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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Non-equilibrium particle number fluctuations Enns, Ernest Gerhard 1965
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Title | Non-equilibrium particle number fluctuations |
Creator |
Enns, Ernest Gerhard |
Publisher | University of British Columbia |
Date Issued | 1965 |
Description | The fluctuations of the constituent populations in a multi-level system are considered as applied to a three and four level semiconductor model. It is shown that under certain conditions the fluctuations of the electron numbers in the conduction band can be super-poisson, defined as having a variance greater than the mean number. The general autocorrelation function for a three level system is obtained. For certain limits, this correlation function is the sum of two damped sinusoidal terms. This would indicate there is an oscillatory interaction between the population numbers. The photon distribution as a function of position within a one-dimensional active medium is derived. When two partially reflecting mirrors are situated at both ends of the medium, the stationary photon distribution obtained is a function of the cavity amplification and the mirror reflectivities. The distribution of photo-electrons emitted from a detector of area A and resolving time T when illuminated by an incident light beam is derived. By using a binomial rather tha a deterministic quantum efficiency an additional term is obtained in the autocorrelation function. The resulting spectral density of the photo-current fluctuations is shown to be the sum of a Poisson particle noise and wave interference term. Several examples of different spectral line shapes are discussed. Also considered is an intensity modulated light beam. |
Subject |
Photons Photelectricity Semiconductors |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-10-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0085529 |
URI | http://hdl.handle.net/2429/38380 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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