Feedback, Synchronization, and Communication in … Synchronization, and Communication in Chaotic Time-Delay ... 4.4 Conclusion ... A.2 Quantum Mechanical Two Level System

Feedback, Synchronization, and Communication in Chaotic Time-Delay

Opto-Electronic Feedback Circuits

A Thesis

Presented to

The Division of Mathematics and Natural Sciences

Reed College

In Partial Fulfillment

of the Requirements for the Degree

Bachelor of Arts

Christopher J. May

May 2008

Approved for the Division

(Physics)

Lucas Illing

Acknowledgements

I would like to thank my family, Janet and Jim, my advisor Lucas Illing, and all those

who in some way contributed to this thesis and the corresponding work: John Essick,

Johnny Powell, Robert Reynolds, Joel Franklin, Darrell Schroeter, Britt Long, Drew

Atwater, and Eric Schmitt and Rick. A special thanks to Johnny Powell and John

Essick, without their guidence I would not be the person I am today.

Table of Contents

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Single Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Chapter 2: Experimental Design . . . . . . . . . . . . . . . . . . . . . . . 7

Equipment Calibration and Measurement . . . . . . . . . . . . . . . . . 7

2.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Feedback Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Temperature(1) and Current Controller(2) . . . . . . . . . . . 9

2.2.2 Butterfly Diode(3) and Mount(4) . . . . . . . . . . . . . . . . 10

2.2.3 Polarizer (5) and Mach-Zehnder (6) . . . . . . . . . . . . . . . 14

2.2.4 Detector(7), Electrical Splitter(8), and RF Modulator(8) . . . 23


2.4 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 Chaos Masking . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.2 Chaos Modulating . . . . . . . . . . . . . . . . . . . . . . . . 34

Chapter 3: Chaos: Model and Theory . . . . . . . . . . . . . . . . . . . 37

Chaos: Theory and Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Model Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 sin2 Map Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 4: Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . 53

Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1 Feedback Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


4.3 Chaotic Communication . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Appendix A: Laser Rate Equations . . . . . . . . . . . . . . . . . . . . . 67

A.1 Wave Equation in Matter . . . . . . . . . . . . . . . . . . . . . . . . 68

A.1.1 Wave Equation Solutions and Perturbations . . . . . . . . . . 72

A.2 Quantum Mechanical Two Level System . . . . . . . . . . . . . . . . 78

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Abstract

We investigate feedback, synchronization and applications to communication of opto-

electronics by general chaos at time scales of up to 10 GHz. We experimentally

examine the dynamics of a single circuit, built using a nonlinear, optically pumped,

opto-electronic device in an amplified time-delay feedback loop. An experimentally

obtained bifurcation plot shows a route to chaos from simple periodic behavior to

chaos, and is indicative of known routes to chaos[1]. These dynamics can be approx-

imated using a one-dimensional map emphasizing the important qualitative aspects

of chaotic dynamics.

We achieved synchronization between two such coupled circuits for both periodic

and chaotic dynamics on the time scales of 100 and 10 ns with high correlation

coefficients, .81 and .84 respectively. Communication using a chaotic carrier wave

between two synchronized circuits showed a high level of signal recovery. Finally, we

suggest modifications to the current experimental set-up to reduce noise and improve

results in future research.

Chapter 1

Introduction

1.1 Motivation

Time-delay feedback occurs in a variety of man-made and naturally occurring sys-

tems; the study of these systems has been adapted to a large number of fields. Since

1994, economists have employed chaotic dynamics in modeling the real business cycle

and the effects of technological advancements on productivity and log(GDP) out-

put [2, 3]. In biology, the modeling of disease transmission has been greatly improved

by chaotic dynamics, the most important of which is the SEIR model, which has had

great success modeling the spread of measles in less developed countries. Population

growth modeling has long used simple one-dimensional chaotic maps to approximate

population growth, starting with May’s study of insect populations in 1976 [4]. An-

thropology has made use of chaotic dynamics in modeling human interactions in

relation to random meetings and information propagation [5].

For my thesis I sought to create a physical system which models a general class of

2 Introduction

chaotic feedback systems, and then specialize to study synchronization and commu-

nication between two such systems. Given the available equipment and work space,

I chose to build a fast time scale opto-electronic feedback circuit using a nonlinear,

optically pumped, opto-electronic device (Mach-Zehnder modulator) in a time-delay

feedback loop to create a system capable of exhibiting chaotic dynamics. This system

has several advantages, the most important of which is the high level of parameter

control over each device in our system, which allows us to study very specific as-

pects of dynamics which might otherwise be obscured. All of the circuits which I

used in this thesis were first developed by the Larger group in their 2002 and 2005

papers [6–8].

Chaos has been studied in the auditory range, where the dynamics consist of a few

well-understood attractors, with no real hope for viable encrypted communication [9].

However, time delay systems are difficult to study due to the fast signal propagation

and slow circuit operational times. Fast time scale circuits are the most natural

medium for studying time delay systems, due to operation times on the same scale

of signal propagation. Fast time scale circuits also show great potential in both the

fields of cryptology and of random number generation[7].

This thesis was broken up into three stages: the construction and dynamics of

a single opto-electronic device; the coupling and synchronization of two feedback

devices. Finally the use of synchronization for communication between two opto-

electronic devices and communication between two coupled devices.

1.2. Single Circuit 3

1.2 Single Circuit

The opto-electronic circuits serve as a medium for studying time-delay feedback sys-

tems, a topic first examined by Ikeda at the University of Japan in the late 1970’s

[10]. Experimental designs for studying opto-electronic time-delay feedback circuits

began in the 1980’s using LiNbO3 electro-optical modulators [10]. The full form of

the dynamical equations governing this system were derived in 2002 by the Larger

group paper on synchronization [7].

A single time-delayed feedback circuit is constructed from an optical driving com-

ponent, a constant wave laser, and a Mach-Zehnder optical modulator, where optical

output power depends nonlinearly on a voltage applied at an RF input port. The

Mach-Zehnder is set-up in a feedback loop, so that the optical output of the MZ,

converted into voltage and amplified, determines the voltage at the RF input port.

We make the simple approximation that the dynamics of this system can be

modeled as a map with an appropriately scaled time delay which is proportional to

the nonlinear output of the Mach-Zehnder. Neglecting prefactors, our model is given

as xn+1 = β sin2(xn + φ) [11]. We use this model to illustrate long-run dynamics of

our system by examining attractors and eventually bifurcation diagrams. Bifurcation

diagrams are plotted for several different values of our parameter φ, varying over β.

We then compare these theoretical bifurcation plots to our experimental data and

show that our map appears to have a route to chaos which seems indicative of our

experimental system.

4 Introduction

1.3 Synchronization

Synchronization of two circuits is accomplished by biasing the nonlinearity of the

Mach-Zehnders in two different circuits using the same signal. These two circuits, the

transmitter and the receiver, are coupled with a non-periodic optical signal biasing

nonlinear elements. Coupling is accomplished using electronic or optical splitters,

each scheme with its respective advantages and disadvantages.

Synchronization of chaotic systems was theoretically realized in the early 1980’s,

but not experimentally demonstrated until 1990 [12]. These early circuits operated

at a slow time scale, with traditional circuit elements. Synchronization in an opto-

electronic circuit was not achieved until 1995 by Celka [13]. This work was nicely

summarized in 1997 by several researchers working at the US Naval Research Lab-

oratory [14]. Finally, high-speed synchronization in an opto-electronic circuits was

achieved in 1998 by Larger and Goedgebuer [15]. We exclusively examine optically

coupled systems, as in the schematic design put forward by Lager [6, 8].

We show experimental schemes for coupling two circuits and the resulting coupled

feedback. We then determine the level of synchronization by examining the correlation

coefficient of the output voltage for the transmitter and receiver.

1.4 Communication

Communication is achieved by optically encoding a message signal into into the shared

chaotic biasing signal and then using the output from the receiver Mach-Zehnder

modulator to recover the message by subtracting it from the incoming biasing signal.

1.4. Communication 5

This of course requires the proper time-delay between the receiver Mach-Zehnder

output and optical chaotic biasing signal from the transmitter loop.

It has been proven both experimentally and theoretically that not only is commu-

nication between synchronized chaotic oscillators possible, but it can be used for en-

cryption purposes. The earliest demonstration of encryption using chaos was in 1992,

and did not feature any type of feedback or optical splitting, but rather used unsta-

ble system dynamics of chaotic electrical circuits for message encoding [16]. Subse-

quent experiments continued to use electronic splitting, but improved communication

through feedback and the chaos masking technique [17]. In 1995, chaos-modulated

transmission was developed and employed for the first time in opto-electronic circuit.

[13, 18]. Chaos-modulated communication differed from previous and traditional

communication schemes by incorporating the message into the feedback as well as

the carrier wave. This significantly improved signal recovery and synchronization for

the purposes of communication. Fast time scale (up to 1GHz) synchronization and

optical splitting was achieved in 2000, and improved in 2002 with the introduction

of the Mach-Zehnder modulator and constant wave laser source [7, 19]. Subsequent

experiments have demonstrated high levels of synchronization up to 5GHz and trans-

mission as well as signal recovery over distances greater than 120km [6].

We coupled and synchronized two opto-electronic circuits, and showed that a peri-

odic message can be transmitted and recovered using the chaos-modulated technique

and optical coupling.

Chapter 2

Experimental Design

2.1 Outline

This chapter is divided into three sections. The first section outlines a single feed-

back circuit and the equations governing the important elements. The second section

outlines the set-up for synchronization and the third outlines the set-up for commu-

nication and signal recovery. This section will describe the set-up, calibration, and

measurement data for all equipment in a single feedback loop. All measurement and

configuration information will be displayed only for the laser labeled Kira and the

Mach-Zehnder Modulator titled MZ3.

2.2 Feedback Circuit

This section will describe the set-up, calibration, and measurement data for all equip-

ment in a single feedback loop. In Fig. 2.1, elements 1 through 5 provide an optimized

8 Experimental Design

LD Dr

iver

TEC Dr

iver

TSET

TEC Controller10 K

I LIM

LD Controller

Output to Scope

FeedBack

5

3

6

7

8

4

9

t1

2

1

Figure 2.1:1: Thorlabs TED 200C temperature controller2: Thorlabs LDC 210C laser diode controller 1A3: Sumitomo Electric SLT5411-CC butterfly laser diode4: Thorlabs LM1452 butterfly laser diode mount5: Thorlabs Polarizer6: JDSU 10Gbs Mach Zehnder Z5 modulator, custom modified7: MITEQ DR-125G optical receiver and New Focus 200KHz 2011-FC photoreciever8: Picosecond Pulse Labs Model 5331 6dB Power Divider9: JDSU H301 RF modulator

t1 is the propagation time from the output of the Mach-Zehnder Interferome-ter(6) to the electronic splitter(8). All following schematic diagrams use the samenumbering scheme. Blue lines correspond to optical signals, and red lines to electricalsignals.

2.2. Feedback Circuit 9

constant source for the Mach-Zehnder and, elements 6 through 9 form the feedback

loop.

The most important element in this loop is the Mach-Zehnder Interferometer (6)

that has a nonlinear optical response, proportional to the voltage applied at the RF

input port. For feedback, this RF port is connected to the RF modulator/ amplifier

and electronic splitter.

2.2.1 Temperature(1) and Current Controller(2)

The temperature of the butterfly laser diode is controlled using a Thorlabs TED

200C temperature controller. The action of lasing and the changing of current pro-

duces heat, a parameter which can affect both output wavelength and functionality

of equipment. Controlling temperature allows for wavelength stabilization, proper

modulation and tuning, and noise reduction.

The TED controller uses a thermistor internal to the butterfly diode as a tem-

perature sensor. Depending on what the internal thermistor value is, the TED has

two ranges in which it can operate; 10Ω- 20kΩ and 20kΩ− 100kΩ respectively. The

desired temperature is set through using the TSET adjust, while the actual resistance

of the thermistor is given by TACT . The temperature as a function of the resistance

measured is given by the equation:

R(T ) = R0eBval(1T− 1

To) (2.1)

Maximum level of temperature control is achieved 10 minutes after power-up due


to warm-up time for internal control elements.

Laser current is controlled by a Thorlabs LDC 210C Laser Diode Controller (1A)

plugged into the laser diode mount. The front panel knob allows for current adjust-

ment, while a variable 10 turn resistor adjusts the upper current limit, labeled ILIM

on the front panel. The current limit can be set to protect power sensitive devices

from overload against inevitable human error. This is a significant concern, as the

maximum output possible for our laser is much higher than the tolerances for either

the fast or slow time scale detector.

2.2.2 Butterfly Diode(3) and Mount(4)

A 14 pin universal butterfly mount, the LM 14S2, is designed to work specifically with

telecommunication laser diodes. A 14 pin general optoelectronics device is fastened

in place with four mounting pins. Pin contact is maintained by two seven-pin Zero

Force Insertion (ZIF) clamps.

Cooling/heating is directed through the DB9 jack at the rear of the unit labeled

TED (Thermal Electric Device) driver, while current is controlled through a similar

jack labeled LD Driver. Four additional mounting pins exist for attaching a Bias T

adapter for a RF modulator, up to 500MHz.

Three constant wave, 50mW, single mode, fiber coupled, class IIIb lasers were used

in this experiment, one in the single feedback loop, two in the synchronization circuit,

and three in the communication circuit. These lasers are type II telecommunication

lasers in type I configuration, operating at 1553.0±.7 nm, with a FC/UPC (Ultra


Polished Connector).

Like all lasers, these lasers have two regimes of input current operation, lasing and

non-lasing. The current at the transition point between these two regimes is called the

threshold current. The existence of these two regimes can be seen by deriving three

coupled first-order differential equations for the electric field, E , population inversion

N , and the polarization, P . These equations are derived from examining first-order

perturbations to the solutions to Maxwell’s equations in matter, and from considering

polarization to be an observable in a two level quantum mechanical system written

in terms of the population inversion, N .

These coupled differential equations are known as the Maxwell-Bloch laser rate

equations and can reduced to two differential equations relating the population in-

version N and the electric field E .

These two reduced differential equations can be written as:

E = −Γ

2E + (1− ıα)

g

2NE (2.2)

from Eqn. A.30 and

N =− γN +J

ed− εon

2

~ωcgN |E|2 (2.3)

g is defined to be the rate of gain, a constant, Γ is the loss of photons (primarily

through output), N is the population inversion, γ is the loss of population inversion

due to non-lasing processes, and Jed

is the current density over the active region, in

this case . 0.1µm. For a full derivation the Maxwell-Bloch equations, see Appendix


A.

Consider that power incident upon a detector is proportional to the photon density(S).

Photon density is given as the energy density, divided by the energy of a single photon,

~ω.

S =n2εo2~ω

EE∗ (2.4)

Using Eqn. 2.2, the rate of change of photon density can written as:

S =n2εo2~ω

(EE∗ + EE∗)

= −ΓS + (1− ıα)gNS + (1 + ıα)g

2NS

= −ΓS + gNS

(2.5)

Eqn. 2.3 can be written as a function of S using Eqn. 2.4

N = (−γN +J

ed− 2gNS) (2.6)

We now wish to examine when the rate of change of the photon density is constant,

S = 0, which has two solutions corresponding to the lasing and non-lasing solutions.

If S = 0, then the system is not lasing. Eqn. 2.5 is solved, and Eqn. 2.6 yields

N = Jγed

. This is clearly a non-lasing solution as the photon density is zero.

S = 0, N =J

edγ(2.7)

Alternatively, Eqn. 2.5 can be solved by setting N = Γg. The full form of this


solution can be written as

S =1

2edΓ(J − edγΓ

g), N =

Γ

g(2.8)

This is the lasing equation solution, where the photon density is not zero. In

addition, we can see that the equation predicts a negative photon density S below

the threshold density, edγΓg

, which is clearly not a physically possible solution. Above

this threshold, the lasing solution predicts a linear relationship between S and J ,

where the population inversion becomes coupled at N Γg.

As J increases the population inversion increases. When the population inversion

reaches the point N = Γg, J = edγΓ

g, the threshold level, the non-lasing solution

becomes unstable, and the lasing solution stable. Stability will not be proven here,

however the claim may be verified by calculating the stability coefficients for Eqns.

2.2 and 2.3.

The threshold and current at 40mW are given in manufacture standards for each

laser. However, the current-power relation can be affected by many different param-

eters, and should be experimentally verified for a sample range of currents values.

These values can be used to fit a linear trend over the full range of currents. The

manufacture and experimentally fitted trend line coefficients are displayed in Fig. 2.2

and fit the form: power = Coefficient× Current+ Intercept

There is an additional loss term in the laser output power due to the mismatched

PC/APC at the laser output. This was experimentally measured to be a fifty percent

optical signal strength loss. Optical isolators built into the laser diode prevent back


10 20 30 40 50

Current mA

2

4

6

8

Power mWCurrent-Power Measurments for Kira

Untitled-2 1

Printed by Mathematica for Students

Figure 2.2: Fitted curves for Kira, plotted in Mathematica. The green line is theexperimentally fitted data, while the pink dashed line is the manufacture providedfit. Note the experimentally fitted curve is above the manufacturer curve. Theexperimental threshold current agrees with the manufactured standard, given as 9.2mA.

reflections from the connector mismatch from causing instabilities in the laser itself.

2.2.3 Polarizer (5) and Mach-Zehnder (6)

Light exiting the SLT5411 laser diode has some initial polarization which is dependent

on the orientation of the output fiber. The Thorlabs FPC (fiber polarization con-

troller) 560 controls polarization before it reaches the Mach-Zehnder, using a quarter,

half, and quarter-wave plate. The first quarter-wave plate linearizes the input polar-

ization, the half-wave plate rotates the linearized light, and the second quarter-wave

plate transforms this linearized rotated light into a new arbitrary polarization.

For reasons to be made clear momentarily, it is ideal to adjust the polarization

such that the Mach-Zehnder optical output is maximized. This ensures that the


Mach-Zehdner splits the constant wave input light equally between the two paths.

The JDSU Z5 is a Mach-Zehnder interferometer(MZ modulator) designed for use

in telecommunication circuits, encoding electrical signals into optical fibers. A MZ

interferometer is designed to split light based on polarization into two paths where

the index of refraction of one path is sensitive to an applied voltage. The change

in the index of refraction increases/decreases the optical path length, and effectively

phase-shifts this light relative to the light of the other path. Commercially, a Z5

uses a CW input and encodes binary messages by applying voltages corresponding

to constructive and destructive interference at the path dependent source. Thus,

a 1 corresponds to an applied voltage resulting in constructive interference, and 0

corresponds to an applied voltage resulting in destructive interference.

To clarify terminology I will call the phase-shifted light, path 1, and the non-

phase shifted path 2. The optical length changing voltage is composed of a DC signal

and the feedback signal, denoted Vfeedback. This voltage is applied at a RF (Radio

Frequency) voltage input port with a GPO connector type located on the side of the

MZ modulator.

Let us consider the electrical field of these two paths, E1 corresponding to path

1 and E2 to path 2. The amplitude and phase of the electrical field for the two

respective paths can be denoted:


E1 =1

2√

2Eeıω(t+τ) + C.C. (2.9)

E2 =1

2√

2Eeıωt + C.C. (2.10)

where C.C. denotes the complex conjugate of both E and the exponential. The co-

factor term 12√

2is there for convenience in dealing with trigonometric and exponential

identities.

We assume polarization is controlled such that equal light passes through both

paths, that the magnitude of the electrical fields of both paths is equal: |E1| = |E2|.

Therefore, complete destructive and constructive interference can be achieved by

applying voltages corresponding to a phase shift of π2, and π respectively.

The electric field amplitude and phase of the MZ output is therefore the sum of

electrical fields of the two paths:

Eout = E1 + E2

=1

2√

2Eeıω(t+τ) + C.C.+

1

2√

2Eeıωt + C.C.

(2.11)

We can now use the voltage output from a detector measuring the MZ optical

output to bias the optical path length of path 1 at the RF input port. This is the

heart of the feedback circuit, and it will be now shown that the voltage corresponding

to the optical output from the MZ modulator is proportional to a nonlinear function.

Consider that the power at the MZ output detector is proportional to the square


of the electric field, the electric field multiplied by its complex conjugate.

P =|Eout|2

=Eout(Eout)∗

=(E1 + E2)(E∗1 + E∗

2) = (E1E∗1 + E1E

∗2 + E2E

∗1 + E2E

∗2)

(2.12)

Eqn. 2.12 can be simplified by assuming that E can be written such that the

complex part can be absorbed into the phase term. Using Eqn. 2.9 and 2.10, Eqn.

2.12 becomes:

P = E2 1

8(4 + 2e2ıω(t+τ) + 2e−2ıω(t+τ) + 2eıωτ + 2e−ıωτ ) (2.13)

Eqn. 2.13 can be simplified by making what is known as a rotation wave approx-

imation; in this approximation, all terms e2ıωt are assumed to be fast oscillating with

a mean of zero. This is a reasonable approximation given that laser frequency is on

the order of 190THz.

We may therefore neglect these terms, assuming that over the entire period we

consider, t, their contribution will be negligible. Eqn. 2.13 can be further simplified

using the identity cos(u) = eıu+e−ıu

2, and reduces to:

P = E2 1

8(4 + 4 cos(ωτ)) (2.14)

which can be rewritten using the identity cos2(u2) = 1+cos(u)

2:


P =E2 1

2(2 cos2(

ωτ

2))

=E2 cos2(ωτ

2)

(2.15)

Thus the output power of the MZ modulator is a nonlinear function. This model

can be further modified to elucidate how we will study the dynamics of the system.

First we will consider τ , then the nonlinear function itself, and finally the prefactor

E2 and how it relates to the system control parameter.

We can now consider that τ , the phase shift, is proportional to the voltage applied

to the RF port, plus an additional voltage which can be added to the feedback signal,

labled VDC : τ ∝ Vfeedback + VDC . VDC ranges over ±15V and is applied to the third

input pin of the MZ modulator. Both Vfeedback and VDC are related to phase by b,

where b = π2Vπ

, and Vπ is the DC half wave voltage. We can now write τ in terms of

this factor b and Vfeedback andVDC .

P =E2 cos2(bVfeedback + bVDC) (2.16)

Now consider the nonlinear cosine squared function to be written as a sine squared

function plus a constant phase shift that is present and varies from device to device.

P =E2 sin2(bVfeedback + bVDC + φ) (2.17)

Both the slow and fast detectors, used at different times in the experiment, relate

input optical power to an output voltage by a constant factor, call it α. For the

fast time scale detector, α = 53

VmW

. Thus we will write the MZ-output power is


proportional to a constant α P = αVDC . In addition, the power of the electric field

is directly proportional to the laser current. E = γλ, where γ is a constant relating

current to voltage.

With these two simplifications, Eqn. 2.17 can be written as:

Voutα = γαλ sin2(bVfeedback + bVDC + φ) (2.18)

Eqn. 2.18 relates Vout to Vfeedback, where we have not clearly defined what Vfeedback

is, or how it is determined. The feedback voltage applied at the RF input port is

equal to the MZ optical output converted into a voltage. However it is not the case

the the output of the MZ modulator immediately affects the RF input, but rather

the MZ output must propagate around through the electronics of the detector and

the splitter as well as the optical connector cables. We call this propagation time, t1

and it can be seen in Fig. 2.1. Therefore, we say that the voltage at the electronic

splitter is given as Vfeedback = V (t− t1) Now Eqn. 2.18 can be written as:

Vout(t) = γλ sin2(bVout(t− t1) + bVDC + φ) (2.19)

We can make one additional simplification by writing bVDC + φ as a single phase

term φ′

Vout(t) = γλ sin2(bVout(t− t1) + φ′) (2.20)


It is clear from Eqn. 2.20 that the MZ output is a nonlinear delayed feedback

equation.

Alternatively, we can use our conversion factor b to write both the voltages on the

left and right side as some units in radian:. x = V b. With β = bγλ. Eqn. 2.20 can

alternatively be written as;

xout(t) = β sin2(xout(t− t1) + φ′) (2.21)

Eqns. 2.20 and 2.21 are the final form of our nonlinear voltage feedback function.

As will be discussed in the theory and model chapter, we will be examining a

simple chaotic model where the band pass filter is ignored and we only deal with

the optical output as a recursive nonlinear map with time-delay scaled to be one,

(t1 = 1). While this simplification does not accurately describe the dynamics of

our system, it does show important aspects of chaotic dynamics and shows period

doubling bifurcation which appears to be similar to the route to chaos taken by our

system.

Experimentally, the nonlinear output of the MZ modulator can be mapped out

using an open loop circuit with no feedback. By inputting a low frequency ramp signal

into the MZ phase shift bias VDC and measuring the output using a slow, < 200KHz

detector on a 2 GHz digital oscilloscope, we can map the nonlinear output as a

function of the applied phase shift voltage.

Fig. 2.3 is the resulting plot of the MZ nonlinearity as a function of the applied


2 4 6 8 10

VfHRampLHVL

2

4

6

8

Vout HVL Vout Vs. Vf for MZ 3

ramp_output.nb 1


Figure 2.3: Output-Ramp voltage plot for MZ 3 optical output. This is measuredusing the slow time (200KHz) detector. The above is an example measurement of theNonlinearity of the MZ modulator using a ramp signal as the RF input. The rampinput voltage is plotted along the x-axes, and the corresponding output voltage onthe y-axes.

phase shift voltage.

We now fit the model we developed in the previous section: Eqn. 2.20 to the

output from the MZ modulator, with VDC = 0. This implies that φ′ = φ. The MZ

detector output is given the blue sinusoidal curve in Fig 2.3.

Vout = λγ sin2(bVramp + φ) (2.22)

The parameters λ, b, and φ, are fitted to the detector output data using a nonlinear

lev-mar sub VI in Labview 8.

Additionally, Z5 modulators are temperature-sensitive such that the nonlinear

output varies until thermal equilibrium with the heat sink has been reached. The

measurements of the nonlinear model coefficients over a period of 60 minutes is pre-


sented in Table 2.1.

Table 2.1: Model coefficients for the MZ3 modulator. Each of the entries presentedbelow is actually the average of three trials where the coefficients were calculated ateach of the specified time intervals.

t (min) b radV

φ(rad.)0 1.104 0.5773710 1.1203 0.57864320 1.1144 0.575630 1.12041 0.5758640 1.10537 0.5751250 1.1086 0.5756960 1.1075 0.57551

These values stabilize at t=60 min, with b and φ respectively as b = 1.1075, and

φ = 0.57551 rad respectively.

The phase corresponding to an applied voltage can therefore be calculated using as

the argument of the sin2 function, bVramp+φ. The VDC voltage and the corresponding

phase are presented in Table 2.2.

Table 2.2: Phase and Corresponding voltage for the MZ3 modulator. The phaseranges from 0 − 2π in steps of π

4. Note that these values were calculated using b as

given above, 1.1075

Phase voltage (V)π4

2.360π2

6.6463π4

10.932π 15.2185π4

19.5043π2

23.7907π4

28.0762 π 32.362

Note that it is necessary to optimize this system such that equal light is passing


through paths 1 and 2 of the MZ modulator before any of these calculations can be

done. This is done by adjusting the polarization until the output signal has been

maximized. In addition, there is an optical loss associated with light passing through

the MZ modulator. This loss has been experimentally measured to be fifty percent

of total input power.

The biasing voltage at the RF port is not just a function of previous optical

outputs, but also of the electronics through which the feedback is filtered. This

concept will be discussed more in the following sections.

2.2.4 Detector(7), Electrical Splitter(8), and RF Modula-

tor(8)

Two different ranges of detector are used for calibration and feedback. The New

Focus 2011 200KHz has a higher maximum input power, 8mW, and is therefore used

for calibration of the laser diode current curve and the MZ nonlinearity. The 2011

is powered internally by two 6V batteries, and has a variety of gain settings for the

output located on the front panel. The time scale of the 13GHz MITEQ detector

does not restrict the dynamics of the system, and thus is used in the feedback loop.

The MITEQ detector has a maximum input power of 3mW. In comparison to the

2011, the 13GHz detector is powered by a +12 V external source and requires a heat

sink to operate effectively.

The output of the fast time scale detector is split electronically using a Picosecond

Pulse Labs Model 5331 6dB Power Divider. The two attenuated outputs connect


respectively to the RF modulator for feedback to the MZ modulator, and to the

oscilloscope for monitoring of the feedback signal. The 6dB splitter is a passive circuit

element, insensitive to which of the three ports is used for input. Each connector port

has a 50Ω resistance.

Voltage passing through the RF modulator to the path 1 RF biasing port on the

MZ modulator also passes through the MZ RF input ports high-pass filter. These

two filters combine to make an electronic bandpass filter for the feedback voltage.

In the frequency domain, the output voltage of a bandpass filter can be described

as some complex function of the angular frequency ω times the input voltage.

Vout(ω) = H(ω)Vin(ω) (2.23)

This complex function H(ω) is called a transfer function.

Similar to Simpson, the transfer function for a bandpass filter is given by [20].

(Note that Simpson derives the absolute value of the bandpass filter|H|, where as we

are only examining H.)

H(ıω) =ıω ·∆

ıω ·∆ + (ω2o − ω2)

(2.24)

where ω2o = ω+ ∗ω− and ∆ = ω+−ω− are defined as the resonant frequencies and

the full width at half maximum of the gain.

The generic expression for a transfer function, Eqn. 2.23, can be solved for the

Vin using the transfer function for a bandpass filter, Eqn. 2.24.


Vin(ω) = Vout(ω) +ω2o

∆

1

ıωVout(ω) +

ıω

∆Vout(ω) (2.25)

Vin(ω), is the voltage corresponding to the nonlinear optical output of the MZ

modulator, Eqn. 2.20. The Vout(t) described in Eqn. 2.20 is the MZ optical output,

which the input voltage, Vin(t), for the bandpass filter. However, Eqn. 2.20 is a func-

tion of t, whereas the bandpass transfer function is written in terms of the frequency.

Therefore, we must inverse Fourier transform Eqn. 2.25 to study the dynamics and

apply our nonlinear input voltage. It can be seen that the second and third terms of

Eqn. 2.25 are Fourier transformations of derivatives and integrals respectively.

The Fourier and inverse Fourier transformation, x(ω) and x(t), are given as

x(ω) = F [x(t)] =1√2π

∫ ∞

−∞dtx(t)eıωt (2.26)

x(t) = F−1[x(ω)] =1√2π

∫ ∞

−∞dωx(ω)e−ıωt (2.27)

It can be shown that for well-behaved, continuous functions of t and ω, the Fourier

transformation of an integral and derivative respectively [21];

F [dx(t)

dt] = −ıωx(ω) (2.28)

F [

∫ t

−∞dlx(l)] =

x(ω)

−ıω+ πx(0)δ(ω) (2.29)


Now, applying the inverse Fourier transformation to both sides of Eqns. 2.28 and

2.29, we find:

F−1[−ıωx(ω)] =dx(t)

dt(2.30)

F−1[x(ω)

−ıω+ πx(0)δ(ω)] =

∫ t

−∞dlx(l) (2.31)

The third term in the inverse Fourier transformation of Eqn. 2.25 can be written

in terms of a temporal derivative. For the sake of clarity, let V be voltage in the

frequency domain, and V be voltage in the time domain.

F−1[Vin(ω)] =F−1[Vout(ω) +ω2o

∆

1

ıωVout(ω) +

ıω

∆Vout(ω)]

Vin(t) =Vout(t) + F−1[ω2o

∆

1

ıωVout(ω)] +

dx(t)

dt

1

∆

(2.32)

The second term in Eqn. 2.32 can be written as an integral in the time domain,

provided that πV (0)δ(ω) = 0. For a band pass filter, it must be the case that

V (0) = 0 [20].

Eqn. 2.32 now becomes:

Vin(t) =Vout(t) +ω2o

∆

∫ t

−∞dlVout(l) +

dVout(t)

dt

1

∆(2.33)

Finally, let us note that Vin in Eqn. 2.33 is actually the current optical output of

the MZ modulator, Vout(t) from Eqn. 2.20. i.e. Vin(t)bandpass = Vout(t)MZ . Writing

Vout(t) as just V (t), Eqn 2.33 simplifies to


λγ sin2(bV (t− t1) + φ′) =ω2o

∆

∫ t

−∞dlV (l) +

1

∆

dV (t)

dt+ V (t) (2.34)

or alternatively, with respect to our dimensionless variable x, using Eqn. 2.21:

β sin2(x(t− t1) + φ′) =ω2o

∆

∫ t

−∞dlx(l) +

1

∆

dx(t)

dt+ x(t) (2.35)

Eqn. 2.35 is the full form of our dynamics [6]. To give an idea of the range in

which this circuit operates, ω− = 75kHz × 2π and ω+ = 10GHz × 2π. The H301 is

the most limited bandwidth circuit element of the system. The other quantities ∆,

and ωo are defined as ∆ = ω+ − ω− ≈ ω+, and ω2o = ω−ω+. Therefore the high-pass

filter term in Eqn. 2.35 with a 1∆≈ 1

ω+coefficient limits the fast time scale oscillations

to the order of 100ps. The low pass filter term in 2.35 with the ω2o

∆ω− limits slow scale

oscillations to occur on a time scale of 10µs. Experimentally, the parameter β was

varied from 0 to roughly 60, with λ ranging from 0-30 mA for a single feedback loop.

I will use this equation in the theory and model chapter to develop the simplified map

which I will be studying. It is important to note that our theoretical model assumes

infinite bandwidth and will not accurately reflect the oscillation limitations imposed

by the dynamics of the bandpass filter. In our model, voltage can change instantly,

however experimentally we are limited by our fast oscillation time of roughly 100ps.

As previously stated, the H301 RF modulator and MZ modulator RF input port


comprise a bandpass filter. The H301 RF amplifier is powered externally by -5 V and

+8 V sources. To activate this unit, first turn on the negative -5V supply and then

the +8V supply. To power down, deactivate external supplies in the reverse order.

We will assume the roll-off of the RF modulator output does not affect the dynamics

of the system, but rather that there exists perfect proportional voltage amplification.

It remains prudent to be aware of such experimental limitations. Saturation of the

amplifier occurs at roughly .55 V input for all values of frequency input, as can be

seen in Fig. 2.4.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

VINHVL

1

2

3

4

Vout HVLRolloff Curve RF Modulator at 1, 120, and 240 MHz

Untitled-4 1


Figure 2.4: The input voltage and output voltage for the H301 amplifier. Saturationoccurs at .55 Vin. Red corresponds to 1MHz, blue to 120MHz and green to 240MHzinput frequency.

We have now covered the general equations governing each of the elements of a

single feedback circuit. The next section will give a general introduction to synchro-

nization and provide a synchronization scheme for two similar circuits.

2.3. Synchronization 29

2.3 Synchronization

Synchronization in general is the coupling of two or more elements of either periodic or

chaotic nature, such that the dynamics of one element completely govern or ”lock ” the

dynamics of the other elements. By ”lock ” we mean that knowledge of the dynamics

of one element absolutely determines the dynamics of the other elements [22]. In this

experiment we are attempting to achieve complete unidirectional synchronization

through the coupling of two opto-electronic circuits.

Coupling of two periodic or chaotic systems occurs by transmission of dynamics

between the two systems. The most famous example of a bidirectional coupled oscil-

lator is the Huygens clock, where two pendulums are coupled weakly by connecting

the respective clocks to a mutual coupling, such as a beam. As long as the small angle

condition for initial conditions is met, these two oscillators will couple such that they

are π out of phase. In analogy to our system, the pendulums are our opto-electronic

circuits, and the coupling is a mutual chaotic signal rather than a beam. Also, our

system is unidirectionally coupled rather than bidirectionally.

The locking element of our system is a closed-loop feedback circuit, the trans-

mitter, and our locked element is an open-loop circuit, the receiver. For complete

synchronization we are attempting to achieve identical chaotic outputs from the MZ

output in both circuits.

The closed loop configuration of the transmitter has the MZ Modulator path 1

feedback as a function of the MZ optical output at a time t1. (Propagation time can

be seen in Fig. 2.1 and 2.5.)


Transmitter Receiverx (t)1

t1

3 5 6

7

3 5 6

77

8

9

8

9G Gt1

x (t)2 y (t)1 y (t)2

10tr

Figure 2.5: As in Fig. 2.1, all circuit elements correspond to their previous numericaldesignations, Again, blue and red lines correspond to optical and electrical signalsrespectively. t1 corresponds to the propagation time from the optical output of thetransmitter MZ to the RF input port on both the transmitter and receiver MZ mod-ulator. The circuit element labeled 10 corresponds to a 1 × 2 50/50 optical coupler.This coupler consists of two fiber cores connected to create an evanescent wave of fiftypercent power in both output fibers. x1(t) corresponds to the optical output from theMach-Zehnder, x2(t) is the output from the transmitter splitter, y1(t) the transmittersplitter, and y2(t) is the output from the receiver detector. Again, we are measuringthe dimensionless constant x rather than the voltage.


The open loop configuration of the receiver has the MZ path 1 feedback as some

external function. Call the external signal y1, and the optical output from the receiver

MZ y2. So far all the dynamics and equations that we have discussed take place within

a single closed loop circuit.

To couple these circuits the MZ transmitter feedback signal is optically split,

maintaining the transmitter feedback, while also serving as the receiver external input.

That is, x2 = x1(t− t1) where t1 is the time it takes the signal to propagate around

the coupled feedback loop. It is experimentally valid to assume that the time it takes

for the transmitter MZ output to propagate to the transmitter and receiver splitter

is equal. Therefore the output x2(t) is equal to y1(t). It follows that x2(t) = y1(t) =

x1(t − t1). If we define the transmission time from the receiver MZ output through

the detector, labeled tr in Fig. 2.5, it follows that y2(t) = x1(t− tr) if synchronized.

Any delay seen between x2(t) and y2(t) is due to the difference in the propagation

times, t1 and tr. This difference was experimentally measured to be 4.7 ns.

Note that the electronic splitter in the receiver loop is only to keep symmetry

between the two circuits, and serves no purpose in measuring chaotic waveforms. If

the receiver loop lacked an electronic splitter, then the transmitter feedback would

experience a 6dB attenuation compared to that of the receiver.


2.4 Communication

2.4.1 Chaos Masking

Communication is achieved by embedding a message into the MZ output using an

optical coupler and attempting to recover said signal by time-delay and signal sub-

traction.

Research has focused on two different schemes for communication, chaos masking

and chaos modulation, with the latter being the focus of more recent research. We

examine chaos-modulated communication exclusively due to significantly higher levels

of message recovery. These schemes are worth examining, as they serve to illustrate

major differences between traditional and chaotic communication.

Chaos masking approaches chaotic communication in the tradition method; a

message signal is coupled with the MZ output after it has been optically split between

the transmitter and receiver. Call the output from the Mach-Zehnder x(t), and the

message signal m(t). Again, let x1(t) stand for the transmitter MZ output, and y2(t)

for the receiver MZ output. See Fig. 2.6.

The feedback of the transmitter circuit is identical to that of the previous section,

where

x2(t) = x1(t− t1) (2.36)

where our nonlinear function is really a sine squared function. In contrast, the receiver

external signal is equal to the transmitter nonlinear output coupled with a message.


Transmitter Receiver

y (t) 1

t1

A-Bm(t)

3 5 6 3 5 6

x (t) 2

x (t)1

77

7A 7B

9 9G Gt1

y (t) 2

3A

1110

y (t) 3

Figure 2.6: Chaos masked communication scheme. The element labeled 11 is a 2× 2,50/50 optical coupler. m(t) corresponds to the coupled message, and 3A is the laserused for electronic message encoding. The the detectors, 7A and 7B are for messagerecovery. The optical splitters are arranged in a 1× 2 and 2× 2 style with the opticalmessage being coupled in the 2 × 2 unit. Note, y3(t) 6= m(t − t1), due to lack ofsynchronization. See Eqn. 2.41

y1(t) = x1(t− t1) +m(t− t1) (2.37)

y1(t) 6= x2(t) in chaos masking communication, thus the nonlinear output from

the MZ in the transmitter and receiver will be different. x1new(t) stands for the new

transmitter MZ output as a function of the previous output.

x1new(t) = fNL(x1(t− t1)) (2.38)

y2(t) = fNL(x1(t− t1) +m(t− t1)) (2.39)


Therefore, as the output from the transmitter MZ propagates, y1 becomes y1new :

y1new(t) = x1new(t− t1) +m(t− t1) (2.40)

In an ideal chaotic communication scheme the difference of y1new(t) and y2(t)

would be synchronized such that y3(t) = y2(t) − y1new(t) = m(t − t1). See Fig. 2.6.

However, the difference between Eqn. 2.39 and 2.40 is

y3(t) =fNL((x1(t− t1) +m(t− t1)))− (x1new(t− t1) +m(t− t1))

=fNL((x1(t− t1) +m(t− t1)))− (fNL(x1(t− t1)) +m(t− t1))

y3(t) 6=m(t− t1)

(2.41)

It becomes clear that the MZ outputs for the transmitter and receiver are not

equal, if x1new 6= y2, and therefore synchronization is not preserved.

2.4.2 Chaos Modulating

Unlike chaos masking, chaos-modulated communication biases the phase of both MZ

modulators with the same feedback signal, which consists of the transmitter MZ

output and the message [22]. Thus, x2(t − t1) = y1(t), which implies that y2(t) =

x1new(t), and finally, that y3 = m(t−t1). Again, let us follow this process step-by-step.

From Fig. 2.7 we see that the x2(t) = x1(t− t1)+m. Assuming equal propagation

time in both the transmitter and receiver feedback loops, y1(t) = x2(t) = x1(t− t1).

Eqns. 2.42 and 2.39 become


Transmitter Receiver

y (t) 1

t1

A-Bm(t)

3 5 6 3 5 6

x (t) 2

x (t)1

77

50/50

7A 7B

9 9G Gt1

y (t) 3

50/50

3A

11 10

y (t) 2

Figure 2.7: chaos-modulated communication scheme. Note that the 2 × 2 couplercomes first, so the message is added into the feedback loop of both the transmitterand receiver.

x1new(t) = fNL(x1(t− t1) +m(t− t1)) (2.42)

y2(t) = fNL(x1(t− t1) +m(t− t1)) (2.43)

so x1new(t) = y2(t)

And the new output at y1 is given by

y1new(t) =x1new(t− t1) +m(t− t1)

=fNL(x1(t− t1) +m(t− t1)) +m(t− t1)

(2.44)

This leads to the conclusion that chaos-modulated communication maintains syn-

chronization and therefore is capable of signal recovery.

Fig. 2.9 is a picture of the experimental set-up for chaotic communication


y3 = m(t− t1) (2.45)

Figure 2.8: This is a picture of the experimental set-up for chaotic communication.

Chapter 3

Chaos: Model and Theory

Before we begin to discuss chaos theory I will give a brief history of the subject.

The notion of chaos is closely linked to the idea of sensitivity to initial conditions,

first studied by Henri Poincare in relation to the three body problem in 1890 [23].

Sensitivity to initial conditions continued to crop up in the newly developing field of

dynamical system modeling. A paper put out by Jacques Hadamard in 1898 proposed

the importance of Poincare’s work to more general systems. The first observation of

chaotic dynamics was by Balthazar Van der Pol when examining nonlinearities in

vacuum tubes. Van der Pol was not aware at the time that this was chaotic behavior,

and instead assumed that it was some sort of noise in the system [23]. Sensitivity

to initial conditions was first numerically observed by Edward Lorenz in 1961 with a

simple meteorological model; by changing the precision of the initial condition from

6 decimal places to 3 the same dynamical model arrived at a different numerical

solution [24]. The term chaos was first used in the nomenclature of mathematics by

James Yorke in 1975, discussing the importance of period three orbits as an indicator

38 Chaos: Model and Theory

for chaos [25]. Incidentally, this is an observation which we will use in the qualitative

analysis of our model.

We now turn to simplifying our dynamical model to a one-dimensional map.

3.1 Model Simplification

Recall that the full dynamical model for our system is given by Eqn. 2.35.

β sin2(x(t− t1) + φ′) =ω2o

∆

∫ t

−∞dlx(l) +

1

∆

dx(t)

dt+ x(t) (3.1)

As discussed in the Larger 2008 paper, the dynamics of Eqn. 3.1 are complicated,

occurring in an infinite dimensional space, with chaotic dynamics present for a wide

range of parameters [11]. However, the purpose of my thesis is not to completely

model the dynamics of a single feedback system, but rather to synchronize two such

systems as described in Chapter 2. I will make the simplifying assumption that both

the high-pass and low-pass filters do not significantly affect the dynamics of this

system. Therefore, Eqn 3.1 reduces to

x(t) = β sin2(x(t− t1) + φ′) (3.2)

This simplification leaves us with a one-dimensional continuous map. This model

lacks a differential component, and therefore is no longer a flow. The definition of

flow and map will be discussed momentarily. We may now scale the time interval by

the propagation time of the feedback loop, t1. Therefore, we multiply both indices

3.1. Model Simplification 39

by 1t1

. Eqn. 3.2 now becomes:

x tt1

= β sin2(x tt1−1 + φ′) (3.3)

We can therefore define the number of roundtrips to be n = tt1

where this n is our

discrete integer valued time. Therefore Eqn. 3.3 can be written as:

xn =β sin2(xn−1 + φ′) (3.4)

or equivalently as

xn+1 =β sin2(xn + φ′) (3.5)

Eqn. 3.5 is a non-invertible map, and the final approximation which we will use

to study and compare to the experimental results of our feedback loop.

To further understand this model and the chaotic behavior which will arise in our

system, we examine different types of dynamical models, and the characterization of

chaos in dynamical systems.

A dynamical model is a mathematical formula for evolving a system forward in

time, where time is either a continuous or discrete variable. If time is a continuous

variable, a dynamical model may consist of a set of first-order, autonomous ordinary

differential equations, giving rise to a flow. In contrast, discrete dynamical models

(maps) take time to be an integer valued variable relating future values of the map


to current or initial values. We will focus exclusively on maps, as that is the type of

dynamical model we are using for our system. It is important to note that there are

two types of map to consider: invertible and non-invertible, and different restrictions

for dynamics apply.

Invertible maps, similarly to flows, have the property that chaos can only occur in

spaces of dimension two or higher. In contrast, non-invertible maps have no such di-

mensional restrictions; chaos in fact occurs in any dimensional space [1]. It is of course

clear that our sin2 model is periodic, and therefore non-invertible. Consequently, our

one-dimensional map could exhibit chaotic dynamics.

3.2 Chaos

Chaotic dynamics are characterized by being exponentially sensitive to initial condi-

tions. Consider a model for a system with variable x(t), starting at an initial point

x1(0). The absolute value of the difference in the dynamics between the original point

x1(0) and a perturbed point, x2(0) at any given time for t > 0 is proportional to a

positive exponential. Write the difference in dynamics as ∆x(t) = |x1(t)−x2(t)|. The

ratio of the difference in dynamics can be written as

∆x(t)

∆x(0)' eht (3.6)

where h > 0.

Thus, exponentially-accurate knowledge of initial conditions is necessary to ac-

curately predict dynamics as the system evolves forward in time. Furthermore, the

3.2. Chaos 41

x1

x2

t= t1t= t0

t

x

x( )t0

x( )t1

Figure 3.1: An example of sensitivity to initial conditions in a flow, where the distancebetween the points x1(t) and x2(t) is given as ∆x(t). As can be seen in Eqn. 3.6, thedistance between these two points is proportional to a positive exponential factor.


solutions are bounded and non-periodic, implying that there exist differential equa-

tions that can’t be solved analytically in the typical manner, using a cocktail of various

periodic functions.

Instead of examining the exact analytical solutions to these models, we seek to

discuss and classify aspects of their long-term dynamics. The first step on this path

is examining attractors, a dynamical feature we expect to see in both model and

experimental results [1, 6]. It is most instructive to give a mathematical description

of attractors, and then develop physical intuition and examples.

Attractors are the most common feature of dynamics, and are defined as a subset

of phase space, A, invariant under f for all time t. (By invariant we mean that

the function is closed over the subset A for all t; if xεA then f(x, t)εA for all t.)

The ”basin of attraction” for the attractor A is defined by the space such that the

dynamics of points within that space converge to the attractor A as t→∞.

An attractor can be thought of as the set of points in the phase space of the

dynamical model that always map to themselves as the system evolves forward in

time. A ”basin of attraction” can be thought of as all points such that as their

dynamics converge to the attractor as the system evolves forward in time. Attractors

only occur in dissipative dynamical systems [1].

There are four common attractor types: fixed-point, periodic, quasi-periodic, and

chaotic. Fixed-point, periodic, and chaotic are the most relevant types of attractors

in our system. In this section we will give an example of a fixed point attractor in

two example systems, and then discuss the definition of fixed point attractors with

regard to maps. In the next section we calculate a fixed point and demonstrate the

3.2. Chaos 43

existence of periodic attractors for the sin2 model. Finally, we will briefly discuss

the characteristics of chaotic attractors, and how they pertain to chaos in our model.

Quasi-periodic attractors can occur in the full dynamical model of the system, Eqn.

3.1.

The canonical example of a fixed-point attractor is a pendulum. The point of min-

imum potential energy, when the rod is pointed straight down, is a one-dimensional

attractor. The corresponding basin of attraction consists of all other initial condi-

tions, save a single point; where the bob is pointing straight up. This is known as a

repeller, a point which maps to itself, but has no basin of attraction. Thus if the bob

started exactly straight up, it would not move from this position, but no other point

within an infinitesimal distance of it would map to the repeller. Due to losses of heat

and friction, all other initial conditions will converge to the fixed point attractors as

t→∞.

The pendulum illustrates the importance of observing long-run dynamics as op-

posed to short-run dynamics. Consider a pendulum where the bob is displaced from

the fixed point attractor. For a time, the bob will oscillate until it comes to rest

at the fixed point. The oscillations before converging to the attractor are known as

transient dynamics. We ignore this initial motion by only examining the system after

it has been significantly evolved forward in time.

Attractors and repellers can be seen in another example. Consider the equation,

x = −x2 + 100 (3.7)


The plot of Eqn. 3.7 can be seen in Fig. 3.2.

-20 -10 10 20x

-300

-200

-100

100

x†X

X

Basin of Attraction: point A

AB

C

Basin of Attraction: point C

x(-!, -!)

Figure 3.2: In this model we see coexisting attractors and repellers, where points Aand C are attractors and B is a repeller. The basin of attraction for point C is allpoints x < −10, and the basin of attraction for point A is all points x > −10.

The point labeled A at x = 10 is an attractor; for x > 10, x is negative. Therefore

the particle moves closer to the attractor, and for points |x| < 10, x is positive and

x moves away from x = −10 and close to x = 10. Therefore the point x = 10 is an

attractor. As x approaches x = 10 from both the left and right x becomes infinitely

small, such that x only approaches the point 10 as t→∞. The basins of attraction

are shown in Fig. 3.2.

This plot also shows coexistence of attractors at x = 10 and a repeller at x = −10,

and an attractor at x = −∞.

With this definition of a fixed-point attractor, let us consider a generic, one-

dimensional map M , acting on x, with a fixed set of parameter values, call them r.

3.3. sin2 Map Theory 45

This can be expressed as xn+1 = M(xn, r) A fixed point is defined as some value of

xn, call it xfixed, such that

M(xfixed, r) = xfixed (3.8)

This is exactly in line with our previous definition of an attractor, a subset (single

point) invariant under a function f or a map M . The basin of attraction is defined as

the set of points in x such that such that the iterations of M approaches infinity, the

value of the map approaches xfixed. Notate the pth iteration of the map M as Mp.

limp→∞

Mp(x, r) = xfixed

(3.9)

We now step out of the generic discussion, and focus on examining both fixed

point and periodic attractors of our sin2 map.

3.3 sin2 Map Theory

Recall the model for our system, Eqn. 3.5:

xn+1 = β sin2(xn + φ′)

The sin2 model has a parameter r = (β, φ′), and exhibits a fixed point attractor


for r = (1, .1), and xo = 1. After 15 iterations, Eqn. 3.5 converges to within seven

decimal places of the fixed-point, 0.0126325. Thus we can say that the value x = 1

falls within the basin of attraction for this fixed-point attractor, and that the first

iterations of this map were transient values.

0.5 1 1.5 2 2.5 3

xn

0.5

1

1.5

2

2.5

3

xn+1

period_one1.nb 1


Figure 3.3: This is an example of a ”cobweb” diagram, where xn is graphed againstxn+1. The blue line represents xn+1 = xn, the aqua line the nonlinear sin2 function,and the black represents the transient and eventual fixed-point attractor.

This result can be seen graphically in what is known as a cobweb diagram. A

cobweb diagram plots xn against xx+1 with both the nonlinear function and the line

xn+1 = xn. The dynamics of the system are determined through an iterative process.

Starting at an initial condition, in this case xn = 1, draw a vertical line until it

intersects the nonlinear function. Then draw a horizontal line from this intersection

until it reaches the line xn+1 = xn. Then draw a vertical line until it intersects the

nonlinear function. Repeat this process again for a suitably large number of iterations


(500,1000), until an attractor has been reached, or until it is clear that the dynamics

are not periodic. In Fig. 3.3, this pattern eventually converges to the fixed attractor.

How do our dynamics change with variation in a parameter, say β, while holding

all other parameters constant? As β increases we see a significant qualitative change

in the dynamics, where a single fixed-point attractor bifurcates to a periodic attractor.

A periodic attractor, specifically a one-dimensional period two attractor is defined as

a pair of points, x1 and x2, where x1 and x2 are elements of x such that M(x1) = x2,

and M(x2) = x1. A qualitative change in the long-run dynamics, like this doubling

of periods, is called a bifurcation. The parameter value at which this change occurs

is a bifurcation point [1].

The process of increasing the number of distinct points in a periodic attractor

by a factor of two is called period-doubling bifurcation, and can take place for any

period attractor. For example, period doubling of a period two attractor would result

in a period four orbit. There are other types of bifurcations, such as reverse period

doubling and tangential; a full discussion of which is available in Ott [1].

Note that in period doubling bifurcations, previous periodic orbit(s) are not de-

stroyed, but rather become unstable. By unstable we mean that small perturbations

to a previous periodic orbit do not converge back to that orbit as t → ∞. It can

be shown that the probability of starting a system on any of these unstable peri-

odic points is zero [1]. Therefore, a previous periodic orbit will not be visible in the

dynamics once bifurcation has occurred. Again, Fig. 3.4 shows a cobweb diagram

demonstrating that as the parameter β is incremented there exists a period two at-

tractor. This plot is for r = 2, .1 with xn = 1. The points composing the period


two attractor are 1.99912 and 1.4918, to which the initial condition xn = 1 converges

after 13 transient iterations.

If period doubling continues, eventually there will be an infinite number of unstable

periodic orbits, at which point the system is said to be chaotic. At this point there

exists an attractor consisting of a fractal set of points: a fractal attractor. Thus

chaotic dynamics are accompanied by fractal attractors. Again, the basin of attraction

is all initial conditions such that as the number of iterations, p, of the map M tend

to infinity, the dynamics approach this chaotic attractor.

0.5 1 1.5 2 2.5 3

xn

0.5

1

1.5

2

2.5

3

xn+1

period_two2.nb 1


Figure 3.4: As before, the blue line line represents xn+1 = xn, the aqua line thenonlinear sin2 function, and the black line the transient and eventual fixed-pointattractor. Here we see that a single box has developed, which indicates that thefunction is at a period two orbit, oscillating between two points.

We have now discussed dynamics, chaotic dynamics, fixed-point and periodic at-

tractors, and bifurcations. We are finally ready to put all of our hard work together

and examine a bifurcation plot for our model. This will be the main tool that we use


in comparing and understanding our experimental dynamics.

A bifurcation diagram is designed to show long-run dynamics of a system varied

over a set of initial conditions, and a range of values of a single model parameter. Note

that our parameter space consists of both β and φ′, β being dependent on the laser

current and φ′ being dependent on the VDC voltage applied to the MZ modulator.

To fully map the dynamics of our model, one would need a three-dimensional bifur-

cation diagram. However, for the sake of sanity we will only experimentally examine

dynamics for one value of φ while varying our parameter β.

With the parameter of variation β, a bifurcation diagram is made by choosing

either a single value or range of values as initial condition and plotting the long-term

dynamics for incremental values of our parameter β. Long-run dynamics refer to

500-1000 iterations. The bifurcation plot in Fig. 3.5 is made with a range of initial

conditions, xo = (.5, 4) incremented by 0.004, and φ held fixed at 0.1. This process is

repeated until the range of values for xo and β have been covered. The parameter β

ranges from 0-10 in increments of 0.004, with β on the horizontal axis and xn on the

vertical axis. A range of initial conditions was chosen to ensure that all the basins of

attraction for xn were visible. Choosing a range of initial conditions could potentially

lead to the examination of more than one attractor over the parameter space. This

does not occur for our map.

Fig. 3.5 shows that our model starts out with a single fixed-point attractor,

bifurcates to a period two attractor, which bifurcates again to a period four attractor

and so on until there are an infinite number of unstable periodic attractors [27]. The

points comprising the chaotic attractor are a fractal set. In addition, this map shows


Figure 3.5: This is a bifurcation diagram for φ = .1. xn ranges from .5 to 5, and βranges from 0 to 8. This plot clearly shows a single fixed-point attractor bifurcatinginto a period two attractor, then a period four attractor, etc. until we reach an infiniteperiodic attractor where chaos occurs. The points of clear period doubling bifurcationare notated A [26, 27]. Transients are evident at the first period doubling bifurcation,labeled B. The period three window and the corresponding three orbits are labeledC. The period three orbit indicates that the system is chaotic [1, 25]. From lookingat this bifurcation diagram, it appears that period doubling is the predominant routeto chaos.


the existence of ”periodic windows,” where only certain periodic orbits are stable.

The second periodic window in Fig. 3.5 has a period three orbit, which is indicative

of chaos [1]. We have therefore shown that our model behaves chaotically, and has a

period-doubling route to chaos. We will compare this model with the experimental

results presented in the next chapter.

Synchronization and communication will be discussed in the Results and Analysis

chapter, where we will experimentally verify that we achieved both synchronization

and chaotic communication, without proving that either of these are theoretically

possible. I am unsure if synchronization has ever been theoretically proven for the

full dynamical model.

Chapter 4

Results and Analysis

We will present the results from the three stages of the thesis in order: single feedback

circuit, coupled synchronized circuits, and message transmission and recovery.

4.1 Feedback Circuit

By studying a single feedback circuit, we observed qualitatively dynamical changes

and compared those with our simple model. We compare the experimental and theo-

retical routes to chaos and show that, experimentally, chaos results from bifurcations

with roughly the same period. We contrast these experimental bifurcations with our

theoretical model, which shows only the three basic types.

An experimental bifurcation diagram of a closed loop system is created by holding

φ constant, and acquiring histograms of the nonlinear voltage output for incremental

values of the laser current λ. Peak histogram values represent persistent voltages and

indicate dynamical features, such as bifurcations, and possibly chaos. Graphing the

54 Results and Analysis

normalization of these histograms together in a three-dimensional space is the first

step in creating a bifurcation plot. This three-dimensional graph can be flattened

by assigning a color scale to the normalized peak height, red for one and blue for

zero. The result is a two-dimensional, color-coded plot showing voltage extrema as

can be seen in Fig. 4.1. Experimental data was acquired from 10.5mA to 23.5mA in

increments of .5 mA with φ biased at 3π4

.

C

D

E

F

Figure 4.1: This is a experimentally-obtained bifurcation diagram with panels A-Fshowing voltage-time plots for bifurcation points and chaotic dynamics. Note thatpanels A and B have vertical voltage scales of 1mV per division, where as panels C-Fare on the scale of 50 mV per division. The lines indicate where each of the voltage-time plots occurred in the bifurcation diagram. Panel B shows the transition at theλ = 11mA bifurcation point from fixed-point to sinusoid, which evolves into squarewaves, as seen in panel C. A non-repeating signal with multiple voltage extrema,indicative of chaos is seen in panel F.

Fig. 4.1 shows a single steady state for values of λ < 11mA and a two period


steady state for 16mA > λ > 11mA. Fig. 4.1, panel A, shows the steady state for

λ < 11mA. At the bifurcation point, λ = 11mA, the voltage signal is a low amplitude

sine wave, as seen in Fig. 4.1, panel B. As λ increases, the amplitude and slope of

the sine wave increase while the peaks and troughs flatten, resulting in a square wave

function, Fig. 4.1 panel C. The peak-trough values of the square wave are represented

by two peaks in the voltage histogram, which qualitatively shows a bifurcation for

λ > 11mA.

A second bifurcation is visible at λ = 16mA. The voltage signal associated with

this orbit is shown in panel D of Fig. 4.1; voltage oscillations are similar to the

previous orbit’s square wave, but show addition oscillations in the peak and trough

of the signal. A voltage histogram plot shows four peaks, each peak representing an

oscillation. An additional bifurcation appears at λ = 20.5mA. The associated voltage

plot is shown in Fig. 4.1, Panel E.

When λ = 23mA, the output voltage has a large number of voltage extrema, which

appears in Fig 4.1 as a vertical band of red at the value λ = 23mA. Fig. 4.1 panel

F shows that the voltage signal appears to be non-repeating, indicating a potentially

chaotic signal at this point. For values of λ > 23mA, the output voltage appears to

only have one extrema. This could indicate a periodic window, where a previously

unstable state has again become stable and dominates the dynamics of the system.

It is difficult to determine what types of bifurcations occur in this system; however,

it does appear that qualitatively similar bifurcation plots have been produced by the

Larger group using a pulsed laser system [11].

Recall that in our theoretical sin2 model β ∝ λ, so the corresponding theoretical


bifurcation diagram would be plotted using the map

xn+1 = β sin2(xn + φ′) (4.1)

where φ = 3π4

and β equivalently ranges from 0-10. The resulting bifurcation diagram

is displayed in Fig. 4.2.

Figure 4.2: This is a bifurcation diagram of our model, Eqn. 4.1 with φ′ biased at3π4

, and β varying from 0 to 10. These values of β are equivalent for our values of λ,given the values of α = 3

5and γ ≈ .3 V

mA. The y axis represents the long-run dynamics

of the model.

We can see in Fig. 4.2 period-doubling bifurcation of the sin2 map at approxi-

mately β = 4, with continuing period-doubling bifurcations at β = 5.25 and β = 5.75.

From these observations we can say that the sin2 model has a period-doubling route

to chaos. While we cannot say with certainty that our experimental system displays

the same route, we can say that the two share some common features.

Finally, we examined breathers, oscillations restricted by the high and low pass


filter in the full dynamical model. Breathers are dynamics that on a slow time scale

are periodic, and on a short time scale are chaotic. The slow time scale periodicity

of breathers is evident in Fig. 4.3; however, the chaotic behavior of the faster time

scale oscillations cannot be resolved in this image.

Figure 4.3: This is an oscilloscope screen shot of a breather, showing slow time scaleperiodicity and fast time chaos. The vertical axis is scaled by 100 mV per division,and a horizontal time scale of 5µs. These dynamics occur when the system is biasedat φ = 3π

4, and λ = 25mA.

Fig. 4.4 is a higher resolution image of fast time scale oscillations of the breather

in Fig.4.3. We can see that these oscillations are non-repeating, which is indicative of

chaos. These fast time scale oscillations are determined by the low-pass filter. Again,

bandwidth is determined by the H301 RF amplifier, with an upper limit of 10 GHz.

This frequency corresponds to roughly 10 GHz oscillations in this signal. Similarly,

the slow time scale oscillations are determined by the low-pass filter, which is again

determined by the H301 RF amplifier to be 75 kHz corresponding to ≈ 20µs, Similar


breathers with similar time scales were reported by the Larger group in both 2002

and 2005 [6, 7].

Figure 4.4: A higher resolution screen shot of the fast time scale of a oscillations asseen in the breather in Fig. 4.3. These oscillations appear to be non-repeating andchaotic, appearing on roughly the same time scale seen for synchronization, in Fig.4.7. The vertical axis is scaled in units of 250 mV per division, and the horizontalaxis is scaled by 5 ns per division. As before, these oscillations occur at φ = 3π

4, and

λ = 25mA.

We have shown that our simple model demonstrates similar routes to chaos as our

experimental data, and verified the existence of breathers. Thus we have completed

the first stage of this experiment.

4.2 Synchronization

Two opto-electronic circuits were synchronized using the scheme described in the

Experimental Design Chapter, Fig. 2.5, observing outputs x2 and y2 to determine

synchronization. Synchronization was first attempted using periodic oscillations as


the carrier signal. The raw screen shot and the plotted results are shown below in

Fig. 4.5. In panel A of Fig. 4.5, the transmitter signal is yellow, and the receiver

is green. The larger panel B in Fig. 4.5 shows the Mathematica plot for these

two output signals, where the receiver signal has been shifted vertically for clarity.

These dynamics were observed on a 100 ns time scale, where φ is biased to 9π4

and

λ = 14.87mA.

It is important to note that there was a time shift in these two circuits, not seen

in either the raw data or the Mathematica plot due to an artificial shift in the data

set. This delay is the difference in the propagation times t1 and tr as seen in Fig. 2.5,

and was experimentally measured to be 4.7 ns.

The output signal from the transmitter and receiver on the 100ns time scale is

plotted in Fig. 4.5. This signal was achieved with φ = 9π4

and λ = 14.87.

Plotting the amplitudes of the transmitter against receiver voltage we can graphi-

cally observe the correlation by noting deviation from the diagonal. A linear relation

indicates that the transmitter voltage determines receiver voltage, and deviations

show lack of correlation. Noise also contributes synchronization errors and shows up

as slight deviations from the diagonal. Fig. 4.6 shows this synchronization plot, with

Vtransmitter (Vt) on the horizontal axis and Vreceiver (Vr) on the vertical axis.

Synchronization can be numerically analyzed by measuring the correlation coeffi-

cient of the transmitter and receiver voltages for each element of the time series. The


6 7 8 9 10time H10 nsL

0.2

0.4

0.6

0.8

1

1.2

Voltage H100 mVL

sync2.nb 1


Voltage (100 mV)

Time (10 ns)

A

B

Figure 4.5: Panel A shows the raw data for the synchronization of a periodic signalwith the transmitter in yellow and the receiver in green. The vertical axis is scaledby 25mV per division for the transmitter and 100mV for the receiver. The horizontalaxis is scaled by 100ns per-division. The parameter space for this periodic signalis φ = 9π

4, and λtransmitter = λreceiver = 14.87mA. Panel B shows the experimental

data plotted in Mathematica, where the transmitter signal is purple and the receiversignal is gold. Note that the receiver signal has been shifted vertically for clarity. Thecorrelation of transmitter and receiver voltages was calculated to be .84 using Eqn.4.2.


0.1 0.2 0.3 0.4 0.5 0.6VT HmVL

0.1

0.2

0.3

0.4

0.5

0.6

VR HmVL

periodic_sync.nb 1


Figure 4.6: This is a plot of the transmitter voltage plotted against the receivervoltage for a periodic signal. Synchronization results in a linear relationship betweenthese two voltages as is seen here. If the outputs were not synchronized, the voltage-voltage plot would look like a blob or a smeared line. The correlation coefficient forthis data set was calculated to be .84 using Eqn. 4.2.

correlation coefficient is given by:

ρ =〈(x(t)− 〈x(t)〉)(y(t)− 〈y(t)〉)〉√(x(t)− 〈x(t)〉)2

√(x(t)− 〈x(t)〉)2

(4.2)

where x(t) is the transmitter voltage at a time t, y(t) is the receiver voltage at a

time t, and 〈x(t)〉 is the mean of the respective data sets [19]. This particular definition

of correlation coefficient is insensitive to scaling factors which naturally occur in data

acquired from an oscilloscope. Using Eqn. 4.2, the correlation coefficient for periodic

synchronization was calculated to be .84. These synchronization values fall well within

the accepted range of .77-.91 for this circuit [19].

Now we perform the same analysis for a chaotic coupling signal by increasing our


parameter λ until a chaotic signal is achieved. Holding φ constant at 4π9

, we increase

λ to 22.23mA, where our signal exhibits a sharp decline in amplitude and no periodic

oscillation. (Again, we will not prove this signal is chaotic, but rather show that

it is non-repeating and that exhibits what we assume to be chaotic behavior.) The

resulting output voltages can be seen in the screen shot, Fig. 4.7.

Figure 4.7: This is a screen shot of synchronized, non-repeating output voltage fromthe transmitter (yellow) and the receiver (green). The vertical axis is 50 mV perdivision for the transmitter and 100 mV per division for the receiver. The horizontalaxis is given as 1ns per division. This signal was achieved by increasing the lasercurrent from the previous value of 14.87mA to 22.23mA, while holding φ constant.Just by observing the raw signal we can see that there is a high level of correlation.The correlation coefficient of the transmitter-receiver voltage was calculated to be .81using Eqn. 4.1.

The voltage-voltage plot for this chaotic signal is shown in Fig. 4.8, with Vtransmitter

on the horizontal axis, and Vreceiver on the vertical axis. A higher level of noise is

present in this signal than its periodic counterpart. I believe this is in large part

due in part to the noise/signal ratio increasing dramatically due to the amplitude of


interest decreasing by a factor of 5. Note that in the small amplitude oscillations in

the transmitter signal are on the order of 20-25mV , whereas the system noise floor is

roughly 15mV . The noise floor is calculated by examining the voltage at x2 with no

current applied to the laser.

0.05 0.1 0.15 0.2 0.25 0.3

VT HmVL

0.05

0.1

0.15

0.2

0.25

0.3

VR HmVL

Untitled-3 1


Figure 4.8: This is a graph of the transmitter-receiver voltage for a chaotic signal. Itis a easy to see a linear relationship between the output voltage of these two loops.The blurring from the clear linear relationship shows that there is noise in our system.I suspect that this is due to a connector mismatch in our system. The correlationcoefficient calculated for this data set using Eqn. 4.2 was .81.

The correlation between the transmitter and receiver voltage was calculated to be

.81 using Eqn. 4.2. Again, this falls well within the accepted correlation coefficient

value range.

Thus, we achieved a high level of synchronization between two circuits, and com-

pleted the second stage of this experiment.


4.3 Chaotic Communication

A chaotic signal with an embedded periodic message was used as the carrier signal in

the chaos modulated communication scheme, described in the Experimental Set-up

Chapter, Fig. 2.7. The embedded periodic message optically coupled to the feedback

loop in the transmitter by modulating a diode laser with a 72 MHz, 400 mV peak-

to-peak sine wave. The signal was then recovered in the receiver loop by subtracting

the coupling signal and the receiver nonlinear output signal. Fig. 4.9 shows the input

message (purple), the chaotic carrier signal (black) where the periodic message is not

at all evident, and the recovered message (gold).

2.5 5 7.5 10 12.5 15 17.5Time HmSL

0.2

0.4

0.6

0.8

1

1.2

Voltage H100 mVL

finaldata2.nb 1


Figure 4.9: This is the final data set showing input message in purple, the chaoticsignal in black, and the recovered message in gold. It can clearly be seen that theintermediate transmitted signal is chaotic, while the recovered signal matches theoriginal input signal in both frequency and amplitude.

This has only demonstrated that chaotic communication is possible; we give no

experimental measurement of signal recovery. I would again say that at least part

4.4. Conclusion 65

of the signal distortion is due to back reflections caused by the large number (5)

of mismatched connectors. This noise could be reduced by splicing fibers with the

optical couplers.

Ideally, a non-repeating message would be transmitted and recovered, providing

a measurement of bit error rate. However, I did not have time to complete this

analysis. I plan to continue this work over the summer along with other research.

I believe it is impressive that we were able to achieve chaotic communication, given

the limited time and available equipment. This completes the third and final stage

of this experiment.

4.4 Conclusion

I successfully examined feedback, synchronization and chaotic communication of

chaotic opto-electronic circuits. Each stage of the experiment laid out in my thesis

proposal was completed on time, with results that show qualitatively similar dynamics

to previous work [6, 11]

The obstacles to creating a synchronized circuit which could be used for chaotic

communication were significant. Understanding and calibrating the nonlinearities

of each MZ modulator for synchronization, as well determining losses in the system

due to splittings and circuit elements was my most challenging task. Maintaining a

feedback circuit required near-constant calibration due to polarization sensitivity and

unavoidable shifts in output fiber orientation. Synchronization proved to be relatively

easy with a firm grasp of the nonlinearities and the correct couplers, though the


process of finding and purchasing these devices was not at all simple.

These circuits required a great deal of construction time, split between machining

heat sinks for thermally sensitive devices, calibration calculations for circuit elements

(RF modulator, three diode lasers).

Communication was by far the most difficult experimental state, where both syn-

chronization and new loss terms become important for dynamics. Back reflections

became an increasing problem as the number of splitters increased.

In the future, I believe it would be beneficial to begin by studying a pulsed laser

system and then move to progressively more complicated systems for continuous

feedback, synchronization and communication. In the future, I believe that studying

synchronization of the sin2 model and fractal attractors would be very instructional.

This thesis still offers great theoretical and experimental research opportunities,

for studying both different communication schemes and dynamical models. I would

suggest that a theoretical thesis could focus on creating an accurate dynamical model

for a single feedback circuit, and comparing these numerical results with experimen-

tal findings. For continuing in an experimental thesis, I would suggest attempting

to achieve a bit error rate for non-periodic communication and studying different

communication schemes using different couplers, optic and electronic. It would also

be instructional to examine chaos masking communication and compare with chaos

modulated communication.

Appendix A

Laser Rate Equations

Semiconductor lasers, and in fact most lasers, work through inversion of a confined

population of atoms resulting in stimulated emission of photons. Population inversion

is characterized by having a disproportionately large number of electrons in an excited

state of the system, and is achieved through ”pumping”, or the application of current

to the semiconductor device. Laser power is a function of current occurring above a

threshold level, below which lasing does not occur. Understanding this current-power

relation is essential to the later process of isolation, a particular dynamical behavior

common in both the transmission and receiver loops. This relation can be developed

from the laser rate equations.

The laser rate equations are a system of coupled ordinary differential equations,

interrelating the electrical (E) and polarization (P) fields and the total electron pop-

ulation of the system (N). While the derivation of the relation between the electrical

and polarization fields is classical, a consequence of Maxwell’s equations in matter,

polarization is treated as quantum mechanical in nature and is described by a two

68 Appendix A. Laser Rate Equations

state system with a perturbed Hamiltonian for stimulated emission.

This system of OED’s has two steady state solutions for applied current, which

correspond to the lasing and non-lasing states, the transition between the two be-

ing the threshold level. This is an experimentally measured quantity, necessary for

coupling and tuning specific regimes of chaotic dynamics.

A.1 Wave Equation in Matter

The Maxwell field equation in matter can be written in terms of the dipole moment

and polarization inherently caused by an applied electric field:

∇ ·−→D = 0 ∇×

−→E = −∂

−→B

∂t

∇ ·−→B = 0 ∇×

−→H =

−→J +

∂−→D

∂t

(A.1)

Where−→E ,

−→B are the electric and magnetic fields, and

−→D ,

−→H are the corresponding

auxiliary fields.−→J is the volume current density.

The divergence of both−→D and

−→B are zero due to the neutrality of the semicon-

ductor material and the lack of magnetic monopoles respectively [28].

Furthermore, the following relations hold

−→D = εo

−→E +

−→P

−→B = µo

−→H

−→J = σ

−→E (A.2)

here−→P is the polarization field, σ is the material conductivity, and εo, and µo are

the permittivity and permeability of free space respectively.

A.1. Wave Equation in Matter 69

εo = 8.85× 10−12 C2

Nm2 µo = 4π × 10−7 NA2

In the hopes of generating a second order temporal derivative for our wave equa-

tion, we examine the curl of the curl of−→E :

∇× (∇×−→E ) = ∇× (−∂

−→B

∂t)

= − ∂

∂t(∇×

−→B )

(A.3)

Using−→B = µo

−→H and Eqn. A.1, we find that the curl of

−→B can be written as

∇× 1

µo

−→B = σ

−→E +

∂−→D

∂t(A.4)

So Eqn. A.4 can be rewritten as

∇× (∇×−→E ) = − ∂

∂t(σµo

−→E + µo

∂−→D

∂t)

= −µ0σ∂−→E

∂t− µo

∂2−→D∂t2

(A.5)

We can now simplify the left side of equation A.5 using the identity

∇× (∇×−→E ) = ∇(∇ ·

−→E )−∇2−→E (A.6)

We will now demonstrate that, in the context of Maxwell’s equations in matter,

∇ ·−→D = 0, imply ∇ ·

−→E = 0.

As in Eqn A.2,−→D is written as:

−→D [−→r , ω] = εo

−→E [−→r , ω] +

−→P [−→r , ω]

Let us clarify and write the induced polarization in terms of the electrical field:

In the frequency regime,


−→P [−→r , ω] = εoχM [ω]

−→E [−→r , ω] (A.7)

Where χM is the linear response of a material to an externally applied electric field,

known as the electric susceptibility. The material which comprises semiconductor

laser is isotropic in nature, and therefore has a scalar electrical susceptibility, as

opposed to non-isotropic materials which have a linear response represented by a

tensor.

With χM now well defined, we can rewrite−→D , eqn. A.8 in terms of

−→E and χM [ω].

−→D [−→r , ω] = εo

−→E [−→r , ω](1 + χM [ω]) (A.8)

The divergence of−→D can be written as

∇ ·−→D [−→r , ω] = ∇ · εo

−→E [−→r , ω](1 + χM [ω]) = 0 (A.9)

The left side of this equation must be zero by Maxwell’s equations in matter.

1 + χ[ω] can not be zero for most, or any, frequencies, so ∇ ·−→E [−→r , ω] = 0. This

demonstrates that Eqn. A.6 reduces to ∇× (∇×−→E ) = −∇2−→E and Eqn A.5 can be

rewritten as

−∇2−→E = −µ0σ∂−→E

∂t− µo

∂2−→D∂t2

(A.10)

The first term on the right side of the above equation, µ0σ∂−→E∂t

, can be considered

the mathematical representation of the characteristic loss of the system due to ineffi-


ciencies of the cavity/semiconductor material. This loss is relativity small compared

to the other terms, and in the lowest order approximation can be neglected.

0 = ∇2−→E − µ0σ∂−→E

∂t− µo

∂2−→D∂t2

≈ ∇2−→E − µo∂2−→D∂t2

(A.11)

−→D can be further rewritten using eqn. A.2 again.

0 = ∇2−→E − µo∂2−→D∂t2

= ∇2−→E − µo∂2

∂t2(εo−→E +

−→P )

= ∇2−→E − µo(εo∂2−→E∂t2

+∂2−→P∂t2

)

(A.12)

The temporal derivative with respect to−→P is not well defined at this point; the

polarization we have discussed in eqn.A.6 is only written in the frequency domain,

ω. To discuss this properly requires an inverse Fourier transformation. This Fourier

transformation is difficult due to−→E and χM both having dependence on ω. This

transformation can be simplified by examining the range of frequency space over

which ω ranges.

The wavelength of the constant light source is 1550nm corresponding to 260 THz

frequency. By comparison, the electrical field generated by our chaotic signal is slowly

varying, resulting in a small frequency range 6 10GHz, where the angular frequency ω

is on the order of electricalıfieldıfrequencyopticalıfrequency

' 10−5. Ignoring this small change in frequency,

and considering ω to be a constant, χM has no dependence on frequency. χM [ω] ' χM

The inverse Fourier transform is now of−→E [ω] only.

There is one other feature of χ which has not yet been considered; for semicon-


ductor lasing material, χM can be complex χ[ω]M = Re[χ[ω]] + Im[χM [ω]] However,

Re[χ] Im[χ] so we can ignore the imaginary part of χM .

The inverse Fourier transformation of−→P can now be written as:

−→P [−→r , t] = εoχM

−→E [−→r , t] (A.13)

There is now a shift in notation, where χM = Re[χM ]. In this case, the linear

susceptibility of any material is related to the index of refraction, n, by the equation

n =√

1 + χM . Eqn. A.12 can be rewritten as

0 = ∇2−→E − µo(εo∂2−→E∂t2

+ εoχM∂2−→E∂t2

)

= ∇2−→E − µoεon2(∂2−→E∂t2

)

(A.14)

Eqn. A.14 can be simplified by the usual relations c = 1√µoεo

and vp = cn, where

vp is the phase velocity of light in a medium.

0 = ∇2−→E − 1

v2

∂2

∂t2(−→E ) (A.15)

A.1.1 Wave Equation Solutions and Perturbations

Let us develop an ansatz solution for Eqn. (A.15) on the basis of a wave guide with

unidirectional light propagation. We will therefore assume a solution of the form

−→E [−→r , t] = 1

2

−→E o[

−→r ]eı(kz±ωt) + c.c, where k and ω take on their respective roles of

wave number and angular frequency. This ansatz will also solve for standing waves,

relevant to our cavity.


The ansatz above can now be simplified by decomposing the amplitude of−→E o[

−→r ]

into two parts:

−→E [−→r , t] =

1

2

−→U [x, y]Eoe

ı(kz−ωt) + c.c. (A.16)

−→U [x, y] represents the transverse mode profile, and Eoe

ı(kz−ωt) represents wave prop-

agation in the positive z direction.

Let us now use our solution, Eqn. A.16 as a solution to Eqn. A.15, with the

four-laplacian written as 0 = ∇2⊥−→E + ∂2−→E

∂z2− 1

v2∂2

∂t2(−→E ). We obtain:

0 =Eoeı(kz−ωt)∇2

⊥(1

2U [x, y])− 1

2k2U [x, y]Eoe

ı(kz−ωt)

+ω2

v2U [x, y]Eoe

ı(kz−ωt) + c.c

(A.17)

hich can be rewritten as:

∇2⊥(U [x, y]) = (k2 − ω2

v2)U [x, y] (A.18)

This is a differential equation for U [x, y], that can be solved using boundary conditions

appropriate to the particular geometry. Two of the most common geometries are

circular and rectangular, where we deal with circular in modeling light transmission

in fiber optics. In rectangular geometries, U[x,y] has solutions modes TEmn and

TMmn [28]. These solutions are compositions of sine waves, where the index represents

different modes where the nm integer values appear as part of the sine argument.

Cylindrical geometries have so-called HEmn and EHmn solutions composed of Bessel


functions [29]. Solving for these, or any other definite geometry, is limiting and

obscures the physics of the situation: solutions exist, and we will assume that they

do. From Eqn. A.18, we can also conclude that k is a function of ω, k = fn,m(ω).

To more accurately model our experimental system we will include those terms

we neglected in the first order approximation. In deriving Eqn. A.15, two terms were

ignored: the cavity loss term in Eqn. A.10 (−µ0σ∂−→E∂t

), and the polarization field effect

on the bulk properties of the material due to population inversion. These polarization

field effects are not associated with the bulk properties of the semiconductor laser,

and thus were not accounted for in the original derivation. This term will be denoted

PL, and the bulk polarization will be denoted PM .

Correspondingly,−→D from Eqn. A.2 is now

−→D = εo

−→E +

−→P M +

−→P L

With our new definition of−→D , the general form of the wave equation, Eqn. (A.10)

can be written as:

52−→E = +µ0σ∂−→E

∂t+ µo

∂2

∂t2(εo−→E +

−→P M +

−→PL) (A.19)

The σ and−→PL terms can be considered loss and pump terms respectively, the loss

due to emission.

With the previous definition of−→P M in Eqn. (A.12), Eqn. (A.19) reduces to:

∂2−→E∂z2

= +µ0σ∂−→E

∂t+ µoεo

∂2−→E∂t2

((1 + χM) +∂2−→PL∂t2

)

= +µ0σ∂−→E

∂t+

1

v2

∂2−→E∂t2

+ µo∂2−→PL∂t2

(A.20)

The terms added to Eqn. (A.15) can be consider perturbations to our system.


Therefore, by making a perturbed ansatz for−→E and

−→P , similar to that in Eqn.

(A.16) and solving zeroth order and first order equations, we can develop a first order

condition as a differential relation for these two fields.

The perturbed solution ansatz will be defined as:

−→E [r, t] =

1

2E [εz, εt]

−→U [x, y]eı(kz−ωt) + c.c

−→PL[r, t] =

1

2P [εz, εt]

−→U [x, y]eı(kz−ωt) + c.c

(A.21)

The ansatz for these two fields now has an amplitude which carries slowly varying

spatial and temporal dependence with a cofactor ε to determine which terms are

small enough to be neglected. Solving eqn. A.20 using these ansatzs and discarding

all terms with ε2 dependence we develop a condition for this perturbed wave equation.

This constraint will give a coupled ordinary differential equation for−→E and

−→P .

The first and second order temporal derivatives for−→E are:

∇2−→E =∂2−→E∂z2

+∇2⊥−→E

=k2−→U [x, y]E [zε, tε]eı(kz−ωt) + 2ıεk−→U [x, y]E ′[zε, tε]eı(kz−ωt)

+ ε2−→U [−→x ,−→y ]E ′′[zε, tε]eı(kz−tω)) + E [zε, tε]eı(kz−ωt)∇2

⊥U [x, y]

(A.22)

∂−→E

∂t= −

−→U [x,−→y ]E [zε, tε]ıeı(kz−tω)ω + ε

−→U [−→x ,−→y ]E [zε, tε]eı(kz−tω) (A.23)

The derivatives of P are:


∂2−→PL∂t2

=− ω2−→U [−→x ,−→y ]P [zε, tε]eı(kz−tω)

− 2ıεω−→U [−→x ,−→y ]P [zε, tε]eı(kz−tω) + ε2

−→U [−→x ,−→y ]P [zε, rε)eı(kz−tω)

(A.24)

To zeroth order perturbation, this returns the original wave equation with bound-

ary conditions, Eqn. (A.18).

∇2⊥(U [x, y]) = (k2 − ω2

v2)U [x, y] (A.25)

Again, dependent on the boundary conditions of the particular geometry, U [x, y] can

be solved resulting in the phase velocity being roughly equal to ωk. It is possible that

there is an additional term to the phase velocity that arises due to the transverse in

the electric field, but we will not consider it here.

The first order perturbation of this equation, those terms with ε coefficient, yields

2ıω

v(E ′ + 1

vE) = − σ

εoc2(−ıω)E − ω2

εoc2P (A.26)

Where E ′ indicates ∂E∂z

and E indicates ∂E∂t

.

Reordering 2ıωv, the above can be rewritten as:

(E ′ + 1

vE) = − σ

2kεoc2E + ı

ω

2nεocP (A.27)

We now take the mean field limit, the average of E over the length of the cavity because

we are not interested in spatial distribution We can now assert that the cavity length


is small compared to the rate of oscillation, and that therefore averaging over the

length of the cavity E ′ does not vary. Taking E to denote the mean field, Eqn. A.27

can be written as

1

vE = − σ

2kεoc2E + ı

ω

2nεocP (A.28)

The coefficients of E and P can be simplified by k = ωv, and c2 = n2v2. Let us

additionally clarify that the ω we have been describing is the frequency determined

by the cavity length, and will now be notated ωc.

E = − σv

2kεoc2E + ı

ωcv

2kεocP (A.29)

Which can be simplified to a form that will be easier to work with by using

Γ = σkεoc2

,

E = −Γ

2E + ı

ωc2n2εo

P (A.30)

This is the first of three coupled first order ordinary differential equations, relat-

ing E and P , and the population inversion N . We develop the next two equations

by examining the laser polarization, P , as a quantum mechanical observable in an

imperfect two state quantum mechanical system.


A.2 Quantum Mechanical Two Level System

For a quantum mechanical system, where the initial state is perfectly known, the

quantum mechanical expectation value for an observable is given by:

〈O〉QM = 〈ψ|O|ψ〉 (A.31)

where O is the observable, and ψ is a pure quantum state, a linear combination of

elements from the complete orthonormal set of eigenstates. A pure state can be writ-

ten as so: |ψ〉 =∑

n an|n〉 where n represents the eigenstates of the system.

However, it is experimentally impossible to prepare a system so that the initial

state is known with absolute certainty. We must therefore modify our expectation

value for account for imperfect information of initial states.

The probability of the observable being in a particular pure state now has an

associated classical probability, Pψ. Thus the expected value of the observable is the

product of the observable, Eqn. (A.31), and the corresponding classical probability

Pψ summed over the possible states ψ [30].

〈〈O〉QM〉 =∑ψ

Pψ〈ψ|O|ψ〉 (A.32)

We can use the properties of otrhonormal of eigenstates and classical probabilities

to develop a density matrix.

For our orthonormal eigenstates,

A.2. Quantum Mechanical Two Level System 79

∑n

|n〉〈n| = 1 (A.33)

Therefore multiplication by Eqn. A.33 leaves any probability unchanged. Let us

use this fact to rewrite Eqn. A.32:

〈〈O〉QM〉 =∑n

∑ψ

Pψ〈ψ|O|n〉〈n|ψ〉 (A.34)

Using the identity,|α〉〈α|β〉 = 〈α|β〉|α〉 and changing the ordering of Pψ, Eqn.

A.34 can be written as:

〈〈O〉QM〉 =∑n

∑ψ

Pψ〈n|ψ〉〈ψ|O|n〉

=∑n

〈n|(∑ψ

Pψ|ψ〉〈ψ|)O|n〉

=∑n

〈n|ρO|n〉

= Tr[ρO],

(A.35)

Where the second sum is written as the density operator,

ρ =∑ψ

Pψ|ψ〉〈ψ|. (A.36)

This density operator can be written out as a matrix where off diagonal entries

represent transitions, and diagonal elements represent stable populations. In the case

of a laser, the quantum mechanical representation of electron population is a two level

system, the excited and ground states.


The evolution of the density operator is described by a first order ordinary differ-

ential equation, the Von Neumann equation [30].

The first order differential equation can be found by examining the first temporal

derivative of ρ.

∂ρ

∂t=

∂

∂t(∑ψ

Pψ|ψ〉〈ψ|) =∑ψ

Pψ(∂|ψ〉∂t

〈ψ|+ |ψ〉∂〈ψ|∂t

) (A.37)

This can be simplified by notingH|ψ〉 = ı~ ∂∂t

(|ψ〉) ⇒ −ı~ H|ψ〉 = ∂|ψ〉

∂tand 〈ψ|H =

−ı~| ∂∂t

(〈ψ|) ⇒ ı~〈ψ|H = ∂〈ψ|

∂t

Therefore Eqn. A.37 can be written as

∂ρ

∂t=−ı~

∑ψ

PψH|ψ〉〈ψ|+ ı

~∑ψ

Pψ|ψ〉〈ψ|H

=−ı~

(H∑ψ

Pψ|ψ〉〈ψ| −∑ψ

Pψ|ψ〉〈ψ|H)

(A.38)

Which can be written as the commutator of the density operator ρ, and H,

∂ρ

∂t=−ı~

(Hρ− ρH)

ρ =−ı~

[H, ρ]

(A.39)

Where the commutator assumes the typical definition, [A,B] = AB −BA

However Eqn. A.39 neglects two important aspects: depletion due to system

inefficiencies, and increases in population due pumping of laser current. Depletion

occurs due to imperfections in the semiconductor band region, and can be represented

by a decay term Γij. (Note that this Γ term has nothing to do with the Γ in Eqn.


A.30.) The equation of motion for the density operator now becomes:

ρij =−ı~

∑k

(Hikρkj − ρikHkj)−∑k

(Γikρkj + ρikΓkj) (A.40)

Eqn. A.40 now accounts for losses, which will result in the complete depletion of

the system. We now add a term representing laser current, ensuring that complete

depletion does not occur. We call this term λ, and assume that the laser current

does not effect electrons in transitions, but rather ground or excited states [30]. This

assumption can be enforced by a Kronecker delta coefficient for λ with the same

indices as ρ.

ρij =−ı~

∑k

(Hikρkj − ρikHkj)−∑k

(Γikρkj + ρikΓkj) + λiδij (A.41)

The Hamiltonian in Eqn. A.41 can be considered the sum of a unperturbed and

a perturbed Hamiltonian. The unperturbed Hamiltonian is associated with that of

the electrical field of an atom, and the perturbed is the Hamiltonian associated with

the induced electric dipole. These are denoted Ho and H1 respectively.

H = Ho + H1 (A.42)

Note that the perturbed Hamiltonian, H1 includes the dissipative and additive

terms described above. We assume thatHo is the conservative closed two-level system.

Therefore Ho can be written as


Ho = ~ωa|a〉〈a|+ ~ωb|b〉〈b| (A.43)

We will now derive the first order perturbed Hamiltonian for an applied electrical

field [31].

First let us assume that the applied electrical field is in the form of a planar wave

whose wavelength is large compared to the size of a polarized atom. This allows for

the assumption of negligible spatial variation, which results in an electrical field of

the form:

−→E [−→r , t] = Eo cos(ωt)

−→k (A.44)

We impose the canonical condition that to first order, the perturbed Hamiltonian

must satisfy the equation

En = 〈ψn|H1|ψn〉 (A.45)

Which results in

H1 = −qEoz cos(ωt) (A.46)

where z is the separation distance of the poles, Eo is the electrical field at the

atom, and ω is the frequency of rotation.

This derivation has assumed that the atom is aligned with the applied electric field,


a condition we have no way of enforcing in our system. Instead we must examine the

full form of the electric dipole matrix and simplify through parity considerations [28].

This can be written as the sum of the products of the dipole matrix:

H1 =− qzE[t]

= −q(|a〉〈a|+ |b〉〈b|)z(|a〉〈a|+ |b〉〈b|)E(t)

= −qE[t](〈a|z|a〉|a〉〈a|+ 〈a|z|b〉|a〉〈b|+ 〈b|z|a〉|b〉〈a|+ 〈b|z|b〉|b〉〈b|)

(A.47)

Where q〈j|z|i〉 is the ith, jth entry of the electric dipole matrix, to be further

denoted as µij.

We can consider that the diagonal elements, b2, a2 is an odd function and thus

integrates to zero. Therefore we may assume these elements go to zero [31]. Eqn.

A.47 now reduces to

H1 = −E(µab|a〉〈b|+ µba|b〉〈a|) (A.48)

(Note that E[t] is written as E.)

We assume that while µ is in general complex, the imaginary part can be absorbed

into the density matrix. Therefore it follows that µab = µ∗ba. This simplifies Eqn. A.48

to H1 = E(µ|a〉〈b|+ µ|b〉〈a|), where µ represents µab.

The time evolution of the density matrix can now be fully described by permuting

the indices of Eqn. A.41.


We assume that the loss does not occur during transition, and is the same for both

the ground and excited state: Γaa = Γbb = γ2, and Γab = Γba = 0

ρaa = λa − γρaa − ı~Eµ(ρab − ρba)

ρbb = λb − γρbb + ı~Eµ(ρab − ρba)

ρab = −(ı(ωa − ωb) + γ)ρab − ı~Eµ(ρaa − ρbb)

(A.49)

The population inversion is defined as the difference in number of electrons be-

tween the excited state (a) and ground state (b). Eqn. A.49 now becomes:

N = ρaa − ρbb (A.50)

It follows that the first derivative of the population inversion is therefore N = ρbb−ρaa,

which can be expanded using Eqn. A.49.

N = −γ(ρaa − ρbb) + (λa − λb) +2ı

~Eµab(ρba − ρab)

= −γN +J

ed+ı

~Eµ(ρab − ρ∗ba)

(A.51)

Here Jed

= (λa − λb), where J is the current density, and the coefficient 1ed

in

this term results in the correct dimension for N of number(unit−less)time

. The observable

polarization for a single atom is given by the equation

−→P = nT r[ρµ] (A.52)

where n is the unit vector for the electric field. (The polarization for the ensemble of

atoms is given by scalar multiplication of the above by the number of atoms.)


The Macroscopic polarization is given by the polarization of an ”ensemble of

atoms”:

−→P = nT r[

ρaa ρab

ρba ρbb

0 µ

µ 0

]

= nT r[

ρabµ ρbaµ

ρabµ ρbbµ

]

= n(ρabµ+ ρbaµ)

(A.53)

We have already shown that ρab = ρ∗ba, so Eqn. A.53 reduces to

−→P = nµ(ρab + ρ∗ab) (A.54)

As in the previous section, we consider polarization to have the form

−→P = n

1

2(Pe−ıωct + P∗eıωct) (A.55)

Comparing Eqn. A.54 and A.55 it becomes clear that :

P =2µρabeıωct (A.56)

Let us examine the first temporal derivative of Eqn. A.56.

P = 2µeıωct((ıωc − ı(ωa − ωb)− γ)ρab −ı

~EµN)

= −(γ + ı∆ω)P − 2ı

~µ22eıωctEN

(A.57)


Where ∆ω = ωa − ωb − ωc.

Defining−→E ,

−→E = n

1

2(Ee−ıωct + E∗eıωct) (A.58)

we can simplify Eqn. A.57

P = −(γ + ı∆ω)P − ı

~2µ2Ne−ıωct(

1

2(Ee−ıωct + E∗eıωct))

= −(γ + ı∆ω)P − ı

~µ2NE

(A.59)

The last line of Eqn. A.59 is a consequence of the rotating wave approximation, where

we are considering oscillations on the order of 2ωt are considered to average to zero.

This is the second differential equation for E, P , and N .

We take a similar strategy as above, and examine the time derivative of N,

Eqn.A.51 with our definitions of E, Eqn. A.58 and P Eqn. A.55 as

N =− γN +J

ed+µ

~(Ee−ıωct + E∗eıωct)(−P∗eıωct + Pe−ıωct)

=− γN +J

ed+

1

~Im(E∗P)

(A.60)

The last line in Eqn. A.60 comes from and again making the rotating wave

approximation and setting any 2ω terms equal to zero.

Eqns. A.30, A.59, and A.60 are known as the Maxwell-Bloch Equations [30].


We now eliminate P from these equations and then use the definition of power to

determine the relation to current, J . We will make the assumption that on the time

scales of interest P = 0. This is a reasonable assumption for semiconductor lasers

where the decay of P to steady state is much faster then the carrier lifetime 1γ

or the

photon lifetime 1Γ

Eqn. A.59 reduces to

P =−ı~µ2NE 1

ı(∆ω) + γ(A.61)

Alternatively, Eqn. A.61 can be written as

P =−ı~µ2NE 1

(ı(∆ω) + γ)

(−ı(∆ω) + γ)

(−ı(∆ω) + γ)

=εon

2

ıωc(1− ıα)gNE

(A.62)

Where g =ωcµ2γ

~εon2

γ2+(∆ω)2and α = ∆ω

γ. We now plug Eqn. A.63 into our equations for

E and N , and find:

E = −Γ

2E + (1− ıα)

g

2NE (A.63)

Using Eqn. A.61, Eqn. A.60 reduces to:

N =− γN +J

ed− εon

2

~ωcgN |E|2 (A.64)


Thus, we have eliminated P from the Maxwell-Bloch equations. By examining

the power as proportional to the square of the amplitude of the electric field we find

there exist two steady state solutions, corresponding to lasing and non-lasing state.

This derivation concludes in the Experimental Design Chapter.

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