EE 232 Lightwave DevicesgLecture 19: Noises in Photoconductors RdiYi10 310 5Reading: Yariv10.3-10.5Instructor: Ming C. WuUniversity of California, BerkeleyElectrical Engineering and Computer Sciences DeptElectrical Engineering and Computer Sciences Dept.EE232 Lecture 19-1©2008. University of CaliforniaPoisson DistributionPoisson distribution:a given event occurring in any time interval is distributed uniformly over the interval.The probability of electrons arriving in a period : isnTThe probability of electrons arriving in a period : is( ) !nnnTnepnn−=where is the an verage number of electrons arriving in TProperties of Poisson Distribution:Mean = VariancennEE232 Lecture 19-2©2008. University of CaliforniaVariance = nSpectral Density FunctionRandom variable ( ) consists of a large number of individual eventsitRandom variable () consists of a large number of individual events(e.g., single-electron photocurrent) at random time:() ( ) 0TNitit f t t t T≤≤∑1() ( ), 0Fourier transform: ( ) ( )TiiNTiit f t t t TIFωω==−≤≤=∑∑11()2TiiiFωπ=∞∞=∑() () ()2iiititit itieft te dt fte dt e Fωωωωωπ∞−−−−∞−= =∫∫−∞22 2 2()11() () () ()TTijNNittTTijIF eNFNTFωωω ω ω−∞−−=====∑∑22: average rate of electron arrivalSpectral density function:NEE232 Lecture 19-3©2008. University of California22228(2)() lim 8 (2 )TTIvSv NF vTππππ→∞==Shot NoiseShot Noise: Noise current arising from randomShot Noise: Noise current arising from random generation and flow of mobile charge carriers.Current pulse due to a single electron moving at ( ):()vt()()1Fourier transform: F( )=()aetitev titdevteωω−=∫dt0Fourier transform: F( ) ( )2vtedωπ∫: arrival time, (0) 0, ( ) , Small transit time 1 ~1aaitdttxxtdtt eωω−==→00Small transit time , 1 ~111()= 1222aaatdtt eedx e eFdtdxddt dωωπππ→⋅⋅ = =∫∫22() 8 22eSv N eIIeNππ⎛⎞==⎜⎟⎝⎠=EE232 Lecture 19-4©2008. University of California2() () 2NIeNi v S v dv eIdv===Thermal Noise (Johnson Noise)• Fluctuation in the voltage across a dissipative circuit element (resistor)()• Caused by thermal motion of charged carriersEE232 Lecture 19-5©2008. University of CaliforniaThermal Noise DerivationConsider two resistors connected by aConsider two resistors connected by a lossless transmission line of length :voltage wave: ( ) cos( )Assume periodic condition: 2Lvt A t kzkL mωπ=±=Assume periodic condition: 2Mode density: ( )=EnergykL mLcπρν=EnergyPower flow: Transit Time1PLPLc c==//11BBhv k T hv k Thv hveeννΔ⎛⎞⎛ ⎞Δ=⎜⎟⎜ ⎟−−⎝⎠⎝ ⎠Lc c222211/11BBN Neehv k TRRPkT v i RRR R RRν⎝⎠⎝ ⎠⎛⎞⎛⎞⎛⎞ ⎛⎞=Δ= =⎜⎟⎜⎟⎜⎟ ⎜⎟⎜⎟⎜⎟⎝⎠ ⎝⎠2Equivalent mean square noise voltage: 44BN NNBRR R RRvkTRkTν⎜⎟⎜⎟⎜⎟ ⎜⎟⎜⎟⎜⎟++⎝⎠ ⎝⎠⎝⎠⎝⎠=ΔΔEE232 Lecture 19-6©2008. University of California24Equivalent mean square noise current: BNkTiνΔ=RNoise in p-i-n PhotodiodeNoises in p-i-n photodiodes: shot noise and thermal noise4kTΔ22 2,,4() () () 2Signal:BN N shot N thermalkTviv i v i v eIdvRΔ=+ =+22()Siv I=2Signal to noise ratio (SNR):4ISNRkT v=Δ42Note that the SNR improves with iBkT veIdvRΔ+ncreasing average photocurrent IEE232 Lecture 19-7©2008. University of
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