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Intensity fluctuations and photoelectric mixing of light beams Burwell, Willis Bryan 1964

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INTENSITY FLUCTUATIONS AND PHOTOELECTRIC MIXING OP LIGHT BEAMS BY WILLIS BRYAN BURWELL B.Sc.(Eng), Queen's University, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF : MASTER OF SCIENCE In the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1964 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of • British Columbia, I agree that the Library shall make i t freely available for reference and study* I further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publi-cation of this thesis for financial gain shall not be allowed without my written permission-Department of Physics The University of British Columbia, Vancouver 8, Canada Date September 1964 i i ABSTRACT Photoelectric mixing i n a photodiode i s used In t h i s work as a s t a t i s t i c a l spectroscopic t o o l . A number of experiments were performed to determine the f l u c t u a t i o n spectrum generated by t h i s process and the s t a t i s t i c a l properties of the l i g h t which might be deduced from the data. Due to p r a c t i c a l l i m i t a t i o n s i n attainable temperatures, blackbody sources were not able to produce an observable mixing above shot noise. Experiments were also c a r r i e d out using l i n e spectra from gas discharge lamps. The best source available was a 300 W Xenon lamp, emitting l i n e s i n the red, which under optimum conditions produced excess current fluctuations equal to 60$ of shot noise. The observation of photoelectric mixing losing a gas laser source has already been reported i n the l i t e r a t u r e , but the Gaussian d i s t r i b u t e d e l e c t r i c f i e l d model usually applied does not f i t the experimental r e s u l t s . A new model was proposed i n t h i s thesis which considers the laser l i g h t as a narrow band of coherent l i g h t embedded i n a r e l a t i v e l y broad band of spontaneous l i g h t . Mixing between the signal and the spontaneous emission was considered to be the only observable ef f e c t due to experimental l i m i t a t i o n s . This model appeared to f i t the data and gave some information about the s t a t i s t i c a l properties of the laser beam. v i AQKNOWLEDGMENTS I should l i k e to thank Professor RE Burgess, ray supervisor, f o r his guidance i n the preparation of the material f o r t h i s t h e s i s . As an o f f i c e r i n the Canadian Array (RC Sigs) I should also l i k e to thank the Department of National Defence fo r the posting to UBC which made t h i s research possible. The research was financed i n part by the National Research Council of Canada i n the form of a Bursary, and by a United States A i r Force Grant. i i i TABLE OF CONTENTS Page Chapter 1 INTRODUCTION 1-1 Photoelectric Mixing 1 1-2 Coherence of Light Sources 7 1- 3 Review of Previous Experiments 10 Chapter 2 APPARATUS AND EXPERIMENTAL TECHNIQUE 2 - 1 Experimental Apparatus 13 2-2 The Standard Noise Source 15 2 - 3 Semiconductor Photodiodes 2 - 3 - 1 Theory of Operation 18 2 - 3 - 2 Noise Considerations 20 2 - 3 - 3 Comparison of Photodiodes 21 Chapter 3 EXPERIMENTAL RESULTS (NON-COHERENT LIGHT) 3 - 1 Blackbody Light Source 24 3-2 Mercury Vapour Lamp 26 3 - 3 Xenon Lamp 28 Chapter 4 EXPERIMENTAL RESULTS (COHERENT LIGHT) 4 - 1 Operation of the Gas Laser 32 4 - 2 Experimental Results 34 4 - 3 Interpretation of the Results 36 Chapter 5 CONCLUSION 42 Appendix A Photoelectric Mixing Equation f o r a Spectral Line 4 6 Appendix B Properties of Photomixing Spectrum for a Spectral Line 50 i v Appendix G Photoelectric Mixing Equation f o r a Laser Source 51 Appendix D Non-Central Stimulated Signal 53 Appendix E Output Power of a Laser 51* Bibliography 56 V LIST OF glG-URES Opposite Fi g Page 2-1 Block Diagram of Experimental Apparatus 13 2-2 Transistor Balanced Modulator 14 2-3 C i r c u i t Diagram of Noise Source 16 2-4 Ac Equivalent C i r c u i t of the Noise Source 16 2-5 Photodiode V-I Characteristic 19 2-6 Equivalent Noise C i r c u i t of a Photodiode 20 2-7 1/f Noise i n a TP 50 Photodiode 21 2-8 Spectral Response of Various Photodiode Types 21 2 - 9 Table of Photodiode Charact e r i s t i c s 23 3 - 1 Photodiode Noise Spectra (Blackbody Source) 24 3 - 2 Photomixing Spectrum (Xenon Source) 29 4 - 1 He-Ne Energy Level Diagram (Simplified) 32 4 - 2 Photomixing Spectrum (Laser Source) 34 4 - 3 Current Characteristic of Photomixing Spectrum (Laser Source) 35 4-4 Laser Photomixing Spectrum (log M(f) vs f 2 ) 37 4-5 Simplified Spectral Diagram of the Laser Radiation 39 D-l and D-2 Mixing of Gaussian Line with Non-central Signal. 53 1 CHAPTER 1 INTRODUCTION 1-1 Photoelectric Mixing Photoelectric mixing (or o p t i c a l heterodyning) involves the int e r a c t i o n of two l i g h t beams of di f f e r e n t frequencies i n any non-linear device sensitive to them. I f the frequency spectrum of the l i g h t beam i s continuous the self-mixing signals produced appear as a continuous spectrum with the low frequency portion extending from zero frequency. The same effect has been used i n radio comm-unications to change the frequency of si g n a l s . No d i f f i c u l t i e s are encountered since very monochromatic signa l sources are available up to microwave frequencies. With the possible exception of the la s e r , no such l i g h t sources exist so mixing at o p t i c a l frequencies was hardly considered to be of p r a c t i c a l value. In recent years intere s t has arisen about the possible use of o p t i c a l mixing as a spectroscopic t o o l . A photodiode i s one device that can be used to produce mixing. A semiconductor photodiode, biased i n the reverse d i r e c t i o n , can be fabricated with a response that i s extremely tlinear with l i g h t Intensity. Since the l i g h t Intensity Is proportional to the square of the e l e c t r i c f i e l d strength E of the l i g h t waves, t h i s type of photodiode w i l l act as a square law device. 2 The dc l i g h t current ( l L ) generated i n the phbtodiode i s given by I L=faEE* Although the v a l i d i t y of t h i s square law assumption was doubted, an experiment performed by Forrester (1955) proved that i t i s correct since mixing was observed. The derivation of equations f o r photoelectric mixing has been ca r r i e d out (Alkemade, 1959 & Forrester, 1961) f o r a single spectral l i n e from a l i g h t source producing a Gaussian d i s t r i b u t i o n of the e l e c t r i c f i e l d strengths. The re s u l t s are e s s e n t i a l l y the same as i n a previous paper (Rice, 1944) concerned with the eff e c t of a square law device on a noise spectrum Input. A b r i e f discussion and review of the derivation i s given i n Appendices A & B. The results show that f o r a given l i g h t spectrum of width /iV(cps) the r a t i o M(f) of the excess current fluctuations to shot noise (2eI;L) at very low frequencies (f «-AV) i s given by M(0)= s(7) 2/A-Q-)(l L/e^V) where 7) = the mean wavelength of the l i g h t JL - the s o l i d angle subtended by the source .at the detector A = the illuminated area of the detector s = a numerical factor near unity depending on the shape of the spectrum The same sort of re s u l t was deduced by Mandel (1958) using Bose-Einstein s t a t i s t i c s of the l i g h t photons. I f 3 the a r r i v a l of a photon could be exactly associated with the ejection of an electron the photoelectric current would also have the same s t a t i s t i c s . I f n i s the mean number of photons a r r i v i n g In a time T m the variance of the number of electrons observed i n the same time i n t e r v a l w i l l be var n » n(l^-b) where b i s the mean number of photons per unit c e l l i n phase space. However, as Mandel points out, due to the uncertainty p r i n c i p l e and a quantum e f f i c i e n c y (electrons out per photon in) less than unity, the photons and photoelectrons cannot be d i r e c t l y associated with each other. He derives two results depending on the length of the measuring time. For the case of interest Ay Tm>> 1 the variance of the number of electrons observed becomes n ( l f sb) where again s i s a shape factor. 1 The-first term i s the same as given by Poisson s t a t i s t i c s and i s just the usual shot noise. The second term i s some excess f l u c t u a t i o n that can be associated with the -mixing derived using the wave in t e r a c t i o n model. In Mandel's model the excess current fluctuations are d i r e c t l y related to fluctuations i n the i n t e n s i t y (or number of photons) i n the l i g h t beam which are greater than those given by "Poisson s t a t i s t i c s . The equivalence of these two models can e a s i l y be seen i f a p a r t i c u l a r case i s worked out. Take for example blackbody r a d i a t i o n i n which the average number 4 of photons of energy h y i n a unit frequency Interval and leaving a unit source area per second into a unit s o l i d angle per independent p o l a r i z a t i o n component i s given by Planck's r a d i a t i o n law n ( v ) =• (I/;?) (exp (h iVkT) - l j _ 1 ~ b / ^ where T i s the radiatio n temperature. I f a small portion of t h i s spectrum with a width Ay i s allowed to f a l l on the detector so that n ( V ) i s nearly constant over the range then the dc current generated w i l l be I L ^ A J 2 A > M ^ ) where A as the source area =s the s o l i d angle subtended by the detector Ii the mean frequency of the l i g h t Here It i s assumed, that the quantum e f f i c i e n c y i s unity and there are no losses i n the opt i c s . Note that the value of ASl defined here i s equal to the one defined i n the mixing equation since AJi. i s determined by the product of the source area and the detector area. Therefore, the source and detector can be interchanged without changing the value of AJ2.. Putting t h i s value of the dc current into the mixing equation gives a mixing r a t i o M(0) = s/ £exp(h V/kT) - l ] = sb This i s exactly the same answer as given by Mandel's treatment so the two methods are equivalent. This formula i s the same as the one used by Alkemade (1959) to calculate f o r sunlight (T = 6000 °K) that M(0)=. .3 f o r a wavelength of 1 8 , 0 0 0 A 0 which i s beyond the response of the s i l i c o n and germanium diodes used i n the present experiments. For t h i s must be small enough that the mixing i s uniform over the i n t e r v a l yet large enough that background noise i n the detector i s n e g l i g i b l e . At 7000 A° t h i s r a t i o becomes only .03 and other losses are expected to reduce t h i s s t i l l more. Obviously there i s l i t t l e hope of observing photoelectric mixing with a blackbody source using photodiodes whose threshold l i e s i n the v i s i b l e spectrum. Using a spectral l i n e the mixing r a t i o should increase by at least an order of magnitude. The width of a gas discharge spectral l i n e i s expected to be of the order of 1 0 1 0 cps with the lamps used since no precautions were taken to reduce broadening. A mercury green l i n e at 5460 A° with a Gaussian l i n e shape and a lamp giving ll/kSl-\ A/cm 2-steradian the r a t i o w i l l be . 5 . Again losses have been neglected but t h i s should be observable i f a powerful enough lamp can be found. This implies an equivalent r a d i a t i o n temperature of 2 3 , 0 0 0 °K which Is very high even f o r a gas discharge lamp. The factor IL/ A J 2 . i s a true figure of merit f o r a lamp since i t i s invariant under simple o p t i c a l transformations i f losses are neglected. I f the r a d i a t i o n contains two or more spectral l i n e s the mixing r a t i o f o r the same current w i l l be l e s s . This i s because inter-mixing i s not observed i n low frequency measurements i f the l i n e s are separated by a s u f f i c i e n t l y large frequency i n t e r v a l . ^In f a c t , f o r N i d e n t i c a l spectral l i n e s (same i n t e n s i t y and width), the observed value of M(0) w i l l be a factor l/N less than the the value i f only one spectral l i n e was producing the same dc current. The only advantage of using a number of spectral l i n e s i s i f the extra current i s necessary to make the other background noise n e g l i g i b l e or i f the l i n e s cannot be separated without large o p t i c a l losses. Forrester (1961) discusses at some length the effect of the shape of the l i g h t spectrum on the low frequency mixing. There i s some ambiguity as the d e f i n i t i o n of the width of the spectral l i n e but he points out that i n terms of the observed quantities (l) the dc l i g h t current ( i i ) the upper cut-off frequency of the mixing spectrum the value of the shape factor s can be determined Line Shape ' s Rectangular .5 Gaussian fTnZ/Tp =» • 48 Lorentzian X/rf « .32 It w i l l be very d i f f i c u l t to d i s t i n g u i s h ^ between the f i r s t two cases. Forrester also computes graphs showing the shape of the mixing spectrum f o r these l i n e shapes. i n th i s thesis a Gaussian l i n e shape i s assumed since i t i s probably the closest to the actual shape of a spectral l i n e because of Doppler broadening. 7 1-2 Coherence of Light Sources "Ordinary" l i g h t can be considered as being made up of a large number of waves. Each wave i s produced i n a d i f f e r e n t part of the source at a d i f f e r e n t time and i s therefore unrelated to any other. The resultant l i g h t has a continuous spectrum r e s t r i c t e d to a frequency band determined by the c h a r a c t e r i s t i c s of the source. I f the e l e c t r i c f i e l d strength of the l i g h t can be represented by (Eqn 1, Appendix A) A/ where the ^ are phase angles uniformly d i s t r i b u t e d over the range 0 to 27T the source i s said to be Gaussian random. This i s because as N-*oo the "central l i m i t theorom" (Sec 2 . 1 0 , Rice, 19^4) can be applied. This theorom"states that the sum of a large number of independent random vectors approaches a normal d i s t r i b u t i o n as the number tends to i n f i n i t y . On the other hand i n laser' l i g h t , produced by stimulated emission, there i s a d e f i n i t e phase and wave vector r e l a t i o n between the production of d i f f e r e n t waves. The l i g h t i s then said to be coherent because the number of^independent radiators has been t h e o r e t i c a l l y reduced to one. However, even a laser beam i s not s t r i c t l y mono-chromatic (as Inferred by a single radiator) so that i n that sense i t Is only highly coherent. 8 The degree of coherence i s discussed i n length i n Chapter X of a book by Born and Wolf (1959) i n connection with the v i s i b i l i t y of interference f r i n g e s . Consider two points and P 2 i n the ra d i a t i o n f i e l d at some distance from the source. I f E(Pi,t) represents the e l e c t r i c f i e l d strength at the point P^ and a time t, then a mutual coherence function can be defined where T i s the time delay introduced by the separation of the two points. In the present experiments the detector area i s assumed to be plane and perpendicular to the incident r a d i a t i o n so that 7"'=r 0 and only s p a c i a l coherence i s of i n t e r e s t . In t h i s case //J^cV/-0 unless the two points are very close together and the ra d i a t i o n at the two points Is said to be non-coherent. The area to which ?1 and P 2 must be r e s t r i c t e d i s c a l l e d the coherence area and i s of the same order of magnitude as the area of the d i f f r a c t i o n pattern ( formed by a single source point. Another, possibly better, way of considering the area of coherence i s from the"point of view of the number of independent o s c i l l a t o r s required to duplicate the r a d i a t i o n f i e l d (Forrester, 1956). The number of degrees of freedom' N associated with the ra d i a t i o n emitted i n a time t from an area A into a s o l i d angle Sc. 9 i s In a coherence time t c •» 1/^V t h i s i s just the number of independent o s c i l l a t o r s n se t t i n g up each p o l a r i z a t i o n component of the f i e l d n = AJ2/7>* Therefore, the area corresponding to each o s c i l l a t o r i s In Appendix A an attempt i s made to introduce the coherence area into the inlxing equations i n a more natural manner. To do t h i s two separate assumptions about e l e c t r i c f i e l d vectors not i n the same coherence area must be made (i) there i s no r e l a t i o n between t h e i r ( i i ) t h e y cannot i n t ^ c f n o n - l i n e a r - l y with each other These assumptions are equivalent to considering the photodiode as being broken up into a number of independent detectors each of area /)/JZ. The mean square fluctuations from each coherence area then simply add. I f the detector area A » A C the photomixing spectrum (as compared with the coherent case) w i l l be reduced by a factor 7t/k In t h i s case the source i s said to be resolved by the detector. 10 1-3 Review of Previous Experiments The only experiment reported to date with a single illuminated photodiode and a non-coherent l i g h t source was performed by Forrester et a l (1955). They used the mercury green l i n e as a source but had to use Zeeman s p l i t t i n g to increase the apparent value of M(f) u from 10 to 2. This was accomplished by modulation of of the mixing signals but leaving the shot noise constant by making use of the p o l a r l z a t i o ^ properties ji>f the Zeeman components. Since t h e i r observations were at frequencies near 10^° cps they were not observing s e l f -mixing but instead mixing between the Zeeman components. The only important r e s u l t of t h e i r observations i s that o p t i c a l mixing i n a photodiode does occur, a fact that had previously been doubted, A recent experiment (Bolwijn et a l , 19°3) was performed using a laser source. They t r i e d to explain t h e i r r e s u l t s by the theory f o r Gaussian d i s t r i b u t e d e l e c t r i c f i e l d strengths but assuming 7^/kJL^\ and a very narrow l i n e width. The awkward plot of the r e s u l t s makes i t impossible to t e l l how good an agreement was achieved. In any case, the bandwidth and the low frequency mixing r a t i o observed are not consistent with the equations used. The authors comment on the rather large o p t i c a l bandwidth (19 kc) obtained but t h e i r explanation, 11 a t t r i b u t i n g I t to plasma resonances, appears to be unsatisfactory. In another paper (Cummins, 1963) t h i s rather large o p t i c a l bandwidth i s said to be caused by "an admixture of a large number of o f f . a x i s modes" present as w e l l as the purely a x i a l ones. In t h i s thesis i t i s proposed that the model i s fundamentally incorrect when applied to a laser source. After a l l , a laser can hardly be considered to be a source of Gaussian random l i g h t . The same conclusion "was reached by B e l l i s i o et a l (1964) from the r e s u l t s of t h e i r experiments. They observed three types of laser photomlxing spectrums ( l ) high noise s i m i l a r to Bolwijn's observations ( l i ) spectrum with spikes and l i t t l e excess noise between the spikes ( i l l ) low noise spectrum The l a s t case i s assumed to be the only fundamental one and they observed no excess photocurrent fluctuations above shot noise. However, t h e i r observations are r e s t r i c t e d by enhanced shot noise from the photomultiplier so that they could only set a rough upper l i m i t to the amount of mixing. They attributed the high noise operation to i n s t a b i l i t i e s i n the plasma because t h i s type of spectrum was not observed i n r f excited l a s e r s . For these reasons i t was considered necessary to 12 repeat these experiments and to t r y to develope d i f f e r e n t models f o r the r e s u l t s obtained. At the same time since no one has reported successful observation of self-mixing with an ordinary l i g h t source, these experiments were also performed. i Wave Analyzer Panalyzor or Oscilloscope Pig 2-1 Block Diagram of the Experimental Apparatus Photodiode or Noise Standarr Amplifier 5 13 CHAPTER 2 APPARATUS AND EXPERIMENTAL TECHNIQUE 2-1 Experimental Apparatus A block diagram of the experimental apparatus i s shown i n Pig 2-1. The amplifier i s a Model 103 Keithley Low Noise amplifier with a gain of 1000 and a pass band from 100 cps to a maximum of 100 kc. The noise spectrum was measured using a Quan-Tech Wave Analyzer with a frequency range of 30 cps to 100 kc. Fixed bandwidths of 10, 30, 100 or 1000 cps are avai l a b l e . To give a f u l l scale d e f l e c t i o n a minimum signal of 0.1 mV i s required. A meter time constant of 1 s e c provides a smooth response. A Singer Model SB-12b Panalyzor was used as a continuous monitor of the frequency spectrum from zero to 100 kc. The voltage vs frequency spectrum of the signals i s displayed on a cathode ray tube. The panalyzor operates i n a standard pass band of U50 kc to 550 kc and i n order to operate outside t h i s range a l o c a l o s c i l l a t o r has to be used to translate the signal frequency. The panalyzor was designed f o r radio frequency work where the signal frequency either l i e s i n or above the pass band. Therefore, the l o c a l o s c i l l a t o r frequency usually does not f a l l i n t h i s range so no attempt was made to suppress the c a r r i e r amplitude. However, to display the spectrum desired here a l o c a l o s c i l l a t o r frequency of 550 kc i s required and the extreme magnitude ^wvvwvv - v w w w 7 Lu o S t <"M 0 CM Q" i—vNA/WV* 1 H W W W V -vi/ «3—NN^NN-—wvwvvv-o i d ? yvvvvw vn ^ 4 V V V V V - Y ^ ^ V W MwwVW -WVWAAA/ - W W V V V oo e» -MAAAAAA/-1 > IP 4 Pig 2 - 2 Transistor Balanced Modulator 14 of the c a r r i e r (.3 V) overshadows the frequency spectrum below 10 kc. A l l signals below 20 kc were observed to lose amplitude on the panalyzor display as compared with the same Input signal strength at higher frequencies. Since a great deal of information about I n s t a b i l i t i e s and other harmful effects i s contained i n t h i s portion of the frequency spectrum, some means of removing t h i s had to be devised. A balanced modulator, external to the panalyzor, was b u i l t which allows the signals to be i n the required pass band. I t was found that to remove the undesirable condition only a rough balance to reduce the c a r r i e r amplitude to near the signal l e v e l i s required which i s an advantage since the c a r r i e r can be used as a marker f o r zero frequency. This type of modulator can simply be b u i l t using t r a n s i s t o r c i r c u i t s eliminating the peed f o r transformers and high voltage as well as filament supplies. The c i r c u i t b u i l t i s shown In Pig 2-2. The t r a n s i s t o r s T l and T2 which have equal load resistances i n t h e i r c o l l e c t o r and emitter c i r c u i t s give the phase inversion instead of the usual transformers. The c a r r i e r input Is then applied to two gating t r a n s i s t o r s T3 and T4 i n the emitter leads of the mixing t r a n s i s t o r s T5 and T6. The gating t r a n s i s t o r s give some gain and allow the c a r r i e r to be introduced to the mixing t r a n s i s t o r s i n 1 5 t \ much the same way as with the second g r i d of a vacuum tetrode. The mixed output of the two t r a n s i s t o r s i s taken o f f a common load r e s i s t o r . The c a r r i e r i s suppressed since under i d e a l balance conditions two equal and out of phase components appear i n the output. The mixed signals w i l l however be i n phase. C a r r i e r balance i s obtained by sett i n g the variable resistances R2 and R3 to t h e i r matching values. The c a r r i e r could be balanced to f u l l scale on the log scale of the panalyzor and i s quite stable over periods of hours. Further balance by using precision r e s i s t o r s and matched t r a n s i s t o r s was not considered necessary f o r the present puposes. S u f f i c i e n t improvement was obtained to allow resolution from the c a r r i e r of a 500 cps signa l on the 100 kc bandwidth. There was no decrease i n the amplitude of these low frequency signals. The panalyzor was used as a monitor only and no actual measurements were made from i t s display. For t h i s reason no attempt was made to account f o r losses i n the modulator or to c a l i b r a t e the panalyzor. 2-2 The Standard tyolse Source A dir e c t comparison method was used to compare the noise output of an illuminated photodiode with that from a standard noise diode. In order to make t h i s method more accurate as many components as possible were O T 4/OISE DIODE rhftoGTO • V 8Z0pF — i v -Rx \ 0 . 2 t o AMPLIFIER Pig 2-3 C i r c u i t Diagram of the Noise Source E l /A/ I '/A/ F i g 2-4 Ac Equivalent C i r c u i t of the Noise Source 16 made common to both c i r c u i t s . The c i r c u i t used i s shown i n Pig 2 - 3 . Dry c e l l s are used as power supplies and the heater current f o r the noise diode i s supplied by a regulated 6 . 3 V power supply. The noise diode standard i s a Sylvania 5722 vacuum diode. The ac equivalent c i r c u i t f o r the noise diode i s shown i n Pig 2-4. R(j sr the diode plate resistance G<j = the diode capacitance R^n=.the amplifier input resistance C i n=ithe amplifier input capacitance Using the usual approximations as well as R d » R , R i « R l n and C d « C i n the low frequency cut-off of the c i r c u i t i s given by f L = 1/277-0! (Hi+R) and the high frequency cut-off by f H = (R 1f-R)/27TC l nR 1R The mid-frequency noise voltage spectrum from the c i r c u i t i s S v = r S I [ R 1 R / ( R 1 + R ) J 2 Inserting the values of the c i r c u i t elements used, i t i s found that the cut-off frequencies are about 1 kc and 400 kc f o r R i n = 10 .MJZ. and C i n * 2 0 pP. The low frequency response i s then determined by t h i s c i r c u i t while the amplifier r e s t r i c t s the high frequencies. I f the vacuum diode i s operating i n the temperature li m i t e d region i t w i l l generate f u l l shot 17 noise which i s produced since the current i s made up of discrete electrons. T h e o r e t i c a l l y pure shot noise has a uniform spectrum which means that S j has the same magnitude at a l l frequencies. This w i l l only be true up to frequencies of the order of 1/7" where T i s the electron t r a n s i t time. Since t h i s frequency i s much larger than 100 kc i n a vacuum diode t h i s need not be considered here. The measured spectrum w i l l not be uniform due to c i r c u i t l i m i t a t i o n s . Shot noise from the diode w i l l be given by Sj(f) » 2 e l d c but i f the c i r c u i t response i s A(f) the measured current spectrum w i l l be S ( f ) = 2 e I d c A 2 ( f ) For a load resistance R, a wave analyzer bandwidth £ f (an i n t e r v a l small enough that S(f) can be assumed to be constant over i t ) and an amplifier gain 0 the mean square voltage w i l l be ( V 2 ) N D « 2 e I d c R 2 A 2 ( f ) A f G 2 I f the photodiode current contains pure shot noise and other signals with a spectral density W(f), the measured mean square voltage i n the wave analyzer w i l l be (V 2) p D=[2eI d c-*-W(f)] R 2 A 2 ( f ) A f G 2 18 The r a t i o of the two readings w i l l be I t W(f)/2el d c= 1 -f-M(f) Here the value of the current i n the noise diode i s assumed to be made equal to the l i g h t current generated i n the photodiode. This function M(f) w i l l be completely determined from the experimental data and independent of the c i r c u i t response. Although M(f) has the same frequency dependence as W(f) i t i s a dlmenslonless spectrum referred to the value of shot noise. 2-3 Semiconductor Photodiodes 2-3-1 Theory of Operation The photodiode consists of a p-n Junction which has one side exposed to the incident r a d i a t i o n . The remaining sides are usually surrounded by a non-transparent material. A larger area photodiode may t y p i c a l l y be made by covering one surface of a sheet of n-type material with a t h i n layer of p-type material. A very t h i n layer of non-reflective material may also be applied to the surface to reduce r a d i a t i o n losses and to protect the surface from humidity e f f e c t s . The p-n Junction i s operated i n a reverse bias condition when used as a l i g h t detector i n order to approach a l i n e a r c h a r a c t e r i s t i c response with the l i g h t i n t e n s i t y . I f the reverse bias exceeds a few tenths of holes I 1— i——i 1 P [ vi -e/ectvo/v/s — - = 4 | . £ — Reverse biased p-n junction • I v. (Slope - 1 ) Pig 2-5 Photodiode V-I Chara c t e r i s t i c 19 a vol t the r e s u l t i n g current i s very nearly Independent of the bias voltage. With no i l l u m i n a t i o n there i s s t i l l a current (-I Q, the dark current) which i s the normal diode reverse saturation current produced by the "thermally excited c a r r i e r s . I f l i g h t f a l l s on the Junction a d d i t i o n a l hole-electron pairs are formed by the absorption of a photon and corresponding e x c i t a t i o n of an electron from the valence band to the conduction band. This can happen as long as the photon energy hV exceeds the width of the energy gap (1.1 eV f o r s i l i c o n and .72 eV f o r germanium). Por a well saturated photodiode the l i g h t current I T i s proportional to the rate of photon absorption or equivalently proportional to the l i g h t i n t e n s i t y . The t o t a l photodiode current i s then i = - d G + i L ) A t y p i c a l V-1 c h a r a c t e r i s t i c i s shown i n Pig 2 -5 . The current i n the photodiode depends on the d i f f u s i o n of the minority c a r r i e r s across the Junction. These minority c a r r i e r s may recombine before reaching the junction so that only a small area near the Junction i s e f f e c t i v e i n producing a current. This current w i l l also be extremely temperature dependent due to variations i n the density, mobility and l i f e t i m e of the c a r r i e r s with temperature. F i g 2-6 Equivalent Noiae C i r c u i t of a Photodiode > 20 2-3-2 Noise Considerations Noise i n semiconductor devices and i n p a r t i c u l a r the photodiode i s considered i n d e t a i l by van der Z i e l (1959). The noise equivalent c i r c u i t i s shown i n Pig 2-6 where the current generators represent the mean square fluctuations i n the photocurrent. The l i g h t current I L shows f u l l shot noise plus some other noise represented as a f r a c t i o n of shot noise by M(f) which may be a function of frequency. The detector noise equivalent dark current 1^ i s used to represent a l l c i r c u i t noise sources when there i s no il l u m i n a t i o n Id = xo f 2kT/eR > 2kT d/eR <j Again i n a photodiode with good saturation i t i s expected that R(j»R. The following a m p l i f i e r w i l l also introduce a noise into the c i r c u i t . I f the ampl i f i e r noise figure i s P the equivalent noise current (at i t s input) w i l l be I A= (P-l) 2kT/Re following the usual assumptions about the r e l a t i v e magnitudes of the c i r c u i t resistances. The t o t a l noise spectrum out of the c i r c u i t with an illuminated photodiode i s then 2e [ l d t I A + ( M + l ) I L ] Prom a l l t h i s the excess fluctuations M ( f ) l L must be deduced. I t i s the r a t i o of these excess fluctuations to the t o t a l noise that determines the resolution l i m i t \/oo F i g 2-7 1/f Noise i n a TF50 Photodiode Conditions': I L = 500 /^A •=» 100 cps F i g 2-8 Spectral Response of Various Photodiode Types 21 of the experiments. At low frequencies so-called l / f noise was observed i n many of the photodiodes. At 1 kc t h i s noise might be as high as 20 times pure shot noise. The graph i n Pig 2-7, taken with one of the worst photodiodes illuminated with l i g h t from a tungsten filament lamp, the frequency dependence of the excess noise i s shown to be almost exactly l / f . l / f noise has been generally attributed to surface effects either at the diode surface, grain boundaries or dislocations i n the c r y s t a l (van der Z i e l , 1959). Even with the same material, processed i n the same way, the amount of excess n o i s e " d i f f e r s from diode to diode by a large factor, l / f noise also depends on the bias voltage and operation at voltages near breakdown may Increase the r a t i o by as much as a factor of ten. 2-3-3 • Comparison of Photodiodes In the s e l e c t i o n of a photodiode f o r the experiments a number of d i f f e r e n t properties were considered. In each case at least two diodes of each type were tested. Each material has i t s own c h a r a c t e r i s t i c spectral response. Approximate curves f o r each of the three general types are shown i n Pig 2-8. For the vacuum type t h i s i s l i m i t e d by the material on the 22 cathode and the glass used i n the envelope. The semiconductor types are li m i t e d by the width of the energy gap and the depth of the depletion layer. Vaouum photodiodes were not used i n any of the experiments beoause of t h e i r low quantum e f f i c i e n c y whioh was measured i n the RCA 929 type to be about ,08 and t h e i r low allowable ourrents for r e l i a b l e operation. Currents above the maximum permissible (about 2 0 ^ A) can be used but t h i s leads to very short l i v e s of the diodes. The germanium type (TP § 0 ) used i s quite s u f f i c i e n t f o r most of the experiments. I t s small area l i m i t s i t s usefulness i n some oases but the' main disadvantages are a high dark current and l / f noise. By ohooslng the best diode and operating at as low a bias as possible the l a t t e r oan be reduoed to almost negligable amounts for frequencies above 1 kc, In the range from TOGO A 0 to 1 0 , 0 0 0 A 0 the s i l i o o n diodes are probably the best. They have low dark currents due to the wider energy gap of s i l i o o n . However, the LS 223 s i l i c o n photodiodes could not be used due to t h e i r poor frequency response. There are two main reasons main reasons f o r frequenoy l i m i t a t i o n s i n any diode ( l ) a f i n i t e t r a n s i t time ( i i ) the diode eapaoitimoef The t r a n s i t time probably l i m i t s the frequenoy response Manufacturer TP 50 TI SSR RCA and P10D LS223 10L267 SJ2411 Type* NPS NPD Material Sensitive Area Dark Current Operating Voltage Peak Current Frequency Response l / f noise Ge o 1 mm*1 2 /tA 1 mA S i 1 mm2 1 /<A 2 2 . 5 V 9 V .5 mA S i 25 mmc •3-/<A 22.5 1 V 1mA low i n some low S i 5 mm • 0 7 5 ^ A 9 V 200 ^ A very high 0-10 kc 0-100 kc beyond 100 kc very low negligable * TP Transistor Products TI Texas Instruments SSR S o l i d State Radiations F i g 2-9 Table of Photodiode Ch a r a c t e r i s t i c s 23 of the LS 223 hut with the large area SSR; 10L267 nuclear detectors the capacitance appears to be the l i m i t i n g f a c t o r . Using the usual p a r a l l e l plate capacitance formula C d»<?A/d where e**the dielectric constant f o r s i l i c o n A »the detector area d= depth of the depletion layer and the data given f o r the detector the capacitance of the 10L267 i s calculated to be 50 pjfc. This reduces the c i r c u i t cut-off to s l i g h t l y less than 100 kc. The RCA SJ2411 nuclear p a r t i c l e detectors were the best photodiodes tested. No l / f noise could be observed down to a frequency of 500 ops. Their dark current was measured to be 0.075/<A whioh makes t h i s negligable i n a l l measurements and the diode capacitance i s s u f f i c i e n t l y low that there i s no reduction i n the high frequency c i r c u i t response. The quantum e f f i c i e n c y of these deteotors was measured to be around .8. The reduction from unity is' probably mostly due to r e f l e c t i o n from the surfaces. A table of the properties of a l l the photodiodes considered i s shown i n Fig 2-9. I.o /o.o 1 0 0 ~r v e ^ (kc) F i g 3-1 Photodiode Noise Spectra Source• Tungsten Filament lamp and #700 o p t i c a l f i l t e r Conditions: I L = 75 y^A A f - 100 cps 24 CHAPTER 3 EXPERIMENTAL RESULTS (NON-COHERENT LIGHT) 3-1 Blackbody Light Source A blackbody spectrum i s defined as a continuous spectrum of l i g h t with a frequency d i s t r i b u t i o n obeying Planck's radiat i o n law such as exists i n a vacuum cavity i n thermal equilibrium. Blackbody-type sources were chosen as a s t a r t i n g point for the experiments even though calculations show (Sec l - l ) that no observable mixing can occur. This type of source served as a test f o r the photodiodes and an i n d i c a t i o n that the apparatus was working properly. The lamp used was a prefocused, 150 W, Sylvania DPA type with a tungsten filament. This w i l l a c t u a l l y be a greybody source since tungsten has an emissivity of .45 at 2000 °K. The radiat i o n has the same frequency d i s t r i b u t i o n but only .45 the i n t e n s i t y as that from a blackbody source at the same temperature. The optics used consisted of a simple condensing lens with o p t i c a l f i l t e r s protected by a heat resis t a n t glass. Power fo r the lamp was supplied by the dc mains i n the lab. An Optics Technology Monopass f i l t e r #700 was used to r e s t r i c t the spectrum to a 150 A° band around 7000 A°. The shape of the f i l t e r pass band i s somewhat Gaussian. The transmission of the f i l t e r was about 35$ at i t s peak. The res u l t s are shown i n F i g 3-1 for the three main types of photodiodes used and under the conditions noted. The shot noise from the noise diode at same operating current i s 25 i d e n t i c a l with the curve f o r the SJ 2 4 l l shown. The excess noise at low frequencies for the other diodes i s due to l / f noise and the high frequency f a l l o f f of the 10L267 Is due to i t s high diode capacitance. This l i g h t source was also used to check the quantum e f f i c i e n c y of the photodiodes. In each case a rough c a l c u l a t i o n showed that f o r the semiconductor photodiodes the quantum e f f i c i e n c y i s between 0.5 and 0.8. The best blackbody source available Is probably s t i l l the sun. I t s temperature i s about 6000 °K which i s a factor of three larger than the tungsten filament lamp. The l i g h t could be focused to about 2 mm2 area but due to the rotation of the earth the spot moved considerably during the measuring time. For t h i s reason the largest area diode (1GL267) had to be used. With the same o p t i c a l f i l t e r as before and a s o l i d angle of .02 steradian a l i g h t current of 75/(A was generated i n the detector. The expected r a t i o of the mixing fluctuations to shot noise Is given by Eqn 2, Appendix B to be M(0) = ( 7\Z/A St) (Ir/2^rh) = 2X i o ~ 4 This i s much less than the r a t i o calculated by Alkemade's formula (Sec l - l ) but may be attributed to losses and the shape of the o p t i c a l f i l t e r pass band. This i s much too small to be measured and only the usual discrepancies f o r the diode from pure shot noise were observed. To observe any mixing an improvement of at least a factor of 1000 w i l l be necessary. 26 3-2 Mercury Vapour Lamp An Osram mercury vapour lamp operating at 50 V and 1.2 A was used to generate the mercury spectrum. Again the dc mains i n the lab were used as a power supply. An Electro Products power supply was also t r i e d but the excessive 120 cps ri p p l e prevented measurements. Operation or) dc current proved to be extremely d i f f i c u l t . A great deal of i n s t a b i l i t y was observed at frequencies below 10 kc and d e f i n i t e o s c i l l a t i o n s were observed at 3.9 kc and 750 cps. The 750 cps o s c i l l a t i o n s were traced to fluctuations at t h i s frequency i n the power supply. This was done by i n s e r t i n g the photodiode output on the horizontal plates of an oscilloscope and a sample of the power supply voltage on the v e r t i c a l plates and checking for correlations at d i f f e r e n t frequencies. At 3.9 kc no such c o r r e l a t i o n was found so these must be plasma o s c i l l a t i o n s inside the lamp. Insertion of a 70/(P capacitor d i r e c t l y across the lamp tended to remove much of the i n s t a b i l i t y . The 3.9 kc o s c i l l a t i o n s disappeared and would not return when the capacitor was removed unless the lamp was moved suddenly or the operating point changed s l i g h t l y . The amount of i n s t a b i l i t y was also noted to depend strongly on the operating current. The lamp would extinguish i f the current varied by more than 10$ from the operating value. This was esp e c i a l l y c r i t i c a l during the warmup period which lasted 27 nearly 10 minutes. Using the large area 10L267 detector considerable current could be generated when a single spectral l i n e was i s o l a t e d . The best l i n e was the 5790 A 0 one i s o l a t e d by the #566 f i l t e r . This raonopass f i l t e r has a bandwidth of 100 A° and a transmission of 25$ at 5790 A 0. Measurements at 75/iA showed only f u l l shot noise i n the part of the spectrum undisturbed by i n s t a b i l i t i e s . This i s to be expected since a large s o l i d angle (about a steradian) and a large detector area (25 mm ) were necessary to obtain t h i s Z Q current. A coherence factor 7\/AJZ=10~° thus removes any p o s s i b i l i t y of observing mixing. The other extreme was also t r i e d where the complete spectrum was allowed to f a l l on the photodiode. In t h i s case the s o l i d angle was reduced to such an extent that the same amount of current was generated i n a small area (.1 mm ) TP 50 photodiode. Again only pure shot noise and plasma i n s t a b i l i t i e s were observed. External modulation experiments were also c a r r i e d out by applying a few tenths of a volt s i g n a l across the b a l l a s t r e s i s t o r i n the lamp c i r c u i t . This modulation was picked up by the detector and displayed on the panalyzor. Harmonic free signals up to 100 mV could be generated i n the detector output. This i s about a 4$ modulation of the dc current. The width of the modulation sig n a l was less than 10 cps as shown by the variable bandwidth of the analyzer. 28 When modulation was ca r r i e d out i n the neighbourhood of 3.9 kc, the i n t e r n a l o s c i l l a t i o n s were observed to s t a r t 1 up again. They sometimes remained f o r a minute or so a f t e r the modulation was removed. After t h i s time they died out again i f the capacitor across the lamp was i n the c i r c u i t . The discharge seemed to be somewhat more stable when modulation was applied. This may be due to power being dissipated i n t h i s way and not being allowed to b u i l d up i n other modes. Use of higher power mercury lamps with better optics might lead to observation of some mixing. However, due to the uncertainty i n the widths of the spectral l i n e s t h i s i s somewhat doubtful. An electrodeless r f discharge might provide a better source because the discharge i s expected to be more stable and there w i l l be no darkening of the vapour envelope due to electrode sputtering. Further experiments using other Osram discharge lamps(Cd, Cs and K) were performed but the lower power from these lamps again made observation of mixing impossible. Similar i n s t a b i l i t i e s were also observed i n these lamps. 3-3 The Xenon Lamp 2. AT Due to the appearance of a factor 7)/B ^ A/c A7\ i n the mixing equation (Eqn 2 , Appendix B) i t was decided that i t would be better to use a lamp giving strong r a d i a t i o n i n the red and infrared portion of the spectrum and an xenon lamp was chosen. A Hanovia discharge lamp operating at 20 V • 7 5 -M(f) .5-•4* Q-,25-F i g 3-2 Photomlxing Spectrum Source: Xenon Lamp Photodiode: TP 50 Conditions: I L = 40 XHA £ f =• 100 CpS (00 29 and 15 A was used with a Trygon M36-3O A power supply with .05% regulation and constant current f a c i l i t i e s . The o p t i c a l arrangement used a s o l i d angle at the detector of .01 steradian. A Corning #2-64 o p t i c a l f i l t e r was used to r e s t r i c t the spectrum to above 7000 A°. The current generated i n a TP 50 photodiode was 40 /(.A. Cooling of the lamp was provided by a small fan. This produced a noise spectrum i n the photodiode currnet about 60$ higher than pure shot noise and the res u l t s are shown i n Pig 3-2. At frequencies below 5 kc plasma I n s t a b i l i t i e s were observed to dominate the spectrum but the value of M(f) remained quite constant over the range from 10 kc to 100 kc. The mixing r a t i o also remained r e l a t i v e l y constant over long periods of time but variations from as low as 20$ to as high as 100$ were observed to occur at i r r e g u l a r i n t e r v a l s . This might be explained by changes i n the l i n e width(due to fluctuations i n the temperature and pressure i n the lamp) or possibly due to changes i n the i n t e n s i t y of the spectral l i n e s . Tests were also made to determine i f the excess noise i s due to photoelectric mixing. Using a SJ 2411 detector with 50 times the area only pure shot noise was observed. Again using the TP 50 diode but with the s o l i d angle increased by a factor of 100 no excess noise could be measured. Note that i n actual fact It i s not these changes that decrease the mixing r a t i o since the l i g h t 30 current w i l l increase proportionally but the decrease i s due to the l i g h t attenuators inserted to hold the current constant, No other phenomena could be found which might produce excess noise that would be affected In these,ways. For t h i s reason i t i s concluded that photoelectric mixing was observed with t h i s l i g h t source. There are about nine main spectral l i n e s In" the l i g h t allowed to f a l l on the diode. Their wavelengths and r e l a t i v e i n t e n s i t i e s quoted from "MIT Wavelength Tables, 1939" are ^ (A°) Relative Intensity 10,838 1000 a r b i t r a r y units 9,923 2000 9,799 2000 8,952 ! 1000 8,819 ! 5000 8,409 2000 8 , 3 4 6 2000 8,280 5000 8,231 5000 Which of these l i n e s were a c t u a l l y present and contributing to the mixing was not determined. Assuming a single spectral l i n e at say 8,500 A° the l i n e width required to give a mixing r a t i o of . 6 would be (using Eqn 2, Appendix B which assumes a Gaussian l i n e shape) M(0) = (7)/ATI) (I^effFB) - 0.6 B — 8 .5X 1 0 1 0 cps A?! = 2(?T/c) f 21n2 7 B = 4 . 8 A° 31 I f a l l nine spectral l i n e s had the same i n t e n s i t y and width then neglecting the d i f f e r e n t wavelengths the width of each would be about .55 A° (Sec l - l ) . These values are reasonable f o r the conditions assumed. Due to the variations i n the amount of mixing over which there seemed to be no control, accurate measurements of the dependence of M ( o ) on I L , A or Si could not be made. HELIUM NEON Fi g 4-1 He-Ne Energy Level Diagram (Simplified) 32 CHAPTER 4 EXPERIMENTAL RESULTS (COHERENT LIGHT) 4-1 Operation of the Laser A Spectra-Physics Model 130 gas phase laser was used as a coherent source of l i g h t . I t i s a continuous emission He-Ne la s e r using dc plasma e x c i t a t i o n 0 The theory of gaseous o p t i c a l masers i s discussed by Bennett (1962). The output of t h i s laser at 6328 A° Is plane polarized due to the Brewster window termination of the plasma, tube. The cavity resonator i s formed by a pair of highly r e f l e c t i v e , multilayer d i e l e c t r i c mirrors (one plane and one spherical) mounted outside the tube. This laser i s e s s e n t i a l l y a four l e v e l system with one He and three Ne lev e l s p a r t i c i p a t i n g . A s i m p l i f i e d energy l e v e l diagram f o r the production of the 6328 A° l i n e i s shown i n F i g 4-1. The He atoms are excited by the dc induced plasma discharge to l e v e l #2 which i s 20.66 eV above the ground l e v e l . Since the Ne atoms have an energy l e v e l very close to the same energy (20.66 eV) they can be excited ' by c o l l i s i o n s and absorption of some thermal energy. The Ne atom i n l e v e l #3 then de-excites by one of the possible t r a n s i t i o n s . However, i f a photon with a wavelength of . 6328 A° interacts with i t f i r s t , induced emission of another i d e n t i c a l photon can occur. The Ne atom i n l e v e l #4 then de-excites to the ground l e v e l ready to s t a r t the cycle again. 33 Since the gas pressure i n the plasma tube i s low, Doppler broadening i s expected to determine the width of the spohtaneousv emission l i n e that would occur i f there was no resonant cavity present. AVD - (2/>|) V akTUna)/^' where M N e i s the mass of the neon atoms. Based on a radi a t i o n temperature of 325 °K the f u l l width at ha l f maximum AV^ of the Doppler broadened emission l i n e i s 1400 Mc. The distance between the mirrors Is L = 30 cm giving an i n t e r v a l of c/2L = 500 Mc between successive longitudinal modes. Using a standard Q representation the width of the cavity resonance at h a l f maximum w i l l be (Eqn 2, Appendix E) AV* = cq/4TTX - 800 kc where q ==°.01 Is the f r a c t i o n a l energy l o s t per 'transit of the cavity . The width of the cavity resonance i s then small enough to separate the di f f e r e n t modes and simultaneous o s c i l l a t i o n i n three longitudinal modes within the Doppler width i s then possible. , r ve's-The output beam i s taken o f f one end of the cavity by making that mirror only 99$ r e f l e c t i v e . The power output of t h i s l a s e r i s rated at .2 mW. The beam diameter i s 22 mm at the exit aperature and has a divergence of 1.3 min of arc. This i s equivalent to a s o l i d angle of 1.1 X 10 steradian and a coherence factor 7\ /AJI<^1. /oo — t r .a. -o " o—g—o- J M) 10 AO frec^ (kc) /o.o loo-O F i g 4-2 Photomlxlng Spectrum Source^ Laser Photodiode: SJ 2411 Conditions: I L = 7 . 8 /{A £ f = 10 cps 34 4-2 Experimental Results • Measurements were made on the frequency spectrum, from 500 cps to 100 kc,of the photodiode current generated by the laser r a d i a t i o n . Three d i f f e r e n t photodiodes were used but a l l the re s u l t s are s i m i l a r . The experimental re s u l t s f o r the SJ 2411 detector are shown i n Pig 4-2 under the conditions indicated. This graph shows the v a r i a t i o n of M(f) with frequency plotted on a log-log scale. Both the background noise and the dark current have been subtracted from the data. Plasma resonances at about 40 kc and 80 kc were observed with the l a t t e r probably just the second harmonic of the former. These o s c i l l a t i o n s have a width of about 10 kc and made observation of an Important part of the spectrum impossible. Also 120 cps r i p p l e with harmonics up to the 8'th from the laser power supply made low frequency measurements very d i f f i c u l t . To reduce t h i s e f f e c t a narrow bandwidth of 10 cps was used i n the wave analyzer f o r the measurements. No lenses were used to concentrate the beam since the SJ 2411 detector c o l l e c t e d a l l the l i g h t and i n fact l i g h t attenuators (passing about 10$) had to be used to prevent overloading of the amplifier by the 120 cps hum. Because of the low l i g h t current (about 10^A) a current i n the noise diode that gave about the same amount of noise was used instead of equal currents as previously. The 5oo-F i g 4 - 3 Current Characteristic of Photomixing Spectrum Source: Laser Photodiode: SJ 2411 Conditions > f = 5 kc Af = 10 cps 35 photodiode current was measured with a Keithley Model 409 Picoammeter. When the lase r i n t e n s i t y control was varied the plasma resonances were observed to change frequency, but they always remained i n the range from 40 kc to 50 kc with corresponding second harmonics. Also at c e r t a i n positions of the i n t e n s i t y d i a l other spurious signals were observed on the oscilloscope used to monitor the photodiode current. These, signals appeared to be co-ordinated with the 120 cps hum so may have been a defect i n the power supply. However, t h i s i s quite s i m i l a r to the high noise operation observed by B e l l i s i o et a l ( 1 9 6 4 ) . At other settings the operation of the lase r remained stable over periods of hours once the laser had warmed up. The graph i n Pig 4-3 shows the v a r i a t i o n of M(f) at 5 kc with changes i n the photodiode current. The r e l a t i o n Is quite l i n e a r as expected but at currents above 50 ^  A some levelling^off occurs. This was at t r i b u t e d to overloading of the amplifier by extreme signals at frequencies below the measurements. The d i f f e r e n t currents were obtained by leaving the' l a s e r i n t e n s i t y constant and placing l i g h t attenuators In the path of the beam. Another experiment was performed to determine i f the factor 7)/kJl should be included i n the equations f o r the mixing of lase r l i g h t . The value of t h i s factor for the conditions Is almost exactly unity and w i l l remain so as 36 long as a l l the l a s e r l i g h t Is c o l l e c t e d by the diode (see Sec .1.-1). The diameter of the beam was further r e s t r i c t e d by an aperature so that the area of the photodiode that Vteis illuminated was reduced by a fa c t o r of 10. In t h i s case the value of Ij/AJl. i s expected to remain constant and thus also the value o f M(f) I f the coherence f a c t o r i s included. However, the value of M(f) was observed to decrease exactly In proportion to the decrease In I L . I t must then be concluded that a factor of 7) /AJl >1 does not appear i n the equations f o r the mixing of laser l i g h t . No method could be found to obtain a factor v^AAO with t h i s gas l a s e r . 4-3 Interpretation of the Results The discrepancy of these r e s u l t s from the model suggested by Bolwijh et a l . (.1963) i s • even more s t r i k i n g than t h e i r r e s u l t s . Using a value of B — 40 kc, which i s reasonable from the data, the low frequency value of M(f). f o r a Gaussian random l i g h t source would be M(0) — IL/2efij?B = 3 • 4 X 10 8 which is•greater than the observed value by a factor of 4 / 10°. Obviously t h i s model cannot be applied and a laser i s anything but a Gaussian random source. An a l t e r n a t i v e model was suggested (Burgess, 1964) i n which the output of the laser i s assumed to consist 1 of a su b s t a n t i a l l y monochromatic stimulated sig n a l embedded i n a background of spontaneous emission. A discussion of F i g 4-4 Laser Photomixing Spectrum 37 the ap p l i c a t i o n of t h i s model to photoelectric mixing i s given In Appendices C, D and E. With t h i s model I f r i s the r a t i o of the sig n a l power to the t o t a l power then the t o t a l area under the low frequency mixing spectrum i s (Eqn 1, Appendix G) which from Pig 4-2 i s approximately MX 2aI Lx: f m a x = 7 . 0 X 1 0 " 1 8 A 2 o 8 Therefore ( l -r) ( l - r * ) / 2 = ^ 5 . 7 X 1 0 giving i sig n a l to noise r a t i o f o r t h i s l a s e r (Eqn 3 , Appendix C) S/N = 1/(1 -r) = 1.8 X 1 0 7 This can a l t e r n a t i v e l y be considered as the r a t i o of the l i g h t current generated by the stimulated s i g n a l to that generated by the spontaneous emission background. The low frequency mixing spectrum f o r a symmetrical stimulated s i g n a l i s given by t h i s model to be (Eqn 2, Appendix C) W L(f) - [4(1 -r)lL2/|/£7f B] exp(-f 2/2B 2) or the dependence of M(f) on frequency has the form M(f) - |2(l-r)lL/l/£lf'eBj exp(-f 2/2B 2) The graph In Pig 4-4 shows a plot of log 1 M(f) vs f 2 f o r the same data as In F i g 4-2. For a true Gaussian spectrum of spontaneous emission t h i s graph would be a straight l i n e but the curve becomes f l a t at low frequencies and tends to f a l l o f f too r a p i d l y at higher frequencies. Therefore, 38 either the spontaneous emission l i n e does not have a Gaussian spectrum or t h i s symmetrical signal model does not apply. However, i n Appendix D considerations are given to the p o s s i b i l i t y that the stimulatedisignal may not be centred on the spontaneous emission l i n e . The graphs (Figs D-l and D-2) computed appear to f i t t h i s type of experimental curve. The s o l i d l i n e i n F i g 4-4 shows the t h e o r e t i c a l conditions which best f i t the data. The parameters f o r t h i s curve are B = 27o5 kc g s|^-^|/B =1.0 This means that the spontaneous emission l i n e has a f u l l width at half maximum ^ = 2 y21n2 ' B = 65 kc and the stimulated sig n a l i s displaced 27«5 kc from the central p o s i t i o n . The agreement of t h i s curve with the experimental data, except for the plasma resonances, Is quite remarkable. Similar curves and agreement were obtained using the other types of photodiodes and d i f f e r e n t l i g h t currents. The value of B remained constant within 10$ and the same value of g had to be used i n each case. The width of the spontaneous emission l i n e i s expected to be determined by the cavity Q since the Doppler broadened l i n e width and the mode spacing are much larger than the cavity resonance width. However, the experimental results give a spontaneous l i n e width of 65 kc which i s a MiArt her of \ 1 F i g 4-5 Simplified Spectral Diagram of the Laser Radiation (the r e l a t i v e magnitudes on the v e r t i c a l scale are not intended to have any meaning) 39 a factor of 12 lower than the expected cavity resonance width of 800 kc. This may be due to the fact that the cavity acts as an active device instead of a passive one as assumed i n the derivation of A^.. In Appendix E considerations are given to the output power of a lase r i n terms of the observed data (Eqn 1, Appendix E) P-(S/U)hV qc/2L Inserting the values of these quantities gives P — . 0 3 mW. This i s almost a factor of ten down from-, the rated value of .2 raW, confirmed from the observed maximum photocurrent. The main assumptions i n the derivation of t h i s formula are ^ l i n e a r rate equation and a single mode i n o s c i l l a t i o n . Both of these are probably incorrect, leading to the above discrepancy. However, t h i s does indicate that the observed S/N i s at least i n the correct range of values. A possible explanation f o r the non-central stimulated si g n a l may be the phenomena of "mode p u l l i n g " which Is discussed by Bennett (1962). The basis f o r t h i s i s that under c e r t a i n conditions the cavity i s fo r c i n g an o s c i l l a t i o n at a large frequency i n t e r v a l from the natural resonance frequency of the material. The resonant modes then tend to be pulled towards t h i s c e n t ral frequency. A s i m p l i f i e d spectral diagram of the laser l i g h t i s shown In Pig 4-5. The dashed l i n e i s the Doppler emission l i n e that would occur i f the cavity w a s not present. Here i t i s assumed fo r s i m p l i c i t y that three modes, separated by 40 500 Mc because of the cavity length, are i n o s c i l l a t i o n and that the central mode i s at the actual material resonance frequency . The cavity mode width i s the width of the spontaneous spectrum measured In the experiment to be 65 kc. The spikes on these modes are the stimulated s i g n a l s . To a f i r s t order approximation the displacement of the stimulated sig n a l i s proportional to the r a t i o of the material Q to the cavity Q and the displacement of the cavity resonance from the material resonance. This leads i to an equation f o r the amount of p u l l i n g (Eqn 40, Bennett, 1962) * \yD -v^i A-VL/AVO if A Vc « A V0 An appreciable displacement w i l l only occur i n the outer modes fo r which V \ VD I *D-HI - 5oo Ale giving IVp - ) i ) ^ 2 3 Ac This value i s close to the value of 27 .5 kc deduced from the observations. However, no attempt was made to separate the mixing spectrums of the 3 separate modes assumed In t h i s explanation. This may change the observed value of ~V0 \ s l i g h t l y . Also i n f i t t i n g the t h e o r e t i c a l curve to the data there i s some i n s e n s i t i v i t y (within 10$) to the value of the parameter g. In t h i s e x p l a n a t i o n i t i 3 assumed that the spontaneous emission i n the mode i s not affected by mode 41 p u l l i n g . This seems reasonable since the stimulated signal i s i n o s c i l l a t i o n and i s thus strongly coupled to both the material i n the cavity and the cavity i t s e l f , but once the spontaneous l i g h t i s emitted i t i s coupled only to the c a v i t y . The model has now yielded three separate properties of the laser r a d i a t i o n which are at least within an order of magnitude of the values deduced from purely t h e o r e t i c a l arguments. The discrepancies are probably due to factors not included i n the t h e o r e t i c a l derivations. 42 CHAPTER 5 CONCLUSION The r e s u l t s of the experiments prove that under cer t a i n conditions photoelectric mixing can he observed with a non-coherent l i g h t source emitting v i s i b l e r a d i a t i o n . However, t h i s i s only possible with high power lamps operating i n the long wavelength portion of the v i s i b l e spectrum. The information obtained i s of l i t t l e use since a single spectral l i n e could not be i s o l a t e d . In order to i s o l a t e a single spectral l i n e a higher power lamp, than the one used here, w i l l be necessary so that the generated noise w i l l be s u f f i c i e n t l y above the background noise to give an observable sig n a l to noise r a t i o i n the analyzer. In the experiment, even though the current f l u c t u a t i o n spectrum i s 60$ higher than pure shot noise, t h i s corresponds to an analyzer signal to noise r a t i o (Bee Sec 2-3-2) of only . 3 5 . Another means of increasing t h i s r a t i o f o r a given l i g h t current Is to go to longer wavelengths i n the inf r a r e d because of the appearance of a factor ?\/B i n the mixing equation f o r a Gaussian-distributed source f i e l d (Eqn 2, Appendix B). With a spectral l i n e there are two unknowns, the l i n e shape and width. To determine these unambiguously two independent observations, the low frequency magnitude and the upper cut-off frequency of the mixing spectrum, are required. The upper cut-off frequency i s approximately of 43 the same order of magnitude as the width of the spectral l i n e which i s around 1 0 1 0 cps and beyond easy measurement. I f the spectral l i n e width was known from spectroscopic methods, very accurate low frequency measurements might be used to determine the l i n e shape. However, i t i s expected that the inte r p r e t a t i o n of the data would be d i f f i c u l t i f not impossible. I f both the l i n e shape and width were known a check of the photoelectric mixing theory could be made. In p a r t i c u l a r , t h i s method could be used to determine i f the output e l e c t r i c f i e l d of the l i g h t does In fact have a Gaussian amplitude d i s t r i b u t i o n . With lase r l i g h t i t i s possible to e a s i l y cover the entire low frequency mixing spectrum. Both the l i n e shape and width should then r e a d i l y be determined but the int e r p r e t a t i o n of the data i s s t i l l i n doubt. The model applied i n t h i s thesis i s highly i d e a l i s t i c but seems to f i t the experimental re s u l t s extremely w e l l . In fact the model i s probably much more general than i t appears at f i r s t s i g h t . The e l e c t r i c f i e l d of the stimulated emission was assumed to have a constant amplitude. I f the stimulated l i n e was assumed to have a f i n i t e width (this i s expected to be at most a few cps) and the. e l e c t r i c f i e l d some s t a t i s t i c a l d i s t r i b u t i o n , t h e resultant mixing spectrum would not be expected to change very much. A further mixing spectrum (much stronger than the observed one) would appear at very low frequencies with a cut-off at the 44 assumed width of the stimulated l i n e . Any i n t e n s i t y d i s t r i b u t i o n i s expected to have a time constant T * ' / / ^ ^ . Therefore, any variations i n in t e n s i t y should be observable from fluctuations of the photocurrent since AV^,^ 1 cps. Meter observations suggest that A I / J < .05 putting an upper l i m i t on the in t e n s i t y f l u c t u a t i o n s . The type of experiment performed by Cummins (1963) i s of li m i t e d value i n determining the laser l i n e shape or width. Here two separate modes are mixed together to give a beat spectrum at a frequency equal to the mode spacing. These observations can be used to determine the number of modes that are o s c i l l a t i n g (Bennett, 1962) but there i s no reason f o r assuming that these modes have the same width and i n t e n s i t y d i s t r i b u t i o n . In t h i s case unless detailed knowledge about one of the modes i s known, l i t t l e can be deduced about the other. Also addi t i o n a l factors l i k e the r e l a t i v e frequency s t a b i l i t y of the modes come into the observations. Also note that according to the model applied i n t h i s thesis i t i s expected that only the mixing between the stimulated signals w i l l be observed since t h i s w i l l overshadow any ef f e c t from the spontaneous emission. In another experiment by B e l l i s i o et a l (1964) the complete frequency spectrum from 14 cps to 12 Mc was observed with a number of d i f f e r e n t l a s e r s . They represent t h e i r data i n terms of modulation of the e l e c t r i c f i e l d 45 strength. They used a photomultipller which has a poor quantum e f f i c i e n c y and gives enhanced shot noise thus reducing t h e i r a b i l i t y to detect modulation of less than 1$. For t h i s reason they wex*e not able to detect any mixing when the laser was i n so-called quiet operation. In the present experiment the s e n s i t i v i t y has been Increased so that mixing could be observed quite r e a d i l y . No mention has been made i n the l i t e r a t u r e about using large area nuclear detectors as photodiodes. For the low frequency mixing experiment they have a number of advantages over the ordinary type of diode. Because of the large area they allow some experiments to be performed which previously required vacuum photodiodes. There i s no window on these detectors and the surface Is extremely f l a t . However, t h e i r high capacitance reduces the upper cut-off frequency to well below a megacycle. Detectors with a much higher r e s i s t i v i t y than 1,000 ohm-cm (as used i n the present experiments) would allow the use of a higher reverse bias and because t h i s Increases the depletion layer depth the capacitance Is reduced. There are s t i l l unanswered questions about the properties of lase r r a d i a t i o n and the concept of coherence which cannot be resolved without further experiments and t h e o r e t i c a l work beyond the scope of t h i s t h e s i s . I t has only been shown that photoelectric mixing can be an important t o o l f o r these investigations. 46 APPENDIX A PHOTOELECTRIC MIXING EQUATION FOR A SPECTRAL LINE Critique of Alkemade (1959) and Forrester ( 1 9 6 1 ) Consider a general spectrum of l i g h t with a power per unit bandwidth G(V) at a frequency V . S p l i t t h i s band into a large number of equal Intervals each of width . Let the i n t e n s i t y of the m'th i n t e r v a l be corresponding to an e l e c t r i c f i e l d representation where f — * ^ are phase angles d i s t r i b u t e d uniformly from 0 to 2 77". This assumption along with l e t t i n g M ~-*oo as A ^ ~ * 0 are the conditions necessary f o r the "central l i m i t theorom" which states that a large number of independent random vectors tends to a normal d i s t r i b u t i o n as the number tends to i n f i n i t y . This i s i d e n t i c a l with assuming a Gaussian random source (see Sec 1 - 2 ) . The output current of a photodiode i s given by I s a E 2 (See Sec l - l ) These assumptions make the theory Identical to the "Noise Through a Square Law Device" considered by Rice (Sec 4 . 5 , 1 9 4 4 ) . Up to t h i s point the treatment has been the same as done by Forrester (1961) but at t h i s time It seems to be advantageous to introduce the concept of coherence areas. I f the l i g h t covers a detector area A which Is larger than the coherence area A c =r 7l/J2.s there w i l l be no 47 r e l a t i o n between a on one coherence area and a on another. Divide the detector area into A/Ac = Afl//)z areas of coherence and represent the e l e c t r i c f i e l d by E = ± £ ccn> (-LW-y^-t - Y^cc) In terms of the l i g h t spectrum There are two separate averaging processes here. The time average removes the f i r s t term for a l l values of y ^ O . The ensemble average removes the second term unless <^*.— y^n,fc which only occurs i f a = b. 2 The photodiode current Is r = (*/£>) £Z2Z Cy»cJcK>.[zTr(^W -i!f^ %k)J Taking the low frequency component " V > v v — — ^ or m - n = k Vv The dc l i g h t current i s the ensemble average of I f f ^ ) when 48 In going over to the i n t e g r a l In the l a s t l i n e i t i s assumed that Q ( ^ ) i s n e g l i g i b l e f o r values of V near 0 . I f k^O there are two separate terms i n I(fjj) i ) m> n or k> 0 i i ) m< n or k < 0 These are Identical since cosine Is an even function, fc' » «. p where k > 0 always. The mean square value of the current fluctuations i s i ^t,- * % c * - % * \ When a l l the averages are taken the coherence factor drops out. However, i f i t i s further assumed that mixing cannot occur between two di f f e r e n t areas of coherence t h i s i s not so. This i s the same as saying that the detector i s made up of A/Ae separate detectors and the mean square current fluctuations generated on each detector are Independent of that generated on any other. Therefore where A f Is the bandwidth of the c i r c u i t used to measure the f l u c t u a t i o n s . I f G(y ) i s only non-zero over an Interval AV«V 0(as w i l l be true f o r most o p t i c a l spectra) the coherence factor may be taken outside the i n t e g r a l . The ac spectral density of the photodiode current 49 fluctuations becomes, CO which i s the same answer as derived by Forrester (1961) and Alkemade (1959)• In the derivation of t h i s r e s u l t a number of assumptions were necessary (i) A Gaussian d i s t r i b u t i o n of e l e c t r i c f i e l d vectors ( i i ) The photodiode has a perfect square law response to the e l e c t r i c f i e l d strength ( i i i ) No r e l a t i o n between the phases of the e l e c t r i c f i e l d vectors In d i f f e r e n t coherence areas (iv) G(V) i s n e g l i g i b l e near V = 0 . (v) Each coherence area acts as an independent diode (vi) I f TIVAJL i s greater than unity the whole diode area l i e s within an area of coherence so the faetor should be replaced by unity i n the above equation ( v i i ) AV«)£ where A V i s the f u l l width at half maximum of G(V ) The v a l i d i t y of the above equation necessarily depends on how good these assumptions correspond to actual f a c t . 50 A P P E N D I X B P R O P E R T I E S O F THE P H O T O M I X I N G S P E C T R U M O F A S P E C T R A L L I N E If t h e e n t i r e low f r e q u e n c y s p e c t r u m o f t h e c u r r e n t f l u c t u a t i o n s i s o b s e r v e d t h e v a r i a n c e o f t h e c u r r e n t i s j u s t t h e a r e a u n d e r t h e s p e c t r u m v a r I s j^jj(^) I f t h e m i x i n g s p e c t r u m i s g i v e n fey Eqn 2, A p p e n d i x A t h e n v a r I e 2a 2(7\*/A/2.) ( G ( V ) d V j % ( V f f ) d f 6 s **(%Z/kXL)[ ^a(V ) d ^ g « s ( 7 1 2 / A A ) I L 2 - ( 1 ) T h e r e f o r e , f o r a n y s o u r c e w i t h a G a u s s i a n - d i s t r i b u t e d e l e c t r i c f i e l d o u t p u t , t h e r e l a t i v e v a r i a n c e o f I i s e q u a l t o t h e c o h e r e n c e f a c t o r (jf'/k f i ) . A s s u m i n g i G a u s s i a n l i n e s h a p e ( c o r r e s p o n d i n g t o a D o p p l e r teroadened e m i s s i o n l i n e ) c e n t r e d a t a f r e q u e n c y V© a n d w i t h a s t a n d a r d d e v i a t i o n B <k(V) s (D/flBrfc) exp -)i)8/&B2J t h e n Ii i s § J ? ( V ) dv * aD a a n d t h e low f r e q u e n c y m i x i n g s p e c t r u m i s I f A V i s t h e f u l l w i d t h a t h a l f maximum t h e n A V — g|/ gins 1 1 She r a t i © of t h e e x c e s s c u r r e n t f l u c t u a t i o n s t o s h o t n o i s e a t v e r y low f r e q u e n c i e s (f<<!) i s t h e n M(0) a (?fyk3l)(lt/'&*iffB) — ( 2 ) 51 APPENDIX C PHOTOELECTRIC MIXING EQUATION FOR A LASER SOURCE For a source with some degree of coherence new eqations w i l l have to be derived i n which the 's are not assumed to have a uniform d i s t r i b u t i o n . Only the extreme case i n which the stimulated l i g h t i s represented as a pure sinsuoidal signal (perfect coherence) w i l l be corisidered. A model of the laser radiation might be a stimulated signal embedded i n a spontaneous emission l i n e . Assuming that the e l e c t r i c f i e l d strength of the l i g h t can be represented by ^ E * C or* BTT^-t + £ C^Ccr* (2TTVjr- # J r / t h i s again reduces to one of the Input spectra considered by Rice (1944) i n Sec 4 .5 . For a square law device he derives the low frequency mixing spectrum to be w£(f) « a 2C 2 / o ( f - y>) + G(f + VP)J + 2 a 2j PQ(V )Q(f-hV) d ^ (4 .5-13, Rice) The average l i g h t current i s ^ given by I L = aC2/2 + a^Q(y) d > (4 .5-11, Rice) o = aC2/2 +• aD Integrating the low frequency spectrum gives b = (1 - r 2 ) I L 2 - ( 1 ) where r =s aC 2/2I L i s the r a t i o of the signal power to the t o t a l power i n the laser r a d i a t i o n . 52 A "signal to noise r a t i o " of the laser r a d i a t i o n may be defined as S/tt = C2/2D = r / ( l - r ) This i s the r a t i o of the current generated by the stimulated emission to that generated by the spontaneous emission. Assuming a spontaneous emission l i n e with a Gaussian shape and a c e n t r a l l y located stimulated signal the low frequency mixing spectrum becomes w£(f) = ^l.r)lL2/fpB][(kv/^) exp"-(f 2/2B 2) f ( l - r ) exp - ( f 2 / 4 B 2 ) ] I f the signal power i s much greater than the spontaneous power (I.e. r ^ l ) only the f i r s t term i s appreciable W L(f) = [ 4 ( l - r ) l L 2 / ^ B ] exp -(f2/2B 2) - ( 2 ) and S/N = l / ( l - r ) — ( 3 ) I f the signal i s very weak ( i . e . r^O) the spectrum reduces to exactly the same as derived f o r a single s p e c t r a l l i n e . Throughout t h i s derivation i t has been assumed that the l i g h t f a l l s within one coherence area as t h i s i s the usual case f o r a gas l a s e r . I f t h i s i s not so a s i m i l a r argument to that i n Appendix A may be used. Pig D-l Mixing of Gau33ian Line with Non-central Signal Signal 53 APPENDIX D NON-CENTRAL STIMULATED SIGNAL The equations derived at the end of Appendix C assume that the stimulated s i g n a l i s c e n t r a l l y located on the spontaneous emission l i n e . For a s o l i d state las e r the displacement of the sign a l from a central p o s i t i o n has been observed and there i s no reason to assume that t h i s could not occur i n a gas laser as w e l l . I f the sign a l i s not centred and i f the value of r i s s u f f i c i e n t l y close to unity that the f i r s t term In T W L(f) dominates, the low frequency mixing spectrum w i l l be given by W L(f) = a 2 C 2 (j}(f -K2^) 4- G( Vv - f j j Again assuming a Gaussian l i n e shape centred at a frequency \L f o r the spontaneous emission as i n Appendix B D = J G(V ) dV = I L ( l - r ) / a Define a new parameter g such that g =r l ^ p - ^ l / B representing the amount that the signal i s displaced. The graphs i n F i g D-l and D-2 have been computed for various values of g to show the shape of W L(f). A l l the curves are normalized to have unit area under them. For purposes of comparison the dashed l i n e shows the corresponding Lorentzlan shape from Forrester (1961). 54 APPENDIX E OUTPUT POWER OF A LASER Consider two le v e l s separated by an energy difference E — hX> and enclosed i n a ca v i t y . Let n be the t o t a l number of photons of energy h>* i n t h i s c a v i t y . I f N g i s the population of the upper l e v e l and N 1 the lower, the l i n e a r rate equation f o r t h i s system dn/dt = ANX •+• BnN 2 - CnN 1 - pn where ANi spontaneous emission rate BnN2=r stimulated emission rate CN 1 — rate of absorption pn = rate at which other losses occur In steady state dn/dt ="0 and the^average number of photons n i n the cavity becomes AN 2 •+- BnN 2 — CnNj - i - pn Neglecting a l l other losses i n the cavit y ( i . e . p =0) then n = 1/(CN1/AN2 - B/A) Assuming a Boltzmann d i s t r i b u t i o n of populations N l / ^ 2 — exp - ( h ^ A T ) and that f o r a single mode n =• 1/ [exp (h y/kT) - 1J — b t h i s gives A=B-=-C. The r a t i o of the rate of stimulated emission to spontaneous emission i s then equal to the average number of photons In'the c a v i t y . The output power of the l a s e r i s just the losses 55 incurred by the fact that one mirror i s only 99$ r e f l e c t i v e , pn i s the rate of loss of photons from the cavity and i f q i s the loss per pass and c/2L i s the number of passes per second then p — qc/2L c •=. v e l o c i t y of l i g h t In the medium L = length of the cavity between the mirrors The power output of the laser i s then P = n h-y qc/2L » n h > p The sig n a l to noise r a t i o of the laser l i g h t i s just equal to the r a t i o of the rate of stimulated emission to the spontaneous emission and i s then equal to n. P = (S/N) hVqc/2L — (l) Now f o r a cavity Q — 2 77 P energy stored *"[ — 1) Lenergy l o s t per c y c l e J Therefore A^. = q C/47TL = p/27T — ( 2 ) and P =. (S/N) 27ThV ^ 4 The main assumptions i n t h i s derivation are a l i n e a r rate equation f o r the system and a single mode i n o s c i l l a t i o n neither of whlch'LTnayx'^g^ery good. 56 BIBLIOGRAPHY Alkemade, C. 1959. Physica 25, 1145 B e l l i s i o , J . , Freed, C , and Haus, H. 1964. Applied Phys. Letters 4_, 5 Bennett, W. J r . 1962. Applied Optics (Supplement on Optical Masers), 24 Bolwijn, P., Alkemade, C , and Boschloo, G. 1963, Phys, Letters 4_, 59 Born, M., and Wolf, E. 1959. P r i n c i p l e s of Optics, Permagon Press Burgess, R. 1964. Private Communication Cummins, H. 1963. Phys. Letters 5 , 39 Forrester, A., Gudmundson, R., and Johnson, P. 1955. Phys. Rev. 99, 1691 Forrester, A. 1956. Am, J . Phys. 24_, 192 Forrester, A. 1961. J . Opt. Soc. Am. 51, 253 Mandel, L. 1958. Proc. Phys. Soc. 72, 1037 Rice, S. 1944. B e l l System Techn. J . 23, 282 and 24, 46 van der Z i e l , A. 1959. Fluctuation Phenomena i n Semiconductors, Butterworths 

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