EE 232 Lightwave
g
Devices
Lecture 19: Noises in Photoconductors
R di
Reading:
Y i 10.3-10.5
Yariv
10 3 10 5
Instructor: Ming C. Wu
University of California, Berkeley
Electrical Engineering and Computer Sciences Dept
Dept.
©2008. University of California
EE232 Lecture 19-1
Poisson Distribution
Poisson distribution:
a given event occurring in any time interval is distributed
uniformly over the interval.
The probability of n electrons arriving in a period T : is
n
n e− n
p ( n) =
n!
where n is the average number of electrons arriving in T
Properties of Poisson Distribution:
Mean = n
Variance = n
EE232 Lecture 19-2
©2008. University of California
Spectral Density Function
Random variable i (t ) consists of a large number of individual events
(e.g., single-electron photocurrent) at random time:
NT
i (t ) = ∑ f (t − ti ),
)
0≤t ≤T
i =1
NT
Fourier transform: IT (ω ) = ∑ Fi (ω )
i =1
1
Fi (ω ) =
2π
∞
∫
f (t − ti )e
−∞
∞
IT (ω ) = F (ω )
2
− iωt
2
NT
NT
∑∑
i =1
e −iωti
dt =
2π
e
− iω ( ti −t j )
j =1
∞
∫
f (t )e − iωt dt = e−iωti F (ω )
−∞
∞
= NT F (ω ) = NT F (ω )
2
2
N : average rate of electron arrival
Spectral density function:
S (v) = lim
T →∞
EE232 Lecture 19-3
8π 2 IT (2π v)
T
2
= 8π 2 N F (2π v)
2
©2008. University of California
Shot Noise
Shot Noise: Noise current arising from random
generation and flow of mobile charge carriers.
Current pulse due to a single electron moving at v(t ) :
ev(t )
ie (t ) =
d
Fourier transform: F(ω ))=
1 e
2π d
ta : arrival time, x(0) = 0,
ta
∫
v(t )e −iωt dt
0
x(ta ) = d ,
Small transit time ta , ωta 1 → e − iωt ~ 1
t
d
1 e a dx
1 e
e
⋅1 ⋅ dt =
F (ω )=
dx =
∫
∫
2π d 0 dt
2π d 0
2π
2
⎛ e ⎞
S (v) = 8π N ⎜
⎟ = 2eI
⎝ 2π ⎠
2
I = eN
iN2 (v) = S (v)dv = 2eIdv
EE232 Lecture 19-4
©2008. University of California
Thermal Noise (Johnson Noise)
• Fluctuation in the voltage across
a dissipative circuit element
(
(resistor)
)
• Caused by thermal motion of
charged carriers
EE232 Lecture 19-5
©2008. University of California
Thermal Noise Derivation
Consider two resistors connected by a
lossless transmission line of length L:
voltage wave: v(t ) = A cos(ωt ± kz )
Assume periodic condition: kL = 2mπ
L
c
Energy
Power flow: P =
Transit Time
hv ⎞
hvΔν
1 ⎛L
⎞⎛
P=
⎜ Δν ⎟⎜ hv / kBT
⎟ = hv / kBT
L c⎝ c
−1 ⎠ e
−1
⎠⎝ e
hv / k BT 1
Mode density: ρ (ν )=
⎛ 2 ⎛ R ⎞2 ⎞ 1 ⎛ 2 ⎛ R ⎞2 ⎞
P = k BT Δν = ⎜ vN ⎜
R
= i
⎜ ⎝ R + R ⎟⎠ ⎟⎟ R ⎜⎜ N ⎜⎝ R + R ⎟⎠ ⎟⎟
⎝
⎠
⎝
⎠
Equivalent mean square noise voltage: vN2 = 4k BTRΔν
Equivalent mean square noise current: iN2 =
EE232 Lecture 19-6
©2008. University of California
4k BT Δν
R
Noise in p-i-n Photodiode
Noises in p-i-n photodiodes: shot noise and thermal noise
iN2 (v) = iN2 , shot (v) + iN2 ,thermal (v) = 2eIdv +
4kk BT Δv
4
R
Signal:
iS2 (v) = I
2
Signal to noise ratio (SNR):
SNR =
I
2
4k BT Δv
R
Note that the SNR improves with increasing average photocurrent I
2eIdv +
EE232 Lecture 19-7
©2008. University of California