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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  3108 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 Ln   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, Ln  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

Ln  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
Cp

exp

L




 p  2

2
w  z   w  z  
w
z
w
z







 

23 up  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  ).

CLG
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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