SIGNAL
Direct and Coherent Detection
E
I
ph
DIRECT
SIGNAL
beamsplitter
E
E
LOCAL
OSCILLATOR
I ph
0
COHE RENT
direct detection is linear in power
Iph = σP
and quadratic in field amplitude: Iph = σA ⟨ E 2⟩ /2Z0
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
1
Coherent gain
In coherent detection, signal and local oscillator fields are:
E = E exp i(ωt+ϕ), E0 = E0 exp i(ω0t+ϕ0)
thus Iph= (σA/2Z0) [⟨ E 2⟩ +⟨  E0 2⟩ + 2 Re ⟨ E0 E*⟩ ] =
= (σA/2Z0) {E2 + E02+ 2 E E0 ⟨ cos [(ω -ω0)t+ϕ-ϕ0]⟩ }
or,
Iph = I + I 0 + 2µ √(I I0)
compared to I of direct detection, we find a
coherent gain Gcoh= Iph/I= 1+2µ √(I0 /I)
µ is the coherence factor. When ω=ω0 detection is called
homodyne, while if ω≠ω0 we have heterodyne detection.
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
2
Coherence factor
µ=⟨ cos (ϕ-ϕ0)⟩ ranges from µ= 0 (uncorrelated phases of signal and
local oscill ator) , to µ= 1 (complete correlation).
Now consider homodyne detection (ω=ω0) and write ϕ as the sum
of a mean ⟨ ϕ⟩ and a random part ϕr: ϕ = ⟨ ϕ⟩ +ϕr
Developing µ,
µ = ⟨ cos(⟨ ϕ⟩ +ϕr -ϕ0)⟩
= cos(⟨ ϕ⟩ -ϕ0) ⟨ cos ϕr⟩ − sin(⟨ ϕ⟩ -ϕ0) ⟨ sin ϕr⟩ .
As ⟨ ϕr⟩ =0, also ⟨ sin ϕr⟩ =0 and if ϕr has a regular statistics. So:
µ = cos [⟨ ϕ⟩ -ϕ0] ⟨ cos ϕr ⟩ = cos ∆ϕ µΦ
Beating signal is multiplied by factor cos∆ϕ, that is, homodyne
detection is sensitive to the in-phase component with ⟨ ϕ⟩ =ϕ0; the
in-quadrature component with ⟨ ϕ⟩ = ϕ0+π/2 gives a zero output.
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
3
Phase fluctuations
The random part ⟨ cosϕr⟩ =µΦ describes relative phase fluctuations.
For ϕr small (<<1 rad), cosine is close to unity and its mean is ≈ 1;
for large ϕr (over 2π), cosine spans from -1 to + 1 and mean will be
≈ 0. For small ϕr <<1, developing cosine in series of ϕr:
µΦ = ⟨ cos ϕ r⟩ = ⟨ 1 - ϕr2/2! + ϕr4/4! +...⟩ ≈ 1− σΦ 2/2
we see that µ is connected to phase variance σΦ .
+π
+π
t
0
−π
−π
small ϕ r
t
0
large ϕr
1
1
t
0
0
t
-1
-1
<cos ϕr > ≈ 1
<cos ϕr > ≈0
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
4
Coherent S/N ratio
In direct detection, (S/N)2dir = I 2/ [2e(I+ Id)B + 4kTB/R]
and quantum limit is (S/N) 2dir/q = I /2eB for I>>Id+4kT/R.
In homodyne, signal is (σA/2Z0)2µEE0 , noise is sum of local
oscillator and signal shot-noises, plus Johnson noise of load:
2
[(σA/2Z
)2µEE
]
0
0
______________________________
(S/N) 2hom =
2e[(σA/2Z0)(E2+E02)+Id] B+4kTB/R
=
4µ2I I 0
____________________
2e(I+I0+Id)B +4kTB/R
dividing by I0 and letting IR=2kT/eR,
4µ2 I
_________________
(S/N) 2hom =
[2e(1+(I+Id+I R)/I 0] B
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
5
Quantum limited S/N
for I0>>I0q=I+ Id+I R, the quantum limit is always reached:
(S/N)2hom /q = 4µ2I /2eB
In coherent detection the Q-L condition is on local oscillator
amplitude, not on signal amplitude as in direct detection. Making
local oscillator I0>>I+Id+IR large enough, Q-L is reached, even at
weak signal levels.
Heterodyne detection follows the same arguments, but beating
signal is now at the frequency ω-ω0, so that
Iph = 2√(I I 0 ) cos [(ω-ω0)t+⟨ ϕ⟩ -ϕ0] ⟨ cosϕr⟩
(S/N) 2het = 2µφ2I I 0 / [2e(Ι+Ι0+Id)B + 4kTB/R]
i.e., it has a modest penalty - a factor of 2 (or 3dB) respect to
homodyne but does not require the phase adjustment.
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
6
Condition for coherent detection
Requirements:
H
H
H
phase matching of signal and local oscillator (for homodyne),
or beating will be reduced by a factor: cos(⟨ ϕ⟩ -ϕ0)
phase coherence, or signal will be reduced by: ⟨ cosϕr⟩ =µφ.
superposition of E and E0 on the PD with spatial coherence
or beating will be reduced by a factor:
µsp =∫AE(x, y) . E0*(x,y)dxdy /[∫AE(x, y)2dxdy ∫AE0(x,y)2dxdy ]1/2
superposition of E and E0 with polarization matching or signal
is reduced by:
E.E0/EE0=µpol
(E, E0 = Jones matrixes)
All previous expressions are generalized by using µφ →µφ µsp µpol
H
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
7
S/N, BER and photons /bit
n
N
N
1
0
N/2
threshold S
p(n)
0
"1"
threshold S
"0"
p(01)
ideal direct detection:
BER=e -N
for BER= 10-9: N=10 p/b
(S/N)2=10
ideal coherent detection:
threshold at N/2
and P(01)=P(10)
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
8
Photons per bit and modulation
• homodyne detection of amplitude modulated (ASK) signal:
BER = erfc N/2σN (N=number of photons per bit)
N=2(I .I0)1/2T/e=2(NsN0)1/2 σN=(2eI 0/2T)1/2T/e=N01/2
then,
BER = erfc √Ns
and for
BER=10-9 we get Ns=36 p/b
•homodyne detection of a phase-modulated PSK signal:
BER = erfc 2√Ns, and Ns=9 p/b
•heterodyne detection of a PSK-modulated signal:
BER= erfc√ 2Ns, and Ns=18
homodyne detection of a 4Φ−PSK modulated signal
BER = erfc 2√2Ns, and Ns=4.5
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
9
State-of-the-art receiver sensitivity
-40
-50
-60
er
mpl ifi
a
e
r
lp
EC
85
-30
x NEC88
AT&T 88 x NEC 89
x
FUJ 87x
x
x
x SEL87
NTT 97
1µ
x
bi t
NTT 92 36 ph / t
i
O
h/b
p
s
m
0
e
t
O NTT 89
1 / bi t
O
t sy s
h
dir ec
AT&T 85
4p
SEL 88 O. NTT 85
.O
O.
AT&T
85
ts
BT 85 O
l imi
t
n
er e
c oh
a
opt ic
10 µ
x
0.1µ
10n
1n
RECEIVER SENSITIVITY (W)
x DIRECT
O COHERENT
N
RECEIVER SENSITIVITY (dBm)
-20
0.1n
-70
2
4
10
0.4
1.0
TRANSMISSION RATE 1/T (Gbit/s)
OR TRANSMISSION BANDWIDTH 2B (GHz)
0.1
20
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
40
10
Balanced detector
I1
+
-
S
E1
I2
E
E
SIGNAL
2
E
0
LOCAL
OSCILLATOR
I1 = (1/2)(I 0+I)+√(I 0I) sin(ϕ-ϕ0), I 2 = (1/2)(I 0+ I)-√(I 0I) sin(ϕ-ϕ0)
S = I 1-I2 = 2√(I 0I) sin (ϕ-ϕ0)
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
11
Beamsplitter phaseshift
At a beamsplitter, the continuity condition of electric fields at the separation boundary requires that
the incident Ei is always the sum of reflected Er and transmitted Et fields:
Ei = Et + Er
(I)
where the underlines indicate rotating vectors. Also, in a l ossless beamsplitter power P is
unchanged upon splitting and, as P is proportional to E2, we have:
Ei2 = Et2 + Er2
(II)
To have both equations satisfied, the three vectors must lie on a right-angle triangle, as shown in
the fi gure below. Then, the angle - or phaseshift - be tween reflected Er and tr ansmitted Et vectors
is π/2 irrespective of the actual splitting ratio, while the angle ψ between incident and transmitted
fields increases from 0 to π/2 as Et decreases from Ei to 0 (or, R goes from 0 to 1). We can then
write, for the lossless beamsplitter:
_Er
_E i
_Er
_Et
P
E i
-
E
_t
ψ
Et = √(1-R) Ei eiψ,
Er = √R Ei ei(ψ−π/2)
For a lossy beamsplitter, Eq.(I) still applies, while (II) holds with the ≥ sign; then point P in
the figure shifts internal to the circle and the Er Et phaseshift becomes larger than π/2 (of an angle
p/2√[R(1-R)] where p is the loss).
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
12
Balanced detectors with input subtraction
+V
bb
E
1
-
E
+
2
S
- V bb
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
13
Coherent receiver with polarization diversity
+
-
GLAN
cube BS
BS
balanced
detector
SIGNAL
BS
GLAN
cube BS
-
balanced
detector
OSCILLATOR
+
LOCAL
∆f
+
freq
contr
≈
DEMOD
OUT
f/V
CONV
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
14
Two -frequency heterodyne receiver
beamsplitter
or circulator
ω
frequencystab. laser
two-frequency
source
transmitter
and receiver
telescope
ω+∆ω
+ -
acoustooptical
modulator
ω
balanced
detector
OUT
from:”Photodetectors”, by S.Donati, Prentice Hall 2000
15