rotatory table was set at its zero angle position and the pinhole mechanisms were adjusted so that the pin- holes were centered on the laser beam. This adjustment insures the pinholes to be on a diameter of the rotatory table and at a height corresponding to the volume element viewed by the detection Optics. Now, when the table is rotated through an angle, readable from the table's mechanism, and the incident laser beam (whose scattering is to be studied) is directed through these two pinholes, the scattering angle is known and the scattering volume element is centered appropriately. Collection Optics Alignment The apertures Al and A2, set at their minimum diameters were centered onto the laser beam which defines the Optical axis. Lens Ll was placed 50 cm from the center Of the rotating table and positioned so that the transmitted beam was undeviated relative to the incident Optical axis beam when viewed on a screen near the photo- multiplier tube. An additional criterion for the positioning of L1 was the beam reflected from its first surface. The lens was adjusted until the incident beam was reflected back onto itself. The backside Of apertures Al and A2 as well as the laser's exit port are convenient viewing points for this criterion. It is not necessary, but the 50 cm (lens focal length) distance can be checked by focusing the laser beam 122 to a point at the rotary table's center and examining the light beam transmitted through Ll for collimation, i.e., constant diameter. The alignment Of aperture A is not critical. Its 4 diameter is larger than the diameter of the laser beam coming through Al and A2. It is only important that A4 is adjusted so that it does not block part of this beam. Interferometer Alignment The interferometer mirrors are mounted into their holders; the mirror closest to the photomultiplier tube is slid along the dovetail rail and locked into place at the desired mirror spacing. Several scratches have been made on this rail to facilitate approximate setting of the mirror spacing. The accurate measurement of the mirror spacing will be discussed in the interferometer calibration section. It should be recalled that the mirrors, for ultimate frequency resolution, are Spaced as close together as possible without serious overlapping of the spectra associated with adjacent interferometer modes. The inter- ferometer's fringe pattern may be viewed on a screen any- where on the axis behind the interferometer or, as the author prefers, on the ceiling after reflection by a mirror propped up by the back Of the interferometer housing. Initially, one Observes a train of bright spots corresponding to multiple reflections of the laser beam. 123 This train Of spots is collapsed into one spot, or a ring pattern, by adjusting the gimbal mount's differential screws. The upper differential screw rotates the gimbal mount about a horizontal axis and the lower differential screw rotates the gimbal mount about a vertical axis. Now, if the interferometer is scanned, the various rings will sequentially collapse to a bright spot. Another dim spot, corresponding to the direct passage of the incident beam through the interferometer mirrors, is visible. These two spots are not coincident because the wedge shape of the first mirror perturbs the normal incidence of the laser beam. The adjustments on the interferometer's housing are varied until the ring pattern collapses onto the leakage Spot. The differential screws are adjusted to give a sharp, well defined set of fringes. These two sets of adjustments are not independent, or orthogonal, because of the interferometer's mechanical design and the mirror's wedge shape. Thus, one must systematically repeat these adjustments until the sharpest ring pattern which collapses onto the leakage spot is Obtained. The differential screws primarily control the parallelness of the reflecting surfaces and hence, the definition of the ring pattern. The housing adjustments Primarily control the angle Of incidence, hence the inalative position of the leakage spot and the center of thee fringe pattern. 124 The final alignment of the interferometer is Obtained, after lens L2 and the photomultiplier pinhole are aligned, by viewing an oscilloscope display of the photomultiplier, picoammeter output. The criteria are: (l) narrowest linewidth, and (2) symmetry of the peak; these factors are primarily related to, (l) the mirror parallelness, and (2) normal incidence onto the reflecting surfaces. The axis of the cone of incident radiation is also affected by the alignment Of the collection Optics and their alignment should now be rechecked. Finally, it is noted that no serious problem arises from the fact that the alignment laser is usually a helium-neon laser and the laser for Brillouin scattering experiments is usually an argon ion laser. The inter- ferometer mirrors are broad band and have sufficient reflectivity at 6328 A0 to yield a finesse adequate for alignment. Post-Interferometer Optics Alignment The placement of lens L2 is not critical. L2 is adjusted to cast the center Of the fringe pattern within the range of adjustability of the photomultiplier pinhole. The photomultiplier pinhole was set at a 1 mm diameter while its mounting mechanism was adjusted to center it onto the fringe pattern. Then this pinhole was enlarged to a 4 mm diameter. When set at smaller diameters, the w-‘ 'v 125 symmetry of the spectra peaks are strongly dependent upon this pinhole's alignment. However, this is relieved (also increasing transmitted intensity) at larger pinhole diameters; and, we may do so without reducing the instrumental finesse (see'fluesection discussing the post-interferometer Optics). Aperture A5 is minimized and aligned onto the leakage beam or center of the fringe pattern. A5 is Opened to a diameter of about 1 cm; hence, its alignment is not critical. The Optimum diameter is normally associated withiacompromise in instrumental finesse and transmitted intensity (56). However, in this instrumental system, with high sensitivity of the detection electronics and inherent properties of the collection Optics and interferometer, this compromise is not a strong function Of either of these two factors. Incident Laser Beam Alignment The mechanism for aligning the incident laser beam has already been discussed. The Bridgeport rotary table is rotated to the desired scattering angle and the incident laser beam's direction is adjusted by translating, rotating, and tilting the mirror in the incident Optics until the laser beam is centered onto the two (sample cell) alignment pinholes. Two things will facilitate these scattering angle adjustments: (1) The laser's mounts, presently lab jacks, are adjusted in order that the laser beam is parallel 126 to the Optical table, 10 inches above it. (2) The laser is positioned so that its beam is also parallel to the rail in the incident Optics system. The alignment or calibration of the laser's polarization is very important and is dis- cussed in a following section. Sample Cell Positioning The glass cell containing the sample Of interest is set onto the COpper base plate and pressed hard enough to squeeze out excessive amounts Of the thermal compound. Then the cell is slid (rotated, etc.) on this base until the cell's surface reflections cause both the optical axis laser and the incident light beams (for the scattering studies) to traverse back upon themselves. For a square cell, this criterion is sufficient; however, a further check may be performed by rotating the table in 900 intervals and Observing the surface reflections. When cylindrical sample cells are used, and measurements are made as a function of the scattering angle, it is important that the sample cell be coaxial with the rotating table. This may be investigated by examining the reflected and transmitted light beams for constancy Of their character as the table is rotated. A cylindrical cell's position, etc. can also be checked by an experiment discussed in the section, "Angular Dependence Of Light Scattering". The vertical position Of the scattering volume can be checked visually; if the sample is illuminated by both 127 light beams, they should intersect. Also, when the high intensity argon ion laser is used to excite the scattering, the scattering volume element may be visually inSpected at any point along the detection optics axis. This tech- nique is eSpecially useful if a Ludox solution is used to examine the incident beam's alignment and is replaced later by the sample Of interest. However, an ultimate criterion for the directional adjustments Of the incident laser beam is the narrowness and symmetry of the Rayleigh peak as displayed on an oscilloscope, or ultimately on a recording of the scattered light Spectra. Finally, for optimum performance, all of the aperature, lens, and especially, the interferometer adjust- ments should be Optimized again by viewing a spectra dis- played on the oscillosCOpe. After the thermostatting unit is lowered into place and thermal equilibrium is Obtained, these adjustments should be Optimized again before a spectrum is recorded. Interferometer Calibration Many different, complex calibration techniques have been devised for Fabry-Perot interferometers (see Fabelinskii (97) for a summary); however, they are based on the need for an absolute frequency calibration. In Brillouin spectroscOpy, we are only interested in frequency dif— ferences and only need to know the interferometer's free 128 spectral range. When a Brillouin spectrum is recorded, at least two adjacent mode spectra are recorded. The distance between corresponding peaks is equal to the inter— ferometer's free spectral range. Thus, since the scanning is linear, one only needs to know the free spectral range in order to determine the frequency scale of a recording. The free spectral range Of the interferometer is determined by its mirror spacing, Table 2.1; hence, we only need to measure the mirror separation accurately in order to calibrate our frequency scale. This may be accomplished by using telescoping gauges, set to match mirror Spacing and measuring the gauge's length with a micrometer. If done in the reverse order, the mirrors may be set at a known, convenient separation. These micrometer measurements may conveniently be made with i 0.1 mil accu- racy and precision; this corresponds to a 3.8 MHz uncertainty in the free spectral range (15 GHz) for a mirror spacing of 1 cm. If the spectrum is recorded with a slow interferometer scan, fast recorder speed, the uncertainty in the spectrum's frequency parameters is reduced to an insignificant value. At this point it is convenient to state that no calibration of the intensity scale was undertaken. For Brillouin SpectrOSCOpy, or any of the photometric type of experiments done herein, only relative intensity measure- ments are necessary. 129 Temperature Calibration A rather convenient method of temperature calibration was investigated as a project tangential to this thesis work. As a natural consequence of that project a more rigorous fixed point cell calibration of a platinum resistance thermometer is available. This fixed point method of calibration is well documented, for example see Shoemaker and Garland (98). The fixed point cells for the phenol freezing point, naphthalene freezing pointauuiphthalic anhydride freezing point were used for additional data points in the temperature range Of interest. These cells are discussed by Enagonio, et a1. (99). An Electro Science Inc. portable potentiometer- bridge and platinum resistance thermometry accessories were used to measure the relative resistance of a platinum resistance thermometer at these fixed temperatures. The method is discussed in the instruction manual and was modified only to the extent Of externally using a Keithley model 155 null detector meter. This meter made the meter interpolation for additional digits in the data easier and more precise. The data was analyzed with least square mathematics and the results are in Table 5.1. A chart of Rt/Ro versus temperature is retained in the laboratory. 130 TABLE 5.1.--Platinum Resistance Thermometer Calibration. Fixed Point R' (exptl.) t (std, oC) t(calc.) Mercury Freezing Point 0.84690:0.00028 -38.862 -38.884 Water Freezing Point 1.00000i0.00003 0.000 0.001 Sodium Sulfate Transition 1.1259510.00021 32.382 32.390 Phenol Freezing Point 1.15859i0.00006 40.849 40.844 Naphthalene Freezing Point 1.3094310.00013 80.239 80.247 Corrected Water Boiling Point l.38430:0.00042 100.000 100.013 Benzoic Acid Freezing Point 1.46836i0.00003 122.375 122.375 Pthalic Anhydride Freezing Point 1.50075i0.00026 131.059 131.041 Equation: R' = R0 + at + 8t2 Least Square -6 Parameters: R0 = 0.99999661 t 8.005 X 10 a = 3.91077781 x 10‘3 i 5.522 x 10"7 B =-6.82273637 x 10‘7 i 4.318 x 10’9 RMS Error in t: 3 .147 x 10"3 131 Polarization Alignment and Calibration Introduction Light scattering experiments are sensitive to the polarization alignment Of the incident light beam (assuming a polarized beam as is common in lasers) and the calibration of polarization filters or analyzers in the Viewing optics relative to the incident beam's polarization. In principle, one may initiate polarization alignment and calibration with a light source of known polarization vector or an Optical device whose Optical axis or polarizing effects are known. In many of the classical Optical experiments, for example polarimetry, only changes in polarization are measured and most Of the polarization calibration devices are of the latter type, defining a relative system. However, in many light scattering experiments this relative technique is not sufficient. This is especially evident when one considers studying the character Of a scattered light beam as a function Of viewing angle with a polarized incident beam. Furthermore, in this versatile experimental system, it is convenient to align or calibrate in a polari— zation coordinate system synonymous with the laboratory coordinate system. Thus, the other subsystems, such as the sample cell, are independently aligned in the same coordinate system. In his depolarization studies, Anderson (82) aligned the laser's polarization vector relative to the laboratory 132 coordinate system and then aligned or calibrated other parts using this light beam of known polarization. His technique was based on the reflections from the Brewster window of the laser's plasma tube. It was convenient to Open the laser, allowing the reflections to cast onto the ceiling. It was also convenient to rotate the plasma tube. Neither Of these procedures are possible with most lasers. Even though one could rotate the entire laser unit, the polari- ization alignment and calibration of this instrumental system was accomplished by rotating the laser beam's polarization vector until it was vertical or horizontal using a half wave plate. Polarization analyzers, etc. were orientated for maxima or minima transmission of this polarization aligned beam. The criterion for the laser beam's polarization being vertical or horizontal is based on Brewster angle reflection from a glass plate external to the laser. Theoretical Discussion Consider a laser beam whose polarization vector is rotated a radians counterclockwise from the vertical x axis, is traveling in the z direction (Figure 5.1), and is incident upon a glass plate whose plane of incidence is coincident with the xz plane. The reflectivities Of the two polarization components are different and a function Of the angle of incidence. For example, the component parallel to the incident plane has zero reflectivity for 133 / / a LASER BEAM 2 . Y Figure 5.1. Coordinate System for Discussion of Polarization Alignment Technique. 134 the Brewster incidence angle whereas the component perpen- dicular to this plane is finite. Thus, when the polarization vector is rotated until it is parallel with the incident plane, or equivalently, until it becomes vertical, the intensity Of the reflected light goes through a minimum. Qualitatively, the incidence angle dependence of these two reflectivities are illustrated in Figure 5.2. At most angles, such as 91, the reflectivity of the parallel component is less than that Of the perpendicular component. The polarization vector of any incident light beam may be decomposed into parallel and perpendicular components. Since the reflectivity of the perpendicular component is favored ascxapproaches zero, the total reflected intensity should go through a minimum at d equal to zero. Similarly, the total reflected intensity passes through a maximum value when a equals 90 degrees. The beam incident onto the glass plate may be given by: I. =Rl (5-1) where Is is the Stokes vector for a laser beam prOpogating in the z direction of the coordinate system in Figure 5.1. lijis arotation matrix; [in is the stokes vector for the incident beam whose polarization vector is rotated by d from the vertical. IRO is given by: INTENSITY LOO r .05 135 00 BI 9b 90° ANGLE 0F INCIDENCE Figure 5.2. Reflectances of a Dielectric. 136 r _ l 0 0 0 R0 = 0 cos 20L -sin 2d 0 (5‘2) 0 sin 2d cos 2d 0 0 0 0 ' 1_ If Is is given by: P11 |S = 1 (5-3) then -— l q I. = cos 2d (5-4) in sin 2d L O 4 In the Mueller Jones system, reflection may be represented, via Fresnel's law, by: m - wherein: sin (0 - 0') s E (5-6) sin (0 + 0') tan (0 - 0') rt- IN (5-7) tan (0 + 0') 137 The incidence angle, 0, is related to the refraction angle, 0', by Snell's law: sin 0 = n sin 0' (5—8) n being the refractive index of the glass plate. In order to convert Ré into the Stokes system one uses: _. I I I -l _ RF — ”(RF X RF*)n (5 9) and the unitary matrix.“ is given by: l l 0 0 l.I I]: ——- l 0 -1 (5-10) /§ 1 L0 -i i d The result is E W 1 sz+t2 tz-s2 0 0 RF = _ tus'z 82+t2 o 0 (5—11) 2 0 0 ZSt 0 L o 0 o 2sti Now, if S E 52 (5-12) cos (0 + 0') and Q E t/s = (5—13) cos (0 - 0') 138 then 1-l+Q2 Q2-1 T —-— O O 2 2 QZ—l 1+Q2 RF = s ———— ——— 0 0 (5-14) 2 2 0 0 Q 0 0 0 0 QJ If the reflected beam is represented by a Stokes vector, lout’ then: Iout = RF Iin (5_15) and I“ W S (1 — cos 2d) + Q2(l + cos 2d) — _ — r~ 2 _. Iout — 2 61 cos 2d) + Q (l + cos 2d) (5 16) 2Q sin 2d L 0 .._I The total reflected intensity, the first element in the Stokes vector, is: Bl — cos 2d) + Q2(l + cos 2dfl (5-17) H II N '0) thus, 31 —— =—S(Q2 - 1) sin 2d (5-18) 3d ' In order that a minimax point occur, S, Qz-l, and/or sin 2d must be zero. First, 139 sin 2d = 0 (5-19) implies: a = 0, fl, . . . for minima and a = g , g1 , . . . for maxima Using back substitution and the physical restriction, n . . . - 5.: a i , we see that I IS a minimum only when d=0. nus The requirement on S: 2 sin (0 — 0'J S = = 0 (5-20) sin (0 + 0'4 leads to sin (0 — 0') = 0 (5—21) provided sin (0+0') # 0. Equation (5-21) implies: 9 '— 6' =0, TI, 2’”, o o 0 However, since 0' is given by sin 0 = n sin 0', only 0 = 0' = 0 (normal incidence) is possible. Finally, for the Qz-l factor, we Obtain: cos (0 + 0') = 1 (5-22) cos (0 n 0') 140 or cos (0 + 0') = t cos (0 - 0') (5-23) Using trigonometry, this equation rearranges to: cos 0 cos 0' - sin 0 sin 0' = I (cos 0 cos 0' + sin 0 sin 0') For the plus and minus signs, this expression converts into the two expressions, I! 0 sin 0 sin 0' (5-25) cos 0 cos 0' = 0 (5-26) requiring 0=0 (normal incidence) or 0= g-(glazing incidence). However, for these values of 0, the reflectances are the same for the parallel and perpendicular components; that is, the total intensity will remain constant (at (%E%)2 and 1 for 0:0 and i g respectively) as a is changed. There- fore, after considering all three factors of equation (5-18), we see that the only case for a minimum in the intensity of the reflected beam is d=0, a vertically polarized beam. Setting the partial derivative of I with respect to 0 equal to zero leads to 0:0 and i %. These cases have already been disregarded in the arguments above. The exist- ence of a minimum reflected intensity, with respect to variations in a, is independent Of 0, but, perhaps the minimum's quality or sharpness is dependent upon 0. In 141 order to examine the quality aspect of these minima, we only need to consider small values Of a. For these small values of a, we may use the truncated series expression: cos 2d = l - % (20:)2 + . . . and transform equation (5-17) into a quadratic function of d: I = SQ2 - de (5-27) wherein B S(1-Q2) (5-28) It is not easy to analytically determine the value of 0 which maximizes the sensitivity of I to a. However, it is easy to see that B = 52 - t2 and that I will be most sensitive to a when 52 — t2 is maximum. Furthermore, $2 = rS and t2 = rp, so that (l) acknowledging rS and r are the perpendicular and parallel reflectivities and that (2) I is a linear combination Of these reflectances, it is qualitatively reasonable that I would be most sensitive to a when their difference is maximum. I The values Of 0 maximizing rS-rp were determined numerically by the computer program POLCAL and are plotted in Figure 5.3 as a function Of the refractive index of the glass plate. Figure 5.4 illustrates rS-rp versus angle of incidence for a given refractive index, n = 1.50. Figure 5.5 illustrates the Brewster angle as a function of refractive index. Following the definitions for these 142 m m . an H mcHNHEmez named OOQOUHOQH mo mosmwcmmmo xmtsH O>Hpomuwmm xmnz. w>....0wpomswmm xmaz. w>.._.0Hpomsmmm .m.m musmwm xmoz. w>§o SHSOHmE momma 20H comum Hmmmn soH cooum mama Homm> musoumz mama Homm> NHSOHOE oflpmm noepmNHHmaome amoepuw> .Esv spacmam>mz mOHDOm pamflq .msmmcmm How sumcwam>wz msmHm> Oflumm cOHummflmmaome amoeuum>un.m.w mamae 165 The depolarization ratios are plotted in Figure 7.3. Extensive effort was not applied in the data collection or analysis; hence, one cannot conclude confidently relative to these results. However, these data imply the vertical depolarization ratio for benzene, at 25° C, may be a non- linear function Of wavelength and it may be worthwhile to pursue this tOpic more extensively. The experiment is a good illustration Of the use of different light sources (wavelengths) to expand the versatility Of this light scattering apparatus. Polarization Coherency Matrix for Benzene Scattered Light This experiment illustrates the possibility of using this instrument for the determination of the polari— zation coherency matrix (PCM) of a light beam. The light beam studied was the light scattered at 90 degrees from a benzene sample in a square cell, temperature maintained at 25° C, and illuminated with a 5 milliwatt helium-neon laser. Momentarily, ignoring the analyzer optics, the Optics were aligned as discussed in the alignment chapter except the interferometer mirrors were not installed. The picoammeter output was fed into the digital voltmeter; functioning on a ten second base, the digital voltmeter provided averaged data in a digital form. Although many experimental techniques are possible, the straightforward technique (advantageous for demonstration 166 .COHpmNHHMHOQmQ Osmwcmm wo mosmtsmmmo npmcmam>mz .m.n mnsmflm .85 1.520403; 000 000 000 000 00¢ 00¢ 000\. q . a _ _ . AV 0..... I '0. C) NO I .LVZIBV'IOdBO (Nd) 167 purposes) outlined by equations (2-51,54) was adopted. A precision polarizer/analyzer unit was inserted into the beam and the transmitted intensity measured for 0° (vertical), 45°, 90° and 135°. The six intensity values were substi- tuted into equations (2e51,54) and the PCM elements were calculated. Before the sample was measured, a vertically polarized laser beam was sent along the system's optical axis by substituting a pentaprism for the sample. The polarization analyzer's reading, zero or vertical reference point, for maximum transmission was noted. The quarter wave plate was then inserted and its orientation locked at the orientation yielding a maximum transmitted intensity; the vertical polarization vector was not rotated. Now, by comparing these two maximum transmittance values, one obtains a transmittance correction factor for inserting the quarter wave plate. The factor is 0.9723 (average Of five such comparisons) and has been applied to the I(%,%) and 3 . 1(41’1) data in Table 7.6. The entire data set, hence the PCM elements, was normalized by inserting an attenuator between the picoammetor and the DVM and adjusting it to yield a DVM reading Of 1.000 when the scattered beam was incident on the polarization analyzer in its vertical position. The tabulated I(0,€) data points were averaged for the five different experimental sequences and the average 168 mmo.o omm.o oomo.o some.o mmm.o 4moo.H m>m smo.o mmm.o mmo.o 4mm.o smm.o mam.o m mmo.o amo.o mmo.o mmo.o oom.o moo.s 4 meo.o mmo.o mmo.o Hmo.o mmm.o mmm.o m omo.o smw.o Hmo.o mmw.o mmm.o ooo.e N Heo.o mas.o mmo.o m4o.o mom.o emo.a a m a m.w IN N m s>mm.H a :VH A .pmcH .o.:.H .o.evH .o.ocH cum Aw.®cH .msmmcmm MOM meme xflupmz mosmnmsou coemeHHmaomII.m.b mqmde 169 values were used in equations (2—51,54) to calculate the PCM elements: r 1.0024 -O.0002+0.0006i -0.0002-0.0006i 0.2586 L The depolarization ratio 0.258 1.0024 agrees with the values in the previous sections. The degree of coherence I J XyI = 0.0012 /J /J XX yy indicates the polarization incoherence of the scattered light; the scattering centers are uncorrelated over the scattering volume element. Although the PCM Off diagonal elements are known to be zero for light scattered from a liquid, this eXperiment may become informative when applied to crystals, highly oriented polymers, liquid crystals, or Optically active compounds. Perhaps the polarization coherence studies may be augmented with Mallick (102,88) spatial—time coherence experiments. 170 Angular Dependence of Light Scattering As stated in the alignment and calibration chapter, this experiment represents a test for a cylindrical sample cell's alignment. It is an example of a photometric versus scattering angle experiment and a possible technique for determining depolarization ratios. Coumou (90) has published measurements of the Rayleigh factor Of several liquids as a function of the scattering angle. This experiment is based upon one Of Coumou's equation rearranged to L0 l-pu 5" sin.0 ==1A- c0529 (7-4) 90 l+pu L6 is the scattered light flux viewed by the detection Optics at the scattering angle 0. is the depolarization pu ratio for an unpolarized incident beam at a 90 degree scattering angle. Equation (7-4) avoids the need for correction factors, known solid angle of collection cone, calibration Of photomultiplier currents in terms of an absolute light intensity, etc. When one plots the left hand side Of this equation as a function of 0, the symmetry of this plot is a test of the system's alignment, Optical quality Of the cylindrical sample cell and cleanliness Of the sample. If one assumes a value for pu, the calculated right hand side of equation (7—4) may be compared to the measured left hand side. A least square analysis of _w. __. 171 Le Sine/L90 then pv from the Krishnan relationship. versus cos20 will yield pu from the slope and The incident beam was provided by the Siemens LG-64 laser; functioning in its multimode Option, it provided an incident intensity of about 8.9 milliwatts at the sample cell. The incident beam was effectively depolarized by passage through a rotating half wave plate. The cylindrical sample cell and alignment techniques are elucidated in the previous chapters. The suppression network on the picoam— meter was used to suppress the dark current (approximately 7.1 x 10’10 amp); with the damping constant at a maximum setting, the picoammeter was read directly. The readings are recorded in Table 7.7. A fluorescein solution was diluted until it yielded photomultiplier currents comparable to those for the other samples. All three liquids were temperature controlled at 25.0° C. The range of values is limited, by the width of the flat windows, between 12° and 178°; however, due to inconveniences and emphasized effects of sample impurities at these low angles. Measure- ments were only done between 20° and 160°. After appropriate transformation of the data in Table 7.7, a least square analysis ultimately yielded pV values Of 0.01615 and 0.2564 for carbon tetrachloride and benzene reSpectively. These values agree well with those in the previous sections of this chapter. Using these 172 TABLE 7.7.—-Angu1ar Dependence Of Light Scattering Data. Scattering L0’ units of PMT current X107 Angles(degrees) Fluorescein Carbon Solution Benzene Tetrachloride 20 7.37 6.92 3.15 30 5.07 4.50 1.99 40 3.96 3.38 1.39 50 3.38 2.65 1.05 60 2.99 2.20 0.835 70 2.76 1.92 0.686 80 2.61 1.77 0.610 90 2.57 1.73 0.582 100 2.58 1.75 0.606 110 2.71 1.90 0.685 120 3.04 2.18 0.837 130 3.39 2.62 1.07 140 4.00 3.34 1.38 150 5.12 4.52 2.02 160 7.56 6.95 3.18 173 least square values, the right hand side of equation (7—4) was evaluated and is represented by the solid lines in Figure 7.4. Visual inspection of Figure 7.4 illustrates the good alignment of the system. The scatter of the experimental points about the calculated curves is easily accounted for by uncertainty in the picoammeter readings. Landau—Placzek Ratio and Relative Brillouin Shift for Glycerine While the instrument was being Optically diagnosed, the adjustments Optimized and the sample cell subsystem thermally calibrated, several Brillouin spectra were recorded for glycerine samples. Because Of poor frequency resolution (Optical misalignment etc.), uncalibrated Fabry- Perot interferometer and uncalibrated temperature settings, this data was not initially utilized for any purpose other than the determination of instrumental malfunctions. Afterwards, it was deemed that, at least approximate, Landau-Placzek ratios and a Brillouin shift expressed as a function Of the interferometer's free spectral range could be obtained. In addition to providing guidelines for further experiments with glycerine, these results illustrate the instrument's utility in several manners. Therefore, on a qualitative basis, some of the better spectra were selected from the numerous recordings; Landau— Placzek ratios and relative Brillouin shifts were Obtained 174 .msfluoeumom unmflq mo wocmtsmmmo smasmsfi - q 20:31.00 2500me31.... 020N200 mo.¢0...zo<¢hwh 200140 0 q q — Q \\ q .‘q 4. .D .4.s magmas ..m .322 ozimt—Sm on. o: om. On. 0! 3.8. o: oo_ oo oo 2. oo on o... 1 .4. q d I q '0 CI 4 C) ’I q 0 'I 00 ON u a U 0. q 0.. 0.0 om; c. I 0 .m 0.. 175 from them, in a manner described below, as a function of temperature. For the selected spectra, the Spectra Physics model 125 (measured 78 milliwatt output) helium—neon laser was used. A square sample cell was aligned for a 90 degree scattering angle. The interferometer contained 1/100, 95% reflectance mirrors. The pertinent linewidths were all measured as a fraction Of the free spectral range; on the recorded spectra, the linewidths were measured as a fraction Of the distance between two central Rayleigh peaks corresponding to adjacent interferometer modes. The areas, integrated intensities, were measured by "cutting the curve out" and weighting the paper. Then, following the three Observations below, the Landau-Placzek ratios were calculated. First, Litovitz's group (120) has shown that I8 = 2 I+(1+A+/wp) (7-5) where IB is the integrated intensity of a Brillouin peak; I+ is the integrated intensity Of the high frequency half of a Brillouin peak (It is obtained at a lower error than IB because of overlapping from the central line, etc.); A+ is the halfwidth at halfheight corresponding to 1+, of the Brillouin peak; wp is the Brillouin shift. Second, cor— reSponding tO Litovitz's experimental definition of the Landau-Placzek ratio, "the scattering Spectrum due to 176 density fluctuations alone may be determined by subtracting 7/6 Of the depolarized spectrum from the polarized spectrum". Finally, assuming the instrumental profile, measured and true spectral peaks are lorentzian functions, for the purpose Of calculating Landau-Placzek ratios, it may be shown that an approximate deconvolution may be accomplished by: At = Am(1+Fi/Fm) (7-6) wherein At is the area corresponding to the true spectrum, Am corresponds to the experimental spectrum and Pi and Pm are the halfwidths at half height for the instrumental profile and experimental peak respectively. TO summarize, the detailed procedure was: (1) The halfwidths, at half height, A+ and wp were measured as a fraction Of the free spectral range. (2) With the dark current as a baseline, the total area for the polarized spectra was evaluated. (3) The area for the depolarized spectra was Obtained and 7/6 of its value was subtracted from the total area above. (4) The region for 1+ was graphi~ cally isolated; cutting this portion away from the total area, this area was evaluated by weighing the paper. (5) Equation (7-5) was used to Obtain a value for the Brillouin line and twice this value was subtracted from the total to Obtain a value for the central Rayleigh line. (6) The Brillouin and Rayleigh areas were corrected for instrumental 177 broadening by applying equation (7—6); the instrumental profile halfwidth was approximated by the halfwidth of the central peak of a Brillouin spectra of benzene. (7) Finally, the Landau-Placzek ratio, IC/ZIB was calculated from one- half of the ratio of these two areas. Note, in this data analysis procedure, all of the quantities appear as ratios in the calculation of the Landau—Pleczek ratio hence their unconventional units cancel out. The results are tabulated in Table 7.8. The Landau— Placzek ratios are graphed in Figure 7.5 as a function Of temperature where they are compared to Litovitz's results. The relative Brillouin shifts are graphed in Figure 7.6 as a function Of temperature. The utility Of these plots are discussed by Litovitz. These two sets Of Landau- Placzek ratios seem to agree quite well. The relative Brillouin shift data could be converted to actual Brillouin shifts by assuming one value, at one temperature, from some other work as a standard. This standard sample can be any liquid as long as it is also measured with the same Fabry- Perot mirror Spacing. Perhaps this is a convenient calibra— tion technique. The temperature values in Table 7.8, have a 0.2° C uncertainty. The free spectral range, estimated from a measurement of the Fabry—Perot mirror separation at a later date is roughly 21.5 GHz (mirror separation Of 1.395 cm). 178 TABLE 7.8.--Landau-P1aczek Ratio and Relative Brillouin Shift for Glycerine. Temperature, C Landau-Placzek Relative Brillouin Ratio Shift 10 1.90 0.432 20 1.82 0.411 30 1.71 0.400 40 1.56 0.383 50 1.22 0.358 60 1.01 0.331 70 0.70 0.310 80 0.48 0.292 90 0.32 0.283 100 0.33 0.270 110 0.22 0.260 120 0.25 0.251 179 .mcflsmomaw How Oflpmm xmmomHmlsmpsmq mo mocmtsmmmo musvwnmmfime .m.n mnsmflm .0o mm:...0...... l . 0. cu ...zmmwma l O. N) ‘vf 180 .mcenmomao MOM pmflgm QHDOHHHHm ®>praom mo mocmtcwmmo msswmummame flop» mmah0zmzcwm... 0 0: .s.s messes 0.- BAIIV‘IBH A1ISN31NI 183 at its fast chart Speed in order to collect data with a relatively high frequency resolution. In order to deter— mine the instrumental profile, the scattering process was simulated by focusing the incident laser beam onto an Opal glass plate with a short focal length lens. The opal plate was first aligned so that it reflected the incident laser beam along the detection Optics axis. The spectro- photometer alignment was optimized between individual runs. Nine spectra were obtained but only one, chosen because it had the narrowest central line, corresponding to best instrumental finesse, was analyzed in detail. The rms difference between this experimental curve and a best fit curve is 0.63%. The instrumental bandwidth was 3.664 MHz corresponding to a finesse of 40.9. Both the instrumental profile and Brillouin spectrum data points were smoothed using a cubic, five data points, least square polynomial technique; data points at convenient frequency intervals were interpolated with these polynomials. The Brillouin spectrum was deconvoluted and curve fit using the techniques in Chapter III. The resulting spectral parameters (assuming the single relaxation model) are tabulated in Table 7.9. In Table 7.9 is a set Of theoretical Values for the Spectral parameters. They were calculated using the theory outlined in Chapter II and discussed by Nichols and Carome (28). Except for the refractive index, the 184 .NmO to mean: ya mmmmo.o mmmso.o «m4 ones gesoaaeum mo puma oespmfifihwwpsm mo mpnpflamad Hsmsm.o Nmmsm.o was mass casoaaaum mo puma oasumasmm mo manpaamss ammma.o Gmwoa.o Hoe mean amcoaumxmamm to magpflamas mamam.o ammam.o Bus mass amasmse mo mwspaamaa HNHN.~ omom.m HOE snaazmamm seameno amcoanmxmamm omao.o mmoo.o sue spoazmamm mcflaamo amasmee mmme.o mmev.o me buoflsmamm cflsoaaflum moom.v moom.v ms pMHEm GHSOHHHAm m>H50 «amoepmuomse Hoefimm Hmmemumm “Hm pmmm Hmecmfiflummxm .mumumfimumm mnpomgm SHDOHHflHm mpfluoanommpme soenmu HOm mmSHm> Hmesmfiflnmmxm can amoepmuomsall.m.h mqmme 185 material parameters used in these calculations are those in Nichols' Table I. A refractive index of 1.4675 was interpolated, relative to wavelength and temperature, from the literature data (103). Figure 7.8 is a comparison Of curves calculated from the theoretical and experimental parameters in Table 7.9. The curves represent true spectra convoluted with the instrumental profile. Contributions due to adjacent interferometer modes are not included. This Simplification helps illustrate the asymmetric component of the Brillouin peaks and experimentally corresponds to a higher free Spectral range. The rms difference between these two curves is 0.99%, insignificantly larger than the rms dif- ference between the experimental and best fit curve, 0.63%. Another Brillouin experiment (104) on carbon tetrachloride (at 4880 A° incident radiation and at 25° C) yielded a value Of 4.70 GHz for the Brillouin Shift. The standard deviation of those measurements was greater than 4%; therefore, this Brillouin shift does not differ signifi- cantly from the one Obtained in the present experiment. The most important conclusion relative to this experiment is that good Brillouin spectra are obtainable from this instrumental system and they may be analyzed by the techniques Of Chapter III. The small differences between the theoretical and experimental parameters in Table 7.9 should not be over emphasized as a measure of 186 00°01 OO'B .wpwHoHSOMHme connmo H00 mneommm swsoaawum mmumasoamo 1.402.020ka II m>m50 .5“. #000 440.202.1053 I .m.s musmam . mu 0 O OO'OI- lllllllllIllllllllllJJILllllljllljlnlll _ 00.0 0.0 00.0 00.0 0*. 0 00.0 00.0 0h.0 00.0 00. 0 00.. 187 the instrument's precision and accuracy since these data represent the best Of nine spectra. The lack of a larger number Of these "good spectra" is due to the interferometer's instability which should be reduced before one seriously considers the instrument's precision and accuracy Of performance. CHAPTER VIII CONCLUSIONS Summary The instrumental system described in Chapters IV and V has proven to be useful and versatile. The experi- ments described in Chapter VII illustrate measurement Of parameters which characterize the incident light beam, the response of the sample (and its scattering mechanism) and the scattered light beam. With the instrument in a photometric mode, the depolarization ratio of scattered light was measured as a function of the sample's temperature and the wave length Of the incident light beam. The angular dependence Of scattered light intensity was also studied. Spectrophotometrically, the Landau-Placzek ratio and relative Brillouin shift of glycerin were Obtained as a function of the sample's temperature. A high quality Brillouinspectrumcaf carbon tetrachloride was obtained when the instrument was prOperly aligned; thus the primary Objective of this research project to design and construct an instrument for this purpose was met. Illustration of the instrument's versatility is presently being supplemented and eXpanded by the research 188 189 of several colleagues; among the projects are: (a) The depolarization ratio and Brillouin spectra Of several sugars in the temperature region around the glass transition temperature have been determined (87,97). (b) The associa- tion of dimethyl sulfoxide in the liquid state is being studied using Landau-Placzek ratio measurements (87,86). (c) A new spectroscopic technique original with Miller (105 to 108) is being used to determined the molecular weight Of polymers. (d) The Brillouin Spectra (109) of polymer solutions are being investigated. (3) Photometric measurements of light scattering by inhomogeneities of polymers in the amorphous phase (88,110) are being made at low scattering angles with high angular resolution. (f) Stiso (105) has substituted a wave analyzer for the instrument's normal detection electronics, converting the system into a light beating spectrometer. Future Research The most serious deficiencies Of this instrumental system are the interferometer's instability and the limit on finesse resulting from acceptance angle of the collection Optics. The present interferometer's instability may be improved by housing it in a thermostat and reconstructing key portions, with a material Of lower thermal expansion such as Invar. Solid etalon spaced interferometers with pressure scanning can provide a more linear scan (111,112). 190 Recently (113), "a highly stable alignment device using self-aligning bearings coupled withpiezoelectric expansion columns has been developed and applied to a Fabry-Perot etalon. The accuracy of alignment was (reported to be) 7 radians." This is better than twice greater than :10— the accuracy claimed by the manufacturer of the present interferometer. Electronically stabilized, "double— passed" scanning interferometers have been reported recently (114). The collection Optics could be changed to reduce the acceptance angle of the interferometer; however I . alignment difficulties would increase. The acceptance angle could be decreased by focusing a small image Of the scattering volume onto a very small pinhole. The beam transmitted through the pinhole would be a closer approxi- mation to light emanating from a pin point. This Option was considered in the early designs but was rejected because of the associated difficulties in optical alignment. The polymeric foam in the Optical table shows fatigue, thereby causing the optical table to become tilted, requiring realignment. Eventually, this effect may seriously reduce the foam's ability to attenuate acoustic vibrations. It is recommended that air filled inner tubes be considered as a replacement for the foam. Collection Of Brillouin spectra of a high quality has become a routine procedure with the result that data 191 analysis has become the limiting factor of useable output. Since this research was undertaken, Jansson (115) has published a method for "resolution enhancement of spectra" by deconvolution. It is recommended that this method, with analog-to-digital data collection, be given serious consideration. Day (116), Hildum (117) and others have recently discussed a technique useful for the deconvolution of Fabry-Perot profiles when one is willing to assume Lorentzian, Voigt or Gaussian shaped lines. 10. 11. 12. 13. 14. 15. 16. 17. BIBLIOGRAPHY Richter, J. R., Ueber die neuern Gegenstande der Chemie, dd, 81 (1802). Tyndall, J., Phil. Mag., dz, 384 (1869). Lord Rayleigh, Phil. Mag., dd, 447 (1871); Ibid., dd, 81 (1881). Einstein, A., Ann. Phys. Lpz., dd, 1275 (1910). Mie, P., Ann. Phys. Lpz., dd, 3, 771 (1908). Raman, C. V., Indian J. Phys., d, 1 (1928). Debye, P., J. Appl. 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Am., dd, 117 (1968). -.. -.--7,7 .fi—_ ,_ ___._.._____. ‘ __ , ——. _— — .. _ai —" q—w—v __ - -v 7. ’APPENDIX Frequency Stabilized Lasers Short Plasma Tube A laser was constructed with an optical cavity short enough that only a single axial mode lased and the frequency Of this mode was stabilized by the incorporation of a Lansing Research Corporation model 80.210 lock-in stabilizer. The laser is schematically represented in Figure 10.1. The plasma tube is a 1 mm bore, 12 cm long tube with Brewster end windows and is filled with a 7:1 mixture of the 3He and 20Ne gases at a total pressure of 3.6 torr. A Jodon model PS-100 power supply provides dc excitation to the coaxial, hot cathode electrode system. The cathode is Operated at 6.2 volts and 1.46 amperes. The plasma saturation current is 4.2 ma; cathode-anode voltage is 500 volts. This bore diameter, total gas pressure, helium neon ratio and exciting current correspond to 20Ne isotopes Optimum output power. Utilization of 3He and insure a significant Lamb dip in output power versus mirror separation alleviating frequency stabilization difficulties. The tube was clamped into a block of Invar which in turn was bolted to an Invar plate. The mirror 199 200 I-——A/2 __.., L*——A/2 LAMB DIP m 5 In K/ 3 3 o O a. Q I K MI 3: ‘2 S _I l-I |-O L2 l-o LASER MIRROR SEPERATION LASER MIRROR SEPERATION SINGLE ISOTOPE LASERS COMPLEX GAs LASERS (a) (b) .IODAN PS-IOO BATTERY PMT POWER SUPPLY POWER SUPPL OPTICAL P LASER TUBE PMT DETECTOR OUTPUT {—m M I - - LANSING RESEARCH LOCK-IN STABILIZER MODULATION OUTPUT I-IV CORRECTION SIGNAL IN ...L M, =LAsER MIRROR M2: LAsER STABILIZATION CONTROL MIRROR P: PIEZOELEcTRIc TRANSDUCER o= MODULATION To P b=Dc CORRECTION To P (C) Figure 10.1. Frequency Stabilized Short Plasma Tube Laser. 201 holders were also bolted to this Invar plate so that the mirrors were 15 cm apart. Note that mirror spacing is the parameter dominant in causing the laser to lase in a single axial mode. The small tube diameter does not allow other Off axis modes to lase. Unfortunately, this Single mode technique compromises output power; when properly aligned, the laser's output power exceeds 230 uwatts. Both high precision gimbal mounts were purchased from Lansing Research. One mount has an adaptor for a one— inch diameter mirror; the other mount has a two input Option, piezoelectric translator (model 21.8.4) with an adaptor for a one—inch diameter mirror. Both mirrors are Oriel A/20, one-inch diameter, two meter radius mirrors. One mirror is coated for a reflectance as close to 100% as is practical; the output mirror is coated for 99.5% reflectance at 6328 A°. Optical alignment is achieved by using another laser to define an Optical axis coincident with the bore of the laser tube. The far—end mirror is mounted and adjusted until the incident beam reflects back onto itself as closely as can be judged visually. Then the other mirror is mounted and adjusted so that the beam reflected from the mirror's back surface is coincident with the incident beam. Since the mirrors are slightly wedge—shaped, the present status does not correspond to correct alignment; however, it is close enough that with systematic rocking Of the micrometer's, the system can be brought into sufficient alignment to cause 202 lasing. To optimize, all four micrometers are systematically varied to maximize the Optical output power. Now frequency stabilization is accomplished by following the extensive procedure explained in the lock-in stabilizer's manual. Two frequency selective elements are present in every laser: (1) The laser medium exhibits a useful gain over a frequency range centered near the Optical transition frequency 00. (2) The optical cavity which exhibits high Q modes at integral multiples of c/2L. c is the phase velocity of light in the Optical medium and L is the length Of the Optical resonator, that is, the mirror spacing. This c/2L spacing is analogous to the free spectral range of a Fabry—Perot interferometer under the approximation Of large radius spherical surfaces to plane surfaces over a small region. Several resonator mode frequencies lie within the frequency range over which the laser medium's Optical gain exceeds Optical loss; therefore, the laser may oscillate at several frequencies. For inhomogeneously broadened optical transitions, e.g. DOppler broadening, the laser will oscillate simultaneously at each mode fre— quency for which the gain exceeds the loss. The simultaneous oscillations are not truly independent; through the action of Optical saturatiOn Of gain about each oscillating mode, the resonator modes couple to stimulate atoms to emit. The resulting mode competition introdues instability. Also, 203 in multimode Operation, the available output power is distributed among the competing modes. Since the Q Of the Optical resonator is typically much higher than the Q of the medium's emission line, the frequency of the resonator mode determines the frequency of oscillation, to within first order. High order effects arise from disperison in the medium and "mode pushing"- However, a resonator functioning near 00 is very nearly independent of these high order effects. This is the condition achieved for laser stabilization. Since the Optical resonator frequencies are a strong function of the resonator's length, laser stabili- zation may be accomplished by the stabilization of the Optical path length of the resonator. While many schemes for stabilization have been devised (see Tomiyasu's bibliography) only two were practical until very recently: One is "stabilization to the Lamb dip" or maximizing the output power and is used for this short plasma tube laser. The second scheme is the locking of a resonant cavity interferometer (internal or external) to a laser line. It was used for the long plasma tube laser. The latter technique is very similar to the technique adapted in the very recently, commercially available frequency stabilized lasers. Even though not presently practical for many applications, another method of stabilization worth consideration for some applications . . .-. 1. - . . --- ---- Wm _. _ .- ._ _ .. H .. ,_ -. _ _ _y . __‘w—_ 1..—outw— - Ir'n'T-V‘W ‘TT"? *— "':‘ ’2. ...a.‘ 204 is the "four mirror stabilization of long lasers" dis- cussed by Smith (118). When c/2L exceeds the Doppler width, only one optical resonator mode has enough gain to sustain oscil- lation. The laser's Optical output power will vary as shown in Figure 10.1(a) when L is varied by A/2 so that the resonator mode sweeps over the gain region. The dip in laser power at vo is the Lamb dip and is a very significant feature in low power, single isotope lasers. When this dip does not exist, the stabilization system can be locked to a power peak. A system for stabilizing the laser to the Lamb dip frequency, v0, is illustrated in Figure 10.1(c). The position of mirror 2 is sinusoidally modulated (A voltage corresponding to a small fraction Of A/2 is applied to the piezoelectric mounting element P.) thereby modulating the Optical resonator length and hence the optical resonator's mode frequency. The laser output power is thereby sinu— soidally modulated at this modulation frequency with the following characteristics, provided the modulation amplitude is small: The modulation index is zero when the mode fre— quency is 00 since the slope of the laser power output versus 0 is zero for v equal to 00. The phase Of the modulated component Of the laser power relative to the modulating voltage applied to P is Opposite for v > 00 relative to the phase for which v < v0. The modulated 205 component of the laser power is detected, preamplified and phase sensitively detected in the Lansing Research model 80.210 lock-in stabilizer. The phase sensitively detected signal, after filtering, provides a discriminator signal which is zero for resonator lengths corresponding to a mode frequency 00. It is positive on one side of 00 and negative on the other side of 00. This discriminator signal is dc amplified and applied to the piezoelectric translator P with a phase appropriate for stable negative feedback. The result is that the resonator's length is controlled in a manner insuring Optical oscillation only at 00. The mode structure Of the stabilized laser was studied with the use Of the Spectra-Physics model 420 Optical spectrum analyzer plug—in unit and a Tektronix 564B oscilliOSCOpe. It was truly a single mode laser. Instantaneously, the laser's bandwidth was about 35 MHz and the stabilizer held the center frequency constant to within about 48 MHz for several hours before it would seriously drift. The stability may be improved with a temperature controlled housing for the laser. However, because Of the low output power, and other problems of the instrumental system, research and application Of this laser was abandoned. Even if the low intensity could be tolerated with photon counting detection, alignment difficulties arise from the non visibility Of a scattered beam along the Optical detection train. 206 Long Plasma Tube A second laser was constructed to provide a single mode output Of greater intensity. Stabilization was to be achieved by locking an external interferometer cavity, functioning as a filter, to one of the laser's frequency modes which had been stabilized by AM or FM mode locking. Tomiyasu (81) and Crowell (119) provide a good explanation Of these mode locking schemes. The frequency locking Of the laser to an external resonant interferometer cavity was investigated. It is the second technique referred to in the previous section and a good introduction is Met's "Working with Etalons" (56). Briefly, this technique amounts to the locking Of a narrow bandwidth Optical filter to one of the laser's oscillating (axial) modes. An interferometer with piezoelectric control Of one of its mirrors was used in a manner similar to that described in the previous section. The laser's output was passed through this external inter- ferometer cavity (a Tropel model 240 interferometer normally used as a spectrum analyzer) whose piezoelectrically mounted mirror was modulated. The resulting correction voltage was applied to the piezoelectric mount in a manner which maximized the transmitted power. Thus, the inherently multi-mode laser was filtered to yield a single frequency output. 207 This laser design is illustrated in Figure 10.2. The laser tube is a commercial 3 mm bore, 90 cm long tube filled with a helium—neon gas mixture to yield a maximum output of 58 milliwatts functioning in a multimode fashion. It has a hot cathode in a bulb connected to the center Of the tube and anodes near each end. Brewster windows were attached Via ground glass joints to allow them to be aligned relative to one another. Magnets are mounted along the tube utilizing the Zeeman effect to suppress laser oscillation at 3.39 u which competes significantly with the lines 6328 A° for long plasma tubes. A Sorensen model QB6-8 dc power supply was used tO excite the hot cathode with 6.3 volts and 3.4 amp. The anode-cathode voltage (1800 volts) which Optimized the output was supplied by an Alfred model 233A regulated high voltage power supply. The total current was 32 ma. The tube was mounted at two points, with modified micrOSCOpe x—y positioners, to an optical rail. The mirrors were both Oriel A/20, one-inch diameter, two meter radius mirrors. One mirror was coated for nearly 100% reflectance and the other mirror was coated for 97% reflectance. They were spaced about 95 cm apart and mounted in Lansing Research model GS-203 gimbal mounts which in turn were mounted onto the Optical rail. After the laser was aligned using the technique described in the previous section, it was stabilized using 208 (0‘5! .Hmmmq mesa mammam 0coq OmNHHHQOpm mocmswmnm .m.oa Ounmflm 20h011(00_m 2(88000 >4¢a30 1030a 0002< «05205.0 2. 0.00.. 20040002 02.0243 >Jauam 1030.. 0007;40 0...; «00¢; 0012.! xmhwlozuuzwkz. Juaomk ”mph—4A0 3400 209 the Lansing Research lock-in stabilizer coupled to the Tropel interferometer. After Optimizing the laser's parameters, its multimode output power was 38 milliwatts; when "filtered"a maximum output power Of 4.1 milliwatts was obtained. The bandwidth Of the "filtered" laser beam was 7 Mc, instantaneously. However, the center frequency of this band drifted seriously because the laser was not yet mode locked. During the investigation of this long laser tube‘s prOperties, two difficulties with the plasma tube became apparent. The tube tended to twist with time causing the Brewster windows to misalign and reduce the laser's output power. Secondly, the tube was warping, decreasing the effective volume Of the active media hence decreasing the output power. PP