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אביב-אוניברסיטת תל

Tel-Aviv University

Faculty of Engineering

הפקולטה להנדסה

School of Electrical Engineering

בי"ס להנדסת חשמל

High order Laser beam

modes

Project identifier: 14-2-1-839

Final Report

Author:

038022869 Guy Shtief (Sztajf)

:מנחה

Instructor:

Tel Aviv University

גיא שטייף

Shoam Shwartz

שוהם שוורץ

The study took place at:

a) The laboratory of Prof. Shlomo Ruschin, Tel-Aviv University.

b) The laboratory of Dr. Michael Golub, Tel-Aviv University.

Table 1: Table of contents

High order Laser beam modes ........................................................................ 1

Chapter 1: Overview .................................................................................... 3

Chapter 2: Foreword ....................................................................................... 4

Chapter 3: Theoretical Background ................................................................ 5

Chapter 3.1 Fourier optics ................................................................ 5

Chapter 3.2 Laguerre-Gaussian beams ........................................ 9

Chapter 3.3 Jacobi-Anger Expansion and Bessel functions .. 12

Chapter 3.4 Bessel Functions and Bessel-Gaussian beams 15

Chapter 4. Simulation approaches and implementations 18

Laguerre-Gaussian beam modes simulations............................................. 21

Reference ...................................................................................................... 46

Chapter 1: Overview

This study consists of simulation methods and a computerized design of an

optical system, with an emphasis on designing a multi-channel optical

matched filter.

The study will explore the multi-mode optical system portrayed abstractly

by the block diagram in, and more specifically, the realization of that system

portrayed in Figure 0–21.

Transmission

medium

Transmitter

(Multimode

Figure 0–1: A block diagram of anfiber)

optical communication system

Receiver

Figure 0–2(Color online) Experimental optical setup for MM free-space optical communication

system. SFS: spatial filter system; BS1 and BS2: beam splitters; S1 and S2: beam samplers; D1, D2,

D3: DOEs; C1 and C2: choppers; MO: micro objective; CMOS:CMOS camera

1

Will be discussed in detail in

Chapter 2: Foreword

Chapter 2: Foreword

Study goals

The goals of this study are the following:

1) Establish a robust MATLAB algorithm code which allows simulations

of optical wave propagations within an optical system.

2) Using said simulations to verify the validity of a multi-mode optical

matched filter that was planned theoretically and compare those

simulations to those obtained by another study1 in a laboratory.

Study motivation: Multiplexing multi-mode optical communication

Nowadays, the optical communication field dominates the long-distance

information transferring networks, such as the internet, telephones etc.

As time went by, the information transfer rate increased as it‟s price dropped

(for example: The internet), however, this increasing rate is approaching an

upper boundary[i][ii], which causes a bottleneck phenomenon in single-mode

optical fibers (The current technology is approaching a theoretical boundary

of 100Tb/s[iii]).

The approach to overcome the information rate boundary

Strenuous efforts are made and invested both in the industry as in the

academy to remedy that boundary, and one of the optimal solutions is the

realization of multi-mode optical fibers (as opposed to the, commonly used,

single-mode ones).

Analogy between this study and another

This study was initiated to provide a tool of validation to support the study

conducted by P.hd candidate Shoam Swartz, as the latter was conducted

almost exclusively based on theoretical design and lab experimentation,

while the former is a theoretical/physical simulation of the physics involved

1

Shoams Study

Chapter 3: Theoretical Background

In this chapter we would like to reintroduce two popular concepts of optics

and communication systems, those are Fourier optics and Bessel functions.

Chapter 3.1 Fourier optics

The wave equation is the mathematical basis for all the phenomena of a

wave.

It can be obtained by Maxwell‟s Equations ( Eq.1-4 ) as follows:

B

1 E

( E is an electric field, B is a magnetic influx density,

t

is the vector operator of , , )

x y z

D

2 H J

( H is a magnetic field, J is a current density,

t

D is an electric influx density)

in linear

isotropic

media

3 D ; D E ( is the spatial electric charge density,

is the dielectric coefficient of the media)

in linear

isotropic

media

4 B 0 ; B H ( is the permeability coefficient of the media)

Solving those equations for homogeneous media that lacks charges and

currents, and applying the Lagrange vector identity of 2 , yields the

wave equation (eq.5),

1 2

5 E 2 2 E

c t

Where c 3108 m / s is the speed of light in free space, 2 is the Laplace

2

operator, and t stands for time in seconds (or nano-seconds).

Limiting the discussion to fields of the mathematical form of:

U r, t a r exp j r exp j 2 t ,

the wave equation, then, degenerates into Helmholtz equation (Eq.6

1

2 E k 2 E 0 (k is the wave number and it satisfies k 2 2 )6).

c

c

1

6 2 E k 2 E 0 (k is the wave number and it satisfies k 2 2 )

c

c

Now, given some abstract distribution of an electro-magnetic field at plane

1

z=0, u x, y;0 , it can be shown , using the Helmholtz equation (Eq.6) , that

the distribution at plane z=z‟, u x, y; z ' must satisfy,

z

; x

, y ,

2

2

2

A , ;0 exp j 1

7 u x, y; z

exp j 2 x y d d

whereas,

8 u x, y;0

A , ;0 exp j2 x y d d

where, sin x

ky

kx

, sin y

k

k

and, A(νx,νy; 0) is referred to as, the Angular Spectrum of u(x,y;0),

9 A , ;0 u x, y;0 exp j 2 x

y dxdy

Alternatively:

2

10 u x, y; z A , exp j

1 2 2 z

Angular

Spectrum

H , ; z transfer function of free space

of u x , y ;0

where in Eq.10 is the 2 dimensional Fourier Transform such that for any

2-dimensional function, g, it‟s Fourier transform is given by G

1

1 2 2

11 G x, y g ,

g , exp 2 j x y d d

Utilizing the convolution theorem1 for Continuous Fourier Transform,

12 u x, y; z u x, y;0 * exp j

2

1 2 2 z

The denotation of „*‟, is the 2 dimensional convolution operator, which is to

be taken as following,

13 u * w u , w x , y d d

At the form of Eq.7, it is said that u satisfies the Rayleigh-Sommerfeld

propagation.

Even though Eq.7 provides a closed formula to evaluate wave propagations,

it is still challenging to evaluate it, both analytically, and numerically,

therefore we will rely on approximations to it such as Fresnel-Kirchhoff,

AKA the Paraxial approximation for wave propagation.

Given the angles of the wave propagation (relative to the optical axis) are

small, we may use the first terms of Taylor expansion for 1 2 2 , such

that,

14

1 2 2 1 x y 1 0.5 x 0.5 y

2

2

2

2

using this paraxial approximation, H, simplifies into,

15 H x , y exp j

2

1 2 2 z ~ exp jkz exp j z x2 y2

which, in the xy plane, has the form of,

16 h x, y 1 H x , y

exp jkz

exp j

x 2 y 2

j z

z

( 1 is the inverse Fourier Transform2)

exp jkz

exp j z x2 y2 u x, y;0 * h x, y; z

j z

17 u x, y; z u x, y;0 *

or, much more compactly:

1

f g F * G

2

1 H ,

H , exp 2 j x y d d

j

2 z

2

2

z exp j exp j z x y

18 u x, y; z

2

2

2

x y d d

u , ;0 exp j exp j

z

z

or, ultimately, in the form that will be carried out in the simulations, and will

be referenced many times more, throughout this paper:

j

2 z

2

2

z exp j exp j z x y

19 u x, y; z

2

z u z , z ;0 exp j z 2 2

in

This final form is very conducive to our simulations cause, as we will show

in later chapters.

Another approximation, which we will use, is the Fraunhofer Far Field

approximation, which is fairly accurate as z (even though in our set of

parameters, z ~ 4 m is pretty much enough), and also, z max 2 2

k

2

This approximation would simplify Eq.19 into,

20 u x, y; z

j

2

2 z

exp j

x 2 y 2 z uin z , z ;0

exp j

z

z

Notice that Eq.20 gives the Fourier Transform of the input wave, multiplied

by a z dependent phase, and a parabolic phase.

In fact Eq.20 implies that in order to get the Fourier Transform of an optical

signal, it should be allowed to propagate to some "long" distance, and

measured by intensity (which will ignore the parabolic and constant phases).

Chapter 3.2

Laguerre-Gaussian beams

The Generalized Laguerre polynomials1[iv] are denoted Ln x and they

satisfy the ODE (Ordinary Differential Equation) of,

21 xy '' 1 x y ' ny 0

which has few forms of solutions, one of them being:

n xi

22 Ln x 1

i 0

n i i!

n

i

The first four Laguerre polynomials, Ln x n 0 ,are presented in Table 2.

3

Table 2: The first four Laguerre polynomials

n

0 1

1 x 1

2 x2

2 x

2

3

Ln x

2 1

2

x 3 x 2 3 x 1 2 3

6

2

2

6

3

2

Our interest in those polynomials arises from the solution of the paraxial

wave equation (which was obtained using ) given a cylindrically symmetric

problem.

In a cylindrically symmetric problem, Eq.19 gives a solution in the form of,

r 2

r 2 2r 2

LG w0

Cp

exp

L

p 2

2

w z w z

w

z

w

z

23 up r , , z

r2

exp jk 2 R z exp j exp j 2 p 1 z

where, r, , z , are the cylindrical coordinates (sometimes denoted ,, z ).

CLG

p is chosen such that,

2

24

u r, , 0

p

2

rdrd 1

0 0

as a power normalization constant, which yields,

1

Also, the Associated Laguerre polynomials

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