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Non-equilibrium particle number fluctuations Enns, Ernest Gerhard 1965

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The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of ERNEST GERHARD ENNS B.Sc, The University of B r i t i s h Columbia, 1961 MONDAY, APRIL 26, 1965, AT 3:30 P 0M„ IN ROOM 10, HEBB (Physics) COMMITTEE IN CHARGE Chairman: I. McT„ Cowan R. E, Burgess G„ P. Erickson ' J . W. Bichard S. W0 Nash F„ L, Curzon G„ B„ Porter External Examiner: K, M„ van V l i e t Department of E l e c t r i c a l Engineering University of Minnesota NON-EQUILIBRIUM PARTICLE NUMBER FLUCTUATIONS ABSTRACT The f l u c t u a t i o n s of the constituent populations i n a m u l t i - l e v e l system are considered as applied to a three and four l e v e l semi-conductor model. It i s shown that under c e r t a i n conditions the fl u c t u a -tions of the electron numbers i n the conduction band can be super-poisson. The general auto c o r r e l a t i o n function for a three l e v e l system i s derived. For ce r t a i n l i m i t s t h i s c o r r e l a t i o n function i s the sum of two damped sinusoidal terms. It i s speculated that the phenomena of o s c i l l a t i n g chemical reactions can be explained by t h i s c o r r e l a t i o n function. The photon d i s t r i b u t i o n as a function of posi-t i o n within an act i v e medium i s derived. A loaded ca v i t y width i s defined and shown to have a lower bound consistent with the usual cavity width c(l-R) /2 nL. The loaded cavity width i s found generally to be a function of the cavity a m p l i f i c a t i o n and of the mirror r e f l e c t i v i t i e s . The d i s t r i b u t i o n of photoelectrons emitted from a detector of area A and resolving time T due to an incident l i g h t beam i s derived. By using a binomial rather than a deterministic quantum e f f i c i e n c y , an addit i o n a l term i s obtained i n the auto-correlation function. The r e s u l t i n g spectral density of the photocurrent fluctuations i s shown to be the sum. of a Poisson p a r t i c l e noise and wave i n t e r -ference term. Several examples are discussed i n c l u d i n an i n t e n s i t y modulated l i g h t beam. F i e l d of Study: Physics Elementary Quantum Mechanics F. A. Kaempffer GRADUATE STUDIES Waves R. W. Stewart Electromagnetic Theory Quantum Theory of Solids Noise i n Physical Systems G. M. Volkoff R„ E. Burgess R. Barrie S t a t i s t i c a l Mechanics R. Barrie Related Studies: Transients i n Linear Systems E. V. Bohn NON-EQUILIBRIUM PARTICLE NUMBER FLUCTUATIONS BY ERNEST GERHARD ENNS B.SC.(Hons.) U.B.C. 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study* I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission* Department of PHYSICS The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 89 Canada Date A p r i l 28, 1965 - i i -ABSTRACT The fluctuations of the constituent populations i n a m u l t i - l e v e l system are considered as applied to a three and four l e v e l semiconductor model. It i s shown that under c e r t a i n conditions the fluctuations of the electron numbers i n the conduction band can be super-poisson, defined as having a variance greater than the mean number. The general autocorrelation function for a three l e v e l system i s obtained. For c e r t a i n l i m i t s , t h i s c o r r e l a t i o n function i s the sum of two damped sinusoidal terms. This would indicate there i s an o s c i l l a t o r y i n t e r -action between the population numbers. The photon d i s t r i b u t i o n as a function of pos i t i o n within a one-dimensional active medium i s derived. When two p a r t i a l l y r e f l e c t i n g mirrors are situated at both ends of the medium, the stationary photon d i s t r i b u t i o n obtained i s a function of the cavity amplification and the mirror r e f l e c t i v i t i e s . The d i s t r i b u t i o n of photo-electrons emitted from a detector of area A and resolving time T when i l l u m -inated by an incident l i g h t beam i s derived. By using a binomial rather tha a deterministic quantum e f f i c i e n c y an a d d i t i o n a l term i s obtained i n the autocorrelation function. The r e s u l t i n g spectral density of the photo-current fluctuations i s shown to be the sum of a Poisson p a r t i c l e noise and wave interference term. Several examples of d i f f e r e n t spectral l i n e shapes are discussed. Also considered i s an i n t e n s i t y modulated l i g h t beam. - i i i -TABLE OF CONTENTS Page Chapter 1 1-1 1-2 1- 3 Chapter 2 2- 1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2- 9 Chapter 3 3- 1 3-2 3-3 3-4 INTRODUCTION Number fluctuations i n M u l t i l e v e l 1 systems Incoherent emission from an active medium 3 Photon Counting 4 NUMBER FLUCTUATIONS IS MULTILEVEL  SYSTEMS Derivation of Spectral Densities and 5 Covariances Two l e v e l system The spectral density and auto-c o r r e l a t i o n function for a three l e v e l system O s c i l l a t i n g chemical reactions The second moments of a three l e v e l system and an example Super-poisson fluctuations Bounds for the second moments Magnitude of cross-correlations Four l e v e l system INCOHERENT EMISSION FROM AN ACTIVE MEDIUM Derivation of photon d i s t r i b u t i o n Calculation of the f i r s t and second moments Equilibrium fluctuations Non-equilibrium photon d i s t r i b u t i o n function 10 11 13 15 18 20 22 22 27 31 32 34 3-5 Cavity l i n e width 40 - i v -Chapter 4 PHOTON COUNTING 4-1 Introduction 44 4-2 The f i r s t and second moments of the 46 photon and photoelectron fluctuations 4-3 Derivation of the photon d i s t r i b u t i o n 50 4-4 Spectral density of the photoelectric 55 current 4-5 Photon counting for a modulated 64 l i g h t source. Chapter 5 CONCLUSIONS 68 Appendix I Equations for second moments for a 70 four l e v e l system Appendix II Derivation of a p a r t i c l e generating 71 function Appendix III Proof of the s t a b i l i t y of any s t a t i s t i c s 74 when subjected to a binomial process Appendix IV Derivation of 4*^)a n d S x & ) 75 Appendix V Modulated poisson source 80 Bibliography 83 -v-LIST OF FIGURES figure opposite page 2-1 N+l l e v e l system including several 5 t r a n s i t i o n p r o b a b i l i t i e s 2-2 Langevin source term 8 2-3 Steady state chemical reaction rates 13 2-4 C y c l i c t r a n s i t i o n s in a pumped 16 photoconductor 2-5 Graph showing the minimum value of 21 var n/no with respect to k as a function of q. 2-6 Tables of and f \ r for 22 various l i m i t s 2-7 C y c l i c t r a n s i t i o n s i n a four l e v e l 23 photoconductor 4-1 Superimposed spectra for a modulated 67 l i g h t source with a dominant wave interference term ACKNOWLEDGMENTS I should l i k e to thank Professor R.E. Burgess, my supervisor, for his invaluable guidance and and assistance during the course of my studies. I am indebted to the National Research Council of Canada for f i n a n c i a l support in the form of three studentships» 1 CHAPTER 1  INTRODUCTION 1-1 Number fluctuations i n M u l t i - l e v e l systems; The fluctuations of a c urrent passed through a semiconductor can be attributed to a combination of thermal e f f e c t s and to population fluctuations i n the e l e c t r o n i c energy l e v e l s . The thermal noise due to the v e l o c i t y d i s t r i b u t i o n of the electrons can be calculated from Nyquists theorem. The transport noise i s due to the fluctuations of the numbers of electrons i n the various e l e c t r o n i c energy bands. These fluctuations are completely determined by the t r a n s i t i o n p r o b a b i l i t i e s between the, various energy l e v e l s . We w i l l be concerned with determining the variances and covariances of the fluctuations i n the l e v e l populations. In p a r t i c u l a r we w i l l be interested i n the non-eguilibrium steady state when the system i s driven by some external source. The general theory for the fluctuations of numbers in m u l t i - l e v e l systems has been derived by van V l i e t et a l (1956, 1965) . We have used the Langevin approach to derive the second moments and the spectral density of the number f l u c t u a t i o n s . We believe t h i s gives the required r e s u l t s more d i r e c t l y without having to expand the generalized Fokker-Planck equation as done by van V l i e t . 2 Previous to an a r t i c l e by van V l i e t (1964), i t was believed that the variances of the numbers of electrons and holes could never be super-poisson, defined as being greater than the mean number. Van V l i e t showed that for a p a r t i c u l a r three l e v e l model, super-poisson fluctuations were possible. We have extended h i s treatment of t h i s model by analyzing the fluctuations of the trapped electrons and the fluctuations of the holes i n the valence band. The c o r r e l a t i o n between the conduction electrons and the trapped electrons i s examined and a simple c r i t e r i o n for the onset of super-poisson fluctuations i n the conduction band i s obtained. The analysis was extended by adding an add i t i o n a l impurity l e v e l into the model. Because of the complexity of the expressions for the second moments, the f l u c t -uations were only evaluated i n two s p e c i a l l i m i t s . The general auto-correlation function for a three l e v e l system has been derived. It i s shown that under c e r t a i n conditions i t contains damped sinusoidal terms. This implies there i s an o s c i l l a t i n g i n t e r a c t i o n between the l e v e l s under consideration. For the semiconductor systems considered, t h i s condition can never a r i s e . It i s speculated that the phenomena of o s c i l l a t i n g chemical reactions can be explained by t h i s c o r r e l a t i o n function. It i s shown that i n a c e r t a i n l i m i t for the chemical k i n e t i c s assumed the condition for o s c i l l a t i o n can indeed a r i s e . 3 1-2 Incoherent emission from an active medium In chapter 3, a one dimensional homogeneous medium i s assumed capable of absorption, induced emission and spontaneous emission of photons. A d i f f e r e n t i a l d i f -ference equation i s derived for the j o i n t p r o b a b i l i t y of having n and m photons t r a v e l l i n g i n opposing d i r -ections respectively within the medium. By introducing p a r t i a l l y r e f l e c t i n g mirrors at both ends of the medium we form a cavity enclosing an active medium. An expression for the fluctuations of the photon numbers as a function of p o s i t i o n within the ca v i t y i s obtained. In t h i s analysis we are j u s t dealing with photon numbers and do not account for the phase and coherence of the r e f l e c t e d waves. Our r e s u l t s therefore do not include the coherent emission l i n e . A general expression f o r the*cavity width of the emission l i n e i s evaluated using the usual d e f i n i t i o n of the Quality factor "Q" of the c a v i t y . I t i s shown that t h i s reduces to the usual cavity widths assumed i n the l i t e r a t u r e f o r the case of almost p e r f e c t l y r e f l e c t i n g mirrors. Upper and lower bounds for the cav i t y width are also determined f o r various mirror r e f l e c t i v i t i e s . 4 1-3 Photon Counting The d i s t r i b u t i o n of the photoelectrons emitted from a detector with response time T i s evaluated. Handel (1958) derived the d i s t r i b u t i o n for the number of photo-electrons emitted i n a time i n t e r v a l T. In a further paper with Wolf (1961) he derives the autocorrelation function for the photoelectron fluctuations in a time i n t e r v a l T. In both of these derivations a deterministic quantum e f f i c i e n c y i s assumed. By using a stochastic binomial quantum e f f i c i e n c y we obtain an additional term i n the autocorrelation function and show i t corresponds to the pure shot noise spectrum of the photoelectric f l u c t u a t i o n s . The spectral density of the photoelectric current i s therefore the sum of a shot noise and a wave interference term. This wave interference spectrum which i s the convolution of the spectral l i n e shape with i t s e l f was f i r s t derived by Forrester(1961) Examples of the photon counting spectrum of various spectral l i n e shapes are considered. We also investigate the s p e c i a l case of an i n t e n s i t y modulated l i g h t beam and the r e s u l t i n g spectrum. The photoelectron d i s t r i b u t i o n calculated i s shown to be the boson d i s t r i b u t i o n for C c e l l s i n phase space, C i s shown to be the product of the number of coherence areas on the detector surface and the number of c o r r e l a t i o n times i n the measuring i n t e r v a l T. \ i n . * \ P»KI p3>Hv| FIGURE 2-1 N + 1 LEVEL SYSTEM INCLUDING SEVERAL TRANSITION PROBABILITIES CHAPTER 2 ; NUMBER FLUCTUATIONS IH MULTILEVEL SYSTEMS 2-1 Derivation of Spectral Densities and Covariances The fluctuations of c a r r i e r numbers in a semiconductor are important i n determining the fluctuations of the current under the influence of an e l e c t r i c f i e l d . In t h i s chapter we w i l l be concerned with the fluctuations of electrons or i " ' holes due to t r a n s i t i o n s between e l e c t r o n i c energy bands. We w i l l not deal with v e l o c i t y fluctuations or scattering within a band as t h i s gives r i s e to the thermal noise which i s calculable by Nyquist's theorem. The fluctuations of numbers i n a m u l t i l e v e l system due to inter-band t r a n s i t i o n s has been generally solved by van V l i e t and Blok(1956) for an N+l l e v e l system. We w i l l summarize h i s method and r e s u l t s and give an a l t e r n a t i v e d e r i v a t i o n . Consider an N+l l e v e l system (refer to figure 2-1) where nt i s the occupation number of the i th l e v e l and jpij&£) i s the p r o b a b i l i t y of a t r a n s i t i o n from l e v e l i to l e v e l j i n a time i n t e r v a l t,t+^t. van V l i e t and Blok wrote down the generalized time dependent Fokker-Planck equation and by assuming that a l l numbers n-t under consideration were large, l i n e a r i z e d the t r a n s i t i o n p r o b a b i l i t i e s . This i s v a l i d provided the fluctuations about the mean values are small. Then bya matrix transformation they were able to show that the 6 j o i n t p r o b a b i l i t y d i s t r i b u t i o n for the number of p a r t i c l e s i n the various l e v e l s was a multi-variate gaussian d i s t r i b u t i o n , Taking the l i m i t as time t - » o o they eliminated the exponential or transient terms to obtain the stationary d i s t r i b u t i o n function. The covariances r e s u l t i n g from t h i s c a l c u l a t i o n are then found from the matrix equation: AC + CA = -B where A = [a.-^ B and C = {<^^\ The transpose i s denoted by ay =. The various matrix elements are then: ^ ^ ( % ^ - | f f \ L.\ = 1.2, ... A/ J p u . 0 The subscriDt "o" means the expression i n the preceding bracket i s evaluated at n,- = rfj , the mean occupation number where the bar implies an ensemble average. Using the above r e s u l t s , van V l i e t and Passett(1965) derived the spectral density matrix of the c a r r i e r f l u c t -uations: S(f) « 2Re (A + jiul)"' B (A - Jul)"' where A and B are the matrices previously defined. We w i l l derive the above spectral density matrix d i r e c t l y by the Langeyin approach. Then by a simple i n t e g r a l theorem we w i l l f i n d the covariances. 7 I f we have N + 1 l e v e l s as i n figure 2-1, then by n e u t r a l i t y ^ nc = constant and we w i l l only have N indepen-dent v a r i a b l e s . The k i n e t i c equations for the stochastic variables are where p^t*) i s again the t r a n s i t i o n p r o b a b i l i t y from l e v e l i to l e v e l j i n the time i n t e r v a l t,t+At. For a stationary process p^  w i l l not be an e x p l i c i t function of time. Thus pvj w i l l be a function only of the N independent n;. Lin e a r i z i n g the above equation by expanding i t about the mean values of the n; (denoted by the subscript "o") and adding the Langevin random fl u c t u a t i n g source terms, we I f we denote the Fourier transform by IF and a n d J ^ - f c ^ - ^  then J^L<LLK%* + $.tl0 i» 1,2, ••• N In matrix notation, where A = and 1 - /*A > 4. Define the vectors then i to FIGURE 2-2 LANGEVIN SOURCE TERM 8 The spectral density matrix i s defined by: ^ - z Re tR^tf B ^-4^1)" where we have l e t B = - ^L«J ^> thus = £ ( p M - f J ( p . j - p J m \ the product of the Fourier transforms of the stochastic source terms. The relaxation time constants, obtained from the macroscopic mass action equation are much longer than any time constants contained i n the stochastic source term where _ Z S . ^ - f y) = F t f > The spectrum of the source term can therefore be considered as white f o r the purposes of our c a l c u l a t i o n . This also implies there w i l l be no measurable c o r r e l a t i o n between d i f f e r e n t elements of the source term. A graph of what F(t) might look l i k e i s plotted i n figure 2 - 2 . A p o s i t i v e or negative d e l t a function pulse indicated the increase or decrease of nt by one electron. The exaggerated f l a t plateau between events indicated whether pjt^>^ — pyCfl*) i s p o s i t i v e or negative. Thus a p o s i t i v e plateau indicates that a downward t r a n s i t i o n i s s l i g h t l y favoured and v i c e -versa. This maintains our steady state. Since the f l u c t -uations of the n K are small, the graph of F(t) i s b a s i c a l l y j u s t a series of p o s i t i v e and negative delta functions. The a.c. spectrum of F(t) i s invariant to whether the 9 pulses are a l l p o s i t i v e , a l l negative or both. The mean rate of occurence of an event i s the constant^p:^ + > therefore the spectrum of ^ j - p ^ - t i s and: Simplifying B, one obtains: i Kl since the net increase I J E . ^ r l - p t r ) of any l e v e l i s zero for a stationary system. The choice of notation i s now evident, as the matrices A and B defined i n t h i s derivation are the same as those quoted i n the previous r e s u l t s . The spectral density matrix i s thus obtained d i r e c t l y without a knowledge of the covariances. The second moments can now r e a d i l y be obtained i f we use the i d e n t i t y : m - IT/% & -^f- m^ < 0 where M = V^ij l i s a n y r e a ^ square matrix independent of UL) . I f we define: Or (0 « S#) 4.4 S&) - £ Re.«tJujfj'B flf-j-ol? But ^ = C a s previously defined. 10 Therefore: •» ^ . AC - $ « o £ 6 C ) ^ - - "8 /2 o c.8 + s-fcto* - - B A The negative sign on the r i g h t hand side was used because: since p^ ;. i s a decreasing function of n; and pij i s an increasing function of n i . Adding the two equations we obtain the desired result: f\t + Cti - - 3 ; 2 - 2 Two l e v e l system For a two l e v e l system we obtain Burgess'(1956) well known g-r theorem. In t h i s case: The spectrum i s given by: The auto-correlation function defined by: and the variance: 11 2-3 The spectral density and auto-correlation function  for a three l e v e l system. The complexity of the spectra and covariances increases notable with the addition of every new l e v e l . We w i l l therefore l i m i t our discussion to two and three l e v e l systems and a few p a r t i c u l a r cases of a four l e v e l system. Consider the general three l e v e l system where P^ Ai.Ka.) and n (+ nx± n 3=constant. For t h i s system our spectral density matrix w i l l be: Z T where 4 s "31 and and 12 The c o r r e l a t i o n function matrix where ^ £r) — fi;* ty*^ - "«[ Then f i n d : ^ ^ where -,- a. _4 = to + and vs a« a** -Z f» - + We have written i n the above fashion to i l l u s t r a t e the fac t that g may be r e a l . Thus f o r g to be r e a l , we must have: (Oil - < -Generally t h i s inequality i s not v a l i d and q i s a pure imaginery number making the sum of two exp-onential terms. I f q i s r e a l however, t h i s would indicate we have an o s c i l l a t i n g i n t e r a c t i o n between the three occupation numbers. FIGURE 2-3 STEADY STATE CHEMICAL REACTION RATES 13 2-4 O s c i l l a t i n g chemical reactions Bak (1963), i n h i s review of recent contributions to the theory of chemical k i n e t i c s discusses the inadequacy of the present theories to deal with o s c i l l a t i n g chemical reactions. The k i n e t i c equations are the same as those under discussion. In t h i s case the d r i v i n g force would l i k e l y be an external heat source as the reactions would n a t u r a l l y be endothermic. An example of a system where q could be r e a l would be the following. (Refer to figure 2-3) Let n N , n ^ and n 3 be the number of molecules of the three constituents of i n t e r e s t . Again n,+n8-»-n^ » constant and we have only two independent v a r i a b l e s . The reaction changing n, to n a and nz to n 3 etc. w i l l be assumed to be favoured over any reverse reaction. This would be the case when a by-product formed i n the reaction n, to n e was used i n the i n t e r a c t i o n n A to n A . We w i l l ignore by-products and catalysts which are un-doubtedly necessary to make the reaction work, but do not enter e x p l i c i t l y into the k i n e t i c s . We w i l l assume the we w i l l have de t a i l e d balance. S i m i l a r l y for the Other two t r a n s i t i o n s . We w i l l examine the case when we have c y c l i c t r a n s i t i o n s (Ui-Vi) > Q and r e a l q r e s u l t i n g -in an o s c i l l a t i n g reaction. net rate changing n, to n ^ as 14 In the steady state: where a l l n, # n z and n 3 used hereafter are mean values. Then the matrix elements are: and r / \* \ / For the second term i n t h i s expression to be dominant and negative, we must have n i * < ^ N - n t ) > which implies we are far from equilibrium and d e t a i l e d balance does not hold. Then f o r q to be r e a l we can have T\x «* f\5 and « n» then q? « *X iH 3 The o s c i l l a t i o n frequency can be determined from the power spectrum: Maximizing S(f) with respect to 10 , we f i n d that: where the c h a r a c t e r i s t i c frequency i00 i s : 15 For the case under discussion, the frequency of o s c i l l a t i o n for a l l n-t i s the same since fy^ » ^ p a _ ^ ) and (&)(f-f) » Cf+f) • Thus a>l ^  - £f?-<f) In t h i s p a r t i c u l a r case we f i n d : L00 i s therefore a function of the molecular concentrations and of the t r a n s i t i o n rates. We have thus shown that q can be r e a l , r e s u l t i n g in an o s c i l l a t i n g reaction about some c h a r a c t e r i s t i c f r e q -uency u20 . We w i l l not discuss other l i m i t s for which q may be r e a l as the purpose of t h i s discussion was merely to show that indeed q can be r e a l . 2-5 The second moments of a three l e v e l system and an, example The covariances for a three l e v e l system are: More e x p l i c i t l y , they become: A oi n 0 0 0 0 0 0 0 0 0 0 0 n + l FIGURE 2-4 CYCLIC TRANSITIONS IN A PUMPED PHOTOCONDUCTOR 16 We w i l l now consider a three l e v e l system where the variance of the fluctuations i n one l e v e l can be much greater than the mean value. We w i l l c a l l these super-poisson f l u c t u a t i o n s . Consider the model of a three l e v e l photoconductor i n the steady state by f a r from equilibrium. The t r a n s i t i o n p r o b a b i l i t i e s p , S ) p a , and p a a . w i l l be considered n e g l i g i b l e compared to a l l other t r a n s i t i o n s . The t r a n s i t i o n p r o b a b i l i t i e s for t h i s semiconductor model are depicted i n figure 2-4. The t r a n s i t i o n i s o p t i c a l l y induced and i s a function of the source i n t e n s i t y and absorption e f f i c i e n c y of the staple only, n, i and p are the c a r r i e r numbers i n the conduction band, the trap-ping l e v e l and the valence band respectively. I i s the number of traps i n a unit volume. S and X are pro-portional to the capture cross-sections for the n and p c a r r i e r s , and w i l l be constant i n the steady state. This p a r t i c u l a r model has been discussed by van Vliet(1964) and Cole(1965). Van V l i e t introduces an approximation into h i s analysis which invalidates several l i m i t s he discusses. (Van V l i e t , 1964: figure 4 i s i n c o r r e c t . The lower bound for the fluctuations never approaches zero) We w i l l correct t h i s error and also discuss the c o r r e l a t i o n and fluctuations i n the trapping l e v e l . Cole, i n h i s c r i t i q u e of van V l i e t ' s work, derived the correct expression for the fluctuations of n from a biv a r i a t e difference equation. This difference equation I i s however j u s t a p a r t i c u l a r case of the general Fokker-17 Planck equation. Cole has unknowingly performed the same l i n e a r i z a t i o n van V l i e t generalized for N l e v e l s . Thus Cole's claim that h i s method i s superior to van V l i e t ' s i s i n v a l i d . The k i n e t i c equations are: ^ r = o t - S n f i c - i ) dt The l i n e a r i z e d matrix elements are: where o*. » S n 0 (j-L^) =• ^Ufn^C) Then f i n d that the fluctuations of the conduction band c a r r i e r s are given by: v ^ n / n e = 1 + Using the r a t i o s = k and 1 ^ , - <J-we have i n van V l i e t s notation: V<trn/n = 1 + -kcfj^z) +fCf+\)  We have written the variance i n the above form to f a c i l i t a t e recognizing conditions for super-poisson and 18 sub-poisson f l u c t u a t i o n s . S i m i l a r l y : M«*i)[Wf%3fr+3) 4. k ^ 4 a ^ 4 ? ) Written i n t h i s form, i t i s immediately obvious that the fluctuations of i and p are sub-poissonian. Also the co r r e l a t i o n between n and p i s always p o s i t i v e . 2-6 Super-poisson fluctuations Super poisson fluctuations occur i n the conduction band when: ( This i s also j u s t the condition for the covariance cov(n,i) to be p o s i t i v e . Thus when VVLr(Yf) > n„ then Oov(n,L) >D and when V<tr(w) < n 0 then t o V ^ , i ) <0 When n i s super-poissonian, an increase i n n tends to increase i and when n i s sub-poissonian, an increase 19 i n n w i l l cause i to decrease. From the steady state rate equations: we obtain r \ N b key =-the r a t i o of the capture cross-sections of electrons and holes by the trapping l e v e l respectively. For super-poisson fluctuations therefore: y w i l l thus always be less than one-half when v a r n y ^ l . When the traps are almost f u l l ( ^ - 1 ^ ) « iB and when the density of c a r r i e r s i n the conduction band i s much less than that i n the trap l e v e l i0 , then the electrons in the conduction band w i l l e x h i b i t giant fluctuations, which we define as WLt- n » h© In t h i s case V«/ir ry^ <* _L since « | This can be i n d e f i n i t e l y great subject to the l i m i t -ations of our l i n e a r i z e d theory. Thus we can write: Vcur n - L0 When the conduction c a r r i e r s e x h i b i t giant f l u c t -uations, namely when .^-« I and 1<«.) , then: V^r IA k5 20 Depending on the r e l a t i v e magnitude of q and k we obtain the following two l i m i t s : i ) when k « q : ** p/P. - i - 0£) Thus the trap c a r r i e r fluctuations tend to zero as the traps become f i l l e d and the hole fluctuations i n the valence band tend to become poisson. i i ) when k>>q l/u - * - %<K) In t h i s l i m i t the variance of both i and p approaches one-half the mean value as the number of electrons i n the conduction band decreases. 2-7 Bounds for the second moments Another case of i n t e r e s t i s when CL» | and k » i va-r - i - i/f - y k For t h i s l i m i t the "bottleneck" e f f e c t of the traps i s minimized and a l l fluctuations become poisson. GRAPH SHOWING THE MINIMUM VALUE OP VARn /n„ WITH RESPECT TO k AS A FUNCTION OF q 21 van V l i e t asserted that the conduction c a r r i e r fluctuations tended to zero in. the high input region, namely when (^ » \ , On examination of t h i s l i m i t , we found the lower bound V a r ( n ^ o > 0*7% This may be seen as follows: Minimizing with respect to the independent variable k requires that: , , __ ^ Thus - 1 - — where There i s only one p o s i t i v e root for t h i s quadratic and i t i s bounded by l < k, < «*> • For any value of q t h i s equation then gives us the value of k that min-imizes the fluctuations of n. (Refer to figure 2-5) The minimum value of 0.75 i s obtained when the input l i g h t i n t e n s i t y i s great, , and the traps are h a l f f i l l e d , k=l. I t i s e a s i l y v e r i f i e d that the fluctuations of i and p have the following bounds: 0 < ^ i - < i and 2. fo k « 1 k = 1 k » 1. q « 1 1 ! 0 . 7 ] ^ 0.71qJ5(l-2qk) A q = 1 0.71^ -0.06 -0.26 q » 1 (k/q ) (1-kq) -0 .41q_JS -q ^ FIGURE 2-6a TABLE OF tVfc FOR VARIOUS LIMITS k « 1 k = 1 k » 1 q « 1 1 2.12qJ5 q = 1 1 0.83 0.72 q » 1 1 1 1 FIGURE 2-6b TABLE OF r-np FOR VARIOUS LIMITS 22 2-8 Magnitude of cross-correlations I t i s of in t e r e s t to examine the c o r r e l a t i o n coef-f i c i e n t s t-^ and t n o where we define: ' P By the Schwarz inequality | t» 4 \ t-^ i s a measure of the degree of c o r r e l a t i o n between the two stochastic variables x and y. Refer to figure 2-6 for a tabulation of tv,, and for various l i m i t s of q and k. From the table of we note that there i s complete p o s i t i v e c o r r e l a t i o n between n and i only when vfcjr ry^ 0 » For a l l other cases the c o r r e l a t i o n between n and i i s r e l a t i v e l y small. From the table of we observe that there i s complete p o s i t i v e c o r r e l a t i o n between the conduction electrons and the holes i n the valence band when eithe r the traps are v i r t u a l l y empty or when the density of conduction c a r r i e r s i s much greater than the density of trap c a r r i e r s . 2-9 Four l e v e l system We w i l l now consider some special cases of a four-l e v e l photoconductor. (Refer to figure 2-7) Again we w i l l discuss a driven system, one i n which a steady state has been reached but where we are so far from thermal e q u i l -/8n (M-m) V ® ® ® ® 0 0 O /N 06 8 0 8 0 0 0 0 I - L 0 O O 0 0 FIGURE 2-7 CYCLIC TRANSITIONS IN A FOUR LEVEL PHOTOCONDUCTOR 23 ibrium that detailed balance does not hold., We w i l l assume the traps are s u f f i c i e n t l y d i f f u s e to eliminate the p o s s i b i l i t y of p A i and p a a t r a n s i t i o n s . This w i l l also eliminate the p o s s i b i l i t y of "hopping" from one trap to another within an impurity l e v e l . A l l t r a n s i t i o n rates not l a b e l l e d i n the diagram are assumed to be much smaller than those l a b e l l e d . The p«n t r a n s i t i o n i s pumped and the conduction electrons now have two possible paths to return to the valence band. For the above k i n e t i c s to be s t r i c t l y v a l i d we should also state that the d r i f t v e l o c i t i e s f o r the electrons and holes are assumed to s a t i s f y the following i n e q u a l i t i e s respectively. Vy, » J a " * i . dz and d3 are the mean distances between traps within l e v e l s 2 and 3 respectively. This i s to ensure that the electron or hole e f f e c t i v e l y "sees" a l l the available traps before making a t r a n s i t i o n . As a consequence of s t a t i o n a r i t y : where the subscript "o" indicates the most probable value which i s very near the inean value for large numbers. 24 The l i n e a r i z e d matrix elements i n t h i s ease ares Solving f o r the matrix elements of C using? AC + CA = -B we obtain s i x equations i n s i x unknowns since C i s symmetric. I f we lets the r a t i o of the net fl u x through the two possible return paths. The s i x equations thus obtained can be found i n Appendix I. These equations are too complicated to solve exactly, thus we w i l l only consider two special cases. i) k » l > &n<L s i o f > r 1 p In t h i s case the t r a n s i t i o n through l e v e l 3 i s dominant. The traps In l e v e l 3 are almost f i l l e d and 25 the density of trap 3 c a r r i e r s i s greater than the density of conduction electrons. I f the t r a n s i t i o n through l e v e l 2 were missing, t h i s would correspond to the giant f l u c -tuation case of the three l e v e l system., On solving the s i x equations we f i n d : C» - v«ur in » n0fk/$ C,t = c-o^C^m) = n 0$-/s I f q and s are of the same magnitude, then the fluctuations of n are super-poisson, and the fluctuations of i are sub-poisson. I t i s i n t e r e s t i n g to note that when the traps i n l e v e l 2 are mostly empty corresponding to t-< I , then m i s also super-poissonian. This Would suggest that l i k e i n the three l e v e l case, the f i l l i n g up of the traps i n the major t r a n s i t i o n route causes the super-poisson fluctuations i n the conduction band» The fluctuations i n l e v e l 2, the minor t r a n s i t i o n route have l i t t l e e f f e c t on the conduction electrons but themselves follow the fluctuations i n the conduction baado. The c o r r e l a t i o n between the n and m, and n and i electrons ejjpressed i n terms of the c o r r e l a t i o n coef-f i c i e n t s i s : 26 Again i f Y-< I and p « | , the c o r r e l a t i o n between l e v e l s 1 and 2 could be considerable as expected i f the fluctuations i n l e v e l 2 are to follow those i n l e v e l 1. tvw , however, i s small i n t h i s case while i t was large i n the 3 l e v e l model. i i ) k « l > ± ^ > % ? In t h i s case the t r a n s i t i o n through l e v e l 2 i s dominant due to the r e l a t i v e l y small capture cross-sections between l e v e l s 1 and 3 and l e v e l s 3 and 4. For a l l sub-cases considered, namelys A) ^ p « 1 B) ^ p > 1 a) C^.\- » k.ps b) <pm Vps i t was always found that V&or 2 6 1 Thus i n t h i s case the fluctuations i n the conduction band are v i r t u a l l y independent of the fluctuations i n the minor t r a n s i t i o n trapping l e v e l . 27 CHAPTER 3 INCOHERENT EMISSION FROM AN ACTIVE MEDIUM 3-1 Derivation of photon d i s t r i b u t i o n Consider a substance, homogeneous i n i t s bulk properties, i n inte r a c t i o n with a radiatio n f i e l d . Depending on whether our system i s a gas or a semiconductor, l e t N a denote the number pf gas atoms or electrons per unit volume respectively i n the upper energy l e v e l E ^  . Si m i l a r l y l e t N, denote the constituent density i n the lower energy l e v e l E, .We w i l l only consider t r a n s i t i o n s between l e v e l s E, and Ez , assuming these are the only two l e v e l s i n the system capable of supporting an inverted population. P h y s i c a l l y both energy l e v e l s E x and E& w i l l have a f i n i t e width AE, and A E a which w i l l be a contributing factor to the width of the emission spectrum. We w i l l examine the c h a r a c t e r i s t i c s of both the in t e r n a l and emitted radiat i o n f i e l d s . Consider a one dimensional system of length L, cross-sectional area A and thickness d«.L so that we may ignore angular e f f e c t s . We can then write a Fokker-Planck equation f o r T*^ >Tn^  , the p r o b a b i l i t y of having n and m photons of energy £ } ^E^- E^)^ t r a v e l l i n g i n a po s i t i v e and negative x-direction respectively i n a volume A^x) at a po s i t i o n x where 0 — x - L„ The trans-i t i o n p r o b a b i l i t i e s , namely absorption, spontaneous emission and induced emission w i l l be not only functions 28 of (Ax)hut also of the electron or gas density i n the appropriate l e v e l s . We w i l l only consider a stationary system i n the sense that at any p o s i t i o n x, the average photon density i s not a function df time. Thus we w i l l not consider transient "switch on" or warm up" e f f e c t s . We w i l l also assume that the steady state values of the densities N, and N^ are s u f f i c i e n t l y large to _be v i r t u a l l y independent of the photon density at any p o s i t i o n x. In t h i s analysis we are j u s t dealing with photon numbers and do not account for the phase and coherence of the r e f l e c t e d waves. Our r e s u l t s therefore do not include the coherent emission l i n e . We are now able to derive a d i f f e r e n t i a l difference equation for "R (W>^.. ~ R , * V > however, i s a function of both " T t - 4 * < and K r^rh'M.-t-.fc-*' To f a c i l i t a t e w riting Tn>m>T<: w e w i l l transform to a time scale using c The "R,*!,^ so defined i s therefore completely determined be "R.w^-At • Thus we are dealing with a f i r s t order Markov process. For the sake of mathematical symmetry and the p o s s i b i l i t y of other applications, we w i l l include i n the following l i s t of t r a n s i t i o n p r o b a b i l i t i e s , several not r e a l i z a b l e i n the two systems aforementioned. Subscripts of + or _ w i l l r e f e r to the x d i r e c t i o n of the p a r t i c l e under consideration. Multiple subscripts w i l l r e f e r to the d i r e c t i o n of the incoming p a r t i c l e and 29 the emitted p a r t i c l e or p a r t i c l e s respectively. Consider the following t r a n s i t i o n p r o b a b i l i t i e s for an incident p a r t i c l e on an increment of width C^-3^)-Absorption: *L+ ^ oL- C^c) This w i l l Include bulk absorption and surface losses due to angular s c a t t e r i n g . Spontaneous Emission: ^ fit**) Induced Emission: , f&+. <W) ^ Transmission: I - VW*£> > I-Reverse Scattering: . . _ . v Keeping only f i r s t order terms i n , we can write: 4 6 - « ^ C 4 ^ - ^ y . ^ Y l - ^ C & ^ | - £ _ ^ - ^ T ^ . f c In the l i m i t as 0, we have i n the space co-ordinate frame: 30 I f we introduce the b i v a r i a t e generating function: G(r,s?x) = ^ "ftv* Then the above d i f f e r e n t i a l difference equation can be written as a p a r t i a l d i f f e r e n t i a l equation i n G.(r,s;x). Multiplying the above equation by £>" l-™ and summing over a l l m and n , we fi n d that: I f we substitute: y = s - 1 and z = r - 1 and l e t P (y,z;x) = In G(y,z;x) 31 where do. = £|6 + - +^++. + and V v + . + +• ^ 4 - + - + •+- • = 3-2 Calculation of the f i r s t and second moments: Generally the e x p l i c i t evaluation of t h i s p a r t i a l d i f f e r e n t i a l equation i s not possible <> The various moments of n and m however may be found using: 'IPX =TFC /HL\ = TIC ' 4 The average value of any function f(ra,m?x) i s defined f(n,m;x) — ~*Tn*«-)C ^ ^wi ->c A l l moments of n and m can therefore be found from the p a r t i a l d i f f e r e n t i a l equation by d i f f e r e n t i a t i n g the appropriate number of times with the variables i n questiono For s i m p l i c i t y , consider the general p a r t i a l d i f -f e r e n t i a l equation of the above form,. where (\0o - "2>0o - Coo 0 32 D i f f e r e n t i a t i n g with respect to y or z, the l e t t i n g y = z = 0, we obtain the following two d i f f e r e n t i a l equ-ations for ~W*. and ~Vv* respectively, the average values of n and m at p o s i t i o n x. S i m i l a r l y we obtain the d i f f e r e n t i a l equations for the second moments i n terms of: v - v&r w - TT* and k = 60V ft*^*^ Therefore: For a p a r t i c u l a r system, these coupled equations could r e a d i l y be solved using Laplace transforms. 3-3 Equilibrium Fluctuations; For the equilibrium case each increment of volume A(dx) w i l l emit as many p a r t i c l e s as i t absorbs. By deta i l e d balance the above statement holds for either 33 d i r e c t i o n . The above equations w i l l thus be independent of x i n equilibrium. Then i n Jacobian notations and where and Letting |^ = 0 :0 0y a: o Then the matrix equation f o r the covariances i s : +. & M = -/fi In equilibrium, the t r a n s i t i o n p r o b a b i l i t i e s w i l l be the same for p a r t i c l e s propagating i n the pos i t i v e or negative x d i r e c t i o n . In that case s thus 2 H „(. = _ jQ The moments of intere s t are thens 34 where ^ m The fluctuations of the t o t a l number of p a r t i c l e s i n a volume V = A(dx) i s s var(n + m) = var n 4- var m + 2cov(n,m) ./ • • • For our system i n equilibrium t h i s reduces tos where (ax-CL^) -=- o i + 4^ ++- + i s p o s i t i v e i n equilibrium. I f we write var n = n (1 + k^ and var (n + m) = 2n (1 + k 2) then k, = W, 4 £ £ ^ < 1 k z w i l l equal k, only when there i s no c o r r e l a t i o n be-tween the n and m p a r t i c l e s as i s the case for photons. 3-4 Non-eauilibrium Photon d i s t r i b u t i o n function For photons i n a gas or semiconductor, the t r a n s i t i o n p r o b a b i l i t i e s J£>A— and must be zero as the inducing photon does not undergo any t r a n s i t i o n . Also for a highly collimated beam, the p r o b a b i l i t y of reverse scattering into a small s o l i d angle i s n e g l i g i b l e . The d i r e c t i o n of the induced emission w i l l also be the same as the d i r e c t i o n of the incident r a d i a t i o n . {Fowler, 35 1929? Sobel'man and Tyutin, 1963) Thus and + _ roust also be zero. In t h i s case the cov(n,ra) = 0 and the p a r t i a l d i f f -e r e n t i a l equation i s separable. In the o r i g i n a l G(r,s;x) notation we haves where we have dropped the multiple subscripts. Let G (r, s ;x) = G , (s;x)G a(r;x) For a generating function V^.& i f : then , ^ ^ fr • ' where ^>^4s m a Y found from ^ j^-T and H(s) i s the input photon d i s t r i b u t i o n , namely For the derivation of these relationships r e f e r to Appendix I I . Solving f o r G (s;x): then Qt^) - - 6u + 36 Thus Also : S ^ ^ V ^ ' = B ^ f ^ \ ) { — ^ — _ I f we have an input photon d i s t r i b u t i o n with e f f e c t i v e temperature "TT0 for frequency V1, then the generating function for the boson d i s t r i b u t i o n i n N, c e l l s i s where Thus -M The required solution i s therefore: 1 «. ^ ^ i - ^ ) 6 - « p - ^ - A ^ * j J X 3 7 N i s the number of c e i l s i n boson phase space of the input photons. Since i t i s proportional to the volume i t w i l l not change throughout our system. Therefore N = and Where now b,(x) i s the e f f e c t i v e boson factor at p o s i t i o n x of the n photons. S i m i l a r l y for the m photons t W = kxu exp For a highly attenuating medium, namely when oL± » the e f f e c t i v e boson factors quickly lose a l l information about the input and f i n a l l y reduce tos where we define k,m and Law, as the e f f e c t i v e boson factors of the medium. Also the gain of the medium to p o s i t i o n x may be written ass Then we have with Burgess (1961)s k , ( * ) = U o ^ . ^ t ) 4- k.» O-^'to) k*6*) - k,u ^ 6^) 4- W« 6 -38 where (Jed*) < 1 > 0 l4 ^ >/S and jj;6*} > 1 «»«L Wto <0 °L< Thus for an amplifying medium the induced emission must exceed the absorption. This corresponds to the medium being represented by a negative temperature„ In a gas or semiconductor l a s e r for example, and ^ » d L — w i l l have to be equal. This i s because the medium i s homogeneous and the doppler broadened width of the emission spectrum w i l l be the same as viewed from ei t h e r end of the device„ This i s of course assuming the doppler broadening i s greater than the Lorentz broadening. I f the energy difference between the two l e v e l s under consideration i s K , then the incoming photon must be of frequency p c \ ^ f t or W„ \ \-£ depending on whether the photon i s t r a v e l l i n g i n the same o r opposite d i r e c t i o n respectively of the i n t e r a c t i n g electrons o r atoms. ^=vny^. where V i s the v e l o c i t y of the electron or atom and n' i s the r e f r a c t i v e index of .the medium. I f the v e l o c i t y d i s t r i b u t i o n of the p a r t i c l e s i s Maxwel-l i a n , then "7* = kT/m where m i s the mass of the 1 electron o r gas atom. I f j£>«\. , then the doppler width w i l l have equal contributions from p a r t i c l e s . t r a v e l l i n g in the same or opposite d i r e c t i o n to.the i n t e r a c t i n g photons. Thus (AWXS , = -2WOJV\]LL 39 I f we have a superimposed d r i f t v e l o c i t y V B , for example due to an e l e c t r i c f i e l d applied along a semi-conductor specimen, then the resultant v e l o c i t y i s iL- 1T± Va> as seen by an n or m photon. Then W ^"T3 +• « YX_ ^ Vt* Again we have equal contributions to from electrons t r a v e l l i n g i n the same or opposite d i r e c t i o n to the i n -ducing photons. In t h i s case: and the l i n e i s broadened. The generating function derived f o r the photons i n the medium i s s t r i c t l y correct only for a single frequency. I t i s however a good approximation for a narrow spectral l i n e where the t r a n s i t i o n p r o b a b i l i t i e s are independent of the frequency. To show t h i s , l e t "P^ n") be the p r o b a b i l i t y of having n photons i n a narrow frequency i n t e r v a l . I f we consider the discrete case where we have k d i f f e r e n t frequencies i n ^ V J ) ^ then: k. subject to the constraint that < f - n<•~ A Then the generating function for " P ^ i s : For bosons: (y-U^) ~ (l4" k i -Wrs") For (rfe) t o have the form CrL^)- d + k - k s ^ 40 a l l roust be the same t>,- k k> and E = Therefore for the above analysis to be s t r i c t l y v a l i d , we must be able to approximate the spectral l i n e shape by a rectangular shape. 3-5 Cavity Line Width Let us now introduce p a r t i a l l y r e f l e c t i n g boundaries into our system at x= O and ^= U , thus forming a cavi t y . Let p, and p 2 be the p r o b a b i l i t i e s that a photon i s r e f l e c t e d at x = 0 and x = L respectively. When photons are re f l e c t e d or transmitted at a boundary they undergo a binomial process. Any s t a t i s t i c a l d i s t r i b u t i o n of p a r t i c l e s remains invariant i n form o r stable when subjected to a binomial process. (Refer to Appendix III) Thus bosons subjected to a binomial process are s t i l l bosons. I f p i s the p r o b a b i l i t y of r e f l e c t i o n and the incident photons have a boson factor b, then the r e f l e c t e d and transmitted photons w i l l have boson factors pb and (l-p)b respectively. Let then When the system has reached a steady state, then? .LM . * pi W 4 1 Solving for V»Xte and b\U , we finds \ _ p .p. . &xp-te,+G^)U For s i m p l i c i t y , we w i l l only consider the symmetric case, when ©"=©,= and ^ =• = Then / v / \ Uu = p,exp (-gift ^ I ~ p.ja* ^xp 6-2© l i ) For an attenuating medium, &>0 while for an amplifying medium Q<0 . ^xo a s i d > however, must always be p o s i t i v e . Thus 1 - p,p k *xp^ae.i}> >0 or > -^-A^>,p^ In the steady state t h i s w i l l be the lower bound f o r & la or the upper bound for the amp l i f i c a t i o n . I f was less than the above minimum, for example i n the i n i t i a l warm up period i n a lase r , then the photon density would b u i l d up u n t i l the losses became s u f f i c i e n t l y large such that > -i-Zn^.pl) . 4 2 The boson factors for the output photons from the cavity w i l l be? At x = 0 L 0 = O-p^bao x = L W - 6 -The q u a l i t y faetor or "Q" of the^eavity may be found from the usual d e f i n i t i o n . Gi — ^° = (energy stored i n the cav i t y V energy, l o s t / c y e l e For a narrow speetral l i n e eaeh photon w i l l have energy approximately equal to Ku0 where ViQ i s the central frequency. Then? ^ The energy stored i n the cavity = The energy l o s t / c y c l e = ^ W_ I LD+IDI where n' i s the r e f r a c t i v e index of the medium. The cavity width (ku)^  i s thens ' C W + ^  where Thus finds 43 When p, = p., = p , we have equal outputs from both ends of the cavity and £&w)c s i m p l i f i e s to? P where A The function i s a monotonically decreasing function of GU , where S^)=L S(o)=i %(&~o The upper and lower bounds for (&M}c, are therefores {Ave!) i s therefore generally dependent on the active medium, i t s lower bound being the usual cavity width 44 CHAPTER 4 PHOTON COUNTING 4-1. Introduction In t h i s chapter we w i l l be concerned with c a l c u l a t i n g the fluctuations of the photoelestron numbers emitted from a detector due to an incident l i g h t source. We w i l l also calculate the spectral density of the photoelectric current which would be the quantity of int e r e s t i n any experimental measurements„ We w i l l i l l u s t r a t e our results by assuming various spectral l i n e shapes for the incident l i g h t bearru We w i l l use the theory of an a l y t i c signals to describe the e l e c t r i c f i e l d on the detector surface. (Deutsch, 1962) Thus consider a quasi-monochromatic l i g h t wave incident on a photodetector of area A„ Let E^ denote the e l e c t r i c vector on the detector surface at time t= Then where u ± and v-t are H l l b e r t transform p a i r s , namely Thus i f u.0*) i s the Fourier transform of u t » then the Fourier transform of v^ is§ P denoting the Cauchy p r i n c i p a l value at t = t . 45 Vt^ = uO>Q uD<0 oO=-0 The Fourier transform of E t i s therefore? - Z uJjD) uS)>Q o u3<0 Thus using a n a l y t i c signals £*o has p o s i t i v e frequency components only» We w i l l assume the e l e c t r i c vector components are gaussian distributed<> For black body radiation t h i s can be found by .considering an ensemble of harmonic o s c i l l a t o r s . This has been generally j u s t i f i e d for "random waves" by various authors by appealing to the Central Limit theorem. We w i l l only consider plane polarized l i g h t i n t h i s discussion. I t i s shown i n Appendix IV that f o r un-polarized l i g h t we have equal and additive contributions to the spectral density from the two resolved components of the e l e c t r i c f i e l d vector. A p a r t i a l l y plane polarized source w i l l also give additive contributions to the spectral density as we can consider the source as a superposition of an'unpolarized and a polarized boam, Thevarea of coherence, defined as the area of the detector over which the radiati o n f i e l d i s c l o s e l y correlated i s given by where A is the wave-46 length of the incident radiation and —Q~ i s the ..solid angle subtended bj^ the source at the detector. (Forrester, 1961) I f C, i s the number off coherence areas,on our detector surface of area A, then (L>\ = (\SL^A We w i l l be concerned with the photon..and photoelectron s t a t i s t i c s i n the following an a l y s i s . Thus i f there are n photons incident on the detector i n a time T, then ft = "^-Tw. where mi are the number of photons incident i n the i-** 1 coherence area. Since the m in d i f f e r e n t coherence areas are uncorrelated, we have the following averages s < » = M<V,;> We need therefore analyze only one coherence area, a l l areas being additive i n the cumulants of which the mean and variance are the most important for present work, Thus consider a beam of plane polarized l i g h t incident on one coherence area of the detector. Define the i n t e n s i t y of the l i g h t on the detector at time t as I-t = E^E^ = u + v*. We w i l l assume there i s no r e f l e c t i o n at the detector. 4-2 The f i r s t and seeond moments of the photon, and photoelectron fluctuations •If J^^liJ) i s the p r o b a b i l i t y that m photons are Incident on omie coherence area of the photodetector i n a 4 7 time i n t e r v a l t - T, t; where T i s the detector resolving time, then s , , \ -r * i p, Li, d-t) = =L I U t where JL - RhU and ,- coherence area R= i^'/t^ impedance of space h. Planck ' s constant p frequency of the input l i g h t On the assumption of a Markov process: dh thus s -fc-T We w i l l assume the detector response has a uniform memory or sampling time T„ The mean value of n i n a single system of an ensemble i n the time i n t e r v a l T iss +• -t-T IHfer w i l l i n general be a flu c t u a t i n g number, the fluctuations being determined by the r e l a t i v e magnitude of T and , the c o r r e l a t i o n time for the input spectral l i n e defined bys (Mandel, 1959) 48 where % & » < I t I t + Y / > - < £ ? i s the autocorrelation function of the i n t e n s i t y We w i l l denote by a <^ ^ > an ensemble average, thus the averaged quantity? < V T > - <Jw> - < * ^ ' ^ > = -<• < i > T S i m i l a r l y the second moment for a single photon counting system iss •fc -t -b Taking the ensemble average, we finds -t-T -t-T o We thus have super-poisson fluctuations, the second term i n the variance due to the wave nature of the incident l i g h t o I t i s shown i n Appendix V that t h i s super-poisson character of the variance can be simulated by modulating a beam of poisson p a r t i c l e s . The above expression i s therefore the variance of the number of photons incident on the detector area A / ^ 49 i n a time i n t e r v a l T, given the autocorrelation function of the incident l i g h t . This expression i s s i m i l a r to that derived by Mandel(1958) for the fluctuations of photo-electrons. His expression i s not s t r i c t l y v a l i d as he assumes a deterministic rather than a stochastic quantum e f f i c i e n c y . Using the r e s u l t s of Appendix I I I , we can re a d i l y determine the fluctuations of the emitted photoelectrons. The generating function f o r p*U;r) for a single system iss I f the detector has a quantum e f f i c i e n c y ^ and assuming- that for every incident photon we have eithe r zero o£ one photoelectron emitted, then we can writes G v * ) = frp 6-\± where (S-QL*) i s the generating function, for /£>"r) the p r o b a b i l i t y that m photoelectrons are emitted i n a time i n t e r v a l t - T, t . Thus? From the well known properties of the generating function, we obtain f o r the ensemble averagess 4-3 Derivation of the photon, d i s t r i b u t i o n The d i s t r i b u t i o n of photons i n an observing i n t e r v a l Let (%) be the generating function f o r p-rdW) ^ e n £ ^ _ t I f we l e t M « £ I * ' ^ ' where P(y) i s the p r o b a b i l i t y d i s t r i b u t i o n for the stochastic variable y. We w i l l derive P(y) for the two limits? and T » % ' i) ~rv< % In a time 1± i s a slowly varying function and we can write y. = IT. I f we f i r s t only consider one coherence area, then* where Q a The average i n t e n s i t y on one coherence area i s I c — 51 I f we l e t z, = u 2 and z<i= v 2 , , then ^ p i -and X Therefore " P t f ) = <sxj=> [-X/ l 0 ] the i n t e n s i t y d i s t r i b u t i o n on one coherence area. For C ( coherence areas s c> where T£lt) * ^ x p X ' 1 - / 1 ^ I Therefore since a l l areas are independent: subject to the constraint - ^ l X ; - X I f we take the Laplace transform of P ( I ) : then S . i ^ V £ P f t i)«<p^ « . I i V l t - O+u-I i s therefore the Laplace transform of the gamma d i s t r i b u t i o n . Therefore when 52 Thus: where <V\-r^ is the mean number of photons incident on one coherence area i n a time T„ i i ) T » In t h i s case there w i l l only be c o r r e l a t i o n between photons a r r i v i n g within a time of each other, where k i s a constant to be determined,, Therefore: where ~ i f u= krt then «o t = (\ 4- i^IcT) In t h i s case: £ T (%) = ^<£* p 1) 2] T 5 ^ ) Ji Thus We therefore obtain the boson d i s t r i b u t i o n of having n photons d i s t r i b u t e d among C = C \ C s c e l l s , namely 53 where \ i = °LT0 k ^ l # the mean number of photons i n one c e l l . The fluctuations of the n T photons can be written Consequently V^-f- m T *= <^ vv\T> ^  I + r^ fc>) The super-poisson fluctuations of the photons thms reappear s l i g h t l y reduced i n the fluctuations of the photoelectrons. From the second moments previously derived, we deduces 2 L © J I f our observing time "V«- %. , then tyfrh « <X> throughout the domain of integration. Therefore! and d. - or Ca = 1 Thus as expected for very short measuring times a l l photons a r r i v i n g i n a single coherence area occupy the same c e l l i n phase space. We w i l l evaluate C e x p l i c i t l y for various (x) i) I f the input spectral l i n e shape i s gaussian, then 54 The number of c e l l s C i s given by; where As expected, when *Ya , then C = Ct . As our observing length increases beyond our coherence length, then only photons within a distance L t = c T t of each other are correlated. F i n a l l y when T»T^. , we obtain £ » CiT^f^ . Therefore C*.- ~^^rVt. or k = 1. C i s therefore the product of the number of coherence times i n our measuring i n t e r v a l T and the number of coherence areas on our detector surface, i i ) S i m i l a r l y i f our l i n e shape i s Lorentzian, then 1fe :M=<i> iexpI- a|r|/r t.] and 6- C ( ^ - ^ ( i - ^ W r . S ) ] " 1 Again we f i n d that when "T«Te. , then C = and when T>>T4. , then £ * . In t h i s case also C i s the product of the number of coherence times i n our measuring i n t e r v a l T and the number of coherence areas on our detector surface. 55 4-4 Spectral density of the photoelectric current Physical measurements on an incident l i g h t beam can be made i n d i r e c t l y by examining the spectral density of the fluctuations i n the resultant photoelectric current i n the detector c i r c u i t . I f Ht^ -r i s the number of photons incident on one coherence area of the detector i n the time i n t e r v a l t - T, t; then l e t WU,T be the corresponding number of photoelec trons emitted within the same time i n t e r v a l . The photocurrent averaged over the i n t e r v a l of duration T terminating at t i s therefore; * - r where e i s the e l e c t r o n i c charge. " W e U t <Ju.T n«r,T> - =• and <ww l T M w , T > - <»» T ^ - Q tv) < T t T ^ > - - t & - $ ^ 4 to We must now r e l a t e the photoelectron autocorrelation function to the photon autocorrelation function. This requires the consideration of the two cases, namely when | T | T and when I'H ^ "T i) When ^ "T" , we are averaging the product of two separate samples of photons or photoelectrons. Let "P(vw*n») ke the p r o b a b i l i t y of having n t photons i n the sample t, t + T and n A photons i n the sample t-t-Tk -t+T+T. Let •Q,6r*»>*»») D e the corresponding p r o b a b i l i t y for the photoelectrons. Then since we are dealing with two non-Then 56 overlapping samples, we w i l l haves Is the binomial d i s t r i b u t i o n . Then -^2L (^) = (lr ^ +I*)" where ]^  i s the quantum e f f i c i e n c y of the detector, The b i v a r i a t e generating functions for T ^ * , and £ £ 6 * i , are defined as followss Then f i n d thats Then from the properties of the generating function; Therefore i f |T| >T In t h i s case: . where p^», 6k>T) i s the Poisson d i s t r i b u t i o n o r i g i n a l l y defined. Therefore: T J 57 where the two T ' s i n h^T^T^^ represent the two upper l i m i t s of integration respectively and i s s e l f evident. i i ) When |V| ^ T , we are averaging the product of two overlapping samples. Let fl, •<= and n ^ - where n, and n A are as previously defined. S A i s the number of photons contained i n the overlapping volume of the two samples. S i m i l a r l y for the photoelectrons, l e t irv>| = ^ a and r n ^ * % A+• ^ 3 Then agains . ^  . ^ ^ -<m.>^ <W»r^ and From the generating function on the previous page: Therefore VdJr W r = r p V ^ n T + < f l r > ^ 6 " 0 This i s the expression f o r the fluctuations of the photoelectrons as a function of the photon f l u c t u a t i o n s . Evaluating the covariances i n 4 (y) , one finds : 58 your and I t i s r e a d i l y v e r i f i e d that: Therefore: Since ^ V Sj") = f L ^ we f i n d : ^ £ f t « ^ [ l o ^ T , ^ + <5g ,>J 4-<^ >^ 6-*{) Therefore the general photocurrent autocorrelation function i s : ;4 M * T Mandel and Wolf (1961) obtained the term involving hfrVT\RP) a s v a l i d f or a l l *V due to using a deterministic rather than stochastic quantum e f f i c i e n c y . The spectral density of the photoelectron fluctuations due to an incident l i g h t beam i s therefore: 59 The f i r s t term of %^ (y) v a l i d for a l l 'V can be s i m p l i f i e d to: _^  —r where Y6r) i s t l i e normalized c o r r e l a t i o n function fior the e l e c t r i c f i e l d defined i n Appendix IV. Thus: T - T By Parseval's theorem: and F(f) i s the normalized spectral l i n e shape derived i n Appendix IV. S i m i l a r l y ; The spectral density becomes: 60 Or r Frequencies of in t e r e s t w i l l be much less than the inverse of the detector response time. Thus we can write for (tot)« 1, <=>o o The shot noise term may be associated with the p a r t i c l e f l u c t u a t i o n nature of the l i g h t while the convolution term i s associated with the wave interference nature of the incident l i g h t . The second or photoelectric mixing term of ^ was derived by Forrester (1961) using somewhat implausible arguments. As stated the above derivation i s v a l i d only for l i g h t incident on one coherence area. Due to the i n -dependence of the fluctuations i n d i f f e r e n t areas we can write: where < : C j i / > * s *~he average photocurrent contribution from the I ^  coherence area and <^/> i s the t o t a l mean photocurrent. Thus the spectral density for the t o t a l photocurrent i n the detector c i r c u i t i s : 61 To determine the r e l a t i v e magnitude of the two terms, consider the case of black body radiation i n a frequency i n t e r v a l , f o r example a s p e c t r a l , l i n e from a gas discharge. The wave interference term divided by shot noise i s therefore: where Therefore For a source temperature of ID, 000° K and wavelength and a quantum e f f i c i e n c y of unity we f i n d This i s also v a l i d f or unpolarized l i g h t because of the additive nature of the spectral density. For thermal sources i n the o p t i c a l range therefore the wave interference term w i l l be undetectable. Since the advent of lasers, we have access to l i g h t sources with very high e f f e c t i v e temperatures. This has made possible spectral density measurements where the wave interference contribution i s dominant compared to the shot noise. 62 We w i l l define a bandwidth of our input spectral l i n e as: d O J where F(f) i s the normalized input spectrum. I t i s r e a d i l y v e r i f i e d , see Appendix IV, that ( ^ / r ) 1 ^ . — 1 I f our input l i n e shape i s : i) Gaussian: then F#) - ^ g - V ^ " ^ * ] where <&4) = 2$rc cr and f 0 i s the central frequency of the l i n e . We assume that I f B i s the bandwidth at h a l f i n t e n s i t y , then Thus: i i ) Rectangular: Then = fetf w M . | = 0 For -V ^  there w i l l be only the pure shot noise component present. 63 i i i ) Lorentzian where B i s the bandwidth at half intensity. In this case I T V + In a gas laser, the mirror spacing is usually sufficiently large to allow the excitation of several longitudinal modes within the doppler broadened line width. If for example we had N longitudinal modes each with a gaussian lineshape, then where fii determines the relative intensity of each line. Then /&i = -1 since F(f) i s the normalized line shape. Sri «= central frequency of the mode (^Jrt) - fl i B t h e l i n e w i d t h o f t h e 1 mode From the definition of we thus obtains For two lines of equal intensity and widths 64 When the l i n e s are w i l l separated, £A 4) i s j u s t the sum of the two in d i v i d u a l l i n e widths. The spectral density i s then: r a" When i = j , the wave interference term i s j u s t the sura of N i n d i v i d u a l self-mixing spectra. The i ^  j terms correspond to the photoelectric mixing between d i f f e r e n t modes and are centered at the corresponding difference frequencies. 4-5 Photon counting for a modulated l i g h t source I f we modulate s i n u s o i d a l l y the input l i g h t i n t e n s i t y , then the i n t e n s i t y at the detector i s : where I has a l l the properties previously defined. a(l+m) and m are less than unity and ^ = Sir*, i s the modulating frequency. For a single spectral l i n e t h i s modulation has the e f f e c t of producing two sidebands each with i n t e n s i t y (^/z) X e where I e i s the i n t e n s i t y of the central l i n e . For the resolution of these three l i n e s , the modulation frequency the linewidth of the incident beam. The average photocurrent i s therefore: 65 The wave interference autocorrelation function of the photocurrent a f t e r uniform averaging over T i s : fa1? 1 - T The spectral density i s : Again using Parseval's theorem: where [|( ft , -» ^ Cos ^6"-*?) Sd*W) «, ' — ate and I U & . * ) » * ^ mT^** U t f t ^ * ) as a function of the input l i g h t spectrum i s t h e r e f o r e i , 66 Using the properties of F ( f ) , we f i n d : + ^ F M F ^ + - A r o Thus: +• ^ F £ v ) F 6 r * V uzr SL Again for frequencies of i n t e r e s t , UDT« 1 and: I f the input lineshape i s gaussian, then again F - & where £4) = * W <T The s p e c t r a l density of the photocurrent i s : FIGURE 4-1 SUPERIMPOSED SOURCE WITH SPECTRA FOR A MODULATED LIGHT A DOMINANT WAVE INTERFERENCE TERM 67 The t h i r d term i n the spectrum i s the sum of the self-mixing terms of the o p t i c a l l i n e and i t s two side-bands. The l a s t two terms correspond to the mixing of the input spectral l i n e with i t s two o p t i c a l sidebands. Since the three spectral l i n e s are derived from the same input spectrum, a l l components of frequency difference 4-, are completely coherent. This r e s u l t s i n the delta function at the modulation frequency i n the detector c i r c u i t . (see figure 4-1) When the wave interference term i s .negligible as compared to the shot noise, for example from a.black body source, then the photocurrent spectrum iss Thus a modulated black body source can be used for the transmission of information. 6 8 CHAPTER 5  CONCLUSIONS An al t e r n a t i v e derivation of the number fluctuations in a m u l t i l e v e l system has been obtained using the Langevin approach, van V l i e t s analysis of a three l e v e l semiconductor model has been extended and a lower.bound of 0„75 was obtained for the r a t i o of the variance to the mean number of the conduction electrons. A special case of a four l e v e l system was discussed„ I t was shown that i n c e r t a i n l i m i t s super-poissoa fluctuations of the conduction electron numbers could occur. A general expression f o r the autocorrelation function for a three l e v e l system was derived„ I t was demonstrated that i n certa i n l i m i t s t h i s was the sum of two si n u s o i d a l l y damped terms, A general c r i t e r i o n for the appearance of o s c i l l a t o r y c o r r e l a t i o n i s a p r o f i t a b l e topic for future i n v e s t i g a t i o n . Another important area for farther work i s the examination of non-stationary systems . This would reguire a modification of the present theory i n terms of the time-dependent Pokker-Plaanek formulation. The fluctuations of the photon numbers as a function of p o s i t i o n within a cavity enclosing an active homo-geneous medium were derived. The width of the emission l i n e was obtained and for small mirror r e f l e c t i v i t i e s 6 9 shown to be a function of the c a v i t y a m p l i f i c a t i o n . For r e f l e c t i v i t i e s almost equal to unity, the width i s v i r t u a l l y independent of the amplification .and ..reduces to the unloaded cavity width*  c-^-^)/a.irn'\i • Further analysis would require the consideration of wave interference e f f e c t s and some s p e c i f i c coupling between the photons and gas atoms. The spectral density of the fluctuations i n a photo-e l e c t r i c current was derived by considering a-plane p o l -arized l i g h t beam incident on a detector of area A and resolving time T, By using a stochastic quantum e f f i c i e n c y , a term additional to that derived by Mandel (1958) i s obtained i n the autocorrelation function. The spectral density derived from t h i s c o r r e l a t i o n function i s then the sum of a p a r t i c l e noise and wave interference term. The d i s t r i b u t i o n of photons incident on a detector of area A i n a time i n t e r v a l T has the boson d i s t r i -bution for C c e l l s i n phase space. C i s found to be the product of the number of coherence areas on the detector surface and the r a t i o of the observing i n t e r v a l T to the r e c i p r o c a l of the l i n e width of the incident r a d i a t i o n . Further work would consist o f deriving the spectral density of the photocurrent fluctuations for any a r b i t r a r y p o l a r i z a t i o n of the incident l i g h t . 70 APPENDIX I The s i x equations f o r evaluating the covariances for the four l e v e l system ares ^ k - i ) pdp4-^) £,» - \<b+ ft Lf+f+t) 4-p ^ a a - f=> - "™• + ^  >) 71 APPENDIX II We w i l l extent the analysis of B a r t l e t t (1951) for a Markov Chain to include spontaneous emission. F i r s t of a l l , however, we w i l l b r i e f l y review B a r t l e t t ' s r e s u l t s . At t=0, we have a one p a r t i c l e input into our system. Each p a r t i c l e can generate a d i s t r i b u t i o n of p a r t i c l e s , whose generating function i s G(s)». I f events occur at dis c r e t e times apart, then i s the generating function of the system a f t e r the ft ^ event, namely at a time ft Thus S %L*)= S i m i l a r l y % As A"fc-*0 we can write Expanding i n powers of „ and We therefore obtain B a r t l e t t ' s results? 0^ A 72 To f i n d S f c s # we then simply evaluate The former analysis has not included the p o s s i b i l i t y of spontaneous emission at any time t . Let F(s) be the generating function of the d i s t r i b u t i o n of p a r t i c l e s generated spontaneously i n a time • Let us now denote the t o t a l generating function of the system a f t e r the n1^- event as V^s. Then again for a one p a r t i c l e inputs Y 0 ^ » s Expanding i n , we obtain i n the l i m i t as the d i f f e r e n t i a l equation. where we have useds . Ffe> - 1 - Mfcfc^) To solve e x p l i c i t l y f o r V+js w e must proceed as follows: ^ f r v S , ^ F ^ t h u s V™ i i ) = ft TT F or taking the logarithms 73 Again using F^s) ~~ \ ~ and also* log(a+b) = log a + b/a i f a» b we f i n d upon expansion i n \ ^ S t H § ^ * In the l i m i t as ^t-*0 we find •fc 1 6 Obviously for no spontaneous emission/ f(s) = 0, and — ^ t s . as previously derived. I f instead of a one p a r t i c l e input, we have an input d i s t r i b u t i o n generating function H(s), then: thus . v .y , v Taking logs and expanding as before, we find that: Y * t o = H ^ s * ^ * ^ p S * C * * r V * where as before s a t i s f i e s the d i f f e r e n t i a l equation 7 4 APPENDIX III Proof of the s t a b i l i t y of any s t a t i s t i c s when subjected to a binomial process. If the d i s t r i b u t i o n of the input p a r t i c l e s i s P(m) and the d i s t r i b u t i o n of the output p a r t i c l e s i s Q(n) afte r having been subjected to a binomial process, then where i s the binomial d i s t r i b u t i o n : " 8 " 6 ^ • ft) f t - 1 - ^ ' * where p i s the p r o b a b i l i t y for an m p a r t i c l e to become an n p a r t i c l e . I f : and &6L6^)- ^ then or Thus the generating functions remain invariant i n form. The moments are therefore: function, then I f £rp(%) i s a boson d i s t r i b u t i o n generating t h U S ^ - O ^ t p - L p . ^ and the e f f e c t i v e boson factor b has become pb, 75 APPENDIX IV Derivation of ^ fa) and ($) : The instantaneous i n t e n s i t y - £.,.»£t where E+, = ^ + 4 v * and and are H i l b e r t transform p a i r s : Also _S> ^ -r> ^ = ~£ i>7fr + ^ 'Pi* where and _[ are unit vectors i n the x and y d i r e c t i o n s r e s p e c t i v e l y . We w i l l assume the l i g h t i s incident i n the z d i r e c t i o n . Also and V£ are gaussian random variables with zero mean. Therefore i f E, and E ^ a r e the x and y components of the e l e c t r i c vector on the detector, then: E** =• ^  +4^-fc Define an e l e c t r i c f i e l d c o r r e l a t i o n c o e f f i c i e n t : W r " ) - <Et* E v V > /(<L>4.>)* = ^ttit, +4 wity^<t+r -J or* where ^ I i ^ i s the i n t e n s i t y due to l i g h t polarized i n the I d i r e c t i o n . From the conjugate properties of u and v , we have: (Mandel and Wolf, 1961) 76 Therefore: Also; and V t t M « Ya.Vf) < I t I + + ^ > « < ^ ^ fai* + + ^ W ) ^ > I f the zi (1=1,2,3,4) form a multivariate gaussian d i s t r i b u t i o n , then we have the following i d e n t i t y . Thus <Cn^U^y ~ < k * ><^> +• 2 <U\p tl*f4ti> The other products may be s i m i l a r l y reduced. Again using the conjugate properties of U-t and V+T , we The autocorrelation function: i s then ^ . ( r ) = ^ ^ [ < ^ W r > * 4- <*l* W t $ ] The spectral density defined by: 77 O O - Z ^ X 1 ^ ^  ^  M Cr) J dr i s Then using Parseval's theorem: " " ^ - J o -Jfc> L e t * <W-U^K-t4.r> then using the properties of an a l y t i c signals (Deutsch, 1962) » 0 ^ 0 S i m i l a r l y = ^ Therefore: -o * For unpolarized l i g h t V»i . and: 78 The i n t e n s i t y on the detector i s I t = ^a-t where <!,> * <E^> = <!*> YRK<£) » ^ R K « ) t f Thus F|| t*r) and are the input spectral l i n e shapes for the two p o l a r i z a t i o n components. For unpolarized l i g h t FH/^ ) — F"L» d£) and we have two equal contributions to the spectral density. For a plane polarized beam i n the x-direction we have (dropping the s u f f i c e s ) : where = <C&t E£T^ /<$* Y ^ =- 0 = 1 Therefore as expected since the in t e n s i t y i s exponentially d i s t r i b u t e d when considering a single coherence area. The wave interference spectrum iss • where F(f) i s the normalized input spectral l i n e shape. Proof that - 1 We defined f£ s ^XL^T / %Q) and P ^° . , ' 79 Using the r e l a t i o n s h i p between the autocorrelation function and the spectral density, namely: then x. « ^ «**00 r ^ and < ^ f ) T i = i 8 0 APPENDIX V Modulated poisson input: Given a source which emits p a r t i c l e s at random, the p r o b a b i l i t y of an emission i n a time i n t e r v a l i^-t-zrfcbeing . Then the s t a t i s t i c s of the number of p a r t i c l e s emitted i n a time i n t e r v a l T i s poisson. Then inse r t a chopper into the path of the p a r t i c l e s such that i t al t e r n a t e l y transmits the p a r t i c l e s for a time "Tj and absorbs them for a time . A detector a f t e r the chopper counts the p a r t i c l e s . By examining the f i r s t and second moments of the number of p a r t i c l e s incident on the detector i n a time T we can show that the r e s u l t i n g fluctuations are always super-poisson. We w i l l examine two cases: 1 . TJ and Ta. are constant 2. T, and are stochastic 1. The p r o b a b i l i t y that n p a r t i c l e s are emitted i n a time T i s : The generating function for "FrO*^ i s : it Then 14 X a T + l k # then where K) - a n d t, and t A are the remaining p a r t i a l i n t e r v a l s that were included i n the time T. denotes the greatest integer less than or equal to x. 81 I f Q T ^ ) i s the p r o b a b i l i t y of detecting m p a r t i c l e s i n a time T, then the generating function f o r Q.-rl^) i s : ° R UT W - *X |» CW ^  ^T'+ The moments of m for t h i s p a r t i c u l a r measuring i n t e r v a l are: r - i TFT = w LMT; + -fcj t w i l l , however, depend on where we s t a r t our measuring i n t e r v a l . I f we take an ensemble average over a l l possible measuring i n t e r v a l s , then: I f we l e t ^ = T- WT0 and T, < j then f i n d : <ij> = ^T'/To and V<wt,] = ^ " ^ ' l T o - S Since V<Lv»"fc\* va>"t* the case for "T^ Tj i s e a s i l y evaluated. 2. T^ and T a are stochastic: I f A.idr i s the p r o b a b i l i t y that the pulse T w i l l terminate between t and -fc+fct i f i t i s on at t; then the p r o b a b i l i t y of having a pulse of duration T i s : 82 S i m i l a r l y "PCTC) - • A***P' i-XT») Then using the previous approach: where ^ _ T t -^TIi = T ¥ and U T^- "T&T.i - ^ p ^ s ^ l ^ ] Therefore: - c r - u "Z> T-rf\ — W " L " Again taking an ensemble average and noting that TL and Tj are" independent i f \ < W i > - g M < T , > where K|. T / £ T , > + < r . > ) We obtain: V a > IVI - < W > + p * N i T , or s In both of these esamples, the modulation increases the r a t i o of the variance to the mean. This modulation has the e f f e c t of sending packets or bunches of p a r t i c l e s to the detector. Thus the super-poisson nature of bosons i s a ttributed to bunching. At present i t i s not known whether there i s any possible operation that performed on the poisson p a r t i c l e s would produce sub-poisson fluctuations i n the output p a r t i c l e s . 83 BIBLIOGRAPHY Bak, T. 1963. Contributions to the theory of Chemical  Kin e t i c s , p. 31, W.A.Benjamin Inc. N.Y. Ba r t l e t t , M.S. 1951. Proc. Camb. P h i l . Soc. 47. 821. Burgess, R. E. 1956. Proc. Phys. Soc. B, §9, 1020. Burgess, R. E. 1961. J . Phys. Chem. So l . 22, 371. Cole, E.A.B. 1965. Proc. Phys. Soc. 85_, 135. Deutsch, R. 1962. Nonlinear Transformations of Random  Processes, chapt. 1, Prentice-Hall, N.J. Forrester, A.T. 1961. J . Opt. Soc. Am. 51# 253. Fowler, R.H. 1929. S t a t i s t i c a l Mechanics, p. 483, Cambridge University Press Mandel, L. 1958, Proc. Phys. Soc. 22L, 1037. Mandel, L. 1959, Proc. Phys. Soc. 2A> 233. Mandel, L., Wolf, E. 1961. Phys. Rev. 124, 1696. Sobel'man, 1.1., Tyutin, I.V. 1963, Soviet Physics Uspeckhi, 6_, 267. van V l i e t , K. M., Blok, J . 1956. Physica 23., 231. van V l i e t , K. M. 1964. Phys. Rev. 123., 1182 van V l i e t , K. M., Fassett, J . R. 1965. Fluctuation Phenomena i n Solids p. 268, Burgess, R. E. Ed., Academic Press, N.Y. 

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