1 Moshe Nazarathy Copyright Lecture VI Introduction to Fiber Optic Communication Optical amplifiers (ch. 16 – part 3 “Notes”) …in which we study Optical

1Moshe Nazarathy Copyright

Lecture VIIntroduction to Fiber Optic Communication

• Optical amplifiers

• (ch. 16 – part 3 “Notes”)

…in which we study Optical Amplification

Moshe Nazarathy All Rights Reserved

Ver1


Optical Direct Detection


The quantum limit for ideal OOK optical transmission


The quantum limit for ideal OOK optical transmission (II)


The quantum limit for ideal OOK optical transmission (III)


The quantum limit for ideal OOK optical transmission (IV)


The quantum limit for ideal OOK optical transmission (V)


The quantum limit for ideal OOK optical transmission (VI)

=10

Ten average photons per bit

Twenty photons in the “one” bit

3 (1)10 0.5 ; 6.21 ; 3.1 /en

T TE n N N photon itP b =


Photon counting detection of optical binary transmission with two non-zero waveforms (I)

“On-pulse”

“Off-pulse”

Self-study


Photon counting detection of optical binary transmission with two non-zero waveforms (II)

Log-likelihood ratio Self-study


Photon counting detection of optical binary transmission with two non-zero waveforms (III)

Self-study


…where Poissonian Photon Statisticsand Gaussian Johnson noise meet…


I&D Detection in shot noise approximated as gaussian

0

( )T

dt

I&D filter

0

ˆ( ) ( )tN

d kk

i t eh t

TeN

1/ eTN

[ ]T TN Poisson 0

( )T

T t dt

Shot Noise

( ) ( )t P t

Approximate the Poisson as Gaussian

[ ] [ , ]T T T TPoisson N

2T T T TVarN N

approx.


Optical Amplification


Physical principles of optical amplification

• Optical Amplifier Types:– Erbium Doped Fiber Amplifiers (EDFA)– Semiconductor Amplifiers– Raman Fiber Amplifiers

• Principle: A collection of quantum objects (atoms, electron-hole pairs, etc.) are “pumped” by an external means (optical, electrical) into excited energy states.

• Stimulated Emission: Incoming photons induce additional photons of the same direction, frequency and polarization. The excited entities go down to the ground state

• Amplified Spontaneous Emission (ASE): the excited objects randomly relax to the ground state while emitting photons.

• Same effects as used in lasers (laser = amplifier + feedback)

16Moshe Nazarathy CopyrightMecozzi, ECOC ‘05

Optical Power in two-sided bandwidth Bin both quadraturesand one polarization

=

0 022

hh

( )outBs za HeP t

( ) ( )out inss GE tE t

( )insE t

+OA

( )aseE t

2( )ase

B HzE t

GOptical Power Gain

Effective Noise FieldRedefine E-field units such that P=|E|^2


2-sidedOptical

All statements here are per one polarization - The two polarizations fluctuations are i.i.d. Total power in both orthogonal polarizations is double

ase


Noise statistics of the optical amplifier

) ( )( )( inos ses

ta

u GEt tE Et

( )insE t

+

OA

( )aseE t

1 12 2

2

0( ) ( ( 1))outase Hz Hz

ase spP t E t n G h

1G

12

2

0( ) ) (( 1)aseE s sp

za e

HS E t n hG

Spontaneous noise factor

SIGNAL AWGN

Two-sidedPSD of the field CE(both quadratures)

*01

( ) ( ) ( ) ( 1) ( )aseE ase ase spHz

t t t t h t tE n GE

[Watt/Hz]

1spn

When E is in units of sqrt[Watt] then PPSD coincides with the PSD of the field: PSD=Var in 1 Hz; Var = mean sq of ZM r.v. When E is in units of [ (Volt/Meter)2/Hz ] the PSD and PPSD are merely proportional

G

Photonic Power Spectral Density (PPSD):

The ASE optical field noise process is approximated as white (and gaussian):

high gain amps

Effective Noise Field

0spn Gh



2

01 1( ) ( ) ( 1) ( )

H

out outase ase asz Hz

e spP P t E Gt hn

12

0 2

0( ) ( ) (1 1

2()

21)

2( )re im

ase ase aseE ase spE E Hz

NS S S E t n hG

12

0

2

0 0( ) ( ) ( 1)ase H

E ase sp spz

G GS n h hN E t n

Two-sided PSD of the real optical field and of each quadrature component

[Watt/Hz+]

) ( )( )( inos ses

ta

u GEt tE Et

( )insE t

+

OA

( )aseE t SIGNAL AWGN

G

Two-sided PSD of the optical field rms complex envelope:

Photonic PSD

00 ( ) ( ) ( 1)( )re imase ase ase

E spE EN S S hGS n

One-sided PSD of the real optical field as well as of each quadrature component

One-sided PSD rms complex envelope of the optical field:

2

001

2 ( ) ( ) 2 ( 1)aseE a e p

zs s

HN S E t G hn

summary

See next slide to justifythe I&Q stmts


Moshe Nazarathy Copyright

00 0( ) ( ) ( )Re 2 2( )2 cos sinj t

n t n t t tere imn t tn

0 / 2N

PSD0N

PSD0 / 2NPSD

Spectra of narrowband white noise and its I&Q and complex representation

0NPSD

( )C tN0NPSD

( )S tN( )n t02NPSD

0 / 2NPSD

Quadrature representation

From T1 ModCom


Optical Amplification spontaneous noise factor derivation



Optical Engineering characteristics of the optical amplifier (I)

we view the optical amplifier... as an amplifier, striving to derive some of its optical engineering characteristics, such as its noise figure and its RIN, as well as its CNR contribution to the output in a link which includes multiple optical amplifiers.

The spontaneous spectral optical power (mean optical power inWatts per one-sided Hz) at the output of the OA in one polarization is given by:

amplified emission

2-sided

2


2 0

20

0

( 1)

1(

1

)

1

(

1)

( )

outase

sp

G h

GG

P

h

G hnG

Self-study


Self-study


Optical Amplification signal-spontaneous and spontaneous-

spontaneous noise contributionsderivation


Optical Filter Electrical FilterPhoto-diode

oFlat B+eFlat B2

G

Canonical Optical Amplification Problem:Canonical Optical Amplification Problem:White Gaussian ASE, Flat o- and e- filters, CW signalWhite Gaussian ASE, Flat o- and e- filters, CW signal

24( ) ( 1) rectin spy

c e

ff

eS GP n G

h B

2 2

0 00

( ) 2 ( 1) tri rect 2 ( ) ( 1)sp spn nye

S n G e B n G eBB

f ff

Bf

c

e

h

Ideal responsivity:Set in the model of last slide:

2( ) rect[ / ]o of fH B

2( ) rect[ / ]e ef fH B ( ) ( )insS GP

CW Signal:

( ) ( 1)sp cnS n G h

ASE PSD:

( )s t( )n t

inP

s nPSD of terms:

PSD of term: n n

( )y t

elec

tr.

PS

DeB 0B0 / 2B

f

n n

s n[Olsson, `89]

Withoutproof



oFlat B+eFlat B2

G

Canonical Optical Amplification Problem:Canonical Optical Amplification Problem:Signal-spontaneous noise dominates over shot-noiseSignal-spontaneous noise dominates over shot-noise

24( ) ( 1) rectin spy

c e

ff

eS GP n G

h B

c

e

h

Ideal responsivity:

( )s t( )n t

inP

s nPSD of terms:

( )y t

elec

tr.

PS

D

eB 0B0 / 2B

f

n n

s n[Olsson, `89]

Its flat in-band level:

21 4

( 1)y

sin sp

c

eN GP n G

h 4 ( 1)in sp

sI

e GP n G 2 2 ( 1)

SHOT

s sp

N

eI n G

2 ( 1)SHO spTyN N n G

The Sig-spont. PSD is stronger than the output shot-noise PSDthat’d have been generated by the same CW input signal going throughan ideal OA with the same gain, then photo-detected with the same responsivity

2 ( 1)spn G

2( ) rect[ / ]o of fH B 2

( ) rect[ / ]e ef fH B

COMPARE SIG-SPONT. vs. SHOT NOISES


Conventional Derivation of s-sp and sp-sp noise

one-sided

00 /spN n Gh g one-sidedis


Derivation of s-sp and sp-sp noise

-



)



jointly




Derivation of s-sp and sp-sp noiseagain…

1 / 2ase

sES

12 2 / 2ase

ss ES rect

B

super

( )aseES

( )s sp

S

s n alias names

for the signal beating with the noise


Signal-spontaneous beat noise

Effective IM noise source…Multiply by the responsivity to yield the s-sp induced photocurrent noise



1 / 2ase

sES

Convolve this with itself

( )aseES


Now we must evaluate the autoconvolution:

Signal-spontaneous and spontaneous-spontaneous beat noises (VIII)

The one-sided spontaneous-spontaneous noise spectral density is then

| |

n n alias namesfor the noise beating with itself

2

2

1 / 2ase

sES

( )aseES

B

22 2

1

B

Area under this graph (multiplied with itself)

2B


Signal-spontaneous and spontaneous-spontaneous beat noises (IX)

| |

its peak value

2B

2B 2B


OA

Signal-spontaneous and spontaneous-spontaneous beat noises (X)

Consider an optical transmission situation involving an optical amplifier.

2B 2B


Optical Amplifier CNR and Noise Figure

THIS SECTION: SELF-STUDY(its content is partially treated in TA class)


Optical Amplifier CNR (I)

The OA internal noise figure of the amplifier is defined as the ratio of internal input and output CNRs

The input CNR (carrier-to-noise ratio) spectral density (in 1 Hz two-sided bandwidth) is referred to the ‘purest’ light, coherent light afflicted only by its inherent shot noise:

OA

Consider an optical transmission situation involving an optical amplifier.

without the effect of coupling


Here we have specifically labelled the CW power with the superscript “in”, and have expressed a generic optical shot noise as

the shot noise associated with the photon stream carrying total energy “charge”

To formally justify the optical shot noise expression , consider the photon arrivals’ process

Optical Amplifier CNR (II)

P


Optical Amplifier CNR (III)

----


Review the derivation why spectral density (=arrivals rate) for filtered Poisson process

Optical Amplifier CNR (IV)

----


Optical Amplifier CNR (V)

2B


Again the beat noise will only show up on the photo-detector, however we choose to represent it as if it is already existent in the optical domain. So, in a sense these are photocurrent mean square quantities, all referred back to the input optical domain by division by the responsivity.

The output CNR then consists of the signal spectral density over the output shot noise plus output s-sp contributions:

taking the ratio between the input CNR, and the output CNR….

Optical Amplifier CNR (VI)


Optical Amplifier noise figure (I)

----


This indicates that 2 2spF n 1spn When ideally

i.e. the ideal large-gain amp noise figure is 3 dB

Optical Amplifier noise figure (II)


PAM Detection

with optically pre-amplified receiver


Beat-noise-limited OA

• Let the ASE (amplified spontaneous emission) be the dominant noise component

• All other noise sources in optical detection may be neglected.

• Realistic assumption for a receiver with an optical pre-amplifier with sufficient optical gain and reasonable noise figure

• the thermal noise photocurrent is negligible relative to the signal-spontaneous noise component in the photocurrent.

• The ASE noise is usually modelled as AWGN


Conventional OSNR definition is not very precise…

(though related)


OSNR – relation to Symbol-SNR

_ _ _ _

_ _ _ _1_ _ 2.5

Avg power over symbol periodOSNR

Noi of both pose Variance inl GHz

(1) (0) (1)

_ _ _ _

/2 2

s s s

P Avg power over symbol period

TT

0_ _ _12.5 refNoise Variance in GHz N B12.5refB GHz(1) (1)

0 02

/ 2 1

4s s

ref ref

TOSNR

N B TB N

“Logical 1” symbol energy

Symbol period (inverse of baudrate)

2 factor due to two pol.

in one pol.

See next page for

OOK

ASE PWR in a reference 0.1 nm band

00 ( 1)sp G hN n

=0.1 nm at 1.5um




2

01 1( ) ( ) ( 1) ( )

H

out outase ase asz Hz

e spP P t E Gt hn

12

0 2

0( ) ( ) (1 1

2()

21)

2( )re im

ase ase aseE ase spE E Hz

NS S S E t n hG

12

0

2

0 0( ) ( ) ( 1)ase H

E ase sp spz

G GS n h hN E t n

Two-sided PSD of the real optical field and of each quadrature component

[Watt/Hz+]

) ( )( )( inos ses

ta

u GEt tE Et

( )insE t

+

OA

( )aseE t SIGNAL AWGN

G

Two-sided PSD of the optical field rms complex envelope:

Photonic PSD

00 ( ) ( ) ( 1)( )re imase ase ase

E spE EN S S hGS n

One-sided PSD of the real optical field as well as of each quadrature component

One-sided PSD rms complex envelope of the optical field:

2

001

2 ( ) ( ) 2 ( 1)aseE a e p

zs s

HN S E t G hn

summary

See next slide to justifythe I&Q stmts


two pol.

OSNR – relation to Symbol-SNR (II)2

(1) (1) 1

0 0

(1)

0

(

4

[

42

] / 2 1 [ ]

( 1)

)ss s

ref refre spf

dtJ T J TOSNR

W W WTB BN B N hHz H

G

t

H

E

z zn

(1)

0

4srefSymbolSNR TB OS

NNR

1010logdBOSNR OSNR

sp

insK

n

Shown Later

(1) (1) 2( ) [ ]ss E Jt dt

00 0( 1)sp spNW

h hHz

n G Gn

This one-sided PSD formula for the the real field in one polarization(=2-sided PSD for the CE fieldin one polarization)

but beware – the OSNR denom has noise for two polarizations

one pol.

related to NF

[ ]W

# of photonsper (1) symbol at OA input

1/ 2[ ]W

Redefine E-field units such that:


The pulse energy is then

Modeling optically-amplified direct detection

10

2

01

( ) ( ) ( 1) )(ase

outase ase spHzE

HzN S P t E t n G h

20 1 102 21

2

( )2 ase spHz

sided

NE t n Gh

Let the optical field (noiseless signal) response at the OA output due to a lone transmitted symbol

be expressed in terms of the unit symbol response

from the transmitted symbol to the OA output

0A

( )h t

0( ) ( )outs h tAE t

2 2 2

0( ) ( )out outss dt hE tAt

( )insE t

( )outsE t

+OA

( )aseE t

GFIBER CHANNEL

OPTTX

0A

( )h tUnit symbol response:

Two-sided PSD of each ASE quadrature


Modeling Optically-Amplified Direct Detection( )in

sE t +OA

( )aseE t

G

( )outsE t

FIBER CHANNELOPTTX

0A OF

( )f tinsK photons

per pulse

The symbol SNR at the OA output equals the ratio of the mean photon count at the OA input and the spontaneous noise factor:

0 sp

it nous sK

N n

OADD

2

0

0

0

2

012

( ) ( )

( )

1( )

/out outouts s

sp sp sHz

in

sid

s

s i

ed

p

n

s

ase

dtdt dt h h

Gh

E t P t

N n n

t

nE t

P

Proof:

2( ) ( )out out

s sE t P t

( ) ( )in outs sGP t P t

( )( )

outin s

s

P tP t

G2

( ) ( ) /out outs sE t P t g

2( )out out

s sE t dt THEOREM:

0 / 2Nis the two-sidedPSD of each ASE quadrature

( )in

sP twatts

more preciselyG-1: corr. factor G/(G-1)

G/(G-1)


Modeling Optically-Amplified Direct Detection

( )insE t

+

OA

( )aseE t

( )outsE t

G

FIBER CHANNELOPTTX

0A OF

( )f tPhotonsper pulse@OA input

0sp

ss

n

K

N

0A

2( )out

sE t dt“energy”in the communicationsense(mean square integralof the amp. waveform)

( )aseE t

( )outsE t

0A

OF

( )f t 0A( )h t 2

photo-det

sq. env-det

=“Symbol SNR”@ OA output

“Electrical” equiv. comm. system:

If OF is a matched filter: *( ) ( )f t h t

krje ~ [0, 2 ]Unif

For stat. analysis purposes we may set 0

Phase uncertaintyin laser source and fiber optical path length

( )h t

+

A responsivity gain would not matter:signal & noiseequally amplified

No electricalfilter assumedhence squaring& sampling may commute

0 ( )A h t

With MF, SamplerSNR=2*SymbolSNR

2

0/ 2 /k k sr r NVar

2

2

2

sp

s

k

k

nn

r K

kk nr

“Sampler SNR”at MF output:


( ) ( )jkk

s t Ae h t kT

(0)j jk k kk k

k

e A p k k T An ne pr

( ) ( ) ( )p t h t f t

re imk kn jn

2~ [0, ]N Indep.

2

0 / 2f N( )s t ( )f t

( )n t

PAM

( )h t( )kA

krje

2

ASYNCH (NON-COHERENT / ENVELOPE) DETECTION

( )s t ( )f t

( )n t

PAM

( )h t( )kA

kr

Coherent/Non-Coherent Bandpass channel – complex representation

SYNCHONOUS (COHERENT) DETECTION

ISI-Free

From S3

Goto S3->……->Return

kq


(0)kj

k kpr ne A

( ) ( ) ( )p t h t f t re imk kn jn

2~ [0, ]N Indep.

2

0 / 2f N

( )s t ( )f t

( )n t

PAM

( )h t( )kA

krje

2

ASYNCH (ENVELOPE) DETECTION


ISI-Free

a

Signal space (complex plane)

rekn

imkn

kr

(1)(0)Ap

(0) (1){ 0, 0}kA A A

OOK modulation format:

2

k kr

2

kk na

On:

Off:

On:Off:

2

k kn

s

22 2/q a

*(1) ( )h tA matched OF

Evaluate “sampler SNR” (Q-sq.):

†(0) ( ) ( ) ( ) (( ) )0 h f h t f t dt h f tp t

*( )h t *( ) ( )a b a t b t dt

2† †(1) (1)( ) ( ) ( )Ah t h t hA t (1) (1) 2 (11) 12 )( ( )( )) ( )(0 sa A A Ah t hp tA

constant gain factors in f(t) do not matter


(0)kj

k kpr ne A

( ) ( ) ( )p t h t f t re imk kn jn

2~ [0, ]N Indep.

2

0 / 2f N


ISI-Free

a

Signal space (complex plane)

rekn

imkn

kr

(1)(0)Ap

2

k kr

2

kk na

On:

Off:2

k kn

s

(1)sa

OOK

† 22 (1)0

2

0)

0(1/ 2 / 2 / 2sN N NAf h

( )s t ( )f t

( )n t

PAM

( )h t( )kA

krje

2

*(1) ( )h tA

2(1) (12( )

0

)

2 12

0( )

22 2

/ 2s s

s

inons

sp

qN N

KaK

n

"Sampler SNR” (Q-squared factor)=twice the “symbol SNR”(also from matched filtering theory): 0 sp

it nous sK

N n

Received effective # of photons at OA input in the on-pulse

per quadraturedimension



arekn

imkn

kr

(1)(0)Ap

2

k kr

2

kk na

On:

Off:2

k kn

s

OOK

( )s t ( )f t

( )n t

PAM

( )h t( )kA

krje

2

*(1) ( )h tA

2(2

2)2 onq

aK

2

2

" "

" "

k

k

b ON

b OFF

Decision law:

( ) 2Pr | Pr |k kon

eP b on b on

| ~k kon Ricena

( ) Pr | 1 ,kon

e

a bP b on Q

Right tail of Rician distrib.

Marcum-Q function

a b

Pr{ } ,k

a bb Qna

Rice pdf

(

0

1)2 s

N

shown last slide



arekn

imkn

kr

(1)(0)Ap

2

k kr

2

kk na

On:

Off:2

k kn

s

OOK

( )s t ( )f t

( )n t

PAM

( )h t( )kA

krje

2

*(1) ( )h tA

2(2

2)2 onq

aK

2

2

" "

" "

k

k

b ON

b OFF

Decision law:

( ) 2Pr | Pr |o fk k

feP b off b off

| ~kk off Rayl ign e h

2

22( ) ( / ) /22Pr | 0,f bk

bo f

e

bP b off Q e e

(Rice with a=0)

Alternatively, , is chi-squared with two DOFs, i.e. exponentially distributed k



arekn

imkn

kr

(1)(0)Ap

2

k kr

2

kk na

On:

Off:2

k kn

s

OOK

( )s t ( )f t

( )n t

PAM

( )h t( )kA

krje

2

*(1) ( )h tA

2(2

2)2 onq

aK

2/ / 2( ) boff

eP e ( ) 1 ,one

a bP Q

For high SNR it may be shown that the optimal threshold tends to half-waybetween on and off in the amplitude domain

/ 2b a

( )2 onqa

K

( ) 1 , / 2oneP Q q q 2 2/2 /2( ) /8qoff q

eP e e

2( ) ( ) /80.5 0.5 0.5 1 , / 2 0.5on off qe e eP P P Q q q e

/ / 2b q

( )( ) ( ) /40.5 1 2 , / 2 0.5onon on

eKP K KQ e



arekn

imkn

kr

(1)(0)Ap

2

k kr

2

kk na

On:

Off:2

k kn

s

OOK

( )s t ( )f t

( )n t

PAM

( )h t( )kA

krje

2

*(1) ( )h tA

2(2

2)2 onq

aK

2/ / 2( ) boff

eP e ( ) 1 ,one

a bP Q

Assume general threshold b a

( )2 onqa

K

( ) 1 ,oneP Q q q

2( ) / 2offeP e 2( ) ( ) ( ) /20.5 0.5 0.5 1 , 0.5on off q

e e eP P P Q q q e

/b q /b a

2 ( )( ) ( )0.5 1 2 , 2 0.5onon o K

enP Q eK K



( )( ) ( ) /40.5 1 2 , / 2 0.5onon on

eKP K KQ e

-940 ph/bit @ BER=10K

( )insE t

+OA

( )aseE t

( )outsE t

G

FIBER CHANNELOPTTX

0A OF

( )f tKphotonson average

0A

photo-det

( ) 9e80.2878 ph/bit P 1.00002 10onK

Cf. OOK with ideal photon counting: -910 ph/bit @ BER=10K More precise threshold optimization (file PhIM.nb & next page):

-939.6 ph/bit @ BER=10 @ 0.52K b a Literature quotes -938 ph/bit @ BER=10 K as it uses a discrete photon arrivals model for the photodetector (and evaluatesOA output statistics) rather than assuming a quadratic detection model like here



( )( ) ( ) /40.5 1 2 , / 2 0.5onon on

eKP K KQ e

( )insE t

+OA

( )aseE t

( )outsE t

G

FIBER CHANNELOPTTX

0A OF

( )f t( )onKPhotonsin “on”pulse

0A

photo-det

( ) 2onK K

/ 20.5 1 2 , 0.5eKP KQ eK

This may be alternatively derived applying the Normalized Q-factor Model(next slide):


ASYN PAM PWEP- Normalized Q-factor (NQF)

2

80.5 0.5 1 [ , / 2s

e OOK ASYNP BER e Q s s

( ) ( )

0 0 0

2 2(0)ˆ 2 2as s s

k

p aq q q

N N N

(i) ASYN On-Off Keying (OOK):

( )

2 22

ˆ 2/ 2(0 ) / 2

A a ad

aaA

…continued…

Assume MF: †,1

f h

†

( )

,ˆˆ 1 2 2a

f hq d

Review from S4Modern Comm

0

0

2 2s K

s

N

KqN

/ 20.5 1 2 , 0.5eKP KQ eK

2

8

2

0.5 0.5 1 [ , / 2q

e

q K

P e Q q q


• Exercise: Evaluate the BER for a two-level

ASK modulation format:

The upper level has K photons at the OA in

The lower level has photons

where is the extinction ratio.0 1 K


Example: duobinary with optical amplification

Exercise: Derive an expression for the BER of duobinary detection with optical amplification.Cf. [Bosco, et. al]


OVERFLOWS


Optical Engineering characteristics of the optical amplifier (III)

So we can relate the optical spectral power to the (two-sided) spectral density of the field:

“1Hz” always 2-sided in these slides

+OA

( )aseE t

G

spGnfor large gain

in effaseK

In a noiseless amplifier of gain if we inject a beam with Poisson statistics with two-sided spectral density

thus with the same average count per second (see next slide)then the ideally amplified input noise generates at the output the same average count as the actual noisy amp,

1

in effase H spz

K n G

in effas pe sn

1ase Hz spK Gn

Ideally, for the input referred effective uncertaintyrepresenting the ASE output noise is one photon per sec

1spn

Also think in terms of an effectivenoisy input field

( ) /aseE t G

2-s 2-s


Reminder: PSD of a homogeneous Poisson impulsive process

1

( ) ( )tN

ii

t t

e.g. a photons arrivals process for a CW beam(each photon arrival marked by an impulse)

Stationary (homogenous) Poisson process with rate (mean number of arrivals per second)

We have shown (in “Random Signals”) that the spectrum of this process is white. Assuming whiteness, its particular PSD constant level is readily derived as follows:

Let us filter the process through an LTI “spectral-analysis” filter with unity gain and 1 Hz two-sided bandwidth, or equivalently with unity energyThe PSD level equals the variance at the output of this filter.

220 0/ 2 / 2h N N

E.g., a one-second integrator, with impulse response

[ 1/ 2,1/ 2]( ) [ ] ( )h t rect t t 1has unity energy, unity DC gain, 1 Hz effective noise bandwidth.

Therefore its output variance, when is input into it, equalsBut the integrator output also describes the number of arrivals in 1 sec, hence is Poisson distributed, i.e. the mean count equals the variance.On the other hand, the mean count at the output equals the process rate

( )t 0 / 2N

2

0/ 2N We conclude: Two-sided PSD level = the mean count rate


Alternative derivation of the optical amplifier noise figure

THIS SECTION: SELF-STUDY






Optical Engineering Application: Optical Amplifiers Cascade

THIS SECTION: SELF-STUDY






Statistical optical mixing of stationary independent light

sources A useful analysis for optical measurements

Here it provides an alternative derivationof the signal x spontaneous beat noise

THIS SECTION: SKIP


Beating two light beams (I)Expressed in terms of the analytic signals and complex envelopes, the electric fields are:

1 2dE E E


Beating two light beams (II)


Beating two light beams (III)


Beating two light beams (IV)


Second-order statistics of the superposition of two light fields (I)

The resulting theory will be useful in the analysis of optical measurements as well as in figuring out the noise properties of optical amplifiers.


Second-order statistics of the superposition of two light fields (II)


Second-order statistics of the superposition of two light fields (III)

2


Second-order statistics of the superposition of two light fields (IV)

Stationary whenever the fields are

The time-averaged autocorrelation of the mixing term of the irradiance for atotally general superposition of two independent fields simplifies to:


Second-order statistics of the superposition of two light fields (V)

At this point, let us specialize to two independent stationary fields, in which case we have seen the fourth moment of the field breaking down into a product of two t-invariant autocorrelations. Application of the time averaging then has no effect in this case, yielding the following expression for the mixing term of the irradiance generated by the superposition of two stationary independent fields:

* *1 1 2 2

1Re ( ) ( ) ( ) ( )

2 a a a aE t E t E t E t


Second-order statistics of the superposition of two light fields (VI)stationary


Second-order statistics of the superposition of two light fields (VII)stationary


Second-order statistics of the superposition of two light fields (VIII)

repeat…

deterministic cross-correlation of two waveforms


Second-order statistics of the superposition of two light fields (IX)

0 0 0

Opt. freq.

0 RF freq.


Second-order statistics of the superposition of two light fields (X)

The total irradiance of the superposition of the two independent stationary fields.

We identify nine terms in its autocorrelation:

a a

*


Second-order statistics of the superposition of two light fields (XI)


Second-order statistics of the superposition of two light fields (XII)

independent

1

1-T/2

T/2

Self-study

The paper also treats the case when one or both of the beatingfields is nonstationary, as is the case when one or both of the fields are subject to modulation.


Signal-spontaneous beat noise derivation (I)

We have worked hard in section ? to derive the heterodyne beating term between two mutually incoherent optical signals that superpose together. Applying those results to the problem at hand,we view the amplifier output as the superposition of the signal and ASE fields:The total photocurrent is given by

sig-spont. noisespont-spont. noise


To begin with, let us take for simplicity

It is convenient to revert to the respective irradiances in the optical domain, doing away with thescaling constants

These should be considered as representations of the respective photocurrents, linearly mappedback for convenience to the irradiance domain.

Signal-spontaneous beat noise derivation (II)


??

Signal-spontaneous beat noise derivation (III)


Signal-spontaneous beat noise derivation (IV)


Apply theorem on the spectrum of the irradiance of the superposition of two independent fields:

Yielding for the cross-term:

Signal-spontaneous beat noise derivation (V)


XS ν

MID1

0+XS ν ν

0XS ν-ν 0+XS ν ν

MID2

00 0202 0

Signal-spontaneous beat noise derivation (VI)


Signal-spontaneous beat noise derivation (VII)


OA

Signal-spontaneous and spontaneous-spontaneous beat noises – summary

2B 2B


Signal-Spontaneous and Spontaneous-Spontaneous beat

noise


Signal-spontaneous and spontaneous-spontaneous beat noises (I)

We view the amplifier output as the superposition of the signal and ASE fields:The total photocurrent is given by

sig-spont. noisespont-spont. noise


Signal-spontaneous and spontaneous-spontaneous beat noises (II)

To begin with, let us take for simplicity

It is convenient to revert to the respective irradiances in the optical domain, doing away with thescaling constants

These should be considered as representations of the respective photocurrents, linearly mappedback for convenience to the squared optical field domain (or to the opticalpower domain).

Alternatively, we might work in the optical power domain: ,s sp sp spP P

It is as if there are optical powerfluctuations of these sizes…Effective IM noise terms…


Signal-spontaneous and spontaneous-spontaneous beat noises (II)

•Again the beat noise terms will only show up on the photo-detector. •However we choose to represent it as if it is already existent in the optical domain. •In a sense these are photocurrent mean square fluctuations, all referred back to the input optical domain by division by the responsivity.


Optically Pre-Amplified ReceiverOptically Pre-Amplified Receiver

)( ()s t n t

2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )e o os n on oy sS H S H S H S H Sf f f f f f f f f H f

22 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 ( ) ( )on n e o o en n ny f f f f f fS H S H S H H S df H

Introduce the cross-correlation of deterministic real waveforms:

( ) ( ) ( ) ( )f f fA B A B d

independent

in the general results*( , ) ( ) ( ) ( )o o eH H H Hf f f Set

PSD of term: n n

s nPSD of terms:


|.|2 e f

Ho

H ( )y t

Assume stationary, Gaussian:( )n t

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Documents

1 Moshe Nazarathy Copyright Lecture VI Introduction to Fiber Optic Communication Optical amplifiers (ch. 16 – part 3 “Notes”) …in which we study Optical