The Art of Modeling Optical Systems · 2009. 5. 19. · The Art of Modeling Optical Systems Taught...

The Art of Modeling

Optical Systems

Taught by

Curtis R. Menyuk

University of Maryland Baltimore CountyComputer Science and Electrical Engineering Department

Baltimore, MD 21250

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Short course notes are available for download at:www.photonics.umbc.edu/Menyuk/CLEO_Short-Course_Materials

The Art of Modeling

Optical Systems

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The Creation of Adam Michelangelo

Taught by

Curtis R. Menyuk

University of Maryland Baltimore CountyComputer Science and Electrical Engineering

DepartmentBaltimore, MD 21250

Whether creating something new…

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The Art of Modeling

Optical Systems

Taught by

Curtis R. Menyuk


Baltimore, MD 21250

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Four Horsemen of the Apocalypse Albrecht Dürer

…or solving a problem…

The Art of Modeling

Optical Systems

Taught by

Curtis R. Menyuk


Baltimore, MD 21250

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…it is an art!BUT: there are “recipes”

…it is more like cooking than painting

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created by

C. R. MenyukJ. Hu

M. A. TalukderD. A. CasaleR. Weiblen

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With special thanks to E. Ghillino, R. Scarmozzino, and… the whole Ossining, NY RSOFT design group

Defining a model’s purpose— Predicting a new phenomenon— Explaining an experiment— Designing an experiment— Designing a system

Determine the time and length scales— Lens design vs. optical filter design vs. nanowire

(dimensions vs. λ)— Power models vs. full time-domain models

(System response time vs. data variation time)

Creating a model: The key steps

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3

Defining a problem’s mathematical character— Propagation problems— Boundary-value problems

Defining the equations / algorithms— Partial differential equations; ordinary differential equations— Split-step method; finite-difference; finite-element

Choosing the software— Homegrown vs. freeware vs. commercial

Creating a model: The key steps

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What exactly do you want to accomplish?THIS IMPACTS

— How accurate does it have to be?— How many cases do you need to study?— How quickly do you need it to run?

Some Examples …

I. Defining a Model’s Purpose:

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4

Solitons in optical fibers— The nonlinear Schrödinger equation can be used to model

propagation in optical fibers.— The split-step method can be used to solve this equation.

Soliton formation with birefringence— The coupled nonlinear Schrödinger equation

Self-similar oscillations in hydrogen gases— The Raman equations in gases

Predicting a new phenomenon

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Self-induced transparency modelocking— Maxwell-Bloch equations— A completely new way to do modelocking

Physical explanation is critical.Analytical solutions are important.Low accuracy is sufficient.

Predicting a new phenomenon

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5

Establishing the basic equation (nonlinear Schrödinger equation)

— Effects of dispersion and Kerr nonlinearity affect the propagation

Find an analytical solution (single soliton)

— Physical insight: nonlinearity and dispersion balance⇒ No spreading due to dispersion!

Example: Solitons in Optical Fibers

( ) ( ) ( ) ( )2

2

2

, ,1 , , 02

u z t u z ti u z t u z t

z tβ γ

∂ ∂′′− + =

∂ ∂

( ) ( ) ( ) 22 2 2, sech exp 2 ;u z t A t i A z Aτ γ β γτ′′= =

11

T / T0

z / z0

Pow

er (

a.u.

)

100

10−6−50 50

0 1000

Computational solutions— FFT split-step method— Solitons are generated from

nearby initial conditions

Example: Solitons in Optical Fibers

12

6

Re-polarization in recirculating loops— Experiment: 100 km recirculating loop with polarization controllers— Simulation: coupled nonlinear Schrödinger equation with randomly

varying birefringence

Spectrum of super-continuum generation in optical fibers— Experiment: High-power laser; special fibers— Simulation: coupled nonlinear Schrödinger equation with the Raman

effect

Explaining an Experiment

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Relaxation oscillations in short-pulse lasers— Experiment: Modelocked Ti:sapphire laser; modulated pump source;

RF Spectrum Analyzer— Simulation: Ginzburg-Landau equation with higher-order dispersion

and spectral gain variation; → linearization and reduction to ODEs

(ordinary differential equations)

Models should contain the essential physics (no more, no less)— Verification may be needed (checking against more fundamental models)

Accuracy should be consistent with the measurements

Explaining an Experiment

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7

Standard model (Ginzburg-Landau equation):

Contains:

Example: Relaxation Oscillations in Short-Pulse Lasers

( ) ( ) ( ) ( )2 2

2

2 2 2

, 11 , ,2R sl g

u T t DT i i l g i u T t u T tT t t

θ γ δ⎡ ⎤⎛ ⎞∂ ∂ ∂

= − + − + + + +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ Ω ∂⎢ ⎥⎝ ⎠⎣ ⎦

( ) 2011 ,

S R

g g u T t dtP T

∞

−∞

⎡ ⎤= +⎢ ⎥

⎣ ⎦∫

1 2 3 4 5

6

15

1. phase offset 4. Kerr effect2. chromatic dispersion 5. fast saturable absorption3. spectral gain variation 6. slow saturable gain

+ Effects of continuous radiation(partial differential equation)

BUT: Experiments show

Time (µs) Time (µs)0 40 40

Inte

nsity

(a.u

.)

5.75

5.15 Fre

quen

cy s

hift

(a.u

.)

0

30

4.85 W4.90 W4.95 W

4.78 W

4.78 W

4.85 W

4.90 W

4.95 W

0

0

0

0

0

0

0

0

0

What we learn• We must include gain dynamics.• We must take into account frequency pulling.

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8

A minimal modification to the standard model:

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 3 3

2

21

2 3

21

2,

, ,2 6R sl

r irg iru T t DT i i l i u T t u T

t t tt

T tθ γ δ

⎛ ⎞∂ ∂ ∂+ − −⎜ ⎟⎜ ⎟∂ ∂

⎡ ⎤∂ ∂= − + − + + +⎢ ⎥

∂ ∂⎢ ⎥⎣ ⎦∂⎝ ⎠

( ) 20 1 , ' 'f f S R

g gdg g u T t dtdT P Tτ τ

∞

−∞

−= − ∫

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BUT: Experiments also showContinuum radiation is negligible!

What we learn• We only need solve an ordinary differential equation.ddt

=− ⋅ +v v SA

perturbationcouplingcoefficients

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

θτϖgw

v

energy

gain

central frequency

central pulse time

phase

Assumptions:• central frequency doesn’t drive energy or gain• energy and gain do not affect the other parameters

0 0 00 0 0

0 00 0 00 0 0 0

ww wg

gw gg

w g

w

w

A AA AA A AA AA

ϖ ϖ ϖϖ

τ τϖ

θ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A

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9

S

D

DD

D

S

2.8 nm OBF

AO

LSS

AOTx RxW

FGIS

IS

LSS

Input scrambler

Loop-synchronous scrambler

Polarization controller

SMF

D DSF

S

Designing an Experimental System

Example: Recirculating Loop Design

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this pieceis critical

This model is appropriate for digital systems with optical noiseloading !

Example: Recirculating Loop Design— Receiver Design

Each box corresponds to a physical model (with access to key parameters)

Performance measure: BER at the receiver (or system Q)

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10

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Example: Dual Injection-Locked Opto-electronic Oscillator

RFAmplifier

RFAmplifier

RFFilter

Photodetector

RFFilter

Photodetector

Laser

Laser

OpticalModulator

OpticalModulator

RFCombiner RF Phase-

Shifter

Long Optical Fiber

RF Output

ShortOptical Fiber

Master OEO

Slave OEO

AGAIN: Each box corresponds to a physical model

Performance measures: phase noise, spur height

High accuracy is needed for the performance measure— not the components

Efficient algorithms are critical

Validation is essential and one of the purposes of the experiments

So, how do you do this …

Build it up over many years (recirculating loop)

Get someone to pay for it (opto-electronic oscillator)

Use already-available software (non-commercial; commercial)

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11

Optical Communication Systems

Atomic Clocks

High accuracy of performance measures is critical

Well-verified code suites are used

An experienced design team is essential

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Designing a Commercial System

A critical and often neglected step !!

II. Determine the Time and Length Scales

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Neglecting this step can often lead to wasted computer timeand personal time!

Carrying out this step not only saves time, but can leadto valuable insights!

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Basic solution procedure:1. Hold fixed; solve for 2. Average over ; solve for

The method dates back to the 19-th century— was first used in celestial mechanics

See H. Poincaré: Les méthodes nouvelles de la mécanique céleste

dθ1dz

= f (θ1,θ2 );dθ2dz

= εg(θ1,θ2 )

A critical and often neglected step !!

1θ2θ

1θ 2θ

II. Determine the Time and Length Scales

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SJ

M

Length ScalesOptical communication systems

Light wavelength1 µm10 µm

100 µm1 mm

10 mm100 mm

1 m

100 m10 m

1 km

100 km10 km

1 Mm10 Mm

100 Mm

Core diameter

Pulsedurations

Polarizationbeat length

Attenuation length

Nonlinearlength

Fiber correlationlength

Dispersionlength

FLAGtrans-Atlantic

Manakov-PMDapproximation

Slowly varyingenvelopeapproximation

Maxwell’s equations

land link

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Length ScalesLight guiding structures (lenses, mirrors, waveguides)

1 m –100 mm –10 mm –

1 mm –100 µm –

10 µm –1 µm –

100 nm –10 nm –

1 nm –

light wavelength

ray optics

average index approximation

quantum-well structures

Maxwell’s equations paraxial wave equations

holey fibersnano-wires

meta-materialsstandard optical fibers

semiconductor waveguides

/ 0.01n n ≤/ 1n n ≈

E(x)

n(x)

E(x)n(x)

lenses and mirrors

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Example: The paraxial approximation( )

Starting point: Helmholtz equation ( )

Exact for TE-waves in slab waveguides

Next step: the slowly-varying envelope approximation

2 22

02 2

( , ) ( , ) ( ) ( , ) 0E z x E z x k n x E z xz x

∂ ∂+ + =

∂ ∂

n(x)

k

zkxk

propagation along zz

/ 0.01n n∆ ≤

0 0 /k cω=

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0( , ) ( , ) exp[ ( ) ]E z x u z x i zβ ω=

2 22 2 202 2

( , ) ( , ) ( , )2 [ ( ) ] ( , ) 0u z x u z x u z xi k n x u z xz z x

β β∂ ∂ ∂+ + + − =∂ ∂ ∂

Length Scales

x

z

14

29

2 22 2 202 2

( , ) ( , ) ( , )2 [ ( ) ] ( , ) 0u z x u z x u z xi k n x u z xz z x

β β∂ ∂ ∂+ + + − =∂ ∂ ∂

0

Key physical insight: | | | |x zk k

Final step: the paraxial wave equation

22 2 202

( , ) ( , )2 [ ( ) ] ( , ) 0u z x u z xi k n x u z xz x

β β∂ ∂+ + − =∂ ∂

Using this equation leads to larger z-steps!

Length Scales

n(x)

k

zkxk

propagation along zz

x

z

Time ScalesExample: Average power models in optical fiber

communications systems

1 2 3 N

…

λ

Wavelength-division-multiplexed channels

In each wavelength channel, the signal (10 Gb/s) varies in 100 psThe amplifier (EDFA) responds in 1 ms and couples the channels.

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g(t)

1 ms

Amp

100 ps

signal EDFA

Simulation of both behaviors simultaneously is not feasible

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Time ScalesExample: Average power models in optical fiber

communications systems

1 2 3 N

…

λ

Wavelength-division-multiplexed channels

Use the average power in each channel to calculate the gainUse the calculated gain to determine the evolution of the bits

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g(t)

1 ms

Amp

100 ps

signal EDFA

The solution:

Time Scales

A Classic Example: Stokes ParametersPolarization states often change rapidly compared to detector/amplifier response times

Astronomical sourcesCommunication signals

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16

Time Scales

A Classic Example: Stokes ParametersGiven the field vector

we average over time to obtain four quantities

and define the degree-of-polarization

ˆ ˆ( ) ( ) cos[ ( )] ( ) cos[ ( )]x x y yt A t t A t tδ δ= +E x y

2 20

0

1lim [ ( ) ( )] ,T

x yTS A t A t dt

T→∞= +∫

2 21

0

1lim [ ( ) ( )] ,T

x yTS A t A t dt

T→∞= −∫

20

2lim ( ) ( ) cos[ ( ) ( )] ,T

x y x yTS A t A t t t dt

Tδ δ

→∞= −∫

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2lim ( ) ( )sin[ ( ) ( )] ,T

x y x yTS A t A t t t dt

Tδ δ

→∞= −∫

2 2 21 2 3 0( ) / 1S S S S= + + ≤

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These time-averaged quantities are often all that can be measured… and all that should be calculated!

Example: Optoelectronic oscillators

LaserOptical

Modulator

Photodetector RFfilterRF

Amplifier

RFCoupler

Long Optical Fiber

Master OEO

SlaveOEO

Four time scales:Period of light: 5×10−15 s (1.5 µm light)

Period of RF: 10−10 s (10 GHz)

Period of round-trip: 3×10−5 s (6 km loop)

Scale of phase noise: 10−3–1 s

One must average over the first two scales

It is often useful to average over the third

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Time Scales

17

III. Defining the Mathematical Character

Propagation ProblemsLinear Boundary-Value problems (mode-solving)Nonlinear Boundary-Value problems (iterative)Problems with randomness

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Optical fibers with nonlinearity— Split-step Fourier transform methods

Waveguides with small index differences— Beam propagation methods

Waveguides with large index differences or complex 3-D structures— Finite-difference time-domain methods— Finite-element time-domain methods

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Propagation Problems

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In all cases, you will:1. Specify the initial conditions in space or time2. Discretize your equations3. Propagate repeatedly over small steps

Example: Finite-difference beam propagation in a slab waveguideBasic equation: The paraxial wave equation

22

02

( , ) ( , )2 [ ( ) ] ( , ) 0u z x u z xi k n x u z xz x

β β∂ ∂+ + − =∂ ∂

1. Specify the initial conditionsExample:

n(x)

d

37

2 2( 0, ) exp( / )u z x x w= = −


38

1 1 1 12

2 2 20

( , ) ( , ) ( , ) 2 ( , ) ( , )( )

[ ( ) ] ( , ) 0

l m l m l m l m l m

m l m

u z x u z x u z x u z x u z xiz x

k n x u z x

β

β

+ − + −− − ++∆ ∆

+ − =

2. Discretize:Example:

,z l z x m x→ ∆ → ∆


3. Propagate repeatedly over small steps

Cautionary note: There are much better (but more complicated) iterative schemes, in particular Crank-Nicholson.

[ ]1 1 1 12

2 2 20

( , ) ( , ) ( , ) 2 ( , ) ( , )( )

[ ( ) ] ( , ) 0

l m l m l m l m l m

m l m

i zu z x u z x u z x u z x u z xx

i z k n x u z x

β

ββ

+ − + −

∆= + − +

∆∆

+ − =

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Linear Boundary-Value Problems(Mode-Solvers)

In all cases, you will:1. Specify the solution on the boundary

You will then:2(a). Discretize your equations

OR2(b). Substitute a set of basis functions into your equations

In either case:3. You solve the resulting matrix equations

39

Example: Modes of a slab waveguide

Basic equation:

1. Specify the solution on the boundaryends of your simulation

2(a).

2(b). Substitutematch u(x) and du(x)/dx at the slab boundaries

3. In either case, you have an eigenvalue or matrix problem:— These problems have been extensively studied— Highly efficient algorithms exist— Use canned routines

22 2 202

( ) [ ( ) ] ( ) 0d u x k n x u xdx

β+ − =

( , ) 0,u z x x± ±= =

2 2 21 102

( ) 2 ( ) ( ) [ ( ) ] ( ) 0( )

m m mm m

u x u x u x k n x u xx

β+ −− + + − =∆

2 2 2 1/ 20( ) exp{ [ ( ) ] };u x k n x xβ= ± −

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Linear Boundary-Value Problems(Mode-Solvers)

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Raman amplification in optical fiber communication systems— Power is specified at both ends of the fiber— Shooting or relaxation method must be used— Computational cost is 10×–20× larger than for propagation in one

direction

Modelocked laser pulses— Pulse should reproduce itself after one roundtrip— Can propagate for many round trips— Acceleration methods can reduce the computer time

Iterative methods must be used in all cases!

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Nonlinear Boundary-Value Problems

Time scales: Short vs. LongExample: Optical Fiber Communications Systems— Short: amplifier spontaneous emission noise

Each bit is affected in a different way

Noise power N2 (spontaneous emission)∝

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IV. Dealing With Randomness

... RxTxAmp

N2

N1

Amplifier produces gain G

Gain = (stimulated emission) – (stimulated absorption)

12 NNG −∝

Amp

21

Example: Optical Fiber Communication systems— Long: randomly varying birefringence

leads to randomization of the polarization state of light— scale length = 1–100 meters

Nearby frequencies are randomized differently— scale length = 10–10,000 km

The differential randomization leads to PMD!

)10~/( 7−∆ nn

Each bit is affected in the same way

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Time Scales: Short vs. Long

Why does this matter?Short time scale:

Long time scale:

The second type is usually much harder to deal with!

Ergodicity appliesYou can sum over bits

No ergodicityYou must sum over fiber realizations

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Time Scales: Short vs. Long

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Communications systems:The amplifier noise is white over the bandwidth of interest

Oscillator systems (including lasers):The noise include flicker (1/f ) noise and its integrals

White phase : α = −2Flicker phase: α = −1White frequency: α = 0Flicker frequency: α = 1Random-walk frequency: α = 2

Modeling 1/f noise is non-trivial!

Freq

uenc

y no

ise

(loga

rithm

ic)

Frequency

α = –2

α = –1α = 0

α = 1

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Noise Types: Environmental vs. White

α = 2

Modeling 1 / f Noise

Two basic approaches:Use an FIR filter on white noiseSum a set of white noise processes

Flicker or 1/f or pink noise appears to be universal in electronic and oscillator systems!

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23

Monte Carlo (Statistical):+ Includes all the physics, including nonlinearity+ Straightforward to implement− Can require many realizations for convergence; slow

Deterministic: + Can lead to analytical results or run quickly− Can only include simple nonlinearities− Require considerable insight to be effective

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Monte Carlo vs. Deterministic Methods

Monte Carlo Methods

48

Standard:+ Every realization has equal weight+ Easy to monitor convergence as the number of realizations increases− Can require many realizations; rare events are hard to detect

Fixed-Bias Importance Sampling: + Can require orders-of-magnitude fewer realizations− The variance must be carefully monitored to avoid errors− Efficient implementation requires physical insight

Adaptive Importance Sampling (Multi-canonical): + Requires less a priori knowledge than fixed-bias methods− Implementation is non-trivial− Error-monitoring is difficult

24

Importance-sampling methods are under-utilized in optics!

They enable the solution of problems that cannot be solved in any other way.PMD-induced outage probabilities in optical communications systemsNonlinearly-induced increases in error rates in optical communications systems

49

Monte Carlo Methods

Deterministic Methods

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These require specially adapted techniques in all casesExamples:

Optical fiber communications systems— additive white Gaussian noise approximation— neglect all nonlinear signal-noise interactions in transmission

Beam spreading in a turbulent atmosphere— Linear, but complex multi-dimensional calculations— Approximations are used to calculate the integrals

Effect of noise on modelocked laser pulses— transform to a “soliton” basis in which the noise impact on the pulses

is linear (energy, central frequency, central time, phase)

25

Verification with Monte Carlo simulations is important!… and often missing!!

Verification is one of the key uses of biasing Monte Carlo simulations.

…and one of the key uses of deterministic methods is to verify both standard and importance-sampled Monte Carlo simulations

Mutual verification makes both approaches reliable!

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Deterministic Methods

V. Verification vs. ValidationVerification: Checking your code or theory against

another code or theory

Validation: Checking your code or theory against anexperiment

52

They are not the same!They are both important!One is not a replacement for the other!

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V. Verification vs. ValidationVerification: Checking your code or theory against

another code or theory

Validation: Checking your code or theory against anexperiment

53

Verification answers the questions:Are you really solving the equations that you think that you are solving?Are your algorithms appropriate?Do you have bugs?

Validation answers the questions:Are the equations that you are solving right?Do you have all the important physics in your model?

Verification Procedures

54

Compare to simple limits / analytical solutionsExamples:

Solitons in the nonlinear Schrödinger equation— single solitons should propagate without change— special initial conditions: should return

periodically to the same shape

Probability density function for 1s and 0s in a receiver— with additive, white Gaussian noise, these are -distributed

Phase noise spectrum in an opto-electronic oscillator— Lorentzian dependence of the power spectral density

2χ

sech( / )τ=u NA t

27


55

Two independent codesHome-grown vs. home-grown (independently-written!)

— used to eliminate bugs in key code elements— Example: propagation solvers in optical communications codes

Home-grown vs. freeware / commercial— useful… but how sure are you that the comparison software is bug-

free? …and that you are using it correctly?

Commercial vs. commercial?— allows you to check that you are using the codes properly


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Two different algorithmsExample: Deterministic vs. Monte Carlo in random problems

This comparison is typically very important in random problems!

Monte Carlo methods often suffer from poor convergence

Deterministic methods often suffer from inaccurate approximations…and bugs!

28


57

Two different algorithmsExample: Uni-directional vs. bi-directional propagation in

modelocked lasersUni-directional:

+ focuses on narrow time window; computationally efficient

− ignores backward-going waves

Bi-directional: + includes backward-going waves;

feedback− computationally longer;

harder theoretical

M1

Gain element

Pulse

M2Backwardradiation

Time window

Some added points:

58

Bugs that don’t appear in one regime may appear in another!

Check, check and re-check!

Don’t cut corners on verification!

Verification

29

Validation

59

Signal

A

A

AA

Examples: Recirculating loop— initial model ignored gain

saturation in the amplifiers— gain saturation is critical in

explaining stability of the signal

Receiver model— initial model timing jitter

(okay at 10 Gbs)— including timing jitter is

critical at 40 Gbs

PDMod

RF Amps

optical signalelectronic signal

Long fiber

Laser

60

Examples: Ti:sapphire lasers— initial model ignored gain

dynamics— gain dynamics is needed to explain

observed relaxation oscillations

Opto-electronic oscillator— initial model ignored amplifier

saturation— amplifier saturation is critical to

explain the observed noise power

Validation

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VI. Choosing the Software

61

Home-grown vs. freeware vs. commercialA key point: Nobody does everything on their own!

We all use operating systems— Microsoft, Apple, Unix

We all use word-processing programs— MS-Word, TeX/ LaTeX

Most of us use high-level programming languages— MatLab, Mathematica, MAPLE

Choosing the Software

62

Home-grown vs. freeware vs. commercialA key point: Nobody does everything on their own!

The key elements are:Easy to verifyHard to write Let someone else do it!Many users

A less obvious example:Mesh generation in a finite-element solverAgain: Let someone else do it!

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63

Choosing the SoftwareHarder to verify, but widely used:

Matrix solving routines: Linpack, Eispack → LapackFast Fourier transform: FFTWStandard special functions: Bessel functions, Legendre polynomials

Use implemented versions! GSL = General Scientific Library

64

Choosing the SoftwareHarder to verify, and not as widely used:

Mode solversPropagation solversOptical fiber communications systems design softwareContinuous wave laser designLens and mirror design

BUTCommercial implementations exist

What should you do?… Decisions, decisions…

32

65

Choosing the SoftwareSome criteria to consider:

Do you have the time, ability, inclination to write and debug your own software? Do you want to document it? And verify it?Do you have the resources ⎯ time, people ⎯ to write and maintainyour own software? And document it properly?— there are hidden costs (recompiling codes when systems change)— and advantages (complete knowledge of the algorithms, training)

Does the software have enough flexibility to meet current and futureneeds? Can you integrate component and system design if needed?Does the software provider tell you the algorithms they are using?

This is critical for mode solvers and propagation solvers!Ease of use: Do you want a GUI?How much support do you need? …And will you get from the provider?

66

Choosing the SoftwareSome criteria to consider:

Can you wrap in your own software?— e.g., introduce MatLab code or C code

Can you include “black box” parameters?— e.g., S-parameters from measurements

How careful do you want to be about verification?But verification is always crucial!

33

67

Choosing the SoftwareNot commercially available / freeware is hard to find:

Importance samplingModelocked laser design

You are on your own!⎯ A point to consider when deciding what to make or get!

68

Finding the SoftwareUseful WEB sites:

Lens and illumination design:— CW lasers, integrated optics, thin films— based primarily on ray optics

www.optenso.de/links/links.html(link provided by: Optical Engineering Software)

— Other companies in this space:Zemax: www.zemax.comOptical Research Associates: www.opticalres.com

34

69

Useful WEB SitesOpto-electronic devices:

— semiconductors, lasers, solar cellswww.nusod.org/inst/software.html(site provided by: NUSOD/Joachim Piprek)

Waveguide modeling / Optical Fiber Communications Systems:— Optical systems components, system modeling— Solutions of the wave equations (not ray optics)— Input-output modeling of systems

www.optical-waveguides-modeling.net(link provided by: Natalia Litchinitser)

… The alphabet soup !FEM = Finite element method

● This method spatially discretizes Maxwell’s equations of the Helmholtz equation. It is mostly used for mode-solving, although it can also be used for propagation. It can deal with arbitrary geometries.

FDTD = Finite difference time domain● This method spatially discretizes Maxwell’s equations. It is mostly

used for propagation, although it can also be used for mode-solving. It can deal with arbitrary geometries.

BPM = Beam propagation method● This implements the paraxial approximation for either Maxwell’s

equation or the Helmholtz equation using spatial and/or temporal discretizations.

Waveguide modeling software

70

35

Split-step Fourier● a portion of the equation is solved in the Fourier domain

The alphabet soup

( , ) ( , ) exp( )u z u z t i t dtω ω∞

−∞

= ∫

Example: An optical fiber with dispersion and nonlinearity

Time domain:

2

0

( , ) 1 ( ) ( , ) | ( , ) | ( , )2

u z ti B t t u z t d t u z t u z tz

∂ γ∞

′ ′ ′= − −∂ ∫

Fourier transform:

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2

0

( , ) 1 ( ') ( , ') ' | ( , ) | ( , )2

u z ti B t t u z t d t u z t u z tz

∂ γ∂

∞

= − −∫

( , ) (0, ) exp ( )2iu z u B zω ω ω⎡ ⎤= −⎢ ⎥⎣ ⎦

● The first term is solved most easily in the Fourier domain

( , ) 1 = ( ) ( , )2

u z ti B u zz

∂ ω ω∂

● The second term is solved most easily in time domain2( , ) (0, ) exp[ | (0, ) | ]u z t u t i u t zγ=

Time domain:

72

Split-Step Fourier Method

36

Discretizing in space and time, one offsets the time and frequency evaluations

One evaluation per step yields accuracy2( )z∆

This method works well in optical fiber propagation problems because and are smooth in t .( )B t ( , )u z t

Caveat: Fourier methods do not deal well with abrupt changes!

73

Split-Step Fourier Method

Frequency Frequency Frequency Frequency Frequency Frequency

Time Time Time Time Time Time TimeIFT IFT IFT IFT IFT IFT

FFTFFTFFTFFTFFT

∆z

Another possibility…● Time-domain split-step method in optical fiber simulations

— has advantages for long bit strings; abrupt changes— takes advantage of the smoothness and broad bandwidth of the

dispersion curve

74

The alphabet soup

37

Crank-Nicholson● A finite-difference approach, commonly used with BPM that yields

implicit equations2

2 2 202

( , ) ( , )2 ( ) ( , )u z x u z xi k n x u z xz x

β β∂ ∂ ⎡ ⎤+ + −⎣ ⎦∂ ∂

1 1 1 1 1 12

( , ) ( , ) ( , ) 2 ( , ) ( , )22( )

l m l m l m l m l mu z x u z x u z x u z x u z xiz z

β + + − + + −− − ++∆ ∆

1 12

( , ) 2 ( , ) ( , )2( )

l m l m l mu z x u z x u z xz

+ −− ++∆

[ ]2 2 20 11 ( ) ( , ) ( , )2 m l m l m

k n x u z x u z xβ +⎡ ⎤+ − +⎣ ⎦

,z l z x m x→ ∆ → ∆

75

The alphabet soup

Crank-Nicholson● A finite-difference approach, commonly used with BPM that yields

implicit equations

— This approach leads to a sparse, implicit matrix!— Highly efficient techniques exist to solve these matrices!

76

The alphabet soup

38

No endorsements intended!“Full-service” Providers

RSoft (www.rsoftdesign.com)Optiwave (www.optiwave.com)VPI (www.vpiphotonics.com)

— Optical component design— Mode-solvers— Propagation solvers— Optical communications system design

77

Some commercial software providers

Other providersPhoton Design (www.photond.com)

— Optical component and circuit design— Mode-solver; propagation solver

COMSOL (www.comsol.com)— Finite-element method (mode-solver; propagation solver)— Temperature modeling (among other applications)

Lumerical (www.lumerical.com)— FDTD method (mode-solver; propagation solver)

And these are just some of the commercial providers!

78

Some commercial software providers

39

Choosing a method: Propagation through a medium

Are your transverse device dimensions ≥ 10–100 × the wavelength?

Do you have two directions of propagation? e.g., a Bragg grating?

Is your transverse dimension time?e.g., optical fiber transmission

Is your system paraxial?e.g., a waveguide with a small index

difference between the core and cladding

Ray tracing

Use split-step Fourier method

BPM FEM or FDTD

Use a time domain or iterative method

Yes

Yes

Yes

Yes

No

No

No

No

79

A final cautionary noteFor any waveguide modeling software

Increase your resolution and make sure that the simulation converges

For simulations on an infinite domainIncrease your window size and make sure that the simulation converges

For propagation simulationsDecrease your step size and make sure that the simulation converges

For Monte Carlo simulationsIncrease your sample size and make sure that the statistics of interest converge

VII. Verification — Part II

80

40

BUTStart with crude estimates— Check if things make sense— Don’t waste computer time

Check that the physics is right— If you converge to the “wrong” answer, maybe you are looking at the

“wrong” problem.

Caveats to the cautionary note

81

OptSimTM : Single Channel 10 Gbps System

OptSim Schematic

41

OptSimTM : Co-simulation with BeamPROPTM

OptSim Schematic

BeamPROP component

OptSimTM : Co-simulation with MATLAB

OptSim Schematic

Matlab Component

42

Bibliography and Notes

Slide nos. 1–5:

A general references that has become the most important reference for computationalmethods is:

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, NumericalRecipes: The Art of Scientific Computing, Cambridge University Press, 2007(third edition)

This book not only has intelligent discussions of algorithms, but also includes soft-ware examples. Older editions have the software in a variety of different languages(C, FORTRAN, etc.). It is possible to obtain copies of all the software from the pub-lisher. Additionally, the publisher runs a WEB site (www.nr.com) that contains a largeamount of additional useful material, including methods for linking their software toMatLab

A general reference that we have found particularly helpful, because it talks aboutwhat you should not do, as well as what you should do is:

F. S. Acton, Numerical Methods That (Usually) Work, The Mathematical Asso-ciation of America, 1990.

Three other general references that we have found helpful are:

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag,1980

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations,Prentice-Hall, 1971.

L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equationsin Science and Engineering, Wiley, 1982.

There are many WEB sites that provide information on algorithms. Maple(www.maplesoft.com) MatLab (www.mathworks.com), and Mathematica(www.wolfram.com) all offer extensive on-line information and help, as well as linksto additional references. We have also found the ubiquitous Google and Wikipediato be very helpful.

Many general references in the optics literature contain references to numerical meth-ods and software. In the area of optical waveguides and components, references in-clude:

A. W. Snyder and J. D. Love, Optical Waveguide Theory, Chapman and Hall,1991.

D. Marcuse, Theory of Dielectric Waveguides, Academic, 1991.

K. Kawano and T. Kitoh, Optical Waveguide Analysis, Wiley, 2001.

43

G. P. Agrawal, Nonlinear Fiber Optics, Academic, 2007.

G. P. Agrawal, Lightwave Technology: Components and Devices, Wiley, 2004.This includes software from RSOFT.

References focused on optical fiber communications include:

L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Sys-tems, Artech, 1996.

G. Keiser, Optical Fiber Communications, McGraw-Hill, 2000. This includessoftware from VPISystems.

G. P. Agrawal, Lightwave Technology: Telecommunications Systems, Wiley, 2005.

R. Scarmozzino, “Simulation tools for devices, systems, and networks,” in Opti-cal Fiber Telecommunications V B, Fifth Edition: Systems and Networks, I. P.Kaminow, T. Li, and A. E. Willner, eds., Academic, 2008. Chap. 20, pp. 803–863.R. Scarmozzino is the founder and CTO of RSOFT. So, this chapter contains theperspective of a vendor of commercial software.

References focused on lens and macroscopic optical design are:

J. M. Geary, Introduction to Lens Design: With Practical Zemax Examples,Willmann-Bell, 2002.

M. Laikin, Lens Design, Taylor and Francis, 2007.

R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder, Optical System Design, McGraw-Hill, 2008.

In addition, a series that is devoted to macroscopic optical systems, including theirdesign is:

H. Gross, ed., Handbook of Optical Systems, Vols. 1–6, Wiley, 2005–2007.

No set of references to classical optical system modeling would be complete withoutmentioning the classic text:

M. Born and E. Wolf, Principles of Optics, Cambridge, 2005.

Works devoted to laser design include the following. There do not seem to be referencesdevoted specifically to the computational modeling of lasers at this time:

A. Siegman, Lasers, University Science, 1986.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits,Wiley, 1995.

An excellent general reference for oscillators, including opto-electronic oscillators hasjust appeared:

E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge, 2009.

44

Additionally, many of the WEB sites of commercial vendors of optical system softwareinclude references to the literature. Their WEB sites are listed in slides 68, 69, 77,and 78.

Slide nos. 9–12:

The original references to solitons in optical fibers are:

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulsesin dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23,142–144 (1973). This paper has the first theoretical discussion.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Picosecond pulse narrowingand solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). This paperhas the first experimental observation.

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I. Equal prop-agation amplitudes,” Opt. Lett. 12, 614–616 (1987); “Stability of solitons inbirefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5,392–402 (1988). These papers contain the first theoretical discussion of solitonsin birefringent optical fibers.

Two texts that discuss solitons in optical fibers are:

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Clarendon,1995.

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals andApplications, Academic, 2006.

Self-similar oscillations in hydrogen gases are described theoretically in:

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimu-lated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992); C. R. Menyuk,“Long-distance evolution of transient pulses in stimulated Raman scattering,”Phys. Rev. A 47, 2235–2248 (1993). These works contain references to the rele-vant experiments.

The theory of self-induced transparency modelocking may be found in:

C. R. Menyuk and M. A. Talukder, “Self-induced transparency modelocking ofquantum cascade lasers,” Phys. Rev. Lett. 102, 023903 (2009).

Slide nos. 13–18:

The work on repolarization is contained in the following papers:

Y. Sun, A. O. Lima, I. T. Lima, Jr., J. Zweck, L. Yan, C. R. Menyuk, and G. M.Carter, “Statistics of the system performance in a scrambled recirculating loopwith PDL and PDG,” IEEE Photon. Technol. Lett. 15, 1067–1069 (2003); Y.Sun, I. T. Lima, Jr., A. O. Lima, H. Jiao, J. Zweck, L. Yan, G. M. Carter, and

45

C. R. Menyuk, “System performance variations due to partially polarized noisein a receiver,” IEEE Photon. Technol. Lett. 15, 1648–1650 (2003); I. T. Lima,Jr., A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, “Areceiver model for optical fiber communication systems with arbitrarily polarizednoise,” J. Lightwave Technol. 23, 1478–1490 (2005). The first two papers presentthe experimental observations and the last paper presents the theoretical model.

Work on super-continuum generation is reviewed in:

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photoniccrystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).

The work on relaxation oscillations and modeling of Ti:sapphire lasers is contained in:

C. R. Menyuk, J. K. Wahlstrand, J. Willits, R. P. Smith, T. R. Schibli, and S.T. Cundiff, “Pulse dynamics in mode-locked lasers: Relaxation oscillations andfrequency pulling,” Opt. Express 15, 6677–6689 (2007); J. K. Wahlstrand, J.T. Willits, T. R. Schibli, C. R. Menyuk, and S. T. Cundiff, “Quantitative mea-surement of timing and phase dynamics in a mode-locked laser,” Opt. Lett. 32,3426–3428 (2007); J. K. Wahlstrand, J. T. Willits, C. R. Menyuk, and S. T. Cun-diff, “The quantum-limited comb lineshape of a mode-locked laser: Fundamentallimits on frequency uncertainty,” Opt. Express 16, 18624–18630 (2008).

Slide nos. 19–22:

The work on designing recirculating loops may be found in:

R.-M. Mu, V.S. Grigoryan, C.R. Menyuk, G.M. Carter, and J.M. Jacob, “Com-parison of theory and experiment for dispersion-managed solitons in a recircu-lating fiber loop,” IEEE J. Select. Topics Quantum Electron. 6, 248–257 (2000);J. Zweck, I.T. Lima, Jr., Y. Sun, A.O. Lima, C.R. Menyuk, and G.M. Carter,“Modeling receivers in optical communication systems with polarization effects,”Optics Photon. News 14, 30–35 (November, 2003).

The basic design of the dual-injection-locked opto-electronic oscillator is described in:

W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator withultra-low phase noise and ultra-low spurious level,” IEEE Trans. Microwave The-ory Tech. 53, 929–933 (2005).

This work is based on the original proposal of Yao and Maleki, described in:

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am.B 13, 1725–1735 (1996).

Models of the DIL-OEO have yet to appear in the archival literature. Our modelingof single-loop OEOs has appeared in:

E. C. Levy, M. Horowitz, and C. R. Menyuk, “Noise distribution in the radio fre-quency spectrum of optoelectronic oscillators,” Opt. Lett. 33, 2883–2885 (2008);J. Opt. Soc. Am. B 26, 148–159 (2009).

46

Slide no. 23:

There are no good references for the design of complete optical communications sys-tems. All companies keep their detailed design tools and algorithms proprietary.Information may be found in the general references on optical fiber communicationssystems (slides 1–5).

In the case of optical clocks, there are no good references — in part because opticalclocks are very new. Even general references on the design of atomic clocks are difficultto find. NIST in Boulder, CO builds the best clocks in the world, and their publicationsmay be found at the WEB site: (tf.nist.gov). This site also contains descriptionsof timekeeping for a general audience. Two useful general references on this site are:

S. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of timeand frequency at the onset of the 21st century,” Science 306, 1318–1324 (2004);M. A. Lombardi, T. P. Heavner, and S. R. Jefferts, “NIST primary frequencystandards and the realization of the SI second,” Measure 2, 74–89 (Dec. 2007).

A reference book that discusses the modern measurement of time and frequency is:

C. Audoin and B. Guinot, The Measurement of Time: Time, Frequency, and theAtomic Clock, Cambridge, 2001 (translated by S. Lyle from French).

Slide nos. 24–25:

The reference to Poincaré is:

H. Poincaré, Les nouvelles méthodes de la mécanique céleste, tomes 1–3, Gau-thier-Villars, 1892–1899.

More recent mathematical references to multiple-scale techniques are:

A. Nayfeh, Perturbation Methods, Wiley, 1973.

J. Kervorkian and J. D. Cole, Multiple scale and Singular Perturbation Methods,Springer, 1996.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientistsand Engineers: Asymptotic Methods and Perturbation Theory, Springer, 1999.

Slides nos. 26–29:

A careful derivation of the coupled nonlinear Schrödinger equation from Maxwell’sequations, and from that the Manakov-PMD equation and the scalar nonlinear Schrö-dinger equation, may be found in:

C. R. Menyuk, “Application of multiple length-scale methods to the study ofoptical fiber transmission,” J. Engin. Math. 36, 113–136 (1999).

The paraxial approximation is one of the most important and widely used approxima-tions in optical waveguides. It holds in most current waveguide structures, and, untilthe invention within the past decade of photonic crystal fibers and nano-waveguides,

47

it held in almost all of them. The earliest discussion of this approach in the contextof optics may be found in:

M. D. Feit and J. A. Fleck, Jr., “Light propagation in graded-index optical fibers,”Appl. Opt. 17, 3990–3998 (1978).

At an early stage, fast Fourier transform methods were applied to both continuous-wave problems, in which the dimensions transverse to the propagation direction are inspace, and to solitons and other problems with optical pulses, in which the transversedimension is time. Experience has shown that fast Fourier transform methods workbest when time is the only transverse coordinate. Otherwise, finite-difference methodsgenerally work better. Important papers that played a role in clarifying these issuesare:

D. Yevick and B. Hermansson, “Soliton analysis with the propagating beammethod,” Opt. Comm. 47, 101–106 (1983); Y. Chung and N. Dagli, “Assess-ment of finite difference beam propagation,” IEEE J. Quantum Electron. 26,1335–1339 (1990); G. R. Hadley, “Wide-angle beam propagation using Padé ap-proximant operators,” Opt. Lett. 17, 1426–1428 (1992).

Slides nos. 30–34:

In order to efficiently solve for the intensities of the different channels in a WDMsystem when Raman amplification is used, one must determine the pump and signal inten-sities iteratively since they are propagating in opposite directions, and one must averageover the powers of each channel in the time domain. An effective procedure for carryingout these tasks is described in:

B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier prop-agation equations with average power analysis,” IEEE Photon. Technol. Lett. 12,pp. 1486–1488 (2000).

They base their approach on the average power method described by:

T. G. Hodgkinson, “Average power analysis technique for erbium-doped fiberamplifiers,” IEEE Photon. Technol. Lett. 3, 1082–1084 (1991); “Improved aver-age power analysis technique for erbium-doped fiber amplifiers,” IEEE Photon.Technol. Lett. 4, 1273–1275 (1992).

The numerical approach of Min, et al. uses a relaxation algorithm in which the right-hand side of the ordinary differential equation is divided by an independent variable(the power) and is then exponentiated. More recent work shows that exponentiationhas no advantage over a standard second-order Euler method. See:

J. Hu, B. S. Marks, Q. Zhang, and C. R. Menyuk, “Modeling backward-pumpedRaman amplifiers,” J. Opt. Soc. Am. B 22, 2083–2090 (2005).

The relaxation method only works with two-point boundary-value problems. Thiswork also shows that the shooting method, supplemented by Jacobi weighting andcontinuation, can effectively deal with a more general set of constraints, and in partic-

48

ular the average-power constraint, which is important when determining the optimalpowers for gain-flattening. However, this method is more computationally intensivethan is the relaxation method.

The Stokes parameters are defined in many places, including:

M. Born and E. Wolf, Principles of Optics, Cambridge, 2005, pp. 31–33, 630–632.

A complete discussion of the Stokes parameters and other polarization representationsis given in:

W. A. Shurcliff and S. S. Ballard, Polarized Light, Van Nostrand, 1964.

Our work on reduced polarization models in optical fiber in optical fiber communica-tions systems is given in:

D. Wang and C. R. Menyuk, “Calculation of penalties in a long-haul WDMsystems using a Stokes parameter model,” J. Lightwave Technol. 19, 487–494(2001); I. T. Lima, Jr., A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk,and G. M. Carter, “A receiver model for optical fiber communication systemswith arbitrarily polarized noise,” J. Lightwave Technol. 23, 1478–1490 (2005).

A discussion of the different time scales in opto-electronic oscillators and how to takeadvantage of their separation may be found in:

E. C. Levy, M. Horowitz, and C. R. Menyuk, J. Opt. Soc. Am. B 26, 148–159(2009).

Slide nos. 35–41:

Remarkably, there is no good reference that specifically discusses this critical stepin designing a model. Both Press, et al., Numerical Recipes, Chap. 18 and Acton,Numerical Methods that (Usually) Work [see refs. for slides 1–5] discuss how to solveboth initial-value and boundary-value problems and point out that the latter are muchharder than the former. Acton also notes that nonlinear boundary-value problems aremuch harder than linear boundary-value problems (see pp. 173–174). However, theyboth assume that you have already carried out the step outlined in slides 35–41. It isa bit as if they told you how to make Canard à l’Orange and Coquilles Saint-Jacquesfor a fancy dinner party — but didn’t tell you the occasions on which you should cookone rather than the other!

Slide nos. 42–44:

All the books on optical fiber communications that are listed in the references forslides 1–5 discuss how to model noise and the more recent ones discuss how to modelpolarization mode dispersion. The main source of noise in modern-day optical fibercommunication systems is amplified spontaneous emission noise from the amplifiers,since the signal is typically pre-amplified prior to detection. As a consequence receivernoise is typically unimportant. For discussions of erbium-doped fiber amplifiers, see:

49

E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications, Wi-ley, 1994; E. Desurvire, D. Bayart, B. Desthieux, and S. Bigo, Erbium-DopedFiber Amplifiers: Device and System Development, Wiley, 2002; P. C. Becker,N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentalsand Technology, Academic, 1999.

For a discussion of optical communications system receivers, see:

S. B. Alexander, Optical Communication Receiver Design, SPIE, 1997.

For discussions of polarization mode dispersion, see:

J. N. Damask, Polarization Optics in Telecommunications, Springer, 2005; A.Galtarossa and C. R. Menyuk, eds., Polarization Mode Dispersion, Springer, 2005.

Slide nos. 45–46:

There are many references that describe how to characterize and model white noise.References that describe effective methods for characterizing and modeling 1/f noiseare more sparse. A good place to start is:

E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge, 2009.See Figs. 1.4, 1.8, and the discussion that surrounds them.

There are several competing techniques for modeling 1/f noise. The technique thatwe have used is presented in:

N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and1/fα power law noise generation,” Proc. IEEE 83, 802–827 (1995).

This reference also points to competitive techniques.

Slide nos. 47–49:

General references to Monte Carlo methods, with an emphasis on importance samplingand multicanonical techniques are:

G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, Springer,2000; R. Srinivasan, Importance Sampling, Springer, 2002; B. A. Berg, MarkovChain Monte Carlo Simulations and Their Statistical Analysis, World Scientific,2006.

Our own work on the application of importance sampling and multi-canonical methodsto optical fiber communications problems may be found in:

R. Holzlöhner and C. R. Menyuk, “Use of multicanonical Monte Carlo simula-tions to obtain accurate bit error rates inoptical communications systems,” Opt.Lett. 28, 1894–1896 (2003); G. Biondini, W. L. Kath, and C. R. Menyuk, “Impor-tance sampling for polarization-mode dispersion: Techniques and applications,”J. Lightwave Technol. 22, 1201–1215 (2004); A. O. Lima, I. T. Lima, Jr., and C.R. Menyuk, “Error estimation in multicanonical Monte Carlo simulations with

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applications to polarization-mode-dispersion emulators,” J. Lightwave Technol.23, 3781–3789 (2005).

Slide no. 50:

The deterministic methods are as varied as the applications. The additive whiteGaussian noise approximation for noise is discussed in all the optical communicationsbooks, discussed in slides 1–5. Indeed, a difficulty is that this approximation is oftenthe only way to model noise that is discussed! The methods for modeling beamspreading in a turbulent atmosphere are discussed in:

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through RandomMedia, SPIE, 2005.

There are no books that focus on how to model short-pulse lasers. A key reference is:

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. QuantumElectron. 29, 983–995 (1993).

Slide nos. 51–58:

A discussion of the importance of verification — and what can go wrong if it is notproperly done — is given in:

C. R. Menyuk, “Statistical errors in biasing Monte Carlo simulations,” J. Light-wave Technol. 24, 4184–4196 (2006).

In this reference, “verification” is referred to as “validation.” Indeed, there is nouniformity in the literature in the way in which these concepts are denoted. However,the terminology that we are using here has begun to be standardized in the high-performance computing community. See, for example:

R. B. Bond, C. C. Ober, P. M. Knupp, and S. W. Bova, “Manufactured solu-tion for computational fluid dynamics: Boundary condition verification,” AIAAJournal 45, 2224–2236 (2007).

Slide nos. 59–60:

The validation study on the recirculating loop is reported in:

R.-M. Mu, V. S. Grigoryan, C. R. Menyuk, G. M. Carter, and J. M. Jacob,“Comparison of theory and experiment for dispersion-managed solitons in a re-circulating fiber loop,” IEEE J. Select. Topics Quantum Electron. 6, 248–257(2000).

The validation study on the receiver model is reported in:

R. Holzlöhner, H. N. Ereifej, V. S. Grigoryan, G. M. Carter, and C. R. Menyuk,“Experimental and theoretical characterization of a 40-Gb/s long-haul single-channel transmission system,” J. Lightwave Technol. 20, 1124–1131 (2002).

The validation study on the Ti:sapphire lasers is reported in:

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C. R. Menyuk, J. K. Wahlstrand, J. Willits, R. P. Smith, T. R. Schibli, and S.T. Cundiff, “Pulse dynamics in mode-locked lasers: Relaxation oscillations andfrequency pulling,” Opt. Express 15, 6677–6689 (2007).

The validation study on the opto-electronic oscillators has not yet been reported inthe archival literature.

Slide nos. 61–69:

No good books/papers exist for this material!

Slide nos. 71–76:

All but one of the methods described in the “alphabet soup,” have been extensivelydiscussed in the archival literature and are described in the references for slides 1–5.The exception is the time-domain split-step method, which has not been extensivelytreated in the archival literature, but is the basis for RSOFT’s time-domain split-stepsimulator. The original reference for this approach is:

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlin-ear and polarization-related effects in fiber,” J. Lightwave Technol. 15, 751–765(1997).

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