THE DEPOLARIZATION 0F RAYLBIGH SCATTERED LIGHT Thesis for the Degree of Ph. D; MICHIGAN STATE UNIVERSITY ROBERT J. mason 1967 LIBRARY Michigan Stat: Uni~ ersity Thesm This is to certify that the thesis entitled The Depolarization of Rayleigh Scattered Light presented by Robert J. Anderson has been accepted towards fulfillment of the requirements for ' ’ _Ph_-D_-__ degree mm Major prof sor Date July 24, 1967 0—169 . ABSTRACT THE DEPOLARIZATION OF RAYLEIGH SCATTERED LIGHT by Robert J. Anderson The depolarization of light, Rayleigh scattered from pure liquids, binary solutions, and polymer solutions, has been studied by measuring the depolarization ratio, pv’ for vertically polarized light. The theory of depolariza- tion by dense fluids has been reviewed, put into a form consistent with the theory of partial polarization, and extended to include the temperature dependence. The depolarization ratio and its temperature de- pendence were measured using a helium-neon laser as the light source, a phase sensitive detection system, and a photometer designed to take advantage of the laser output. The results obtained are considered to be the most accurate available, since the methods used in this study have re- moved the sources of most of the difficulties previously encountered in these measurements. The experimental techniques are discussed in con- siderable detail, and potential improvements in the system are included. Moreover, an analysis of the errors inherent in this method yields an indication of the overall accuracy Robert J. Anderson of the result, and shows what limitations are to be expected on measurements of this type. The results indicate that the depolarization ratio of vertically polarized light has a temperature dependence somewhere between linear and logarithmic. The theory is found to agree well with the experimental results in the case of polar molecules, while for non-polar molecules agree- ment is poor for those species with a high degree of symmetry. These results have been interpreted as indicating that the hyperpolarizabilities play a significant role in determining the depolarization by pure liquids. In binary solutions, it is found that the results agree well with theory at intermediate compositions, even when the two pure components do not. THE DEPOLARIZATION OF RAYLEIGH SCATTERED LIGHT By .9 Robert JF'Anderson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1967 ACKNOWLEDGEMENTS The author wishes to acknowledge the assistance of the following persons, without whom this study would have been impossible: Mr. James Grumblatt, electronics techni- cian; Mr. Russell Geyer, machinist; Mr. Andrew Seer, glass- blower; Mr. Larry Dosser, for his enthusiastic assistance; and Professor J. B. Kinsinger, for his encouragement and advice. TABLE OF CONTENTS ACKNOWLEDGMENT LIST OF LIST OF LIST OF CHAPTER I. II. TABLES FIGURES APPENDICES INTRODUCTION Historical Rayleigh Scattering . Purpose of this Research THEORY . . . . . . . . . . . Electromagnetic Field Properties Maxwell's Equations . Field Intensity and the Poynting. Vector The Electromagnetic Plane Wave . Coherence Matrix Formalism Introduction Jones' Calculus Coherence Matrices Instrument Operators Coherence Matrices for Cases 00f Special Interest Some Equivalent Representations Theory of Rayleigh Depolarization Intensity of Radiation from an Induced Dipole . . Matrix Treatment of Rayleigh Scattering Rayleigh Scattering from a Perfect Gas. Depolarization of Vertically Polarized Light iii Page ii vi vii U'INl—I' l—‘CDV \l 13 13 14 15 17 18 20 21 21 26 30 33 TABLE OF CONTENTS (Continued) CHAPTER Page Depolarization of Unpolarized Light . . . 36 The Horizontal Depolarization Ratio . . . 38 Depolarization by Dense Fluids of Spherical Molecules . . 39 Temperature Dependence of Depolarization by Dense Fluids . . . . . . . . . 43 Laser Theory . . . . . . . . . . . . . . . . 47 Emission and Absorption of Radiation . . 47 Oscillation and Modes . . . . . . . . 51 General Description of Lasers . . . . . . 55 III. EXPERIMENTAL . . . . . . . . . . . . . . . . 61 Laser Design Considerations . . . . . . . . 61 Introduction . . . . . . . . . . . . . . 61 Power Output . . . . . . . . . 62 Gas Mixtures and Pressures . . . . . . . 62 Resonator Configuration . . . . . . . . . 63 Laser Characteristics . . . . . . . . . . . 66 General Description . . . . . . . . . . . 66 Filling Techniques . . . . . . . . . . . 68 Power Supply . . . . . . . . . . . . . . 69 Laser Alignment . . . . . . . . . . . . . 7O Photometer Design . . . . . . . . . . . . . 71 Introduction . . . . . . . . . . . . . . 71 Light Source . . . . . . . . . . . . . . 72 Photometer . . . . . . . . . . . . . . . 74 Detector . . . . . . . . . . . . . . . . 79 Alignment . . . . . . . . . . . . . . . . 84 System Characteristics . . . . . . . . . . . 86 Measurement Techniques . 86 Temperature Stability and Reproduc1b111ty 88 Photomultiplier Response . 88 Effect of Beam Parameters . . . . . 89 Calibration of the Neutral Density Filters . . . . . 91 Error Analysis and Total Accuracy . . . . 92 Error Sources . . . . . . . . . . . . . . 100 Preparation of Samples . . . . . . . . . . . 106 iv TABLE OF CONTENTS (Continued) CHAPTER Page IV. DATA AND RESULTS . . . . . . . . . . . . . . 108 Pure Liquids . . . . . . . . . . . . . . . . 108 Benzene and Its Derivatives . . . . . . . 108 Hexane and Related Compounds . . . . . . 116 Chlorinated Methane Derivatives . . . . 122 The Chlorinated Derivatives of Toluene. . 126 Summary . . . . . . . . . . . . . . . . . 129 Binary Solutions . . . . . . . . . . . . . . 133 Benzene- Nitrobenzene Mixtures . . . . . 134 Benzene- Carbon Tetrachloride Mixtures . . 141 Summary . . . . . . . . . . . . . . . . . 144 Polymer Solutions . . . . . . . . . . . . . 146 V. CONCLUSIONS . . . . . . . . . . . . . . . . 149 The Present Study . . . . . . . . . 149 Suggestions for Further. Study . . . . . . . 152 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 154 APPENDICES . . . . . . . . . . . . . . . . . . . . . 156 LIST OF TABLES Page Coherence Matrix Representation of Certain Optical Instruments . . . . . . . . . . . 18 The Coherence Matrices of some Special States of Polarization . . . . . . . . . 19 Calibration Factors for Neutral Density Filters . . . . . . . . . . . . . . . . . 92 Limiting Errors for Extreme Values of Depolarization . . . . . . . . . . . . . 104 Depolarization Ratios of Benzene and Its Derivatives . . . . . . . . . . . . . . . 110 Temperature Variation of Q Using the Data for Benzene . . . . . . . . . . . . . . . 112 Comparison of Theory and Experiment for Benzene and Its Derivatives . . . . . . . 115 Depolarization Ratios of Hexane and Related Compounds . . . . . . . . . . . . . . . . 117 Comparison of Theory and Experiment for Hexane and Related Compounds . . . . . . ll8 Comparison of Saturated and Unsaturated Ring Compounds . . . . . . . . . . . . . 120 Depolarization Ratios of the Chlorinated Methane Derivatives . . . . . . . . . . . 123 Comparison of Theory and Experiment for the Chlorinated Methane Derivatives . . . . . 125 Physical Properties of the Toluene Derivatives . . . . . . . . . . . . . . . 127 vi LIST OF TABLES (Continued) TABLE 4.10 4.11 4.12 4.13 4.14 4.15 4.16 7.1 Depolarization Ratios of Ortho- and Para- Chlorotoluene Comparison of Theory and Experiment for the Toluene Derivatives Comparison of Theoretical and Experimental Values of “T . . . . . . . . . . . . Depolarization Ratios at Various Tempera- tures for Benzene-Nitrobenzene Mixtures Values of —Q for Benzene-Nitrobenzene Solutions . . . . . . Physical Properties of Benzene-Carbon Tetra- chloride Solutions Vertical Depolarization Ratios of Benzene— Carbon Tetrachloride Solutions at Various Temperatures Values of -Q for Benzene—Carbon Tetra- chloride Mixtures Summary of Results for the Standard Poly- styrene Samples . . . . . . . The Energy Barrier for Depolarization Difference in Refractive Index Between Salt Solutions and Pure Water vii Page 127 128 135 135 138 141 142 142 147 151 170 FIGURE 2.1 2.2 LIST OF FIGURES The Coordinate System Modes in a Laser Interferometer 'Energy Levels in a Helium-Neon Mixture Outline of a Helium-Neon Laser Resonator Configurations The Polarization Photometer Light Trap The Detection System The Depolarization Ratio of Vertically Polarized Light Versus Percentage Error Plot of Zn pV Versus T for Benzene and Its Derivatives Depolarization Versus Temperature for Benzene Derivatives Plot of Zn pv Versus T for Hexane and Related Compounds Plot of Zn pV Versus T for the Chlorinated Methane Derivatives Plot of the Depolarization Ratio Versus Mole Fraction for Benzene-Nitrobenzene Solutions Plot of fin pv Versus T for Benzene- Nitrobenzene Mixtures viii Page 26 53 58 59 64 76 78 80 105 111 113 119 124 137 139 LIST OF FIGURES (Continued) FIGURE 4.7 7. 7. .9 1 2 Plot of -Q Versus Mole Fraction for Benzene- Nitrobenzene Mixtures . . . . . . . . Plot of Zn pV Versus T for Benzene-CC£4 Mixtures . . . . . . . . . . . . . Plot of -Q Versus Mole Fraction for Benzene- Carbon Tetrachloride Mixtures . . . . The Laser Divergence Percentage Error in ph as a Function of pV ix Page 140 143 145 156 169 LIST OF APPENDICES Divergence of the Laser Output Inherent Resolution of the Polaroid Filters . . . . . Analysis of Errors in the Horizontal Depolarization Ratio Calibration of the Differential Refractometer . . . . Comparison of Results Page 156 158 160 168 171 CHAPTER I INTRODUCTION Historical Recorded light scattering measurements date back to 1802, when Richter1 observed the scattering from a sol of colloidal gold. Tyndall2 obtained an artificial blue sky in a mixture of butyl nitrate and hydrochloric acid, and showed that if the incident rays were plane polarized the scattering was only visible in one plane. This so-called "Tyndall Effect” became the starting point of the light scattering investigations of Lord Rayleigh,3 and in subse- quent papers he‘outlined a complete theory based on induced moments in the scattering particles. Since then, the theo- retical and experimental study of light scattering has been steadily pursued, until today it has become a sophisticated tool for the study of molecular properties. In recent years interest has centered on two dis- tinct areas: linear, or Rayleigh scattering, and non-linear, or Raman scattering. Since the Raman effect is more closely related to spectroscopy, it will not be discussed here, and instead we will concentrate on the treatment of Rayleigh scattering. Calculations on the scattering of a system may be approached in two ways: quadrature summation of the 1 radiation contributions from each scattering particle, or treatment of the scattering as the result of statistical fluctuations in the bulk dielectric constant of the medium. Actually, these methods are not completely independent, and the exact treatment of a scattering system involves consid- eration of both approaches. Quadrature summation was originally used by Rayleigh,3 in the calculation of the scattering from a gas, and has since been extended by Debye18 to the case of a dilute solution of non-interacting high polymers. The fluctuation theory was develOped by Einstein,4 Smoluchowski, and Gans, and has applications in the scatter- ing from pure liquids as well as solutions of interacting particles. Rayleigh Scattering The behavior normally referred to as Rayleigh scattering is actually the sum of three scattering pro- cesses: (l) Scattering of the incident light, v0, with no change in frequency, from entropy fluctuations in the medium, giving rise to a narrow, .Olcm_1, line. (2) Scattering of the incident light at some shifted frequency, v', from density fluctuations propo- gated at the velocity of sound. This scattering corresponds to modulation of the incident light by a DOppler shift, Av, such that where Av is of the order of .Scm-l. (3) Scattering from orientation fluctuations in the 1) background centered about the frequency of the in— liquid, giving rise to a very broad (tens of cm- cident light. This corresponding to modulation of the incident light by a highly damped rotational Raman effect having, again, both Stokes and anti- Stokes components. The first process giving rise to the central un— shifted line, is the actual Rayleigh (or Tyndall) scatter- ing; the two Doppler shifted sidebands comprise the Brillouin doublet; and the broad background is the Rayleigh 19 have shown that the wing. Recently, Cummins and Gammon Rayleigh and Brillouin lines are highly polarized. The quantity which is generally of interest in a light scattering measurement is the Rayleigh ratio, R(6), where 6 is the angle with respect to the incident light at which the observation is made. Using vertically polarized incident light, we may write for pure fluids of small molecules, dn 2 3 + 3pv 2 [__] [_—_————] (l + COS 6) (1-1) 2 2n n R (6) = T RTE[—] V A N edT 3 - 4pV a where subscript v denotes the vertical polarization of the incident light, N is Avogadros number, A the wave length of the incident light, R the gas constant, T the absolute temperature, n the refractive index, E the isothermal com— pressibility, a the coefficient of thermal expansion,_and p the vertical depolarization ratio of the medium. (1-1) V may be rewritten, when 6 = 90°, as _ E _ Rv(90) — [IR + 2IB + 1.75 I I (1 2) O W where I0 is the incident intensity, r is the distance to the detector, IR the intensity of the central, or Rayleigh, line, IB the intensity of each of the Brillouin sidebands, and Iw the total intensity of the Rayleigh wing. Moreover, the scattered radiation contains both a vertically polarized, Vv’ and a horizontally polarized, Hv’ component, such that Vv = IR + 2IB + Iw (1-3) H 0.75 I . v W These terms each contain an inverse fourth power dependence on the wavelength, making it necessary, in the past, to perform scattering measurements using blue or green light, and to utilize large light sources. This practice has given rise to ambiguities due to fluorescence in the medium, un- certainty in the wavelength, multiple scattering due to the source size and wavelength, and Optical and electronic noise. The development of the ruby laser by Maiman,5 and the helium-neon laser by Javan,6 has largely alleviated these difficulties. The laser emission is highly collimated, monochromatic, and extremely intense, reducing both spatial and temporal uncertainties in the light source. In addi— tion, the longer wavelengths and higher intensities avail- able from lasers allow measurements to be made without fluorescence, and with small enough Ixunms that multiple scattering is minimized. Moreover, the extremely narrow band-pass of the laser output permits observation of the individual components of the Rayleigh scattering. That is, using laser sources, and interferometric techniques, the individual Rayleigh, Brillouin, and Rayleigh wing compo- nents may be resolved. In this study, as we are interested in measuring the unresolved scattering, we shall consider the Rayleigh-Brillouin lines, and Rayleigh wing, all to be part of the Rayleigh scattering. Purpose of this Research A preliminary study was undertaken to determine the characteristics of the Rayleigh scattered light, using a helium-neon laser operating at 632.8 nm as the source. These experiments utilized a Brice-Phoenix light scattering photometer in its customary configuration, the only modi- fication being the replacement of the mercury lamp by the helium-neon laser. After careful calibration of the in- strument (neutral density filters, diffusers, refraction corrections, etc.) it was found, using pure liquids, that the results of the measurement were consistent with pub- lished data for this instrument except for the depolarization ratio, pv' The depolarization ratio was observed to be difficult to measure consistently, due to instrumental factors such as uncertainty in the angle of observation, uncertainty in the polarizers and resolvers, stray light within the photometer, and noise in the electronics. More- over, examination of the light scattering literature (see, for example, Staceyzo) showed that the same uncertainty existed in values of pv measured elsewhere. Since (1-1) shows that this value is of considerable importance in the Rayleigh ratio, and since examination of the literature showed that the depolarization ratio had not been studied extensively, it was decided to apply the superior charac— teristics of the helium—neon laser to the study of the depolarization of Rayleigh scattered light. It was decided moreover, to investigate both the theoretical and experi- mental basis for the existing knowledge of the depolariza- tion ratio, and to extend this basis to include the temperature dependence of pv. It is hoped that the results of this study will serve to remove some of the uncertainties in the existing light scattering data, and to form the basis of an ex- panded inquiry into the nature of the depolarization pI‘OCGSS . CHAPTER II THEORY Electromagnetic Field Properties Maxwell's Equations The basic concepts of electrodynamics were formu— lated in the early part of the nineteenth century, and were summarized by Maxwell in a consistent set of equations known as Maxwell's Equations. The state of an electromagnetic field may be rep- resented by five field vectors, viz. the electric vector E, the magnetic vector H, the electric displacement D, the magnetic induction B, and the current density J. These field vectors are then related by: (p being the charge density) Vxfi=—C 3+%afi/at (2—1) VxE': - l as/ (2-2) c at v.1; = o (2-3) V D = 4np (2-4) where, as is customary in Optics, we have used Gaussian units. These relations however, are not sufficient to allow a unique determination of the field, and must there— fore be supplemented by a set of equations which describe the behavior of a material under the influence of the field. These equations are known as the constitutive rela— tions, and are in general very complicated; however, if the material is at rest and is isotropic they take the form: 3 = 0E (2-5) B = 8E (2-6) B = 11H (2'7) where o is the specific conductivity, 6 is the dielectric constant, and u is the magnetic permeability (in vacuum 6 = u = 1). Field Intensity and the Poynting Vector Using Maxwell's Equations,we may develop the energy law for an electromagnetic field. We see from (2-1), and (2-2) that E.V X H-H.V x E = H.B (2—8) file :1 J.E + l E.D + c all—4 where the dot refers to differentiation with respect to time. This result may be rewritten as: fire x a) + (as + ms) + file. :1 J.E = 0, (2—9) (UH--J and multiplying by c/4N, integrating, and applying Gauss's theorem yields gflmms) dV +[J.E dy+§F fExHJh‘ as: 0 (2—10) 3 3 being a unit vector normal to the surface S, of the volume V. Equation (2—10) represents the energy distribution of an electromagnetic field, and is valid for any material. In the case of linear, isotropic media, we may use the con- stitutive relations to find that, — —2 l_ — L _ l — 8(cE) 1 3(eE ) l 3 _ 4n(E°D)‘Z1TE at 'fi‘a't""§fi§f(ED) (211) 1 — L _ l — 3 — _ 1 3 —2 _ l a — — EEO-1°13) -— 4—TT Hug—E- (11H) - ‘7]7—15 (UH ) ‘ _—1T-._t (H-B) (2‘12) Setting -1 —— WE — §E E.D 1 (2-13) WM=§?T-H.B and W =_/(WE + WM) dV (2-14) where WB and WM represent the electric and magnetic energy of the field respectively, equation (2-10) becomes dW/dt +J[3.E dV + EFJf(E X H) .3 d8 = 0. (2-15) 5 We may now define a vector, S, and a scalar Q, by the relations s = 7%; (E x H) (2—16) Q =fJ.E dV =fo‘13'2 dV (2-17) so that the conservation of energy in the field is ex- pressed by dW/dt = -Q fs . H dS (2-18) S 10 The vector S is known as the Poynting Vector, and from (2-18) may be interpreted as the representation of the flow of the energy density. The direction of S is the direction of motion of the field, while the magnitude is the energy density of the field per unit time. The term Q represents the dissipation of energy (as Joule heat) at the expense of the field, and for a non-conductor (o = O) is zero. The compounds to be considered in the following are approximately lossless (i.e. transparent) and may therefore be considered to be non-conductors. In this case the conservation law (2—18) may be written in the form of a continuity equation: dW/dt + V . s = 0 (2-19) which is particularly useful in application to geometrical optics. The intensity of an electromagnetic wave is usually defined as the time average of the absolute value of the Poynting vector, or, 1 = |<§>|. (2-20) Using equation (2-16) we may then write I T = —1r f 27 (E x H) dt (2-21) 2T _Tv I where T is a time interval large compared to the funda- mental period of the wave. Then, writing the expression for the time-harmonic field as E(?,t) = Re (soc?) e‘lwf} Wit) = Re {11063) e'lwt} (2-22) we have that T' ‘ _ _ _ _ ' _ _* _* _ _* _* ' = _E_ _lT E XH e let+E xH +E XH +E XH e21”t dt 167T 2T ' O O O O O O O O ‘T (2-23) or s - C r H* s* a 2 24 < > — fl ( O x O + 0 x O) ( _ ) _ C _ _* = 3? Re (EO x HO) (2-25) Applying a well known result of vector analysis, we now have that _ _ _* I = II = gglEOIIHOI (2-26) The Electromagnetic Plane Wave In the remainder of this work we shall be concerned primarily with electromagnetic plane waves. Not only do such plane waves represent the simplest electromagnetic field, but, as we shall see later, for the phenomena of in— terest they represent a valid approximation to the true field distribution. For such waves we have the well known results: ‘ .— _ _ E- _ — Eo _ ‘V8 5 x HQ > (2‘27) .— _ - .8— — .— HO - ’U s X EO 12 where 3 is a unit vector in the direction of the Poynting Vector, S. Scalar multiplication with E gives Eo.s = Ho.s = 0 (2-28) which shows that E0, HO, and S form a right—handed, ortho— gonal coordinate system. We also have, from (2-27), that VEIHOI =\EIEOI (2-29) Using (2—27) and (2—26) we see that the intensity of the plane wave may now be given by _ c 2 _ I _ ,flgmd . (2 30a) Thus, we have an expression for the intensity of an electro- magnetic plane wave in terms of the spatial portion of the time—harmonic field. In free space, a = u = n = 1 (where n is the index of refraction) so that we may write (2-30) as I — C ‘ J _ 87 [E.E ]. (2—30b) Since a dilute gas has a refractive index close to one, (2-30) represents a good approximation to the intensity in a gas. 13 Coherence Matrix Formalism Introduction In this research we propose to study the interaction of polarized radiation with matter, and subsequently general- ize this interaction in terms of the theory of partial polarization. In what follows, we shall concern ourselves first with the mathematical formalism that describes the polarization properties of a wave field, and shall see that the formalism introduces considerable simplicity when the polarization properties of a complex system are investigated. We restrict the wave field to a plane—wave propagat- ing along the positive z—axis of a right—handed space-set of axes. The effect of a given system on the polarization pro— perties of the wave, may then be described as an operation on the incoming wave field to produce the outgoing wave. We must therefore describe the system uniquely with a transfor- mation of the representation of the incoming wave to the representation of the outgoing wave. 10 was the first to describe the field in Stokes terms of observables, called the four Stokes parameters, which refer to both the total intensity and state of polari- zation of the field. Jonesll reconsidered the problem of monochromatic (hence fully polarized) beams and introduced the use of matrix algebra. Jones' matrix elements are not observables of the field however, so that Mueller12 intro- duced a transformation of the Jones method such that the 14 matrix elements became the Stokes parameters of the field. The more recent treatments employ the use of correlation functions and ”coherence matrices,” which were originated by Wiener and Wolf.12 Jones Calculus Jones considered monochromatic (hence fully polar- ized) fields so that we can describe a wave by its two spatial components, Ex’ and Ey’ where EX and Ey are time harmonic. Using vector notation we then have: E = (2-31) and E is a unique representation of the field. We can then describe the effect of any linear system on the wave field by a linear operation I such that, I: = (2-32) The wave field outgoing from the system is then given by, V E = Its. (2-33) Using equation (2-30) we see that the total intensity of the wave field is given by, (to within a constant) (2-34) 15 For N Operations on the field, the N - Operators may be multiplied together to yield a combined Operator matrix, I, for the system (i.e. I = Enffi_1... Ezfl). In reality how- ever monochromatic wave fields are idealizations, and we must consider the more realistic quasimonochromatic approxi- mation to the wave field. The two component representation is now not sufficient tO uniquely represent the field how- ever, and we must extend the arguments tO a higher order representation Of the field. Coherence Matrices We now formulate the coherence matrix representa- tion Of the wave field by considering a quasimonochromatic light wave Of mean frequency 5 propagating in the positive z-direction. Wolf4 has shown that the effect of an optical instrument is the same for the mean frequency as for all Fourier components Of the field. Let Ex(t) al(t) exp [i(¢l(t) - ZnUtfl (2-35) Ey(t) a2(t) exp [i(¢2(t) - ZnUtfl represent two mutually orthogonal components Of the field, at a point x in the field, at time t. Now, the coherence matrix 3 is defined by the direct product, * * J J _ _ x x x y y J = = = (2-36) 7‘: 7% J Y X Y Y . 16 Where E is the two component column vector with elements EX and By given by equation (2-35), and the vector E7 is the Hermitian conjugate Of E. The <> stands for the time aver- age. Now, according tO the Jones method, when such a beam of light interacts with an instrument described by an Opera- — _' _ _ tor L, the outgoing beam is given by, E = L E, and therefore _t the coherence matrix, J , Of the outgoing beam is I _! _v+> J = (2-38) and since E is independent Of time, —' _ .— — — J = LL+ (2-39) and from (2-36) we have, .1 _ _ _+ J = L J L (2-40) which is the transformation law for the coherence matrix. The total intensity Of the field is given by (2—34) so that we can write I = J + J = Tr J, (2-41) XX yy where I is the intensity, and JXX and Jyy are the diagonal matrix elements of 7. Since the matrix elements Of the coherence matrix are physical observables, the coherence matrix is Hermitian and is sufficient to supply the state Of polarization Of the field. Further, the intensity Of the outgoing beam can be found from (2-40), ) (2-42) r—c ll Tr [(f f)?] (2—43) we can therefore treat a cascade Of instruments as a lumped parameter, Operating on the incoming coherence matrix to ' produce an output Of intensity I . Instrument Operators The instrumentOperator, L, is easily generated from the physics Of any Optical instrument since E Operates di- rectly on the components of the field Ex’ and Ey' Let us consider the case Of a polarizing element such as a Nicol prism, which passes only a particular component Of the field, such as the component making an angle 6 with the x-axis. Then we have the relations 1 E = E cos2 6 + E sin 9 cos 6 (2-44) x x y ' . . 2 E = E Sln 6 cos 6 + E Sln 6 (2-45) Y X Y We then have by inspection that 2 . _ cos 6(51n 9 cos 9 _ L = = P(e) (2-46) sin 9 cos 6)sin2 9 which is Hermitian and real. E is thus represented by a projection operator P(O), and satisfies the relations, 18 _+ _ _ P (e) — P(e) (2-47) as) rice) = ice) We) = We) (2-48) Table 2.1 gives the instrument operators Of the most usual devices as shown in Wolf.12 Coherence Matrices for Cases of Special Interest Wolf has shown that we may express the intensity of a field in terms of 0 and E, the angles of observation and retardation, as, 1(6 E) = J cos2 0 + J sin2 9 + 2 cos 6 sin e Re(J EiE) ’ xx xy xy (2~49) where Re means the real part of the argument. TABLE 2.l--The 2 x 2 Coherence Matrix Representation of Cer- tain Optical Instruments INSTRUMENT 2 x 2 REPRESENTATION COMPENSATOR: Introduces a e16 0 phase difference of 28 C(O) = -i6 0 e ROTATOR: Rotates the plane cos 0 ~sin 6 Of polarization counter— R(6) = clockwise an angle 6 about sin 6 cos 6 the z-axis POLARIZER: Takes the pro— cos ¢ (cos 6 sin O) jection Of the B field P(¢) = making an angle O with the (cos 6 sin ¢) sin2 6 x-axis ABSORBER: nX and n are e—nx o the absorption coefficients A = in the x and y directions nx 19 Now, we can determine the coherence matrix for completely unpolarized light, since it must Obey the condition I(e, E) = Constant (2-50) I I _ _ * _ Th1s 1mpl1es that ny — 0 — Jyx , and Jxx — Jyy’ and the coherence matrix is given by, _I _ J-7 (251) In a similar fashion, the coherence matrices Of other polarization states may be found, and are tabulated in Table 2.2. TABLE 2.2.--The Coherence Matrices of some Special States of Polarization STATE OF POLARIZATION 3 H O Plane of Polarization in xz plane 0 O Plane of Polarization in yz plane 0 l-‘ Plane of Polarization at 135° to xz J J -1 '1] J J ‘ I ' ‘ f—~1 O O H 1 1 Right Circular Polarization [' H H l -1 Left Circular Polarization [_ 20 Some Equivalent Representations If several independent light beams are superimposed, the resulting coherence matrix is the sum of the coherence matrices of the individual beams. Conversely, since we may decompose any matrix into the sum of two or more matrices, we may consider a given beam to be a superposition of two or more independent light beams. In particular we may decom— pose a given 7 in the following way: 3 = 31 + 32, (2-52) where A 0 1 0 31 = = A (2-53) 0 A 0 1 and 3': B D J, = (2-54) D C 2 BC - |D| = 0 (2-55) Wolf12 has shown that the matrix (2—54) and equation (2—55) are the general equations representing a completely polarized light beam, while 31 represents a completely de- polarized beam. We can therefore consider a given beam to be the superposition of a completely polarized and completely unpolarized beam. 21 Theory of Rayleigh Depolarization Intensity of Radiation from an Induced Dipole Let us now examine the characteristics of the field radiated in a vacuum by an infinitesimal linear electric dipole, situated at a point To, and vibrating along an axis represented by a unit vector H. We may eXpress the electric polarization as mat) = puma? - 170)}; (2-56) where p(t) is the dipole moment, and 6 the Dirac delta function. Following Sommerfeld,ZZ ‘we now define an elec- tric Hertz vector, Fe, such that r = p(t I; We) 3 (2-57) where R = I? - fol, and the quantity t - R/c is the retarded time. The Hertz vector obeys the homogeneous wave equation (everywhere except the origin) so that we have V n = $2 F (2-58) c and in addition it may be shown that ml n curl curl Fe (2-59) II n 1 ; — curl n c e 22 Using the vector identity, curl curl = grad div - V2, and the homogeneous wave equation, (2—58), we find that, _ _ , _ 1 L E - grad d1v ne- —2 We C (2-60) — _ l H — E curl fie Now, if we consider the dipole moment to be time harmonic, we find that Bp/ar = -l p, azp/ar2 = $2 p (2-61) c c and using this result and the definition of the Hertz vec- tor we find that div Re = — i§% + iRi 5.?) (2—62) f-\ graddivee=lél§uflp+1a (We- 111.121 a CR R c2R3 R3 cR2 - (2-63) --[115n—— _ 1 - + ( x R) (2 64) cur fie —E§ cRZ n where the square brackets denote retarded values (i.e. values taken at time (t — R/cD. The field vectors now assume the values s: an.n>_3['1._a[1(—,e)e-m.m.["1 R5 CR4 czR3 n R CR2 2%; a: 11%._P_H (HxR) cR c R (2-65) 23 which may be converted tO spherical coordinates R, 6, and o by the transformations R = RTR 5| = (cos e)iR — (sin 6)ie (2-66) where the i are unit vectors. We find now that (H.R) = R cos 6 (HXR) = (R sin e)1¢ (2-67) and, using the above results and (2-65) E = E I + E I H II I H (2—68) Comparing (2—68) and (2—65) term by term, we may identify each of the field components and find for the three non-vanishing terms: R3 cR E = Z{Ufl_+ ifi%} cos 6 Be = {(p]/R3 + [p]/cR2 + [£1/c2R} sin e H¢ = .fi51/CRZ + [p]/c2R} sin 9 (2—69) Of interest in Rayleigh scattering, is the field a long distance from the dipole so that we have 24 R>>c|Bl, R>>c|2| (2-70) p p Examination of (2-69) shows that at large R we may neglect all terms but those in l/R. In this approximation we have then, that ER = 0 (2-71) H¢ = E6 = L%% sin 0 (2-72) c Hence, at large R, E and H are of equal magnitude and per— pendicular to each other and to the radius vector R. The field of the linear electric dipole in a vacuum is then that of a plane wave, and the radius vector R coincides with the direction Of the Poynting vector. Furthermore, at e = 0 and 0 = n, the field is zero, so the dipole does not radiate along its axis. We shall now consider a real molecule, whose dipole moment, p, may be expanded in the form p(t) = pO + mat) + - - - (2-73) where we consider only lossless (transparent) media so that the term pO (permanent moment) is time independent, and E(?,t) is the field inducing a moment in the molecule whose polarizability is 5. According to (2—22) we may write, p(t) = pO + OLEOe—lwt + - - - (2-74) 25 and so fi = -<»Z(E-E0)e_1‘*’t (2-75) p = -4n2v2(a.fio)e'iwt (2-76) Hence, using 2—72 we see that (assuming the size of the molecule to be small compared to c/v) -4n2v2(a.EO)e'lwt E6 = 2 sin 6 (2-77) c R (while Ee may be generated by multiplying (2—77) by a unit vector in the coordinates defined by S Of the scattered light) thus from (2—30), 16n4v4 — 2 . 2 -—-——— lal I = 4 2 I s1n 0 (2—78) c R 6 where I0 is the intensity of the incident field. Then at e = 90°, the scattered intensity per unit solid angle for an ensemble of N0 dipoles is 4 4 16h v NO _ 2 I90 = ————Z———— Iol IO (2-79) c I = K|E|2 I (2—80) 90 o where l6n4v4 N K — 4 (2-81) IV 26 These results are well known from classical scatter- ing theory, and will be used in the subsequent treatment. Actually, (2-79) is correct only for a system Of gaseous dipoles, and for condensed systems must be corrected by a factor, \fE7E: as in (2-30). In addition the polarization properties Of the scattered light are affected by the ten- sor character of the polarizability, as will be discussed in the next-section. Matrix Treatment of Rayleigh Scattering Consider a single scatterer oriented at the origin of a right—handed coordinate system (X,Y,Z), along which an electromagnetic wave is propagating in the +2 direction. The scattered radiation FIGURE 2.1--The Coordinate System will be Observed in the YZ plane at some angle 6, and will define a new coordinate system (X',Y',Z') such that the scattered wave is propagated in the +2. direction. The scattered wave being Observed at some distance, r, along the Z' axis. A unit vector in the primed (scattered) co- ordinate system, Y', is transformed into a unit vector, i, in the unprimed system by the transformation 27 — Y = if (2-82) where l 0 O H = 0 cos 9 —sin 6 (2-83) 0 sin 0 cos 0 We must now concern ourselves with the generation of a matrix, S, (not to be confused with the Poynting vec- tor) which will transform the incident wave 3 into the _I scattered wave J . Therefore, we have from (2—36) that J' = s s1 (2—84) C—Il 01‘ ___I _I —"l' J = (2—85) ___I _I _'+ J = K

(2—86) where K is as defined in (2-81), and using the transforma- tion from primed to unprimed coordinates, (2—82), (2—86) becomes V J" = «nix 51313 (2-87) ' + J =K (2—88) As we are considering a single scattering center fixed in space ——-' __ ._ __ J = K11a6 3* (2—89) and from (2—36) JV 28 J' = KHEJEIH‘L (2-90) Thus, we see by inspection that S = {En-6L" (2-91) where 5 represents the components of the polarizability in the space-fixed coordinate system (X,Y,Z). IO a a I xx xy xz O(N) = ayx Oyy Oyz (2—92) Osz OLzy O‘zzJ Carrying out the Operations as indicated yields O a u xx xy xz ml ‘54 (nycos O-OZX31n 6)(oyycos e-azySIH 6)(Oyzcos 6—a2251n e. . . + . . J(ayxcos e+azx51n 6)(Oyycos 6+ozysin 6)(ayzcos 6 OZZSIn 6, (2-93) and since there is no electric field component on the Z axes Oxx Oxy 75:44? o cos 9 — a sin e a cos 6 — a sin 6 yx zx yy zy J (2-94) This matrix is valid only for the components, ONN,, of the polarizability in the space—fixed coordinate system. We may identify a molecule—fixed coordinate system with axes 1,2,3, such that the components Of the polarizability in this system are 29 21 0‘22 0‘23 (2‘95) Furthermore, we may choose the molecule-fixed axes to be the principal axes of polarizability, so that we now have o1 O 0 5(1) = 0 a2 0 (2-96) 0 0 a3 where the Oi (i = 1,2,3) are called the principal values of the polarizability. A unit vector in the molecule—fixed coordinate system may then be transformed into the space— fixed system by the transformation “x Cxl sz st n1 ny = Cyl Cy2 Cy3 n2 (2-97) nz C21 C22 C23 ”3 where the Hi (i = 1,2,3,) and EN (N = X,Y,Z) are unit vec- tors on the molecule-fixed, and space-fixed axes, and the CNi are the direction cosines of the Euler angles relating the two systems. We then have that the polarizability components in the space-fixed system are given by, 30 3 O‘NN' = 5;; 0‘i CNi CN'i (2'98) The scattering matrix thus generated being valid for a sin- gle radiator, fixed in space. For an ensemble of such radiators we must sum the contributions (taking into account the phase relationships), and average the results over all the possible orientations of the ensemble. Rayleigh Scattering from a Perfect Gas Assuming a dilute gas of independent particles, we see that the Oi will be constants (i.e. there is no need to consider phase relationships due to the distances between molecules), and the summation of scattering contributions is given by the product of the average contribution and the total number Of scatterers, N. From the transformation law (2—84) we see that the terms to be averaged will be of the form: O‘NN' = C‘iCNiCN'i (2—99—a) 'Il.[\/JCN 2 2 aiCNi Z: ajCN'j (2-99-b) 1 j 1 ________ 3 o‘NN0‘N'N' = 3 l (where the bar denotes the spatial rather than the time average), and since the Oi are constants in the averaging we may write 31 ——— 3 2 _ 2‘2“2" O‘NN' ‘ X: O‘iCNiCN'i + Z: O‘io‘jCNiCN'iCNjCN'j i=1 i j (2-100-a) a——a"—— = 3 OZC 2 +Z:a C2 NN N'N' $2: i CNi CN'i 0‘i 0‘j CNi CN'j‘ 1=1 iii (2-100—b) In this approximation (i.e. independent scatterers) the scattering from any physical system may in principle be treated by appropriately averaging the direction cosines (CNi) over all the possible orientations Of the system. In this case, the gas molecules are free to assume all possible orientations, with respect to the Observer's axes, with equal probability. The averaging process then becomes particularly simple because the diagonal term (i.e. N = N') takes the form, 1T N 2 ‘4‘ _ ”‘Z“ _ 1 4 . CNi — cos 6 - 4E jf Jf cos 6 Sln eded¢ (2-101) 0 O C . = 1/5. Then, from (2—97), we see that we may write n..n. — |n?| = c2. + c2. + c2. = 1 (2-102) 1 1 1 x1 y1 21 or 2 ‘4 2 2 |n.| = 3CNi + 6 cNicN,1 = 1 (N+N') (2-103) 32 so that combining (2-101) and (2-103) yields the Off- diagonal term c2.c2 N1 Nfi_= 1/15. (N+N') (2-104) In the same manner we Obtain Inillngl = 3 C2 C2 Ni N'i+ 6 C C = 1 (2-105) Ni N'j where N + N', and i + j, and combining (2-104) and (2-105) we obtain, (32 c2 = Ni N,j 2/15. N + N', i + j (2—106) Finally, using the orthogonality condition, f1“ O-Ii-O = C ‘C ‘ + C “C ' + C 0C 0 = 0 (2-107) 1 j X1 X] y1 yj Zl Zj and |n n |2 = 3 c2 C2 + 6 c C c c = 0 (2-108) i' j Ni Nj Ni N'i Nj N'j we see that CNiCN'iCNjCN'j = -l/30. (2-109) We may now compute the components Of the coherence matrix of the scattered radiation (i.e. 3') using the pre- ceding results, and equations (2-100). The non-vanishing terms are easily seen to be N LN 3 = __ Z a? + _2_ an]. (2-110) 33 3 2 2 _ _ l 2 _ .1 laxyl lale _ _ 1—' X: 0L1 15 OL1O‘j 1:1 l + 1882 (2—157) NN b 1b 2 “‘72—‘— IOIONN,| = 33 (2-158) where we now have that 2 _ 4 1b dc _ B - a Z Z < Z Eij E:ij COS Wbcz wldy)> b c 1 i,j d b (2-159) Benoit and Stockmayer23 have shown that 43 CD 1 + 4nJ’ [g(r) - N/V]r2dr O :: RTE/V (2-160) where R is the gas constant, T the temperature, K the com- pressibility and V the molar volume. In addition, Buckingham and Stephen7 have shown that 82 may be rewritten as 2 2 rMR - MRO a L (2-161) 8 = MR 0 where MRO is the molecular refraction Of the molecule in the gaseous state, and MR the molecular refraction in the condensed fluid. Hence we see that MR - MR 3 O MRO pv = MR - MR0 RT? (2-162) 4 MR0 + 10"?— and we now have an expression for the vertical depolariza- tion ratio Of a dense fluid of spherical molecules. Temperature Dependence of Depolarization by Dense Fluids Making use of the result (2-162) we may now examine the dependence of the depolarization, pv’ on the tempera- ture. First, however, we take note Of the fact that the molecular refraction, MR, given by 2 2 MR ill—1“}- M (2—163) Ln + 1 d 44 where n is the refractive index, M the molecular weight, and d the density, is practically temperature independent, and is a function Of the bonds present in the molecule. We shall therefore make the following assumption, which will be justified by experiment: Assumption: The temperature dependence of the depolariza- tion ratio is due to the diagonal elements of the polari- zability tensor (i.e. the term, lORTR/V), so that we may write 3A pv = 4A + B(T) (2'164) where A refers to a constant, and B(T) = loRTK (2-165) V Now we may write _ —3A (apv/BT) — 2(BB/3T) (2-166) P (4A + B) P or, rearranging terms, _ -___1__ - (8 En pV/BT)p — 4A + B(BB/3T)p (2 167) However, using (2-164), we see that 4pr 45 so that (2-167) becomes _ _ l/B 8B _ v v p or [40V - 3]1(3B) Moreover, from (2-165) we have that, 10R TE — — — 8B T = -=— ":— T T T 2‘17]. ( /3 )1, V [V (EV/3 )p + K + (BK/3 1p] ( ) and 1 8B _ 1 1 a? _ 2.) W(— I. 22222 P P (2-173) Combining (2-170),and (2-172) we have finally, the result 1 (a in E ] T ' 0‘1‘ + 8T P 49V - 3 (3 Kn pV/BT)p = ———§——— (2-174) Hence we now have an expression for the temperature de- pendence of pv’ which depends only on the independently measurable thermodynamic quantities, OT and R, as well as 46 pv. We may further simplify this eXpression by using the well known relation, 2 _ c - c = 'rV/E' (2-175) p v ”T and the approximate result a TV 3?) _ _ — — — _ T [3(Cp-CV)/3T]p — 0 - aTV+OTT(3V/3T)p+2TV(8OT/3T) __?_(§T (2-176) We then have that — _ l l — 2 _ and since 2 (BOT/8T)p 2 OT, (2—178) combining (2-174), (2-177), and (2-178) we finally obtain [40" ' 3H1 1 (8 [VI pV/BT)p = Z ——-—3——— T + (IT (2‘179) This is indeed a very simple relationship between the depolarization and temperature, and involves only the temperature, coefficient Of thermal expansion, and depolari- zation ratio. Equations (2-174) and (2-179) form the basis for the analysis Of the experimental data to follow. 47 Laser Theory Emission and Absorption of Radiation Atomic systems, such as atoms and molecules, may exist in stationary states called energy levels. Each energy level corresponds to a definite value Of energy, and may be described by a suitable wave function. Transi- tions between energy levels may occur with attendant emis— sion or absorption of radiation, or with the transfer of energy in some other fashion. Radiative transitions Obey the Bohr frequency condition; hvl2 = E2 — El’ where vlz'is the frequency of the absorbed or emitted photon, h is Planck's constant, and E and E are the energies of the 2 1 states between which the transition takes place. When radiation impinges on an atomic system, it perturbs the Hamiltonian of the system, and can cause a change from the initial state to some other state. In the transition from one state to the other, the system must then either absorb or emit radiation of the same frequency as the perturbing radiation, depending on whether the sys- tem was initially in the lower or higher stationary state. The case in which the system is initially in the higher energy state is particularly interesting since a photon is then emitted due to the influence Of the perturbing field, and the emitted photon has the same frequency and phase as the perturbing field. This process is called stimulated emission, and has the interesting prOperty that the 48 perturbing photon and the emitted photon are coherent with one another. The Boltzmann distribution describes the way a sys- tem at equilibrium will distribute its population among the available energy levels, and is given by, NZ/Nl = exp [(El - EZ)/kT], (2-180) N2 and N1 being the populations of the states with energies EZ and El’ temperature. The subscript 2 refers to the higher energy k the Boltzmann constant, and T the absolute level. According to the Boltzmann distribution then, a system in a state with energy E2>kT is not at equilibrium, and must therefore tend to lose energy whether there is a perturbing field present or not. The loss of energy in this fashion is called spontaneous emission, and since there is no field present the emission is incoherent. Using time-dependent perturbation theory, it is easily shown that the probability of a transition from a lower to a higher state is given by, p B (2-181) 12 = p12 12’ where p12 is the radiation density of the appropriate fre- quency, P12 is the probability of absorption, and B12 is called the Einstein coefficient for absorption. The proba- bility of transition from a higher to a lower level is given by. P21 = A21 + 021 B21 (2‘182) 49 where P21 is the probability Of transition, A21 is the Ein- stein coefficient Of spontaneous emission, p21 is the radiation density Of the appropriate frequency, and B21 is the Einstein coefficient for stimulated emission. The re- lations between the Einstein coefficients is given by, Bnm = an’ Anm = 8fl233 Bnm (2-183) From these relations it is easily seen that at very low frequencies (eg. radio frequencies), Anm<m, what is the response of the ensemble to n collimated radiation Of frequency v and density pnm? nm’ The number of emissions from n to m is given by, P N = (A + OB ), per second, and the number Of absorp— nm n nm nm tions is given by, P N = p B N . Now since N is less mn m m mn m n 50 than Nm for a system anywhere near thermal equilibrium, the beam will suffer a loss of (Nm - Nn) pnm Bnm photons per second. The Anm Nn spontaneously emitted photons will be radiated in all directions uniformly, and will therefore be lost from the beam. Thus, a beam passing through a material at or near thermal equilibrium will always lose energy, and will therefore not support stimulated emission since pnm will diminish, making spontaneous emission the more probable process. An ensemble can easily be visualized however, in which Nn is greater than Nm’ even though n>m. Examination of the Boltzmann distribution shows that such a material is certainly not in thermal equilibrium and indeed, since we have that, Nn/Nm = exp [(Em - En)/kT] (2-184) the system must exist in a state of negative temperature. This ensemble is then said to contain a population inver- sion, and will act as an amplifier of radiation of the proper frequency, since a beam will be enhanced on passage through the medium by (Nm - Nn) pnm B photons per second. The nm amplified radiation is coherent since the amplification is by stimulated emission, and spontaneous emission of the same frequency will appear as amplifier noise. A laser is, by definition, a device that contains such a population inver— sion, so that it acts as an amplifier of radiation with the prOperty that the output is coherent. 51 In practice, it is more desirable to use the laser as a source Of coherent radiation through oscillation at the appropriate optical frequency. The oscillator is constructed by adding a feedback mechanism, in the form of mirrors, to the light amplifier, so that the laser becomes a saturated amplifier of noise (noise in this case being spontaneous emission Of the proper frequency). Laser action is only possible if the material can be placed in a sufficiently large pOpulation inversion, and if, in addition, a minimum feedback can be established by means of mirrors. The re- quirements on the material are therefore very stringent, since the pOpulation inversion depends on the rate at which excitation is supplied, and on the rates of relaxation and transition through the levels to be used. Oscillation and Modes In their paper on Optical masers, Schawlow and TownesLSpointed out that a Fabry—Perot interferometer may be used as the feedback device for an Optical oscillator. Such a device represents a very large cavity however, and will therefore generate a highly complex electromagnetic field within the amplifier, affecting the properties Of the coherent light emitted. This field may be regarded as the superposition of a large number of plane waves traveling back and forth in the resonator, with oscillation occurring along those portions of the field that form a standing wave. A particular set Of such standing waves, giving rise to 52 oscillation, is a function of the geometry of the resonator and may be referred to as a mode of oscillation. The electric field distribution Of the light emis- sion from such a device is a function of the mode of oscil- lation, and Figure 2.2 shows the configurations associated with the dominant (TEMOO) circular mode as well as a number of higher-order circular modes. These distributions have the cartesian designation TEMmnq which refers to transverse electric and magnetic fields. The subscript m is an integer giving the number of nodes in the radial direction, while n is the number Of nodes in the azimuthal direction. The in- teger q refers to the number of axial modes and is therefore a very large number given approximately by q = 2L/l (2-185) where L is the length of the cavity, and A the wavelength of oscillation. The arrows in Figure 2.2 indicate the phase of the field while lines indicate the nodes. Each mode Of oscillation corresponds to a specific resonance of the Fabry-Perot interferometer, and therefore represents a particular frequency component in the output. The frequency separation between modes is given by: A(l/l) =[l— Aq + 1— Li D (ZmAm + Amz + ZnAn + Anzfl 2L l6 (2-186) where D is the field aperture, and the quantity N = (Dz/1L) is the Fresnel number of the aperture. It can be seen from 53 flu) Ill 41+ TEMoo 41+ 111 TEMZO 22% 2W} TEMlO i +.I I TEMOl TEMZl TEMll , 42691) \‘W 4 TEM02 TEM12 2.2.--Modes in a FIGURE — 54 (2-186) that the frequency difference between two modes of the same type (i.e. same m and n) is given by, A1 = l/q (2-187) or, combining (2—185) and (2-187) we have A1 = 12/21 (2-188) and rearranging terms we now find that, v = Av = C/ZL, (2‘189) A resonator formed by two spherical reflectors of equal curvature is Of particular interest when the reflectors are separated by their common radii (i.e. the confocal con— figuration). The variation Of the electric field over the surface of the reflector is given by, E 2n l l x2 + 2 —§X = H x(——)2 H (Zn/1L)2 ex 'W—————Z— E0 m I n y p Ll (2-190) where Hm and Hn are Hermite polynomials of degree given by the mode integers m and n, and x and y are the coordinates of the point on the surface. From (2-190) we now see that the field in the TEMOO mode is Gaussian, and falls to 1/e Of its maximum value at a radius given by, 1 rs = [LA/N]2. (2-191) The surfaces of constant phase are spherical, with radii of curvature given by 55 (2-192) where R is the radius Of curvature of the reflector, and Z is the distance from the center Of the resonator to the point at which the Observation is made. The resonant wavelength of the confocal system is given by A = ZL/(l + q + m + n). (2-193) General Description of Lasers MaimanS first achieved laser action with a ruby crystal consisting Of a 2 cm long cylinder of pink ruby containing 0.05 percent chronium. The end faces were plane and parallel to a high degree. One end face was made com- pletely reflecting and the other partially reflecting. The pOpulation inversion was achieved by irradiating the crys- tal with a burst of very intense white light from a flash- lamp through which a capacitor was discharged. Total input energy was 1000 to 2000 joules in a pulse of a few milli- seconds duration. The blue-green output of the flashlamp was absorbed by the crystal, and this energy transferred to a narrow metastable state around 694.3 nm. Laser oscillation then occurred from this level to the ground state, with the emission of 694.3 nm coherent light. A simple laser of this type produces several kilowatts per square centimeter flux density, in a spectral line about 0.01 nm wide, centered 56 about 694.3 nm. By way of contrast, a black-body at 417° K has its maximum output at 694.3 nm and emits only 1700 watts/ cm2 in its entire spectrum, while emitting only 0.016 watts/ cm2 in a 0.01 nm pass band at 694.3° nm. Laser action in gases is more difficult to attain since there are no broad fluorescent levels available for optical pumping to a population inversion. Excitation by electron collision may be used however, since, when a dis- charge takes place in a gas, ions and free electrons are formed. The free electrons.are then accelerated by the field that creates the discharge. In low pressuredischarges, the average kinetic energy of the free electrons usually greatly exceeds that of the atoms or ions in the discharge. In a steady discharge, within a short time, the electrons estab- lish a Maxwell-Boltzmann energy distribution among them- selves that is characterized by an electron temperature Te, prOportional to the mean electronic kinetic energy. Inelas- tic collisions between atoms and electrons then occur, in which the atoms distribute themselves among their energy levels according to the Boltzmann distribution, where the temperature is now Te. Then the number Of atoms in state i is given by Ni = Noe-Ei/kTe. When more than one gas is present, excitation is exchanged between atoms of different kinds, provided that they possess energy levels near one another. The probability of such an exchange is given by 57 Pex = exp [-A/kT] (2-194) where A is the energy difference between the levels, and T is the temperature Of the gas mixture. This process is called the resonant exchange of energy, and is particularly interesting when the excited level of one gas is metastable, since resonant exchange provides a means of relaxation from the crowded long-lived state. Javan6 achieved population inversion at 1.15p in neon, by using resonance exchange in a gas discharge through a helium-neon mixture. Figure 2.3 shows the energy level diagrams of helium and neon. The lowest excited state of helium is the 238 which is metastable, and therefore long- 3S state collide with neon lived. When helium atoms in the 2 atoms in the ground state, the excitation may be transferred to one of the ZS states Of neon, which lie only 300 cm’1 be- low the helium level. Radiative transitions then take place from the four ZS levels to the ten 2P levels. The 18 ter- minal levels are quenched by collisions with the walls Of the discharge tube. Whether a population inversion will occur depends on the relative abundances Of helium and neon, the excitation rate, gas pressure, and diameter Of the dis- charge tube. Laser action at 632.8 nm in helium—neon mix- tures is possible because of the degeneracy Of the 218 level in helium with the 38 levels in neon. The general appearance of helium-neon lasers is shown in Figure 2.4. The mirrors are generally spherical, ENERGY IN 3‘7 I I I I I | | | I I l I I | e— impact I I 0 __J______ HELIUM 58 diffusiOn e” impact I | I I I I I NEON FIGURE 2.3.-—Energy Levels in'a Helium-Neon Laser. 59 .eemeq coez-asflfiez a mo eseupso--.e.m mmaoun 60 and coated with a highly reflective dielectric. The windows on the discharge tube are oriented at the Brewster angle to minimize losses and unwanted longitudinal reflections. The Optimal tube diameter and length depend on the application. The discharge may be excited by high voltage dc applied to internal electrodes, or radio-frequency power applied to ex— ternal electrodes. Oscillation in the helium-neon laser occurs at, or near, the peaks of those cavity resonances that fall within the linewidth Of the Doppler broadened atomic line in neon. Cavity resonances occur at frequencies given by, v = v + Av (2-195) where Av = c/2L. Here L is the separation between the mir- rors, and the n refers to the number of nodes in the standing wave pattern. For one meter separation, Av = 150 MHz, and since Doppler line is about 1000 MHz wide, several modes may be excited at one time. The linewidth of a single mode is immeasureably small by spectroscopic techniques, however beat frequency measurements show that it is less than 20 Hz over a short term. The coherence length, measured interferometri- cally, is of the order of thousands Of miles for even un- stabilized gas lasers Of this type. CHAPTER III EXPERIMENTAL Laser Design Considerations Introduction The output characteristics of a laser oscillator are functions Of the properties of the active medium (i.e. atomic linewidth, pressure, gas mixture) and the configura— tion Of the Optical resonator (length, aperture, reflecti— vity, etc.). Since the application involved constrains a number of these parameters to predetermined values, it is necessary to adjust the remaining variables to Optimal positions. Unfortunately, the extreme sensitivity Of the laser output to the cavity configuration precludes the possibility Of a general relationship between the variables of the system. However, since most lasers are Operated in extremes Of either single-mode or highly multi-mode cavities, it is possible to make simplifying assumptions, and arrive at relationships between system parameters that are valid in these limiting cases. Smith14’15 has developed a set Of such relationships describing the output characteristics of the laser in terms Of variables Of the system, hence making possible the Optimization of the output for any desired application. 61 62 Power Output The power output of a helium-neon gas laser in single- mode cw (continuous wave) Operation is 31 5 d 2 P = 7) wO GM[(tw1)Opt/wO GM] (3-1) where (twl) is the maximum single-mode output intensity opt per pass (found graphically in (15)), W0 is the gas satura- tion parameter (found graphically in (14)), GM is the incre- mental gain of the amplifier, given by G = 3.0 x 10‘4 M Z/d (3-2) where K is the discharge length (cm) and d the diameter (cm). In the limit of extreme multi-mode Operation, the output power becomes 2 2 Awo GM(1 1%) (3-3) 30 (watts/cmz) and a is the loss per pass in .21 5 NICL where Aw o the resonator, and is found to be 0.2 percent for the mir- rors and 0.05 percent for each Brewster angle window. Gas Mixtures and Pressures 14,15,16 It has been found that a range of helium- neon mixtures varying from 5:1 to 10:1 yield good results at 63283. Moreover Mielenz and Nefflen report that for a rf-excited discharge, a mixture of 7:1 helium-neon is Opti- mal for most tube diameters. Best results are generally rummmwnnunmv 63 Obtained however, when tubes are filled to a pressure p, given by p = 0.4/d (374) where d is the tube diameter in cm. Resonator Configuration The properties of the laser output are defined to a large extent by the curvature and reflectivity Of the reflectors used in the Optical resonator, as well as by the length and aperture Of the cavity. The reflectors used in Optical resonators are of extremely high surface quality, and are coated with multiple layers of dielectric materials such that they may achieve virtually any desired reflecti- vity in any given spectral region. Generally one Of the reflectors in the resonator is made as reflective as pos- sible, while the other is chosen to have a reflectivity so that the power out Of the cavity is maximized. The choice of the geometry of the resonator (i.e. curvature and length) has a significant influence on both the spatial and temporal characteristics of the output. Moreover, the geometry may affect both the available power output and stability Of the devices. The four most useful and stable configurations are shown in Figure 3.1 and are called the "large-radius,” confocal, spherical, and hemi- spherical resonators. If the mirrors which define the optical resonator have radii of curvature greater than the length of the 64 I Y z I LARGE RADIUS CONFOCAL SPHERICAL HEMISPHERICAL FIGURE 3.l.--Resonator Configuration 65 cavity, the "large radius" configuration is Obtained. This configuration is stable, and yields a relatively high power output provided that the radii are not extremely large (i.e. plane mirrors have r = 00). When the mirror radii are three or four times the cavity length, the curvature of the mir- rors corresponds tO the curvature Of the wave front yielding relatively good collimation as well as stable power output. The confocal resonator (radii Of curvature equal to the cavity length) represents the most common configuration for multi-mode Operation. It is inherently stable and is extremely easy to align, although the power output is not as great as in the "large radius” configuration. If the radii Of curvature of the two mirrors are not identical, the near-confocal configuration is unstable when the cavity length is between the two radii, so in practice the confocal arrangement is modified to keep the cavity length somewhat greater than the radii of the mirrors. If the cavity length becomes twice the radii of curvature of the mirrors, the spherical cavity is Obtained. This represents an extremely useful configuration for single- mOde Operation, although only about one-third the available gas discharge is used. Alignment of the spherical cavity is somewhat more difficult than the confocal, and the con- figuration becomes unstable when the cavity length is greater than twice the radii Of curvature. A special case of the spherical cavity is obtained when a curved mirror of radius equal to the cavity length 66 is used in conjunction with a flat mirror to achieve a hemi- spherical resonator. The properties of this configuration are much the same as the Spherical case, however alignment is much easier, and the output is considerably more stable. The hemispherical resonator is the most common configuration for single-mode output. Laser Characteristics General Description High intensity, beam stability, and a high degree Of collimation are prime requirements when observing Rayleigh scattering. We have therefore constructed two complete laser systems to fulfill the following design criteria: (1) Maximize output power. (2) Stabilize output power. (3) Obtain a high degree Of collimation. (4) Maintain a beam diameter of less than 0.25 inches. Since a very narrow band pass is not necessary, multi-mode (high intensity) Operation was used whenever possible. Radio frequency excitation via external electrodes was chosen since it is convenient aJId provides a relatively stable power output after an initial warm-up period. The mirrors used in the Optical resonators were supplied by Perkin—Elmer Corporation, and were made Of either high quality borosilicate glass or fused quartz. The mirrors were coated with multiple layers Of dielectric material de- posited in vacuum to the specified reflectivity at 6328 R. ““flflflflflfllfllfl! 67 Laser I was constructed entirely Of fused quartz and was supplied by Thermal American Corporation. The dis- charge tube was 85 cm long with an inner diameter Of 0.7 cm and was viewed through high-quality quartz windows (flat to 1/10) fused to the tube at the Brewster angle. The tube was filled with a 7:1 mixture Of helium and neon to a pres- sure Of about 1.5 torr after an initial cleanup and bake- Out procedure. The resonator consisted of two spherical mirrors with radii of curvature Of two meters in a cavity 130 cm long, thereby comprising a "large radius” configura- tion. One mirror was coated to a reflectivity very close to 100 percent, while the output was coupled from the second mirror which had a reflectivity Of 99 percent. The output power for this system calculated from (3—3) was 86 milli- watts for multi-mode Operation. Observation of the output using an EGG Lite—Mike calibrated at 6328 X yielded a power of 50 milliwatts with no observable component at 60 Hz. The output power was stable to i one percent after initial warmup, and the mode structure (observed visually) appeared very stable. The output beam was approximately 0.25 inches in diameter and had a divergence Of less than 2 milliradians. Laser II was constructed with a quartz discharge tube of length 105 cm and inner diameter 0.5 cm, fused by means of graded seals to Brewster angle windows of 7094 pyrex. The tube was supplied by PEK Laboratories and was filled with a 7:1 mixture of helium and neon at a pressure of 1.5 torr. The resonator consisted of a spherical mirror of two-meter 68 radius and reflectivity of 100 percent, and a planar mirror with reflectivity of 99 percent. The mirrors were mounted with a separation of 150 cm, thereby comprising an almost hemispheric cavity. The calculated multi-mode output power for this laser was 90 milliwatts, however the Observed out— put power was only about 20 milliwatts due to the less effi- cient cavity. The beam diameter at the planar mirror was 0.1-inch or less, and had a divergence of about 4 milliradians. Because of its greater output power and lower diver- gence, Laser I was used in most of the depolarization mea- surements. Laser II was mounted on a bench constructed Of Benelex phenolic resin, giving it considerable thermal and mechanical rigidity and allowing it to be moved about for use in measuring refractive indices. Filling Techniques Cleaning and filling of the laser tubes was accom- plished on a high-vacuum system incorporating a Cenco Hyvac II mechanical pump in series with a CVC Oil diffusion pump. Pressure in the vacuum system was measured using CVC ioniza- tion and thermocouple gauges up to a maximum of one torr. Pressures above one torr were measured using an RGI mercury flotation gauge with a range of 10 torr in increments of 0.1 torr. The helium and neon were obtained both as ultrapure gases and as mixtures (7:1 and 10:1), in pyrex bottles with glass break seals, from Linde Division of Union Carbide 69 Corporation. The gases were used without further purifica- tion except for passage through a liquid nitrogen cold trap. The discharge tubes were prepared by sealing to the vacuum system and pumping to a pressure less than 10—6 torr while heating with a heating tape. After a bake-out of 24 hours the tubes were filled to a pressure of about one torr with ultrapure helium, and the gas ionized by radio-frequency energy coupled into the tube via external electrodes. After several minutes Of discharge the tubes were again pumped to 6 a pressure of less than 10‘ torr, after which they were again filled and discharged. Having repeated this cycle un- til inspection with a small hand held spectroscope revealed no impurities being outgassed from the glass, the tubes were finally filled with the Operating mixture of helium and neon. Preliminary measurements Of power output as a func- tion of gas mixture and pressure were made with the laser attached to the vacuum system. Results indicated that a 7:1 mixture Of helium and neon is optimal over a range Of pres- sures from 0.8 to 2 torr (as measured by the flotation gauge), with maximum power output Obtained at 1.5 torr. Having been filled to this pressure and removed from the vacuum system, the tubes were used for several hundred hours before refilling. Power Supply The gaseous discharge in both Lasers I and II was obtained by COUpling radio—frequency energy into the tube by means of external electrodes. The radio-frequency supply 70 was an E. F. Johnson "Viking—Challenger” transmitter, modi- fied to allow continuous operation at 27.1 MHZ as well as to permit continuous adjustment of the input power to the laser. Coupling of the low impedance transmitter output to the high input impedance of the laser was achieved by using a parallel tuned, balanced impedance matching net- work. The matching network and electrodes were adjusted to maximize the power input to the discharge, the power output from the laser then being controlled by adjustment of the transmitter. Radio-frequency excitation has the advantages of long tube life (since there are no internal electrodes) and an extremely stable discharge, with the chief disadvantage being the necessity Of supplying rf shielding. Typically the power output of the laser was maximized with a power in- put of about 80 watts. Laser Alignment The mirrors of the laser resonator were mounted in precision gimbal suspensions supplied by Lansing Research Corporation. These suspensions allow a smooth and contin- uous rotation of the plane of the mirror about two orthogonal axes perpendicular to the longitudinal axis Of the cavity. TO achieve oscillation the mirrors must be made exactly parallel to one another, and perpendicular to the axis of the discharge. Two methods were devised to permit rapid adjustment of the cavity and tube to fulfill these requirements. 71 (l) A small point source was used as an autocollima- tor by placing it at the focal point Of a telescope eyepiece. The collimated light emerging from the eyepiece was projected along the axis of the tube from outside the cavity, and the mirror at the Opposite end of the cavity was then rotated to reflect the light back to the source. The near mirror was then rotated until the images of the primary and secondary beams on the far mirror coincided. The cavity was then well enough aligned to permit oscillation and further mirror ad- justments were made to maximize the power output from the laser. (2) A glass plate was placed inside the optical cavity tO inhibit laser oscillation so that the image of the gaseous discharge could be Observed in the far mirror, by sighting through the bore of the discharge tube. Satisfactory align- ment was assured when multiple reflections from the two mir- rors could be seen in the image at the far mirror. The glass plate was then removed from the cavity and laser action would immediately begin. This was found to be a very simple and rapid means Of obtaining alignment, but requires considerable experience and practice on the part of the experimenter. Photometer Desigg Introduction Light scattering measurements involve the detection and precise measurement of extremely low light levels. Therefore, the elimination or minimization of potential 72 sources Of noise, both electrical and Optical, is of prime importance in the design and fabrication of a system for the measurement of Rayleigh scattering. The system for the precise measurement Of depolari- zation ratios consists basically of three parts: the light source, the photometer, and the detection system. It is convenient to consider each of these subsystems independently before considering the characteristics of the system as a whole. Light Source The light source used in our measurements was Laser I, which was described in the section entitled Laser Charac- teristics. Preliminary experiments showed that there are four sources of amplitude noise in this laser, which give rise to fluctuations in both the output power and the spa— tial intensity distributions. The first noise source was a statistical fluctuation in the gaseous discharge, which superimposed a ”white" noise spectrum on the laser output. This problem was easily over- come by modulating the laser output (using a mechanical chopper) and using a narrow band-pass detector Operating in coherence with the modulator (as will be discussed in the section entitled Detector). The second source of noise was caused by the presence of dust in the air, and manifested itself as a low frequency (ml Hz) fluctuation in the laser output, with a relatively large spectral power density. 73 Mechanical and thermal stresses caused small shifts in the laser alignment, thereby creating a noise spectrum in the region from 1000 HZ to 0.1 Hz, and finally, fluctuations in the power supply output contributed to both high and low- frequency noise. Elimination Of thermal and mechanical stress as a source of noise was accomplished by mounting the entire laser on a table top made of a slab of ”Benelex," a pheno- lic resin supplied by the Masonite Corporation. The slab was 12 feet long, 3 feet wide, 2 inches thick, weighed ap— proximately 650 pounds, and was mounted on a permanent laboratory table of the same dimensions and weight. The combination of large mass and very small coefficient of ex- pansion effectively damped mechanical and thermal oscillations. Drift in the power output of the transmitter was minimized by stabilizing the line voltage, allowing a suf— ficiently long warmup period before operation, and by operating well below maximum output ratings. Moreover, to prevent radiation Of the 27.1 MHZ transmitter output to other system components, the entire laser assembly was en- closed in an aluminum box, and all cables and plugs were fitted with rf filters. The only remaining component of amplitude noise in the light source was that due to dust in the air within the cavity. This was not found to be a serious problem however, and by using a long time constant in the output filter of the detector any noise due to this source was minimized. 74 Photometer The intensity of Rayleigh scattered light, and in particular the vertical component of the scattering, is a function of the refractive index and temperature of the scattering material. It is therefore imperative in the precise measurement Of depolarization that both sample and cell be kept scrupulously clean, and at constant tempera- ture. In addition, the rescattering of Rayleigh scattered light greatly enhances the depolarization, necessitating the consideration Of means to minimize such multiple scatter- ing effects. Since the probability of scattering is directly proportional to the path length, it is necessary to keep the sample small while also keeping the light beam as narrow as possible. Obviously, the use Of the 6328K emission from a helium-neon laser is of great significance in reducing multi- ple scattering effects due to the inverse fourth power wave— length dependence Of the scattering, and due to the high intensity available from a relatively small diameter beam. Elimination of extraneous light in the Observation of Rayleigh scattering is a somewhat more difficult problem as there are several potential sources for such light. The principal sources of extraneous light may be tabulated as follows; (1) Ambient room light leaking into the measurement system. (2) Reflection of the incident beam from the sur- faces of the sample cell. 75 (3) Reflection of Rayleigh scattered light from the walls of the cell or cell holder. (4) Reflection of the incident beam from the beam stop at the end of the light path. and will be considered one at a time in the following. (1) Ambient room light was excluded from the photo- meter by enclosing the entire system in a light-tight box (Figure 3.2) constructed of 0.250 inch aluminum, 10.50 inches long, 9.50 inches wide, and 6.75 inches high, ma- chined to tolerances of 0.001 inch. The light path within the photometer was defined by sets of apertures 0.500 inches in diameter. After assembly Of the instrument, all the joints were coated with a liquid rubber sealant, and the entire interior was sprayed with several coats of flat black Krylon. Light was admitted to the photometer through a stan- dard camera shutter attached to a long (6.0 inch) cylindrical barrel whose interior was painted flat black. After entering the system, the beam passed through an aperture and into the compartment containing the cell holder. The cell holder was constructed of a solid copper cylinder, 2.25 inches high and 2.50 inches in diameter, milled to index the position of the scattering cells, and soldered inside a close fitting copper Sleeve, 5.0 inches in height. The COpper sleeve, into which the scattering cells were set, was wrapped with a coil of 0.125 inch copper tubing, a layer of asbestos, two coils of nichrome heating wire, and several more layers of asbestos. 76 p- .— I" _- Ph PV F j l I---------_-J A Shutter B Sample Cell C Cell Thermostat D D Light Trap Ph Horizontal Polarizer PV Vertical Polarizer r__--_______ F Neutral Density Filters PM Photomultiplier Tube FIGURE 3.2.--The Polarization Photometer 77 This entire assembly was painted flat black, and light was admitted through a set of 0.50 inch apertures. The Rayleigh scattered light was Observed at 90° to the incident beam through a second set Of 0.50 inch apertures in the cell holder. (2) Reflection of the incident beam from the front surface of the sample cell was eliminated as a source of stray light by careful indexing of the cell within the cell holder, and by indexing the cell holder within the photo- meter so that such reflections passed back through the en- trance apertures and could not reenter the photometer. Reflection from the back surface of the cell could not be eliminated, however use of high quality cells minimized the enhancement Of the depolarization due to this source. (3) Reflection of Rayleigh scattered light from the walls of the cell holder was minimized by the diffuse black surface of these walls. Moreover, the reflection along the viewing axis (90°) was further reduced by allowing Rayleigh scattered light (at -90°) to leave the cell holder through a 0.50 inch aperture Opposite the detector aperture, and absorbing it in a black felt light trap. (4) After passage through the sample cell, the in- Cident beam left the photometer through a 0.50 aperture and entered a specially designed light trap, as shown in Figure 3.3. The trap was constructed of pyrex, and was designed t0 fit into a photomultiplier housing in place of the photo- multiplier tube. Measurement Of the incident beam intensity 78 FIGURE 3.3.--Light Trap could then be accomplished by simply removing the light trap and replacing it with a photomultiplier. The geometry Of the trap was such that a beam entering through the 0.50 inch aperture would undergo several reflections from the diffuse surface of the trap and then exit through an aperture, into a closed space with a diffuse black surface. All the sur- faces of the trap were painted flat black, and attempts to measure the amount of light reflected back out the front Of the trap were unsuccessful, indicating that the design was at least reasonably efficient. (5) Temperature Control: Constant temperature in the sample was maintained by cementing a thermistor probe to the wall Of the cell holder and using this termistor as the sensor for a Thermistemp Model 71 Temperature Controller. The controller Operated a Variac Autotransformer connected to the nichrome heating wire on the cell holder. The variac was adjusted to yield approximately equal on-Off cycles at the controller, which achieved temperature regulation of bet- ter than 0.1°C at 25°C. Any desired temperature from approx— imately 10°C to 150°C could be maintained by circulating f1Uid of a slightly lower temperature through the copper 79 coil surrounding the cell holder, and using the nichrome heating wire to regulate the temperature at the desired point. (6) Sample Cells: A number of scattering cells of various sizes and shapes were constructed and used in pre- liminary experiments. It was found however, that best re- sults were Obtained using two commercial cells available from Brice-Phoenix Corporation. These cells were square, 1.18 inches on a side, and were 2.375 inches high, contain- ing a volume of about 40 cms. They were constructed of Optical quality pyrex, and yielded remarkably consistent and identical results. It was found that with no sample in the cell, no scattered light could be Observed by the detection system. Detector The detection subsystem (Figure 3.4) included the polarizers for separating the two linear polarization com- ponents, neutral density filters for adjusting the intensity level, interference filters for blocking unwanted wavelengths, and the photomultipliers and amplifiers. The principal Opti- cal problems in constructing the detection subsystem were maintaining the relative alignment of the Optical components, suppression of stray light and fluorescence, and the choice Of an Optimal geometry. The Rayleigh scattered light (at 90°), after leaving the cell and cell holder, passed through a 0.50 inch aperture TC LT I —Ivvtr'U- H-NI»2>r-U)—L<(fi23I—ZZ-H —__.‘_._._ Z -'I(') )> IZ'fi 80 MN Transmitter Matching Network Laser Mirror Chopper Variac Temperature Controller Sample Light Trap Vertical Polaroid Horizontal Polaroid 50% Neutral Density Filter 10% Neutral Density Filter Interference Filter Photomultiplier Tube Battery Supply for PM Preamplifier Lock-In Amplifier Strip Chart Recorder “<—-><'—- FIGURE 3.4.--The Detection System. ' ’I 'n' thug“. 1T 81 into the compartment containing the detection subsystem. The polarizers and neutral density filters were mounted on four plates Of 0.125 inch aluminum which were free to move in slots perpendicular to the Optical path. Each plate contained two 0.50 inch apertures, one of which was in the Optical path when the plate was at either end Of its slot. A brass rod was connected to each plate, and projected through a rubber ”O"-ring light seal and a double wall Of 0.25 inch aluminum to the exterior Of the box. One optical element was mounted on each slide, so that the element could be positioned in or out Of the light path by pushing on the appropriate rod. The geometry Of the light path was not changed by removing or inserting an Optical component since the plates were indexed very precisely in their slots. The polarizers were mounted in the first two plates so that no subsequent Optical elements could contribute to the depolarization. The neutral density filters were supplied by Baird- Atomic and were Obtained in transmittances of 50 percent, 30 percent, 10 percent, and 1 percent; making available the following transmittances when used one or two at a time: .50, .30, .15, .10, .05, .03, .01, .005, .003, .001. Com- binations Of neutral density filters were placed in the light path during the measurement of the vertical polariza- tion component, to make its intensity comparable to the horizontal component. It was then unnecessary to make any 82 electrical changes in the apparatus during a measurement, as the depolarization ratio was simply given by 0,, = [Sf-ECU, (3-5) where the subscript v refers to vertically polarized inci- dent light, HV and VV are the deflections on the strip chart due to the horizontal and vertical polarization components, and T is the transmittance Of the neutral density filter combination. Obviously it is desirable to keep the ratio HV/VV as near unity as possible, so a filter combination as near pV as possible must be chosen for best results. In ad- dition, the neutral density filters must be calibrated very carefully to obtain the best possible resolution. The photomultiplier assembly was a one piece COpper cylinder, mounted to the photometer with a light tight ”O”- ring seal. The photomultiplier tube was held in place by a nylon bushing, and viewed the scattering volume through 0.50 inch apertures in the photometer wall and the nylon bushing. The inside surfaces of this assembly were also painted flat black, and the nylon bushing was carefully sealed so that light could enter the photomultiplier tube only through the Optical aperture. To eliminate any effects due to broad- band fluorescence, room light, or incoherent emission from the laser, a narrow band pass interference filter was mounted in the nylon bushing which held the photomultiplier tube in place. This filter had a half—width of 5.2 nm and a maximum 83 transmittance of 75 percent at 632.8 nm. This band-width was sufficient to pass all the components Of the Rayleigh scattered light, and yet prevent any fluorescent or extra- neous light from being detected. The photomultiplier tube (RCA 6199 or RCA 7102) was electrostatically and magnetically shielded using mu-metal tube shields, and was wired according to reference data supplied by RCA. Since the RCA 6199 and RCA 7102 are phy— sically identical, they could be rapidly interchanged to provide a broad region Of good response characteristics. Photomultiplier power was Obtained from a variable voltage battery supply to assure stability of Operation and a mini- mum Of 60 Hz noise in the output. In order to reduce the electrical band-width of the detection subsystem to a value small enough to eliminate most of the noise spectrum, it was decided to use a phase- locked amplification system. In a phase-locked detector, the signal is made to appear at a specified frequency and phase. The receiver is then tuned to the frequency and phase of the signal and ignores any other spectral compo- nents accompanying the signal. The receiver is then said to be "phase-locked" to the signal, and will tend to block noise contributions within its band-pass since they are of random-phase. Amplitude modulation of the cw laser output was achieved by mechanically chopping the beam with a Princeton Applied Research Corporation light chOpper Operating at 84 80 Hz, and emitting a reference signal of the same frequency and phase as the modulated light beam. The chOpper was placed at the far end Of the laser and inside the Optical cavity so that the coherent output at 6328 3 was modulated while the incoherent output was not. Hence, only that com- ponent of the signal that was actually due to the scattering of coherent light was amplified. The output of the photomultiplier tube was first preamplified by a Philbrick Associates Model P25A solid state Operational amplifier, and was then applied to the sig- nal channel input Of a Princeton Applied Research Model JB-S Lock-In Amplifier. The reference signal from the chOpper was applied to the reference channel input of the amplifier, and phase-locked to the signal channel. The amplifier output, as detected by a standard strip chart recorder, was then pro- portional to that component of the photomultiplier output appearing with the specified frequency and phase. This tech- nique effectively removed those components of the noise spectrum with a frequency greater than 1.5 Hz. Lower fre- quency noise components, and amplifier noise were reduced by filtering the amplifier output through an RC filter with a 30 second time constant. Alignment Alignment of the entire system was perhaps the most critical part of the experiment. To maintain the relative alignment over both long and short periods, the polarization 85 photometer, as well as the laser, was mounted on the Bene— lex table tOp described on pages 68 and 69 under Filling Techniques. Defining a right-handed orthogonal coordinate system with the direction of light propagation the Z-axis, the X-axis vertical with respect to the laboratory, and the Y-axis horizontal (Figure 2.1), the system was aligned using 21 high quality cathetometer whose axis was parallel to the laboratory floor. The laser was mounted to the table top first, and was aligned such that the Z—axis defined by the laser output was colinear with the Optical axis of the cathetometer. Next, the polarization photometer was mounted such that its Optical axis was colinear with the Z-axis. The table tOp was found to be level to better than a degree over its sur- face, so that the X and Y axes of the laser and photometer were parallel to better than a degree. Alignment of the plane of polarization of the laser was extremely easy since the Brewster angle windows reflect, in the plane Of polarization, a small amount of the light incident on them. Two plumb lines were then dropped from the ceiling to intersect the Z-axis of the system, and the line defined by these two points represented the intersec- tion of thelfilplane with the ceiling. The reflections from the Brewster angle windows were then observed on the ceil- ing and the discharge tube rotated to bring the reflections onto the line defining the X2 plane. In this way the plane 86 of polarization was adjusted to better than 0.5 cm of arc on a 200 cm radius, or to less than 0.0025 radians. The polarizers, which were constructed of high- quality Polaroid Corporation Type HN-32 sheet, mounted in aluminum circles, were aligned by measuring their planes of polarization with a conventional Optical polarimeter (using light at 632.8 nm). This procedure allowed align- ment Of the polarizers to 'within 2° of arc. System Characteristics Measurement Techniques Measurements were made by placing the sample cell in the cell holder and allowing sufficient time for the system to come to thermal equilibrium. Preliminary results indicated that this time was of the order of five to ten minutes, de- pending on the initial and final temperatures. Three modes Of temperature control were available: (1) Cooling alone. (2) Coolant plus heat. (3) Heat alone. Initial runs were made by first cooling the system to a low temperature limit of 10°C by circulating water, cooled by an ice bath, through the cooling coils around the cell holder. Temperature control was then initiated using both coolant and heat. After a measurement had been Obtained, the temperature was increased, generally in increments of five degress centi- grade, temperature control being maintained during the course 87 of a particular measurement. At ambient temperatures or slightly above, the coolant supply was disconnected and temperature control maintained in the "heat alone” mode. The electronic components were Optimized by follow— ing the recommended procedures outlined in the manufacturers' Operating manuals. The photometer shutter was then Opened, allowing light to impinge on the sample in the thermostat: care being taken to insure that all the polarizers and fil- ters were in the Optical path at this time. The vertical polarizer was then removed, allowing the horizontally polar- ized component of the signal to reach the photomultiplier. Phototube voltage, preamplifier gain, and the neutral density filters were adjusted to yield an adequate signal level as indicated by the panel meter on the amplifier, with the output in the "signal" mode; the voltage and gain being balanced so as to yield the best signal to noise ratio as indicated by fluctuations of the panel meter. The amplifier was then switched to the ”output” mode and the input level and phase controls used to ”peak” the signal within the linear Operating range Of the amplifier. The system was then opera- tional, requiring only that the apprOpriate neutral density filters be inserted in the Optical path to bring the verti- cally polarized component Of the signal to an on—scale level. In most cases at least five independent measurements of each component Of the signal were taken at each tempera- ture. Furthermore, after the maximum temperature was reached the measurements were repeated while decreasing the temperature 88 in increments of five degrees centigrade. The data were then analyzed statistically using a least-squares treatment. Temperature Stability and Reproducibility Throughout the temperature range Of the thermostat, the sample temperature was maintained with a stability of better than 0.3°C, as measured by both a thermistor and a sensitive thermometer placed in a sample cell containing water. Reproducibility was a somewhat more difficult mat- ter, requiring calibration of the controls. Using only the permanent dial markings on the controller, the temperature could be set to an accuracy of i1°C; however, upon calibra- tion this was improved to iO.5°C. Temperature reproducibi- lity and stability were definitely areas that could bear improvement, however, as we shall see in the results, they were not the accuracy limiting factors. Photomultiplier Response The photomultiplier tubes selected for the detection subsystem (i.e. RCA 6199, RCA 7102) have been reported to have uniform response characteristics tO various states Of polarization. This was verified by rotating the tube while illuminating the photocathode with the linearly polarized output of a Bausch and Lomb grating monochromator using a tungsten lamp as the light source. Over the wavelength re- gion from 550 nm to 700 nm no polarization bias could be observed in either photomultiplier. 89 In most of the measurements reported here, the RCA 6199 photomultiplier was used because of its higher quantum efficiency at 633 nm and its better noise characteristics. For measurements at 1.15 microns, the RCA 7102 would be the only possible choice due to its 81 photocathode. Effects of Beam Parameters Stacey20 points out that the results of light scat- tering measurements in general, and depolarization ratios in particular, are affected by the spatial characteristics __ .—nn_— nun-rum. I”. Of the illuminating radiation. Consideration of the scat- tering process shows that these effects may be due to one Of two sources; multiple scattering, or uncertainty in the angle of observation. Use of a helium-neon laser tends to alleviate this spatial dependency, in the first case due to the fourth power wavelength dependence Of the Rayleigh scat- tering, and in the second case due to the extreme collimation and coherence Of the source. Preliminary experiments were undertaken to measure the effects of beam parameters using the helium-neon laser, and to compare, when possible, these data with those obtained using a conventional light source (e.g. a mercury lamp). Beam Diameter: Lontie24 reports that the depolariza- tion ratio varies linearly as the beam diameter, the slope being very wavelength dependent. The effects Of beam diam- eter in this system (at 633 nm) were measured by using the laser in multimode operation (large beam), and varying the 90 diameter of the iris diaphragm in the photometer shutter. It was found that no diameter dependence could be Observed using simple molecules Of either high or low depolarization ratio; however, it is to be expected that solutions Of strong scatterers (i.e. high polymers) will exhibit diameter dependence due to multiple scattering, in which case a highly collimated laser Operating in the TEMOO mode would yield the best results. Single Mode Operation (TEMOO): In single mode Opera- tion, the laser emits a single well defined frequency, in a very narrow beam with a Gaussian intensity distribution (hence a very low spatial frequency). The "effective” width of the TEMOO mode is considerably less than its apparent width due to its intensity profile, making this mode of op- eration ideal for use when multiple scattering may be a problem. Since simple molecules exhibit little or no di- ameter dependence, this mode of Operation was used to see if the smaller divergence and smaller band-pass would cause a noticeable decrease in depolarization. Results of prelimi- nary experiments showed that this was not the case, and multimode Operation was therefore acceptable. Beam Shape: The effect Of beam shape on depolariza- tion is basically the same as that of beam diameter since, for a given intensity, changing the beam shape simply changes the effective scattering volume. Hence, for a given power output, a spherical scattering volume (circular beam) will have the smallest effective size. 91 Since many beam shapes may be conveniently Obtained in a laser by simply Operating in an appropriate mode, an attempt was made to observe the affects of changing this parameter. Although no effect could be observed in the scattering from small molecules, it is reasonable to assume that in scattering from high polymers the TEMOO mode should be used exclusively to avoid this problem. Calibration of the Neutral Density Filters The range of depolarizations in this study was such (.5 to .02) that two neutral density filters were adequate to cover the range. The 50 percent and 10 percent trans— mission filters were chosen and mounted in the two slides behind the polarizers. Careful calibration Of the filters is an absolute necessity due to the linear dependence Of the depolariza- tion on the filter transmittance. Calibration was accom— plished by placing a sample in the cell and operating the instrument as discussed in the preceding sections, except that the deflections were measured for the six possible combinations Of filters and polarizers. Over a hundred data were taken in this way under conditions identical to those used in measuring the depolarization. These results are summarized in Table 3.1: 92 TABLE 3.l.--Calibration Factors for Neutral Density Filters F1 F2 F12 Vertical Polarizer 0.120910.0005 0.511:.001 0.0644i.0002 Horizontal Polarizer 0.1222: .0005 0.513:.001 0.064Si.0002 where Fl refers to the 10 percent filter, F2 to the 50 per— cent filter and F12 to the combination Of the two. The un- certainties represent the rms deviations Of individual measurements from the mean. Prior to making a measurement with this apparatus, the calibration factors were checked by making a number Of calibration measurements. It was found that the factors remained constant over the period of time the apparatus was in use. Error Analysis and Total Accuragy Having Optimized the various system parameters, the remaining, and limiting source of error is misalignment of the system components. The error accumulated from this source may be analyzed most conveniently by the methods Of Chapter II, using the coherence matrix formalism. Consider the light scattering system as being oriented on a right-handed coordinate system whose origin is the center of the scattering volume, and whose Z-axis is the direction of propagation of the incident beam (see Fig- ure 2.1). We are now interested in Observing the polarization 93 components of the light scattered at some distance (r) along the Y—axis (0 = 90°), assuming the incident light to be plane polarized in either the X2 (vertical) plane, or YZ (horizon- tal) plane. The potential sources of error may now be identi- fied as rotations of this laboratory-fixed coordinate system about the various axes. Denoting a rotation about the Z-axis by O, about the X—axis by 0, and about the Y-axis by y, we may generate the instrument Operator (E) of the various sys- tem components including their alignment errors. We shall assume the system to consist Of a totally unpolarized light source (laser discharge tube), followed by a polarizer (Brewster-angle window), a scattering system, and a resolver which takes the projection of the scattered wave field along some axis. The coherence matrix represen- tation of the unpolarized source is given by: J = . (3-6) The instrument Operator of the polarizer (Brewster window) is given byITable 2.1), cos2 O sin O cos O sin O cos O sin2 O where O is the angle the polarizer makes with respect to the X-axis. Now, assuming the polarizer to be oriented along the X—axis, with some small error, we have that, 94 1 sin O 1 ¢ V ~ (3-7) sin O sin2 O ¢ ¢2 WI n I where the subscript v refers tO the vertical axis, and O represents the angular misalignment with respect to the vertical axis. The instrument operator for the scattering system is given as before by sxx sxr g = (SXY cos 9 - SXZ Sln e)(SYY cos 9 — SZY Sin 0) (3-8) where SNN' 2 SN'N = K o‘NN' Since we are interested in the scattering along the Y-axis, we shall consider the angle 0 tO be given by, e = 90° 1A6 where A0 is the error in alignment in the Y2 plane. The scattering matrix now becomes XX SXY 3(90) = (3—9) SXYAe ' sz SYYAe ' SYZ Finally, the operators representing the resolvers may be seen (Table 2.1) to be 95 cos 8 sin 8 cos B B 2 sin 8 cos 8 sin 8 WI M (3-10) where B is the angle the resolver makes with respect to the X'—axis. That is, the angle 8 is zero for a resolver tak— ing the intensity component parallel tO the X'-axis, and B = 90° for a resolver taking the intensity component par- allel to the Yv—axis (where the resolver plane is perpendi— cular to the Z'—axis). Then, letting y represent the error in alignment of the resolver we have, for small y: _ 1 Y RV = (3-11) 2 Y Y 2 _ Y Y Rh — (3-12) y l where subscripts h and v refer to the horizontal and verti- cal components respectively, and we have assumed the mis- alignment to be the same for both resolvers. Using the results of the section entitled Coherence Matrix Formalism in Chapter II, we may generate the instru- ment operator Of the entire system of polarizer, scatterer, and resolver since L = Rh 3 PV (3-13) I =R SP. (3‘14) 96 Substituting the appropriate expressions, we Obtain the result: _ YM ¢YM L = (3-15) where M = stx + ¢YSXY + °SXY ‘ sz + ¢°SYY ' ¢SY2 (3‘16) (replacing A6 by 0, where 0 is now a small error angle) and _ N ON LV = (3-17) YN ¢YN where N = Sxx + ¢er + Y°SXY ‘ stz + °I°SYY ‘ °YSYZ (3-18) Now, as was shown in Chapter II, the coherence matrix representation, 3', Of the light Observed by the detector is given by i _ _ _ _ _+ _ Jh — Lh J Lh (3 19) where the horizontal resolver is used, and, _' _ _ _+ J = L J L (3-20) V V V when the vertical resolver is used. 97 Combining (3-6), (3-15), and (3—19) we obtain, Y Jh = (1 2 422m2 (3-21) y l where M is given by (3-16) and the subscript h refers to the fact that the horizontal resolver is being used. Simi- larly, for the vertical component, we have from (3-6), (3-17), and (3-20): _. 1 Y 2 2 J = (1 + O )N (3-22) v Y Y2 where N is given by (3-18). Hence, the intensities of the horizontal and vertical components of the scattered radia- tion are given by, respectively, HV Tr Jh (1 + y2)(l + O2)M2 (3-23) II II vV Tr JV (1 + y2)(1 + ¢2)N2 (3-24) where the subscript v refers to the vertically polarized incident light. The depolarization ratio, pv, for verti- cally polarized incident light, is now found by combining (2-71), (3-23), and (3-24); that is, _ _ 2 2 pV — H /v - M /N (3-25) where M and N are as defined in (3-16) and (3-18). Perform- ing the squaring Operation, drOpping cross products Of 98 off-diagonal terms (SMM,, SNN') which are zero according to (2-115) and (2-116), and dropping terms of fourth degree in y, 0, and O, (3—25) becomes 2 2 2 2 2 2 2 2 _ sxz+Y Sxx+° SYz+° SXY+¢Y6[SXXSYY+SXY] pv ‘ 2 2 2 (3'26) 2 Sxx + sz 1 SXY +¢YG[SXXSYY+SXY] and we have derived the desired expression for pV in terms of the angular errors O, 0, and y. Assuming now, that the third order term, Oey, is small compared tO the second order terms, we have; 2 2 2 2 2 2 2 p _ sz I ° SYZ + Y 8xx + e SXY (3-27) v ‘ 2 2 2 2 2 Sxx + ¢ SXY + Y Sx2 and recalling (2-115), that is, [450.2 + 4BZ]K/45 (N = N') s = (2-59) NN' 2 I [38 /451K (N + N ) we see that = 382(1 + e2 + ez) + (45022 + 487w2 (3-28) V (45022 + 482) + SBZCOZ + Y2) Recalling that the exact value of pV is given by 2 p = 238 (2-128) V 45o + 482 99 we see that the measured depolarization ratio, (pV)M, may be written as 2 2 2 RV + O 0V + 6 0V + Y (ov)M = (3-29) 2 2 l + Y 9V + O pv where pV, is the exact depolarization ratio defined by 1'? I 222-22412 "- 2 (2-128). The difference between the measured and exact depolarization ratios is now computed to be 2 2 2 2 6 9V + Y (1 - ov) + pVO (1 pv) A = (p ) - p = (3-30) v M v 1 + YZPV + ¢va and the percentage error, caused by misalignment, is given by the relationship 2 2 2 2 1004 6 0V + Y (1 - 0V) + pVO (1 - 0V) PE = = 2 2 2 2 x 100 (3-31) pV RV + Y pv + O pv This is the desired result, and may be used to analyze the errors in the system quantitatively. However, (3—31) also shows some interesting qualitative behavior, which may be used to make some general remarks about the accuracy of these experiments. In the first place, since all the error angles appear as squared terms, the depolarization ratio of verti— cally polarized incident light is increased by any misalign- ments. Moreover, the terms y2 and O2 in the denominator of (3—31) tend to decrease the error, thus compensating for the 100 corresponding terms in the numerator. Hence, we may assume that measurements of vertical depolarization ratios tend to be on the high side, and that the error contributions tend to compensate one another. Error Sources We are now in a position to quantitatively discuss the various errors present in the system, making estimates of the error magnitudes based on the design parameters dis- cussed in Chapter III. (1) Rotation of the system components about the Z-axis (direction of propagation of the incident light) cor— responds to the error angle ¢- According to the section on Photometer Design (page 71) there are two potential contributions which must be considered: (A) Misalignment of the plane of polarization of the laser, ¢l’ which we estimate in the section on Alignment (page 84) to be ¢l = 0.0025 radians. (B) The finite area of the scattering volume viewed by the detector, giving rise to an uncertainty, $2, in the plane of polarization of the laser. From the geometry of the photometer, we see that the angle ¢2 is given by the radius of the incident beam divided by the distance from the scattering volume to the photomultiplier (i.e. 12 inches). Thus, the maximum value of $2 is given by $2 = 0.188/12 — 0.015 radians while the average value of .‘r‘..' 2Y¢thw1_ (Z) 101 $2, which is actually the quantity of interest is considerably less (20.004 radians). The total error angle squared is thus given by Z 2 ¢ = C¢l + $2) = 6.25 x 10‘4 (3-32) where we have used maximum angles rather than averages, so that ¢2 is a maximum. The term 6 corresponds to a rotation of the detection system components about the X-axis (through the center of the scattering volume). This error angle has three potential contributions, as follows: (A) Rotation of the phototube about the sample, 61, estimated from the alignment technique (page 84) to be less than 81 = 0.034 radians. (B) Divergence of the incident beam in the YZ—plane, 6 estimated in Appendix under Divergence of the 2, Laser Output to be less than 82 = 0.002 radians. (C) The finite area of the scattering volume viewed by the photomultiplier, 63, estimated in the preceding section to be considerably less than 63 = 0.015 radians. Hence, the total angular error component, 62, is given by 2 _ 4 e — (61 + 82 + 83) where, as before, this must be considered an absolute 2 = 17.6 X 10- (3-33) maximum value for the error. 102 (3) The term y corresponds to a rotation of the system com- ponents about the Y-axis passing through the center of the scattering volume (i.e. the ZI-axis where the prime refers to the scattered wave as in Chapter II). This term contains four significance contributions as follows: (A) Rotation of the resolver about the Z'—axis, Y1, which was estimated in the section on Alignment (page 84) to be Y1 = 2° = 0.040 radians. (B V Rotation of the photometer about the Y-axis, YZ’ which was minimized in the initial alignment pro- cedure so that Y2 = 0.002. r"\ O V The divergence of the incident beam in the X2 plane Y3, estimated in Appendix under Divergence of the Laser Output to be less than Y3 = 0.002 radians. (D) A contribution due to the inherent resolution of the Polaroid filters, Y4, such that Y4 = 0.0026 as may be seen from Appendix under Inherent Resolu— tion of the Polaroid Filters. The total angular error component y2 is thus given by 2 2 Y = (Yl + Y2 + Y3 + Y4) 4 = 22.1 x 10' (3-34) where, as before, (3-34) is to be considered an absolute maximum. 103 We may now obtain a numerical value for the per- centage error, PE, in pV as a function of pv, using the above error components. Combining (3-31), (3-32), (3-33), and (3—34) we find that Z 2 2 Y + DVCG + ¢ 2 2 2 ) - DV(Y + ¢ ) X 2 2 2 2 0V + OVCY + ¢ ) _.£_ _ 2 - PE — pv[0.221 + 0.239pV 0.284pV] (3 35) Furthermore, we may distinguish between two extreme cases in the measurement of pV: the case of large pV (pv = l), and the case of small pV (pV = 0). When 9V is approximately one we find that (3—35) becomes PE = 62 x 100 (3-36) or PE = 0.176%. Hence, the limiting error in the case of large 9V is given by 62, and is caused principally by the uncertainty in the viewing angle, and the diameter of the incident beam. In the case of small pV, we find that 2 PE = l— x 100, (3-37) pV 01‘ PE = 0.221/pv, 104 and the error is now a function of pv. It is now reasonable to ask what is the minimum value of pv that may be measured by this technique. Taking the maximum permissable error as 20 percent, we see from (3-37) that (pv)min. = 0.221/20 = 0.011. Here again, we take note that this is a maximum error, and in our system we should obtain a percentage error of less than 10 percent at this value of the depolarization. We also note here that the resolution of our system is limited by the error angle yz, which is principally due to misalign- ment of the resolvers. Therefore, to increase the resolu— tion of this system, a more sophisticated means of aligning the polariods is necessary. The results of this section have been summarized in Figure 3.5, a plot of PE, the percentage error, versus pV, the vertical depolarization ratio. In addition, Table 3.2 summarizes the limiting cases discussed above. TABLE 3.2.--Limiting Errors for Extreme Values of Depolari- zation v 1.0 pv = 0.01 '0 ll Percentage Error 0.176 20 Major Error Contribution e y lo > > a \ AN a ¢m~.o u a mmm.o + HNN.OV n mm 105 .hophm omnpdooyom map msmpo> unwfiq eoNuHmHoa sHHmUfippa> we ouuam eonumNfisafiomoa wee--.m.m mmauHm TNH Eva -©H er ION 80883 39V1N3383d 106 Preparation of Samples In any light scattering measurements, the elimina- tion of large impurities, particularly dust, is absolutely essential due to the enormous increase in scattering as the physical size of the scatterer increases. Moreover, the elimination of dust is particularly essential to an accurate measurement of the depolarization, since the specular re- flectance from a large particle is highly polarized, giving rise to a decrease in pv. The benzene used in this study was obtained in both spectroscopic grade and reagent grade. The two grades were handled separately, however, both were fractionally crystal- lized, and dried by distillation in vacuum. Examination of the small angle scattering using a microscope, and illuminat— ing with a helium-neon laser, revealed no dust particles after filtering through an ultrafine sintered glass filter under pressure from pure, dry, nitrogen gas. The benzene derivatives were purified by filtration through an activated alumina column, and subsequent distil- lation under vacuum. The samples were used immediately after distillation by filtration into the cells through ultrafine sintered glass filters. Carbon tetrachloride, chloroform, methylene chloride, hexane, cyclohexane, and related compounds were obtained in sealed one pint bottles, as reagent grade, and were filtered through activated alumina and distilled over P205. They 107 were then stored until ready to be used, when they were dis- tilled under vacuum, and then filtered into the cell. The polymers used in this study were the standard polystyrene samples distributed by the National Bureau of Standards, and by Dow Chemical Company. They were used without further purification, however the solvents used, were purified as described above. The polymer solutions were filtered through ultrafine sintered glass filters be- fore use, to remove dust particles. The ultrafine filters were cleaned by baking over— night in an annealing oven, and then flushing with hot aqua regia, followed by repeated rinsing with high purity con- ductance water. The filters were then dried in an oven at 110°C and stored in a vacuum dessicator. The conductance water used in this study was pre- pared by passing distilled water through a commerical deionizer, and then redistilling over potassium permanga- nate. The redistilled water was then distilled once more, and finally stored in a polyethylene bottle until used. The purity of all the compounds used, was checked by measurement of the index of refraction. CHAPTER IV DATA AND RESULTS Pure Liquids The depolarization ratios of molecules small com- pared to the wavelength, were measured using vertically polarized incident light. We shall see that the quantities F = (40V - 3)/3 (4-1) and (l/T + a1), exhibit relatively small temperature depen- dencies in the region of interest, so that (2-174), and (2-179), may be used to predict a temperature dependence of p somewhere between linear and logarithmic. The logarith— v mic dependence is slightly preferable since the slope is then independent of the temperature; however, to avoid ambiguity, the results will be compared at the standard temperature of 25°C (298°K). For completeness, values of Du have been included. These were obtained by calculation from (2-147), where ph was assumed equal to unity. Benzene and Its Derivatives The study of the depolarization by benzene and its monosubstituted derivatives is of basic interest, due both 108 109 to the solvent prOperties of these compounds, and to the degree to which their molecular properties are understood. Moreover, the derivatives chosen represent variations in parameters of importance to the depolarization, such as size and electronegativity. Table 4.1 presents data obtained with the benzene derivatives at a number of temperatures, where T is the absolute temperature, pV the vertical depolarization ratio, and pu the unpolarized depolarization ratio (assuming ph = l). The data for benzene represent the average of four samples: two obtained from spectroscopic grade benzene, and two from reagent grade. The data for the four samples showed no sig- nificant differences in either the total scattered intensity, or the depolarization ratio. Figure 4.1 is a plot of the natural logarithm of the vertical depolarization ratio versus the absolute tem- perature, and the slope is given by; Q = a fin oV/BT = A Zn pV/AT. (4-l-b) The approximate linearity of this plot is obvious, however, we may also plot pv versus T, to obtain an approximately linear graph, and evaluate Q from _ _ £_ - _ Q — 3 Kn pV/ST — pV(ApV/AT). (4 l c) The Q evaluated using a logarithmic plot will be denoted by Q(log), and is seen from (4-1-b) to be temperature indepen— dent; while the Q evaluated from the linear plot, Q(lin.), 110 TABLE 4.l.--Depolarization Ratios of Benzene and Its Derivatives. O 0 Compound T( K) pv pu Compound T( K) pv pu Benzene 293 0.263 0.417 Chloro- 298 0.406 0.578 298 0.255 0.406 benzene 318 0.374 0.544 308 0.244 0.393 338 0.346 0.514 313 0.239 0.385 358 0.311 0.474 318 0.232 0.380 328 0.222 0.363 338 0.210 0.347 Bromo- 298 0.426 0.548 benzene 318 0.395 0.566 338 0.369 0.539 Toluene 298 0.321 0.486 348 0.353 0.521 308 0.312 0.476 358 0.340 0.507 318 0.288 0.447 328 0.275 0.431 338 0.260 0.412 Iodo- 298 0.441 0.612 benzene 308 0.435 0.606 318 0.420 0.592 Fluoro- 298 0.319 0.484 328 0.409 0.580 benzene 308 0.302 0.463 338 0.399 0.571 318 0.281 0.438 348 0.385 0.556 323 0.275 0.431 358 0.370 0.540 328 0.268 0.422 338 0.251 0.401 343 0.243 0.390 Nitro- 298 0.600 0.750 348 0.234 0.379 benzene 308 0.588 0.743 352 0.233 0.377 318 0.581 0.735 328 0.572 0.728 111 -l.60~— ///////////// BENZENE .///////// -l.50r— / //// ////o //// ' —1.40- FLUOROBENZENE ::::;///////// ”E ' TOLUENE °-—1.3o- ,////. Z /. £3 z-quzoh— .//////// [3 gr: / Kn 9v VERSUS TEMPERATURE C) 5: Q. UJ Q -l.]_0 " / ./// -1.oo-— CHLOROBENZENE .////////’ BROMOBENZENE / / -0.90 L. '/ ////////"/////?SBOBENZENE FIGURE 4.1.--Plot of Zn pV Versus T for the Benzene Derivatives. l l l 310 320 330 TEMPERATURE 340 350 112 is obviously dependent on the temperature because of the term, l/pv. Figure 4.2 is a plot of 0V versus T for the benzene derivatives, and illustrates the approximate lin- earity of the temperature dependence. Using equation (2-179), we may examine the tempera— ture dependence of Q, predicted by the theory. Table 4.2 compares the temperature variation in Q obtained using (2-179) with that determined experimentally using (4-l-c) and M-l-b). TABLE 4.2.--Temperature Variation of Q Using the Data for 3 Benzene. Tem 2 3 -1 3 . (°K§ "314pv'31 O‘TXIO T X10 —Q(Theory) -Q(11n.) -Q(log) 298 1.320 1.237 3.356 6.06><10'3 4.61><10'3 4.72x10‘ 308 1.352 1.325 3.247 6.17 " 4.83 H n 318 1.382 1.413 3.145 6.29 " 5.08 " H 328 1.408 1.504 3.049 6.41 " 5.31 " H 338 1.440 1.595 2.959 6.56 " 5.62 H H 105 x (AQ/AT) = 1.25 2.5 0 Hence, the theory predicts a temperature variation in the slope, Q, that is midway between that obtained from a linear and a logarithmic depolarization ratio; thereby implying that the data may be analyzed in either way with approxi- mately equal errors. Similarly, using the data for chloro- benzene, we find that: -0.78 x 10’5 Theoretical AQ/AT -1.43 X 10‘5 (linear) 0. ' (log) Experimental AQ/AT Experimental AQ/AT 113 .mo>fiuw>fipom ocomcom How mQSumpomEoH mamho> :oHpmNfipmHommm--.N.v mmbon mmapoa 0000-00 0000 0000000000 0000 0000-00 0000 0000000000 0000 .mumfluomthSm >p powwowpcfi moocoymmom 00.0 00.0 n 000000000 000 ocmNGon 0 00.0 00.0- 00.0- 00.0- 00.0- 00 000.0 000.0 000.0 -00002 0:00aon 00.0- 00.0- 00.0- 00.0- 000.0 000 0 -0000 mamNGmn 00.0 00.0 00.0- 00.0- 00.0- 00.0- 00 000.0 000.0 000.0 -00000 mGoNGon 00.0 00.0 00.0- 00.0- 00.0- 00.0- 00 000.0 000 0 000.0 -000000 00.0 0 00.0- 00.0- 00.0- 00.0- 00 000.0 0000.0 000.0 0000000 osquon 00.0 00.0 00.0- 00 0- 00.0- 00.0- 00 000.0 000.0 000.0 -000000 00.0- 00.0- 00.0- 00.0- 00.0- 00.0- 00 000.0 000.0 000.0 0000000 0000 0000 00000 00000 0 > .000 .>00 000x0000 000x0000 000x000000 00x000000 000000000000 000x 0 0 00000000 .mm>0um>wamn muH 0:0 acoucmm How unmeflpmmxm 0cm >000:H wo :omflhmmEou--.m.0 mqm -2.0- // BENZEWL/,.0- -l.5"‘ ./_ / _-/ __._.. 300 310 320 330 340 350 TEMPERATURE FIGURE 4.3.--£n pV Versus T for Hexane and Related Compounds. 120 TABLE 4.6.--Comparison of Saturated and Unsaturated Ring Compounds. Ratio Exp. (log) Theory Q(CH)/Q(MCH) 1.03 1.03 Q(B) /Q(T) 0.90 1.23 Q(CH)/Q(B) 1.21 1.47 Q(MCH)/Q(T) 1.03 1.76 CH = Cyclohexane B = benzene MCH = methylcyclohexane T = toluene agreement between theory and experiment for hexane, that the ring closure has altered the behavior, in the liquid state, of the hexane related hydrocarbons. It is also apparent from the measurement of the depolarization ratios, that in the case of benzene, ring closure (and loss of six hydrogen atoms) has resulted in an increased anisotropy as compared to hexane, whereas in the case of cyclohexane, ring closure has resulted in a decreased anisotropy. In either case however, the dipole moment has remained zero (except for toluene which has a moment ®o = 0.36D). Com- parison with the results of the previous section shows a definite error decrease with increasing dipole moment. It is therefore evident that the theory predicts good results for molecules with non-zero dipole moments; however the re- sults are poor when the dipole moment vanishes, regardless of the total anisotropy of the liquid. It is also worth- while to note that the measured Q is smaller than the 121 predicted Q for molecules of zero dipole moment, indicating a relatively large temperature dependence in the magnitude of the horizontal component of the scattered light, and with a slope opposite that of the vertical component. It also seems apparent that agreement with the theory becomes worse as the sphericity of the molecule increases. Thus benzene exhibits an error of 30 percent while cyclo- hexane has an error of 60 percent. It would seem therefore that the relatively good agreement between theory and experi- ment in the case of hexane is due in part to the rod-like character of the molecule. Hence, we are led to the obvious conclusion that, in dense fluids of molecules with a very high symmetry, the anisotropy contribution due to intermolecular interactions of the polarizability is of the same order as the anisotrOpy contributions due to the hyperpolarizability interactions. Thus the hyperpolarizability interactions cause a tempera- ture dependence in the anisotropy such that the anisotrOpic scattering is increased as a function of the temperature. Returning to the section on Temperature Dependence of De- polarization by Dense Fluids (page 43) we see that, with respect to temperature, and assuming a temperature dependent anisotropy, A, (2-170) becomes Q = (8 in pv/BT) = F(8 8n B/aT) — r(a £n A/ar) (4-2) where F is as defined in (4-1). Thus we see that an increase 122 in A with respect to T causes a decrease in Q, as observed in the case of the ring hydrocarbons. The Chlorinated Methane Derivatives The chlorinated methane derivatives are now of par- ticular interest, since the results obtained with the previous series of compounds imply that, due to spherical symmetry, the measured Q for carbon tetrachloride should be considerably smaller than that predicted by (2-174) or (2-179), however, the slopes measured for chloroform and methylene chloride should agree well with the theoretical predictions. The data obtained for this grOUp of compounds are summarized in Table 4.7, where, as before, pu has been cal- culated from (2-147) assuming a horizontal depolarization ratio, ph, equal to one. Figure 4.4 is a linear plot of the natural logarithm of the vertical depolarization ratio, pv’ versus the abso- lute temperature; and Table 4.8 is a comparison of the experimental and predicted results using the values of Table 4.7. It is apparent from the results of Table 4.8, that the anticipated behavior is indeed correct. Moreover, it appears that there is, again, a direct relationship between the dipole moment, spherical symmetry, and slope of the vertical depolarization ratio. 123 TABLE 4.7.-—Depolarization Ratios of the Chlorinated Methane Derivatives Compound Temperature (°K) pV pu Carbon Tetrachloride 298 0.0166 0.0327 308 0.0160 0.0315 318 0.0151 0.0297 328 0.0147 0.0289 333 0.0146 0.0287 Chloroform 298 0.108 0.195 308 0.101 0.184 318 0.0916 0.168 328 0.0847 0.157 Methylene Chloride 288 0.146 0.255 298 0.136 0.239 308 0.126 0.224 Kn 124 -__""g__,,,0 .__“__,,._ fiflflfl,,_.l ' c RBON TETR c E ruflflflfifl,fl,,u00——~———* A A HLORID -4.0—' -3.5— -3.0- -2,50. CHLOROFORM/,,000" _”,,,00»—' --*”””/E METHYLENE CHLORIDE ._"’_flfl__,,_. -1.51 #-/"/ _._.000————'*’”‘ BENZENE -1.0- I I r I I I 300 310 320 330 340 350 TEMPERATURE FIGURE 4.4.--£n pv Versus T for the Chlorinated Methane Derivatives. 125 .o oflesuflgwmoH 0:0 ow pumm000 £003 :oxwu mcoflumfl>om0 00.0 -- 00.0 -- 00.0 00.0 -- 000.0 000.0 00000000 0:000:00: 00.0- 00.0 00.0 00.0 00.0 00.0 000.0 000.0 0000.0 0000000000 00.0 00.0 00.0 00.0 00.0 00.0 000.0 000.0 0000.0 0000000000000 Conhmu 0000 0000 } 0.0000 0.0000 000x .000 .000 00x0000- 00x00vo- 00x0- 00x0- 000N000 00x00 >0 00000000 0 0 0 0 0 0 0 - l . r .mo>0uw>0hmm ocwapoz wopmmfihoano 0:0 00% ucosflhomxm 0:0 >000ne mo acmfihmeOU--.w.0 mqm 00000 00000000000000 000 00 0000--.m.0 000000 020020000002 20000<0d 0002 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 b _ p H _ — _ _ _ 052020000000 u 0 00000000 + 0000000 u >00 0<0z00 "mu MommN H QZMP 0 .0 0 0 .0 ‘0 01 x N011vz10v10030 1 oo <3‘ 138 The natural logarithm of the depolarization ratio has been plotted versus T for the five solutions in Figure 4.6, and the results obtained from Table 4.13 and Figure 4.6 are compiled in Table 4.14. It may be seen from the Table that the curvature of the actual slope, Q, is not large (approximately 3 percent) and that agreement with theoretical values, (4—10), is reasonably good. TABLE 4.14.-—Values of —Q for Benzene-Nitrobenzene Solutions Mole Fraction Nitrobenzene 0.224 0.342 0.464 0.591 0.722 + -Q(logarithmic pV)XlO3 4.32 4.89 4.12 3.05 2.92 3 —Q(1inear 0V)x10 4.06 4.54 3.88 2.92 2.79 3 -Q(theoretica1)X10 4.78 4.06 3.36 2.87 2.32 -Q(idea1)x103 4.05 3.67 3.30 2.90 2.52 Theoretical values from (4-10). Ideal values from (4—9). Figure 4.7 has been constructed from the results of Table 4.14. Several trends are apparent from this graph, and are worthy of note: the theoretical and experimental values agree well; the theoretical values are less than the actual values; and the non-linear, or "non-ideal" behavior is easily discerned. It may therefore be concluded that dipole induction interactions are causing the solutions to behave as predicted, and that the temperature dependent de- polarization ratios may be used to study the behavior of these mixtures. 139 r 298 308 318 328 TEMPERATURE FIGURE 4.6.--Plot of Zn pv Versus T for Benzene-Nitrobenzene Mixtures. 140 _bnw .mOHSHXH—Z OfiwNHHODOHHHZIOEONHHOm HOW COflHUNHm 0H0: WSWHQ> 01 W0 HOHAHII.N...V MMDUHHH MZMszmomHHz zomho 0<000000000 u 00 0000<> 0<103 versus the mole fraction of carbon tetrachloride. The behavior in this case is more obvious than with the nitrobenzene-benzene solutions since the theory predicts a positive linear graph, while the linear extrapolation (”ideal” solution) yields a negative linear graph. The behavior of the measured Q shows that the dispersion and configuration interactions between the two species tend to make the measured values approach the theoretical. Since these forces are small compared to dipole-dipole, or in- ductive effects, the measured values are all smaller than the theoretical predictions. At some intermediate concen- tration, the interaction becomes a maximum, and subsequently the measured values fall off to intersect the linear ”ideal” graph at a mole fraction of one. It is therefore obvious that this technique may be used to study short range inter- actions in the liquid state. Summary The data and results for benzene-nitrobenzene and benezene—carbon tetrachloride mixtures have been found to exhibit behavior indicating the existence of short-range interaction potentials in the liquid state. These forces may be expected to be functions of the polarizability and hyperpolarizability tensors discussed previously. Thus, the results show that binary solutions obey equation (2-179) reasonably well over intermediate concentration ranges, even though the pure liquids may exhibit appreciable error. 145 .mphdpxflz owfiHOHnumhuoH conhmu-ocoacom you :ofiuuagm 6H6: wsmyo> 0. mo “can--.m.v mmauHm unamOJIu 78) would establish the dependence on the dielec- tric constant. In addition, water would be of interest due to its extremely small coefficient of thermal expansion (0LT - 0.207 x 10‘3). 153 Measurement of the temperature dependence of ph, rather than pv’ may be found to yield data relating to the hyperpolarizability, since any temperature dependence in this quantity must be due to terms of higher order than the polarizability (i.e. according to the theory of Chap- ter II, ph must be unity). Measurement of the intensities and line widths of the resolved Rayleigh-Brillouin spectrum as a function of the temperature, may be expected to yield information con- cerning the precise mechanism of the temperature dependence. That is, the dependence of each component of the spectrum may be observed independently, thereby resolving any uncer- tainties as to which scattering processes are being affected. Moreover, the symmetry dependence of each component of the spectrum could be independently investigated. (l) (2) (3) (4) (5) (6) (7) (8) (9) (10) (ll) (12) (l3) (14) (15) (16) (17) (18) J. J. BIBLIOGRAPHY B. Richter, U.d.n. Gegenstande der Chemie 44, 81 (1802). Tyndall, Phil. Mag. El, 384 (1869). Lord Rayleigh, Phil Mag. 44, 447 (1871). A. T. A. ZNO’U > Einstein, Ann. Phys. Lpz. 33, 1275 (1910). H. Maiman, Phys. Rev. 123, 1145 (1961). Javan, W. R. Bennett, Jr., and D. R. Herriott, Phys. Rev. Letters 6, 106 (1961). D. Buckingham and M. J. Stephen, Trans. Faraday Soc. 54, 884 (1957). C. Van de Hulst, ”Light Scattering by Small Particles," John Wiley and Sons, Inc., New York, 1957. Doty, J. Polymer Sci., 4, 750 (1948). G. Stokes, Trans. Camb. Phil. Soc. 9, 399 (1852). C. Jones, J. Opt. Soc. Am. 46, 126 (1956). Born and E. Wolf, "Principles of Optics," Pero- gramon Press, New York, 1964. L. Schawlow and C. H. Townes, Phys. Rev. 12, 1940 (1958). W. Smith, J. Quantum Elec. 3’ 62 (1966). W. Smith, J. Quantum Elec. 2, 77 (1966). . Mielenz and K. F. Nefflen, Appl. Opt. 4, 565 (1965). Rousset, Ann. Phys. Paris 5, 5 (1936). J. W. Debye, J. Appl. Phys. 45, 338 (1944). 154 (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) 155 H. Z Cummins and R. W. Gammon, Appl. Phys. Letters 6, 171 (1965). K. A. Stacey, "Light Scatteringin Physical Chemistry," Academic Press, New York, 1956. J. A. Osborn, Phys. Rev. 67, 351 (1945). A. Sommerfeld, "Electrodynamics," Academic Press, New York, 1952. Benoit and Stockmayer,J. Phy. Radium ll, 21 (1956). B. A. Brice, and M. Halwer, J. Opt. Soc. Am. 44, 1033 (1951). A. Kruis, Zeits, f. physik. Chemie B34, 13 (1936). International Critical Tables D. Tryer, J. Chem. Soc. 105, 2534 (1914). R. E. Gibson and O. H. Loeffler, J. Am. Chem. Soc. 4;, 2515 (1939). Jean Timmermans, "The Physico-Chemical Constants of Binary Systems in Concentrated’Solutions," Volume 4, Interscience Publishers, Inc., New York, 1959. C. P. Smyth, ”Physical Methods of Organic Chemistry,” Interscience Publishers, Inc., New York, 1946. R. C. C. Leite, R. S. Moore, and S. P. S. Porto, J. Chem. Phys. 40, 3741 (1964). J. Ehl, C. Loucheux, C. Reiss, and H. Benoit, Makromol. Chem. 7g, 35 (1964). APPENDIX Divergence of the Laser Output 1" Since the beam divergence is a quantity of some importance in the analysis of the errors inherent in this system, a measurement was made of the divergence, in both single and multi-mode operation, of Lasers I and II de- scribed in Chapter III, the section on The Polarization Photometer (page 76). Using Laser 1, the system used in most of the work reported here, in the "semi"—confocal configuration, it was found that the beam diameter near the output mirror was slightly less than 0.25 inches with a multi-mode output. At 300 feet however, the output had diverged to approxi- mately 12 inches, as shown in Figure 7.1: W [MIN/’1 \12" y <—x—91<————— 3600" FIGURE 7.1.-—The Laser Divergence where 0 is the divergence half—angle, and x is the distance to the point at which the radiation appears to be originat— ing. Obviously, 0 is given by 156 157 20 = 0.25/x = 12/(3600 + x) (7-1) and, rearranging, we obtain I2 x 75 inches (7-2) so that 1.7 x 10'3 (D R radians. (7-3) In single mode operation, the beam width near the output mirror, using this laser, became approximately 0.10 inch. At 300 feet, the width had diverged to about seven inches, yielding a divergence half-angle, 6, of 3 e : 7/(3600 x 2) 2 1 x 10- radian. (7'4) In this case, single mode operation was achieved by simply tilting the mirrors until the TEM00 mode was made to oscil- late due to the small mode volume within the device. Using Laser II, it was found that at the surface of the output mirror the beam diameter was approximately 0.10 inch, while at 300 feet the diameter had diverged to about 28 inches. In this case, as before, the divergence half- angle is given by I? 20 28/3600 (7-5) so that 3 0 2 4 x 10- radians. (7‘6) Obviously this configuration is considerably more divergent than that of Laser 1, and no attempt was made to measure the divergence in single mode Operation. 158 Inherent Resolution of the Polaroid Filters Since the polaroid filters are not ideal, they pass a sma11.component of light polarized perpendicular to the plane of polarization of the filter. This inherent resolu— tion may be considered to be an uncertainty in the alignment of the polaroid and may therefore be represented by the in- strument Operator P, as shown in Table 2.1. Thus, when the polarizer is oriented parallel to the X-axis, that is so as to take the vertical polarization component, the instrument operator is given by 1 sin 6 wl 0 (7-7) sin sin? € where 6 is the angle representing the misalignment, and the subscript v refers to the vertical orientation. When 6 is small, as is the case, we may write _ 1 6 PV 2 2 (7-8) 6 6 and, similarly for a horizontally polarized filter _ 7'62 € Ph =[ (7-9) G 1 where the E in both cases is the same since it is a function of the material only. Now, if two such polarizers are 159 oriented perpendicular tO one another, we may generate the instrument Operator Of the combination as _ _ _ € 1 P1 = Pvph = 26 2 ; (7-10) 5 6 and illuminating the combination with unpolarized light, the coherence matrix of the light passing through the com- bination is given by 1 € — 2 2 J = 46 (1 + G ) (7-11) 1 [6 62 From (7-11) we see that the intensity, 11’ Of light trans- mitted by the combination is 2 11 = Tr ii = 462(1 + €2)2 2 4e . (7-12) If the two polaroids are oriented with their axes parallel, the resulting Operator is 1 6 PH = F F = (1 + 62) , (7-13) VV and illuminating the combination with unpolarized light yields an output whose coherence matrix may be written as 1 € €€2 The intensity Of the transmitted light may now be seen to 3H = (l + €213 (7-14) be IH = (1 + €2)4 z 1. (7-15) 160 Combining (7-12) and (7-15) yields Il/IH = 462(1 + €2)2/(l +6214 = 462/(1 +62)2 01‘ l e . ,4; (7-... This is an extremely useful result, since it relates the inherent angular resolution Of the polarizing elements to the easily measured raio of intensities with the polarizers crossed and the polarizers parallel. Measuring Ill and Ii using the photometer described in Chapter III, and a very intense tungsten lamp with a red filter as the light source, we Obtained the results: -5 I 22x10 1 1 = 0.7 || so that € = 0.0026 radians. Thus, the uncertainty in the plane of polarization of the polaroid is 2.6 milliradians. Analysis of Errors in the Horizontal Depolarization Ratio We have thus far concerned ourselves only with the measurement of the depolarization ratio for vetically polarized incident light. In the case Of horizontally 161 polarized incident light, as was seen in Chapter II, the section on The Horizontal Depolarization Ratio (page 38), the total scattering is considerably reduced in intensity, and has a depolarization ratio of approximately unity. Since the deviation of the horizontal depolarization ratio, ph, from unity may be a quantity Of some interest (and more particularly, its temperature dependence) it is interesting to inquire with what accuracy we may measure this quantity using the photometer described earlier. Approaching the problem in the same manner as in Chapter III, the section on Error Analysis and Total Accuracy (page 92), we shall consider the light scattering system as being oriented on the right-handed coordinate system of Figure 2.1, and shall Observe the scattered light at some point along the Y—axis (the Z'—axis Of the scattered field). Assuming, as before, that the system consists Of a totally unpolarized light source, a polarizer, a scattering system, and a resolver, we may generate the instrument Operator Of the entire system as follows. The polarizer (Brewster window) may be represented by the matrix Ph, where the subscript h denotes the horizon- tal orientation, in the form q _ cos ¢ sin 0 cos 0 Ph = J (7-17) sin 0 cos 0 sin2 ¢ 162 where 0 is the angle the polarizer makes with respect to the X-axis. Since in this case 0 is approximately 90 degrees we may write _ cosz(90 i A0) cos(90 f A0) Ph = (7—18) cos(90 i A0) 1 where A0 is the angular error due to misalignment and is assumed to be very small. (7-18) may be rewritten as sin2 A0 sin A0 P = (7-19) sin A0 1 and dropping the A in the notation (bearing in mind that 0 is now the error rather than the total angle) we see that 424 = . (7-20) 0 l The instrument operator for the scattering system is given as before by equation (3-9), that is = (3-9) S XY X2 YY YZ where A0 is the error in alignment in the YZ plane (i.e. about the X-axis). The resolvers are, as before, repre- sented by (3-11) and (3—12), that is 163 _ 1 Y RV = (3—11) 2 Y Y 2 _ Y Y y l where the subscripts h and v refer to the horizontal and vertical components respectively, and the angle y repre- sents the error in alignment of the resolver (assumed to be small). The instrument operators describing the entire sys— tem are then given by L = R h h 8 Ph (7-21) and LV = RV 5 Ph (7-22) or, substituting (7-20), (3-9), (3-11), and (3-12) into (7-21) and (7—22), _ ¢YQ YQ Lh = (7-23) ¢Q Q where Q = Y¢Sxx + YSXY + e¢SXY ' 85x2 + esYY ‘ SYZ (7'24) (and we have replaced A0 by 0, remembering thate is now a small error angle) and 164 iv = [ (7-25) where w = ¢Sxx + Sxy + eY¢SXY ' Y¢sz + eYSYY ' Y5Y2“ (7—26) Now, as was shown in Chapter II, in the section on Theory of Rayleigh Depolarization (page 21), the coherence matrix representation, 3', of the light observed by the detector is given by I _v I .1. J = Jh + JV = L J L (7-27) where 2 2 ' Y Q YQ 3h = (1 + 42) (7—28) 2 2 YQ Q and W2 sz _l JV = (1 + 42) (7-29) sz Y2W2 W and Q being given by (7-26) and 0-24). Thus, the inten— sities of the horizontal and vertical components Of the scattered radiation are given by, respectively, H, = Tr 3; = (1 + 421c1 + v2102 (7-30) 165 Vh = Tr 3; = (1 + ¢Z)(1 + Y2)w2 (7-31) where the subscript h refers to the horizontally polarized incident light. The depolarization ratio, ph, for horizon- tally polarized incident light, is now found by combining (2-145), (7-30), and (7-31); that is, _ _ 2 2 oh — Vh/Hh — w /Q (7-32) where W and Q are as defined in (7—26) and (7—24). Perform- ing the squaring Operation, dropping cross products Of off diagonal terms (SMM, SNN') which are zero according to (2—115) and (2-116), and dropping terms Of fourth degree in 0, 0, and y, (7-32) becomes 2 2 2 2 2 2 XY + ¢ Sxx + Y SYZ + ¢eYSXY 2 . 2 2 2 2 * 2 XY + ¢ sz + 0 SYZ + ¢6YCSXXSYY + SXY) S D h S2 + YZS (7-33) Assuming now, that the third order terms, 00y, are small compared to the second order terms, (7—33) becomes 2 2 2 2 2 S + 0 S + Y SYZ 9h = S2 +XY252 +XX282 + 92 2 ’ (7‘34) YZ Y XY 8 xz Y2 and recalling (2—115), that is 2 2 [45a 45‘£§—]K N = N' sNN, = 2 (2-115) [38 /45]K N = N' 166 we have that _ 382(1 + Y2) + 62(4562 + 482) ph ‘ 2 2 2 2 2 2 (7‘35) 38 (1 + y + 0 ) + e (450 + 48 ) or 1 + Y2 + (02/9v (7_36) D , h l + Y2 + ¢2 + 62/pv where 0V is the vertical depolarization ratio. The percentage error, PE, is given by (ph)M ' ph ph PE = x 100 (7-37) where (oh)M is the measured value of ph. Since the exact value of p is ver near unit , h Y Y PE = 100((ph)M - 1) (7-38) and combining (7—38) and (7—36) yields 02(1 - 9V) - 62 9V(1 + Y2 + ¢2) + 6 PE = 100 (7-39) 2 This result, (7—39), is the solution to the percentage error in our measurement, as a function of the error angles, 0, 0, and y, and may be used to quantitatively discuss the accuracy of a measurement of Oh. At this point we shall examine the two limiting cases Of (7-39), that is, the percentage error when pv 167 is either unity or zero. When pv is zero, we may write PE = 100(¢2 - eZ)/ez. (7-40) This result is rather interesting in that it predicts an increasing error as 0 is reduced, when 0 is larger than 0; and in that the error is independent of misalignment in the resolvers. A more realistic estimate of errors at small pv is found by assuming a small, non-zero pV so that (7-39) becomes PE = 100(¢2 - 62)/(pv + 02), (7-41) and the quantity 0V in the denominator limits the error when 0 is reduced. Using the values found in Chapter III under Error Sources (page 100) for the error angles, we find that for pv equal to zero PE = -0.1135/.0018 = —63% (7-42) that is, the measured horizontal depolarization ratio is 63 percent smaller than the actual value. Since the smallest 0V measurable with the photometer described previously is pv = 0.01, we may ask with what accuracy we can measure ph in this case. Using (7-41) we find that PE = —0.1135/0.012 = —9.5% (7-43) when pv = 0.01. In the case of large pV we may rewrite (7-39) in the form, 168 PB = -100 02/(1 + 02 + Y2 + 02). (7—44) This result is somewhat strange also, since increasing y and 0 tends to decrease the error. Substituting 0, 0, and Y in (7—44) yields for the large pv, PE = -0.16%. (7-45) Thus we have seen that misalignment errors do not affect 0h as seriously as they do pv, and that the error due to beam parameters, 0, tends to reduce the value Of ph (assuming 02 > 02). Therefore, we may expect effects such as multiple scattering, increased beam diameter, and in- creased beam divergence to reduce ph, whereas these same parameters tend to increase the measured 0V. Substituting values for the error angles into (7-39) yields the numerical result, PE = - (0.113 + 0.063pV)/(pv + 0.0018) (7-46) which is graphed in Figure 7.2. Calibration of the Differential Refractometer In the measurement of the Rayleigh ratio Of solu- tions, it is necessary to know the rate of change of re— fractive index with respect to concentration, dn/dc, very precisely. A number Of instruments are available with which this quantity may be measured, however the differen— tial refractometer is perhaps the most suitable choice. In 15 10 o\° 169 01. 0.01 .10 FIGURE 7.2;-—Percentage Error in 0h as a Function Of pv. this laboratory, a differential refractometer similar to 24 has been used tO the one described by Brice and Halwer measure dn/dc at 632.8 nm. Since it was necessary to Obtain standards with which tO calibrate the refractometer, the data of Kruis25 were interpolated tO Obtain values at 632.8 nm. The inter- polation was accomplished by using a least-squares fit Of the data to a polynomial Of degree n/2, in the wavelength; where n varied from one to (m-l), m being the number Of data points. The best polynomial was chosen using the ”Gauss criterion Of fit,” and was then used tO calculate An at 632.8 nm. The results are presented in Table 7.1, 170 TABLE 7.1.-—Difference in Refactive Index between Salt Solutions and Pure Water. (Temp. = 25°C) Salt Concentration(g/kg) An x 105 K C3 0.6986 9.510 NH4NO3 0.7905 9.687 K CZ 1.0702 14.549 Na C8 0.9421 16.392 ” 1.0371 18.064 NH4NO3 1.6775 20.497 K CZ 2.8117 38.010 NH4NO3 3.4502 41.988 Na CZ 3.3750 58.338 " 5.6274 96.750 NH4NO3 10.4082 125.07 K CK 10.8691 144.83 Na CZ 6.9003 183.54 " 11.3107 192.60 ” 20.5128 345.10 NH4NO3 29.4789 346.41 Na CZ 37.8543 624.98 NH4NO3 60.7311 693.07 Na CK 69.0916 1107.62 ” 108.9536 1688.20 171 and represent the difference in refractive index between the solution and pure solvent (water). The results were checked by graphical interpolation and by measurement on the refractometer. The light source used in the measurements being the helium-neon laser de- scribed elsewhere in this work. Comparison Of Results Comparison Of the results reported here with those Of other investigators is very difficult due to the differ- ing conditions (i.e. wavelength, spatial characteristics Of the source, sample purity, etc.), however Porto31 has re— ported depolarization ratios measured with a helium-neon laser at a single temperature. Table 7.2 compares this data with those reported here, at the same temperature. Comparison Of Results Molecule Temp. pv (Porto) pv (this work) Percent Diff. Benzene 15°C 0.281 0.269 4 Toluene 15°C 0.359 0.336 6 Cyclohexane 225°C 0.0304 0.0188 50 CC£4 z25°C 0.0195 0.0166 14 CHCK3 15°C 0.114 0.114 O The results Of this work are generally lower than those reported by Porto. Assuming no dust present in the samples, this indicates better alignment Of the system in 172 our case. The only major difference is that Of cyclohexane, however Porto did not specify a temperature for this data, and a valid comparison is therefore not possible. Benoit and co-workers32 have measured the depolari- zation Of unpolarized light by Benzene at a number Of tem- peratures. Although their data are not comparable due to the different wavelength, light source and instrument, it is interesting tO note that their values of the depolariza- tion ratio are larger than those reported here (calculated by assuming ph = 1) as is tO be expected, while the slope Of their data is the same within experimental error. ”11111111111011fllflfllfllflilfllflfllfllfilfis