Parity-based measurement and control of the spatial wave ...physics.wooster.edu/Leary/Theses/M_Ahmad_Thesis_12.pdf · Parity-based measurement and control of the spatial wave function

Parity-based measurement and control ofthe spatial wave function of photons

Mohammad Saif Ahmad

Physics Department

The College of Wooster

A dissertation submitted in partial fulfillment of the requirementsof Senior Independent Study in Physics at The College of Wooster

AdviserDr. Cody Leary

AdviserDr. Pamela Pierce

March 16, 2012

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Abstract

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Acknowledgments

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Contents

Abstract iii

Acknowledgments v

1 Introduction 1

2 Fundamental Mathematics 3

2.1 Matrices?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 The Hermite Polynomial . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 Hermite’s Differential Equation . . . . . . . . . . . . . . . 4

2.3.2 Rodrigues’ Formula . . . . . . . . . . . . . . . . . . . . . 7

2.3.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 The Laguerre Polynomial . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Physics Theory 11

3.1 Laser Beam Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Electromagnetic Wave Equation . . . . . . . . . . . . . . . . . . . 12

3.3 Longitudinal Field Distribution . . . . . . . . . . . . . . . . . . . 14

3.4 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Transverse Field Distribution . . . . . . . . . . . . . . . . . . . . 15

3.5.1 Hermite-Gaussian Modes . . . . . . . . . . . . . . . . . . 15

3.5.2 Laguerre-Gaussian Modes . . . . . . . . . . . . . . . . . . 17

3.6 First Order Mode Space . . . . . . . . . . . . . . . . . . . . . . . 17

3.6.1 Linear Combinations of HG modes . . . . . . . . . . . . . 17

3.6.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6.3 Parameterized Electric Field . . . . . . . . . . . . . . . . 21

3.7 1-D Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.8 Intererometer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.8.1 Parity Math . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.9 1 Beam Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.10 2 Beam Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

vii

viii CONTENTS

4 Mathematical Theory 294.1 The Unitary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Simulating the Mathematical Model 335.1 1 Beam Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Experiment 356.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.1.1 Iris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.2.1 Setting up the interferometer . . . . . . . . . . . . . . . . 356.2.2 Borrowing beam from 2D Parity Sorter . . . . . . . . . . 35

7 Results and Analysis 377.1 HeNe Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 2D Sorter Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8 Conclusions 438.1 Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Appendices

A Extra Stuff 45

B Extra Stuffing 47

Bibliography 49

List of Tables

2.1 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Laguerre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 9

7.1 HeNe Interferometer output . . . . . . . . . . . . . . . . . . . . . 387.2 HeNe Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.3 Interferometer PBS output . . . . . . . . . . . . . . . . . . . . . 397.4 HeNe PBS Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 40

ix

x LIST OF TABLES

List of Figures

2.1 The figure shows a plot of the typical Gaussian function witha = 1, b = 0 and c = 1√

2. . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The figure shows a plot of a 2-Dimensional Gaussian functionwith a = 1, x0 = 0, y0 = 0 and cx = 1√

2, cy = 1√

2. . . . . . . . . . 4

2.3 The figure shows a plot of Hermite polynomials of degree n = 0to n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 The figure shows a plot of Laguerre polynomials of degree n = 0to n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 The figure shows a typical Gaussian beam with beam waist radiusw0 at z = 0 and Raleigh range z0 at

√2w0. θ is the divergence

angle of the beam. Image taken from [10]and edited for use here. 13

3.2 The 3-dimensional plots show the transverse electric field ofHG10

and HG01 modes. The 2-dimensional plots show their respectiveintensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 The figure shows transverse modes HG00 and HG10 along dif-ferent values for z. It should be noted that changing the z valuedoes not change the structure of the transverse electric field butsimply the diameter. By keeping the longitudinal portion of thebeam to a minimal displacement within the Raleigh range, thischange becomes negligible . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Some interesting linear combinations of HG10 and HG01 modesare shown in these intensity plots. Considering Eq. (3.47) thefirst linear combination is attained when θ = π

2 and φ = 0 and

the second is attained when θ = π2 and φ = π

2 . . . . . . . . . . 22

3.5 By changing the parameters in the spherical coordinates θ and φone can create any first order mode. All first order modes can berepresented as some point on the sphere, thus this sphere visuallyrepresents the entire vector space described in the experiment. . 23

3.6 3D plot showing the effect of Πx operator on HG10 and HG01

modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

xi

xii LIST OF FIGURES

3.7 The inputs to the interferometer are labeled 1 and 2. The thindiagonal lines represent beam splitters and the thicker lines rep-resent mirrors. GP represents a glass plate that will be used tocontrol the beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.1 Sorting of the HG00 mode. When this mode is input to theinterferometer and phase difference in the interferometer φ = 0then the beam would come out of port B. Conversely if the phasedifference was φ = π then the beam would come out of port A.For any other phase difference, the HG00 would be output ofboth ports with varying intensity. . . . . . . . . . . . . . . . . . . 38

7.2 Output at both ports for polarized input beam . . . . . . . . . . 397.3 Output at both ports for polarized and realigned input beam . . 40

Chapter 1

Introduction

The basic particle of light, photons have always been intriguing to physicists.A photon can act like a wave or a particle, a phenomena known as the wave-particle duality of light [3] leading from the corpuscular and field theories oflight [4]. In beam-like geometries these photons exhibit four degrees of freedom,of which two (color and polarization) have been researched extensively. Theother two degrees of freedom are called the transverse spatial degrees of freedom.These transverse degrees of freedom are defined by a mathematical object knownas the wave function of the photon. This project looked at trying to measureand control the transverse wave properties [7] of the photons in a laser beam.

When thinking of a laser beam one often pictures a single dot where thebeam hits. In reality, this dot contains the superposition of several electricfields that determine the wave function of the photon. Each distribution is asolution to the wave equation electric field distribution and is called a mode.By considering two general types of modes called Hermite-Gauss (HG) modesand Laguerre-Gauss (LG) modes and limiting experimentation to some of thesimplest forms of these modes, the photons and hence the wave functions of thephotons that would be manipulated were put in a form that was easy to manageyet still representative of wave functions with more general functional forms.

The experiment was carried out by building a device called a one dimensionalparity interferometer (a variation of the Mach-Zehnder Interferometer [6]) andrunning it to collect data. This data was then compared to the predictions madefrom simulations of the phenomena using a model created in Mathematica. Theinterferometer worked by accepting one or two input beams of known intensity,which underwent changes in their wave functions due to beam splitting, paritybased reflections about y-axis and interference within the interferometer. Fol-lowing interference, the resultant beams were ejected from one or both of thetwo output ports with varying intensities depending on the conditions set withinthe interferometer (and thus the amount of change induced in the spatial wavefunctions of the photons).

Within the interferometer device, each interaction between a photon and acomponent of the device can be represented mathematically as a calculation

1

2 CHAPTER 1. INTRODUCTION

performed on a column vector of input wave functions representing the inputbeams. The computations representing the interactions occurring within theinterferometer were simplified and represented by using mathematical opera-tors and matrices that acted on the input column vector of light beams. Oncealigned, controlled adjustments to the interferometer directly manipulated thewave function of the output beams. This was mathematically modeled by cor-responding changes in the operator matrix that applied to the input beam.Simulations predicted the output beams based on the input beams and specificparameters within the interferometer.

Chapter 2

Fundamental Mathematics

2.1 Matrices??

This is text. This is bold text. This is text with emphasis. This is “doublequotes”.

Paragraphs are separated by one or more blank lines.

2.2 The Gaussian

One of the fundamental elements of mathematical physics is the Gaussianor the exponential. The gaussian is simply defined as:

f(x) = ae−(x−x0)2

2c2 (2.1)

where a, x0, c > 0 and e = Euler’s number. The graph of this function hasthe characteristic ”bell curve” shape as shown in Fig. 2.1. To further under-

Figure 2.1: The figure shows a plot of the typical Gaussian function with a =1, b = 0 and c = 1√

2

3

4 CHAPTER 2. FUNDAMENTAL MATHEMATICS

stand the parameters defined: a represents the height at which the curve peaks,x0 determines the center of the peak and c determines the width of the mostprominent portion of the bell. For the purposes of the experiment, the typicalGaussian will be expanded to two dimensions:

f(x, y) = Ae−(

(x−x0)2

2c2x+

(y−y0)2

2c2y

)(2.2)

where A represents the amplitude of the curve, x0 and y0 determine the centerpoint and cx and cy determine the x and y spreads of the curve. This functioncan be visualized in Fig. 2.2

Figure 2.2: The figure shows a plot of a 2-Dimensional Gaussian function witha = 1, x0 = 0, y0 = 0 and cx = 1√

2, cy = 1√

2

Thus the Gaussian has been explained, and from this point any time thatthe Gaussian is mentioned, it should be taken to refer to the 2 dimensionaldefinition given in Eq. (2.2) above unless otherwise mentioned.

2.3 The Hermite Polynomial

2.3.1 Hermite’s Differential Equation

A vital component in mathematically describing photons in a beam-like ge-ometry, Hermite polynomials are one of the classical sequences of orthogonalpolynomials. In this section we will look at the derivation of the Hermite poly-nomial and work through the derivation of some of its forms. The Hermitepolynomial is governed by Hermite’s differential equation as defined in [8].

(e−x2

y′)′ + 2ve−x2

y = 0 (2.3)

which can be expanded by differentiating the first term to give

e−x2

y′′ − e−x2

2xy′ + e−x2

2vy = 0 (2.4)

2.3. THE HERMITE POLYNOMIAL 5

from which the term e−x2

can be factored out to simply give:

y′′ − 2xy′ + 2vy = 0 (2.5)

From this form we can develop the solution as presented in [2] by taking

y = a0 + a1x+ a2x2 + ... =

∞∑i=0

aixi (2.6)

and differentiating to get

y′ =

∞∑i=0

iaixi−1

y′′ =

∞∑i=0

(i+ 1)(i+ 2)ai+2xi (2.7)

which we substitute into Eq. (2.5) to get

∞∑i=0

[(i+ 1)(i+ 2)ai+2 + 2(v − i)ai]xi = 0 (2.8)

By considering that the power series will only equate to zero if every coefficientof x = 0 we take

(i+ 1)(i+ 2)ai+2 + 2(v − i)ai = 0 (2.9)

and obtain the recurrence relation

ai+2 =2(i− v)

(i+ 2)(i+ 1)(2.10)

From this we can obtain the polynomial solutions of Eq. (2.5) for every v = nwhere n is a positive integer. Thus from Eq. (2.10) we get that

an+2 = an+4 = an+6.. = 0 (2.11)

For even values of n, Eq. (2.5) leads to the general form

an = (−1)n2

2n2 n(n− 2)...4× 2

n!a0 (2.12)

and this solution is considered a Hermite polynomial of degree n and is writtenHn(x). If a0 is chosen such that

a0 =(−1)

n2 n!

n2 !

(2.13)

then

Hnx = (2x)n − n(n− 1)

1!(2x)n−2 +

n(n− 1)(n− 2)(n− 3)

2!(2x)n−4 + ... (2.14)


where it should be noted that even if n is odd the polynomial solution of Eq. (2.5)would be written as Eq. (2.14) if a1 were chosen such that

a1 =(−1)

n−12 2n!

(n2 −12 )!

(2.15)

where some of the simpler Hermite polynomials are shown in the table

Table 2.1: Table of Hermite polynomials of degree n = 0 to n = 4

Degree PolynomialH0(x) 1H1(x) 2xH2(x) 4x2 − 2H3(x) 8x2 − 12xH4(x) 16x4 − 48x2 + 12

which are plotted as shown in Fig.2.3

Figure 2.3: The figure shows a plot of Hermite polynomials of degree n = 0 ton = 3

2.3. THE HERMITE POLYNOMIAL 7

2.3.2 Rodrigues’ Formula

From Corrolary 3 and the subsequent example in [5] we see that the Hermitepolynomials can be put in the form

Hn(x) = (−1)nex2 dn

dxn(e−x

2

) (2.16)

To confirm this let us take the specific form of Rodrigues’ equation:

De−Q + 2xe−Q = 0 (2.17)

where Q = x2 as stated in [5] and D = ddx . For simplification we write q = e−Q

to get the equationDq + 2xq = 0 (2.18)

we then differentiate this (n+ 1) times using Leibnitz’ Rule to get

Dn+2q + 2xDn+1q + 2(n+ 1)Dnq = 0 (2.19)

where we can use the substitution y = (−1)nDnq to get

D2y + 2xDy + 2(n+ 1)y = 0 (2.20)

By using the substitution u = ex2

y we see that

Du = ex2

(2xy +Dy)

andD2u = ex

2

(D2y + 4xDy + 4x2y + 2y)

such that Eq. (2.20) becomes

D2u− 2xDu+ 2nu = 0 (2.21)

which clearly shows that

u = (−1)nex2

Dn(e−x2

) (2.22)

is a polynomial solution of the form Eq. (2.5)

2.3.3 Orthogonality

According to [8] the weight function for the Hermite Polynomials is:

e−x2

(2.23)

Thus when Eq (2.23) is multiplied by Eq (2.4) we get

e−x2

× (e−x2

y′′ − e−x2

2xy′ + e−x2

2vy) (2.24)


which we will rewrite as

e−x2

2 × (y′′ − 2xy′ + 2vy) (2.25)

where the second portion was shown to have solutions of Hermite polynomialsdefined as Hn(x). We define by the functionFn(x)

Fn(x) = e−x2

2 Hn(x) (2.26)

Differentiating this function we get:

DFn(x) = e−x2

2 H ′n(x)− xFn(x) (2.27)

and differentiating again we get:

D2Fn(x) = e−x2

2 [H ′′n(x)− 2xH ′n(x)] + x2Fn(x)− Fn(x) (2.28)

but from Eq. (2.5) we know that

H ′′n(x)− 2xH ′n(x) = −2nHn(x) (2.29)

so the previous equation becomes:

D2Fn(x) = −2nFn(x) + x2Fn(x)− Fn(x) (2.30)

which gives:D2Fn(x)− x2Fn(x) + (2n+ 1)Fn(x) = 0 (2.31)

To show that the set {Fn(x)} is orthogonal in the range −∞ < x < ∞ wemultiply Eq (2.31) by Fm(x) to get:

Fm(x)D2Fn(x)− x2Fn(x)Fm(x) + (2n+ 1)Fn(x)Fm(x) = 0 (2.32)

where Fm(x) and Fn are interchangeable such that

Fn(x)D2Fm(x)− x2Fm(x)Fn(x) + (2m+ 1)Fm(x)Fn(x) = 0 (2.33)

and represent any orthogonal Hermite polynomial function. By subtracting thelast two equations from the previous equation and integrating we get that

Gn,m =

∫ ∞−∞

Fn(x)Fm(x)dx =1

2(n−m)

∫ ∞−∞

(F ′′n (x)Fm(x)− Fn(x)F ′′m(x))dx

(2.34)which implies that if n = m then Gn,m = 0. If n = m then we get

Gn,m =

∫ ∞−∞

e−x2

Hn(x)Hn(x)dx =

∫ ∞−∞

ex2

Dn(e−x2

)Dn(e−x2

)dx (2.35)

By doing an integration by parts, and using Eq. (2.20) to simplify the integralwe find that

Gn,m = 2nn!√π (2.36)

2.4. THE LAGUERRE POLYNOMIAL 9

and hence it can be deduced that by using the normalization factor

1√2nn!√π

(2.37)

can be used to get functions such that

1√2nn!√πe−

x2

2 Hn(x) (2.38)

which form a complete orthonormal set.

2.4 The Laguerre Polynomial

The Laguerre polynomial is another type of polynomial that will be encoun-tered in the experiment. Like the Hermite polynomial it is one of the classicalsequences of orthogonal polynomials. The Laguerre polynomial will not be dis-cussed as in depth as the Hermite polynomial because it is a minor component ofthe experiment compared to the Hermite polynomial. The Laguerre polynomialis governed by Laguerre’s’s differential equation as defined in [8]

(xe−xy′)′ + ve−xy = 0 (2.39)

which leads to the polynomial solution presented in [2] representing a Laguerrepolynomial of degree n and is written as Ln(x):

Ln(x) = (−1)n(xn − n2

1!xn−1 +

n2(n− 1)2

2!xn−2 −+...+ (−1)nn!) (2.40)

with the first few degrees of the polynomial listed in the table and their respec-

Table 2.2: Table of Laguerre polynomials of degree n = 0 to n = 3

Degree PolynomialL0(x) 1L1(x) 1− xL2(x) 2− 4x+ x2

L3(x) 6− 18x+ 9x2 − x3

tive plots shown in the Fig.2.4Finally the Rodriguess formula for the Laguerre polynomial as seen in [2] is

Ln(x) = exdn

dxn(xne−x) (2.41)

2.5 Linear Algebra


Figure 2.4: The figure shows a plot of Laguerre polynomials of degree n = 0 ton = 3

Chapter 3

Physics Theory

3.1 Laser Beam Optics

Light can radiate in many different forms. The light from a bulb has adifferent structure from the light of a laser beam. This experiment focuses onlaser beam light. In general, laser-beam propagation can be approximated bythe Gaussian. Gaussian beams have specific properties that will be describedsoon but first we must understand to what extent this approximation can beheld true. There are several different types of lasers, but the type of laserused in the experiement will be a typical helium neon laser (abbreviated asHeNe lasers). While it is not necessary to learn the structure of laser itself,some important facts about it are that a typical HeNe laser will output 633 nm(although other variants are available). The output power can vary betweentenths to tens of milliwatts depending on the size and length of the tube.[11].Although the output from a laser can be approximated by the Gaussian, inreality the output of a laser is not truly Gaussian and can vary depending onthe type of laser. This variance is measured but a quality factor known as theM2 factor. For a perfect Gaussian M2 = 1 whereas in real laser beams, itis some value M2 > 1. Fortunately, for typical HeNe lasers 1 < M2 ≤ 1.1which allows the approximation to be quite accurate. Thus for simplification incalculations, the beam will be treated as a pure Gaussian.

Assume that the laser is set so that its output will in the z direction in Carte-sian space. The beam itself is now considered. An ideal beam would consist ofa single ray of photons being emitted from the mouth of the laser. However inreality the beam emitted from the laser will experience diffraction in the lightwaves. In trying to achieve this closeness to the ideal, one is said to collimatethe laser beam. In a collimated beam, the photons are almost parallel and itdiverges very little compared to its point source [15]. Due to the diffractionthe light waves spread transversely as they propogate and it is impossible tohave a perfectly collimated beam. Pure diffraction theory accurately predictsthe spreading of the beam and from it the following equations can be derived

11

12 CHAPTER 3. PHYSICS THEORY

[10]:

w(z) = w0

√1 +

(λz

πw20

)2

(3.1)

and

R(z) = z

[1 +

(πw2

0

λz

)2]

(3.2)

where z is the distance propagated by the light wave, λ is the wavelength of thelight, w0 is the radius of the beam at its waist (minimum beam width), w(z)is the radius of the beam (beam width) at the point where it has propagatedthe distance z and R(z) is the radius of the curvature of the light wave (phasecurvature) as it has propagated the distance z. R(z) is infinite at the pointz = 0, passes through a minimum at some finite value of z and then slowly rizesagain toward infinity as z increases. The beam waist radius of the Gaussianbeam, w0 is attained at the point z = 0. [10]

The divergence in conventional light beams is linear. Gaussian beams how-ever, have non-linear divergence. The beam waist, which is often close to theoutput of the laser beam has a divergence angle that is relatively small. Thissmall divergence is called the near-field divergence. Farther away from the beam,the divergence can be significantly different. The Raleigh Range (z0) is the dis-tance over which the beam radius increases by a factor of

√2 and is defined

as:

z0 =πw2

0

λ(3.3)

The Raleigh range is considered to be the dividing line between the near-fielddivergence and the larger divergence factors. It is the distance from the waistat which curvature of the beam is at a maximum. [10]. As R(z) aymptoticallyapproaches z for large z, the term θ is introduced such that:

θ =λ

πw0(3.4)

This term is called the half-angle divergence and represents the angle between astraight line that originates at the center of the beam waist w0 and is aymptoticto the curvature of the beam. These terms can be seen in FIG. 3.1

3.2 Electromagnetic Wave Equation

A beam of laser light can be thought of as consisting of several constituent“modes” of light, represented mathematically by well-defined functions, that su-perpose to form a single beam. Two important modes considered in the projectwill be Laguerre-Gauss (LG) and Hermite-Gauss (HG) modes. To understandthese modes we must first consider the solution to a general wave equation.Let us consider the solution to the vector wave equation as established in [14]:

∇2E(r, t)− 1

c2∂2tE(r, t) = 0 (3.5)

3.2. ELECTROMAGNETIC WAVE EQUATION 13

Figure 3.1: The figure shows a typical Gaussian beam with beam waist radiusw0 at z = 0 and Raleigh range z0 at

√2w0. θ is the divergence angle of the

beam. Image taken from [10]and edited for use here.

where ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 is the Laplace operator[12], c is the speed of light in

free space (≈ 3×108) and ∂2t is the second derivative with respect to time. Theelectric field, E is a function of space and time where the vector r = xx+yy+zzis the Cartesian position vector. We can rewrite this equation to better modelbeam-like radiation by expressing E(r, t) (which is a complex valued functionof 4 arguments: x, y and z being spacial and t being time) as a product of thetransverse field distribution ET (rT, t) and longitudinal field distribution EL(z, t)[13]:

E(r, t) = ET (rT, z)× EL(z, t)e (3.6)

where rT = xx+yy denotes the Cartesian position vector in the plane transverseto the direction of propagation for the beam and e represents a constant unitvector. In considering a beam of light, each component has its own physicalinterpretation in the beam. The transverse component ET represents what onewould see upon taking a cross section of the beam at any point along its length.The longitudinal component EL represents the color or wavelength distributionproperties of the beam and the unit vector e represents the polarization of thebeam. These three properties combine to determine the electric field of thebeam. Each property will be explored in more depth in the following sections.

What this ultimately leads to is the simplification that the z componentalong the beam is negligible with respect to the properties that will be inves-tigated. We can approximate Eq. (3.5) as shown in [7] to attain the paraxialwave equation:

(∇2T + 2ik∂z)ET (rT, z) = 0 (3.7)

where ∇2T = ∂2

∂x2 + ∂2

∂y2 represents the transverse portion of the Laplacian,k represents the phase constant and ∂z represents the partial derivative withrespect to z.


3.3 Longitudinal Field Distribution

The longitudinal field distribution, EL(z, t) from Eq. (3.6) is the first to beconsidered. From [7] it can be seen that

EL(z, t) =

∫E(β)ei(βz−ωt)dβ ≡ F (z, t)× ei(kz−ωt) (3.8)

where F (z, t) is a term called the longitudinal envelope function and the ex-ponential term determines the wave like motion of the beam. The longitudinalterm EL(z, t) will be neglected in the development of the theory. This is becausethe longitudinal envelope function is approximately constant in our experimentwhich employs a continuous wave Helium-Neon laser.

To better understand this consider that the electromagnetic field within alaser resonator must fulfill certain boundary conditions [1]. One such require-ment is that the field exactly reproduces itself in relative amplitude and phasein one round trip of the laser cavity. Generally for a two-mirror cavity of opticallength L

λ =2L

N(3.9)

where λ is the operating wavelength of the laser and N is an integer. Now sinceλ is directly related to the frequency, f by the equation

λ =c

f(3.10)

where c is the speed of light, we arrive at the equation:

f =cN

2L(3.11)

where the allowable frequencies f will be referred to as longitudinal modes. From[9] it is seen that for helium neon laser operating at 632.8 nm, with a cavityof length 25 cm, the mode spacing is approximately 600MHz. The number oflongitudinal laser modes depends primarily on two factors: the length of thelaser cavity and the width of the gain envelope of the lasing medium. A HeNelaser rated at approximately 633 nm with a 25 cm laser cavity will at any giventime only allow a maximum of two or three longitudinal modes to be presentsimultaneously.

Based on this information, the simplifying assumption that a HeNe laserproduces approximately one longitudinal mode at given time can be made. Thisleads to the inference that the longitudinal envelope function can be treated asa constant, which in this experiment will be taken to be F (z, t) = 1.

The second element in the longitudinal field distribution is the term ei(kz−ωt).There are a number of reasons this term can also be taken as a constant andneglected in the development of the theory and some of these reasons will bemore obvious or better explained in later sections. First, we will consider thatmeasurements are taken at z = 0 and t = 0 which causes the exponential term

3.4. POLARIZATION 15

to simplify to e0 = 1, which is a simple constant. Secondly, it has been proven inthe mathematics section that any |eiα|2 = 1. In the experiment this will indeedhappen when taking intensities of any E(r, t) and once again |ei(kz−ωt)|2 = 1 asimple constant.

Thus from Eq. (3.8), the longitudinal field distribution is simplified as EL(z, t) ≈1, given that the laser in the experiment fulfills the conditions stated above.

3.4 Polarization

Every beam of light has some type of polarization. This is represented inEq. (3.6) by the vector e. The general form for this polarization is the col-umn vector:

e =

(ψxψy

)=

(cos θeiαx

sin θeiαy

)(3.12)

where α represents phase angles in the respective coordinates and θ representsthe polarization angle with respect to the x-axis. When the phase angles αx

and αy (let αx = αydef= α) are equal the equation simplifies to

e =

(cos θsin θ

)eiα (3.13)

which is the general form for a linearly polarized beam. Consider the two specialcases θ = 0 when the beam is purely x polarized and θ = π when the beam ispurely y polarized. A purely x polarized beam is written:

x =

(10

)eiα (3.14)

and likewise for y:

y =

(01

)eiα (3.15)

This means that in general linearly polarized beams can be written as the linearcombination

e = cxx + cyy (3.16)

where x and y represents polarization in the respective cartesian direction andthe 0 ≤ cx ≤ 1 and 0 ≤ cy ≤ 1 constant coefficients. For simplification of thetheory however, the beams will be generally taken to be explicitly y polarized(cx = 0).

3.5 Transverse Field Distribution

3.5.1 Hermite-Gaussian Modes

The general solutions to the paraxial equation in Cartesian coordinates(x, y, z) are known as the Hermite-Gaussian(HGnm) modes and the solutions incylindrical coordinates (ρ, φ, z) gives us the Laguerre-Gaussian (LG`p) modes.


The Hermite-Gaussian takes the form:

ET (rT, z) =Anmw(z)

Hn

(√2x

w(z)

)Hm

(√2y

w(z)

)e− x

2+y2

w2(z) e−i[(m+n+1) tan−1 zz0

+ k2R(z)

(x2+y2)]

(3.17)

where Anm =√

2−(n+m−1)

πn!m! is a normalization constant, w(z) = w0

√1 + z2

z20

parametrizes the beam width, Hq(u) = (−1)qeu2 dq

duq (e−u2

) denotes a qth-order

Hermite polynomial, R(z) = z(1 +z20z2 ) characterizes the phase curvature, z0 =

w2cw

20 is the Raleigh Range and w0 is the minimum beam width. The Laguerre-

Gaussian modes have a similarly complicated definition that can be seen in[7].

To visualize these modes defined by the electric field distribution, it shouldbe noted that the value of |ET (rT, z)|2 (The absolute value squared) gives thetransverse intensity distribution of the photon. This value can be physicallydefined as a region where the photon is most probable to be found.

The area of interest primarily focuses on the transverse properties of thesolutions so that for simplification the modes may be considered at the pointz = 0, which is taken to be the waist of the beam . This simplifies our formulafor the Hermite-Gaussian modes to give:

ET (rT, z) =Anmw0

Hn

(√2x

w0

)Hm

(√2y

w0

)e− x

2+y2

w20 = HGnm (3.18)

HGnm is the shorthand for this mode, where n and m are the order of the xand y Hermite polynomial respectively. The HG modes form a complete basisfor any physically realizable transverse electric field and it will be proven thatany mode can be expressed as a distinct linear combination of the HG modes.

Since the focus is initially going to be on HG modes, the LG modes willonly initially be visually represented or represented as a linear combination ofHG modes. With respect to the HG modes, the focus will primarily be on twospecific HG modes: HG01 and HG10 which are visually represented in 3.2:

The primary concern here is not simply to determine ET (rT , z) but to de-termine the electric field E(r, t) and now that ET (rT , z) can be simplified andwritten as HGnm, the component EL(z, t)e must also be included.

(for simplicity we take F (z, t) = 1). Also, the exponential term goes awaywhen considering the intensity (as proven in Math section). For simplification,the beam in the light is considered to be linearly polarized. Thus the polarizationterm e can be taken to be consistent throughout the experiment.

In the experiment, the underlying assumption is made that the beam widthfluctuates very little over the range of z and is well within the Raleigh range.The significance of this assumption is shown in FIG. 3.3.

3.6. FIRST ORDER MODE SPACE 17

(a) HG10 (b) |HG10|2

(c) HG01 (d) |HG01|2

Figure 3.2: The 3-dimensional plots show the transverse electric field of HG10

and HG01 modes. The 2-dimensional plots show their respective intensities

3.5.2 Laguerre-Gaussian Modes

3.6 First Order Mode Space

3.6.1 Linear Combinations of HG modes

At this point a moment should be taken to consider the mathematical termsinvolved in the HG modes. First, the statement that any first-order mode can beexpressed as a linear combination of two other first-order modes is considered.By definition the modes HG10 and HG01 can be taken as first-order simplybecause the sum of the subscripts n,m = 1. Any other first-order HG modemust be a linear combination of these modes. Also, there are first-order LGmodes that can result as a linear combination of the first order HG modes. The


Figure 3.3: The figure shows transverse modes HG00 and HG10 along differentvalues for z. It should be noted that changing the z value does not change thestructure of the transverse electric field but simply the diameter. By keeping thelongitudinal portion of the beam to a minimal displacement within the Raleighrange, this change becomes negligible

linear combination is defined as:

C10HG10 + C01HG01 = any first order mode (3.19)

where C10 and C01 are simply some complex constants that multiply the modes.Now looking specifically at the elements HG10 and HG01, they can be furthersimplified. Take beam width to be w0 = 1 since this constant does not sacrificegenerality with respect to the transverse portion of the beam. Considering HG10

this would simplify to:

2e−(x2+y2)

√2

πx (3.20)

and similarly HG01 becomes

2e−(x2+y2)

√2

πy (3.21)

This is defined to be a basis for any first order mode. The basis is defined asthe finite subset B = {C10HG10, C01HG01} of the Vector space, V (defined tobe the vector space containing any first order mode) over C. For B to hold trueas a basis, the elements must be linearly independent and also span the vectorspace V .

To show this let us take the two distinct solutions of HG modes that wehave picked to be our basis into consideration. This does not imply that theseare the only first order HG modes since their linear combinations should alsobe first order modes. As a solution of Eq. (3.7), HG10 can be written as

(∇2T + 2ik∂z)HG10 = 0 (3.22)


and similarly HG01 can be written as

(∇2T + 2ik∂z)HG01 = 0 (3.23)

To show that any linear combination C10HG10 +C01HG01 is also a solution tothe equation we write:

(∇2T + 2ik∂z)(C10HG10 + C01HG01) (3.24)

which we can expand to write

∇2TC10HG10 +∇2

TC01HG01 + 2ik∂zC10HG10 + 2ik∂zC01HG01 (3.25)

which can be rearranged to write

C10∇2THG10 + C102ik∂zHG10 + C01∇2

THG01 + C012ik∂zHG01 (3.26)

which is simply

C10(∇2T + 2ik∂z)HG10 + C01(∇2

T + 2ik∂z)HG01 (3.27)

but we know that

(∇2T + 2ik∂z)HG10 = (∇2

T + 2ik∂z)HG01 = 0 (3.28)

and therefore(∇2

T + 2ik∂z)(C10HG10 + C01HG01) = 0 (3.29)

so the linear combination

C10HG10 + C01HG01 (3.30)

is also a solution to the paraxial transverse wave equation Eq. (3.7). Further,we define an LG mode as the linear combination C10HG10 + C01HG01 whereC01 is purely imaginary and C10 is purely real.

Now since the electric field can be written as a product of the longitudinaland transverse wave functions, by plugging in Eq. (3.18) and Eq. (3.8) intoEq. (3.6) the Electric Field E(r, t) of Eq. (3.6) can be written as Enm where

Enm = HGnm × F (z, t)× ei(kz−ωct)e (3.31)

where HGnm is used to represent the HG mode that defines the electric fieldEnm

Consider Etotal = C10E10+C01E01 where E10 and E01 are two special casesof Eq. (3.31) and the coefficients C10 and C01 are complex constants. These canbe written out to see that

Etotal = C10HG10 × F (z, t)× ei(kz−ωct)e + C01HG01 × F (z, t)× ei(kz−ωct)e

By taking the longitudinal and polarization portions of the electric field tobe consistent throughout the experiment (so that all modes in the experimentshare these factors), they can be treated as constants and then


Etotal = (C10HG10 + C01HG01)× F (z, t)× ei(kz−ωct)e (3.32)

It has already been established that any first order mode can be represented bythe linear combination in Eq. (3.30) and that can be applied to Eq. (3.32). Sothe most general first-order solution to the paraxial wave equation form thatwill be encountered in the experiment can be written as:

Enm = (C10HG10 + C01HG01)× F (z, t)× ei(kz−ωct)e (3.33)

where once again it can be taken that in the experiment the polarization e willbe kept fixed so that all beams have the same orientation, hence allowing thiscomponent to be factored out as a constant. The envelope function applies moreso to beam pulses and in the experiment a steady beam output is used so thatF (z, t) = 1. Finally, if we look at any eiα we know that it can be written ascosα + i sinα. In finding the intensity, the absolute squared value of all theseelements are found so any eiα component simply becomes 1 as well.

(proof) |eiα|2 = (cosα + i sinα) × (cosα + i sinα)∗ where (cosα + i sinα)∗

is the complex conjugate (cosα − i sinα). This becomes (cos2 α − i2 sin2 α) =cos2 α+sin2 α = 1 or simply |eiα|2 = 1. Thus for the purpose of this experiment(for x or y linear polarization only) the simplification can be made that

F (z, t)× ei(kz−ωct)e = 1 (3.34)

and the electric field can often simply be represented by the equation:

Enm = HGnm = (C10HG10 + C01HG01) (3.35)

3.6.2 Inner Product

A moment must be taken to introduce some terms that will be essential tobetter understand the nature of the HG modes. The linear combination of HGmodes provide a basis for all first order solutions. This basis has the specialtrait of being orthonormal such that if the basis consists of the elements {ai, aj}then the inner product 〈ai, aj〉 = 0 if i 6= j and 〈ai, ai〉 = ||ai|| = |ai|2 = 1 forany i. Now by considering any normalized first order mode, it is establishedthat ∫∫

|HGnm|2dxdy = 1 (3.36)

which means that the inner product of any first order electric field after nor-malization would be 1. Now by taking the above statement and consideringEq. (3.35) it should hold that∫∫

|Enm|2dxdy = 1 (3.37)

and once again it should be mentioned that |Enm|2 implies that the complexvalued function |Enm| is multiplied by its complex conjugate |Enm|∗. It should


also be recalled that |Enm|2 gives the intensity. If∫∫|Enm|2dxdy = 1 (3.38)

is true, then ∫∫|[C10HG10 + C01HG01]|2dxdy = 1 (3.39)

must also be true. Expanding gives∫∫(C10HG10 + C01HG01)× (C∗10HG10 + C∗01HG01)dxdy = 1 (3.40)

which then becomes

|C10|2∫∫|HG10|2dxdy + [C10C

∗01 + C01C

∗10]

∫∫|HG10||HG01|dxdy

+|C01|2∫∫|HG01(x, y)|2dxdy (3.41)

and it is implied from Eq. (3.36) that∫∫|HG10|2dxdy = 1 and

∫∫|HG01|2dxdy =

1. Now looking at the middle section, as a trait of the HG modes, they are or-thonormal and hence any inner product

∫∫|HG10||HG01|dxdy = 0. Thus the

equation simplifies to|C10|2 + |C01|2 = 1 (3.42)

which is analogous to the form cos2 θ + sin2 θ = 1. Now it has been establishedthat the coefficients C10 and C01 are complex, a moment must be taken toelaborate on these. Every time the term |Cnm|2 is encountered, this impliesthat absolute value of this complex number is taken and squared. For anycomplex number Cnm = x+ iy the absolute value is simply

√x2 + y2.

3.6.3 Parameterized Electric Field

From this point, consider any arbitrary electric field E that can be writtenin the form of Eq. (3.35) as Enm = (C10HG10 + C01HG01) and rewrite thecomplex numbers C10 = r10e

iφ10 and C01 = r01eiφ01 respectively to get

Enm = r10eiφ10HG10 + r01e

iφ01HG01 (3.43)

which is then manipulated as follows:

Enm = eiφ10 [r10HG10 + r01ei(φ01−φ10)HG01] (3.44)

Now in this form, the factor eiφ10 can be removed because in taking the intensity,this value will be squared, which as previously shown implies |eiφ10 |2 = 1 so theintensity becomes

Inm = r210HG10 + r201eiφHG01 (3.45)


where eiφ = ei(φ01−φ10) but this is still analogous to cos2 α + sin2 α = 1 so theintensity can be described by

Inm = cos2(θ

2)HG10 + sin2(

θ

2)eiφHG01 (3.46)

where eiφ represents the difference in the φ values and thus the electric field canbe parameterized to be:

Enm = cos(θ

2)HG10 + eiφ sin(

θ

2)HG01 (3.47)

Where the angle θ2 is used to simplify the intensity function plot since it fully

parameterizes the function in cylindrical coordinates. Some examples of theintensity plots of resulting from different linear combinations of the HG modesare shown in FIG. 3.4

Figure 3.4: Some interesting linear combinations of HG10 and HG01 modes areshown in these intensity plots. Considering Eq. (3.47) the first linear combina-tion is attained when θ = π

2 and φ = 0 and the second is attained when θ = π2

and φ = π2

Thus to simplify the vector space represented by the first order HG modes,one can visualize the Bloch sphere in FIG. 3.5

3.7 1-D Parity

The definition of 1-D spatial parity transformations operators Πx and Πy whichrespectively act on the x and y coordinates of the transverse modes is givenbelow. Parity operations may occur every time an incident beam hits a reflectivesurface. Consider the transverse electric field, ET being acted on by the parityoperators:

ΠxET (x, y, z)→ ET (−x, y, z) (3.48)

ΠyET (x, y, z)→ ET (x,−y, z) (3.49)

Note that ET (x, y, z) is another way to write ET (rT, z) from Eq. (3.6) and waswritten a such to show the significance of the parity operation. For the types of

3.8. INTEREROMETER MATRIX 23

Figure 3.5: By changing the parameters in the spherical coordinates θ and φone can create any first order mode. All first order modes can be represented assome point on the sphere, thus this sphere visually represents the entire vectorspace described in the experiment.

interactions occurring in the interferometer, only the x coordinate parity oper-ator Πx needs to be considered because the beam will be confined to splitting,reflection and interference in a single plane such that only x parity occurs, aswill be shown later. The Πx parity operator can be visualized in Fig. 3.6

It should be noted that the parity operator literally causes the 3D electricfield plot of the various modes to flip about the y-axis. This is the reason that aparity operation on an HG10 mode shows a visible change while an HG01 modeshows no such change. Hence any mode that is symmetrical about the y-axiswill show no visible change upon being reflected while any other mode will.

3.8 Intererometer Matrix

The structure of the interferometer can now be considered. Taking as inputa single beam in an arbitrary first order mode, the structure of an interferometeris illustrated in FIG. 3.7. In the figure, the beam which is a linear combinationof the form seen in Eq. (3.47) goes in at the port labeled 1 where it encountersa 50 : 50 beam splitter. The 50 : 50 beams splitter was chosen for simplificationof calculations as shown in [7]. The two subsequent beams then undergo variousinteractions and then combine at the second 50 : 50 beam splitter resulting in


(a) HG10 (b) Πx HG10

(c) HG01 (d) Πx HG01

Figure 3.6: 3D plot showing the effect of Πx operator on HG10 and HG01

modes.

beams ejected from the outputs A and B. It is interesting to note that althougha single beam was passed in, it was separated according to the respective 1-Dparity components that comprised the beam. Due to the conservation of energy,the sum of intensities of the beams at the output port should equal that of theintensity of the input beam due to conservation of energy. The mathematics ofthe sorter is described in the following section.

3.8.1 Parity Math

The mathematics behind the parity-based interferometer follow a simple formulaas established in [7]. Let the input electric fields be defined as the column vector

Ein =

(E1

E2

)where E1 and E2 represent the electric fields of the inputs 1 and 2

in FIG. 3.7. This represents the general input of the interferometer but let theinput vector E2 = 0 for simplicity. This represents the simpler case of 1 inputbeam going into the interferometer. Another simplification to be made is thatany beam passed into the interferometer will have y as the polarization term

of Eq. (3.6). Likewise the output beam is defined as Eout =

(EA

EB

)where EA

and EB are the electric fields of the respective output beams A and B shown in

3.8. INTEREROMETER MATRIX 25

Figure 3.7: The inputs to the interferometer are labeled 1 and 2. The thindiagonal lines represent beam splitters and the thicker lines represent mirrors.GP represents a glass plate that will be used to control the beam.

FIG 3.7.Within the interferometer, the input beam will be manipulated by a series

of different operations that occur every time a beam is split or interacts witha mirror. These interactions can be simplified into a single operator actingon the input beam defined as O such that the output of the interferometerEout = OEin. Taking a closer look at the operator O, it simplifies the followinginteractions:Upon entry into the system, the electric field of the input beam is subsequentto the first beam splitter. This interaction is governed by Eout = UbEin whereUb is the operator matrix of the beam splitter as defined in[7]. The vectornotation is (

EA

EB

)=

1

2

(1 ii 1

)(E1

E2

)(3.50)

The beam is oriented in such a way that parity based reflections occur only in thex direction as established in the previous section. Since one component of thebeam undergoes a single mirror reflection and the other component undergoestwo mirror reflections as seen in the FIG. 3.7, every reflection can be representedby a parity operation

Π ≡(Π2x 0

0 Πx

)=

(1 0

0 Πx

)(3.51)

Since there is also a glass plate that can impose some arbitrary but uniform


phase shift φ there is a matrix for that as well, which is represented by

Φ ≡

(eiφ2 0

0 e−iφ2

)(3.52)

Finally there is another beam splitter at the output end which is of the formUB so that the total interaction Eout = OEin is represented a simplification ofEout = UbΦΠUbEin which gives:(

EAEB

)=

1

2

(eiφ2 − e−i

φ2 Πx i(ei

φ2 + e−i

φ2 Πx)

i(eiφ2 + e−i

φ2 Πx) −(ei

φ2 − e−i

φ2 Πx)

)(E1

E2

)(3.53)

3.9 1 Beam Input

Now by the mathematical definition given for the basis, any first order modecan be written as some linear combination of the form Eq. (3.47). Since theinterferometer involves parity with respect to the x component, the HG modecan be distinguished as “odd” or “even” depending its structure. The casewhere the n subscript in HGnm has a value of 1, it is odd and likewise whenn has a value of 0, the HG mode is even. This is important because the Πoperator within U affects these odd and even modes differently. An exampleof this was shown in FIG. 3.6 with respect to the HG10 and HG01 modes.For simplicity, the assumption is made that there is only 1 input beam andthus E2 = 0 and that the form of the input E1 is a distinct HG10 or HG01

first order mode (such that either the coefficient C10 = 0 or C01 = 0 in thelinear combination established in Eq. (3.35) that defines the electric field). Thiswas done to simplify the action of the Π operator since this would allow thereplacement ΠxE1 → (−1)1E1 → −E1 where an HG10 or ”odd” mode isencountered or ΠxE1 → (−1)0E1 → E1 where an HG01 or ”even” mode isencountered.

Considering the odd mode for input E1 and since E2 = 0 the multipliedEout becomes (

EA

EB

)=

1

2

([ei

φ2 + e−i

φ2 ]E1

[i(eiφ2 − e−i

φ2 )]E1

)(3.54)

which can then be rewritten by multiplying the second element by ii and bring

the fraction 12 into the equation to get:

(EA

EB

)=

eiφ2 +e−i

φ2

2 E1

i2 (eiφ2 −e−i

φ2 )

2i E1

(3.55)

which is of Euler’s form for Cosine and Sine respectively. Thus substitutingthese values gives: (

EA

EB

)=

(cos(φ2 )HG10

− sin(φ2 )HG10

)(3.56)

3.10. 2 BEAM INPUT 27

Likewise, considering the even mode for input E1 and since E2 = 0 the multi-plied Eout in this case becomes(

EA

EB

)=

1

2

([ei

φ2 − e−i

φ2 ]E1

[i(eiφ2 + e−i

φ2 )]E1

)(3.57)

which is treated similarly to the first matrix by multiplying the first element byii and bringing in the scalar 1

2 from the outside to get:(EA

EB

)=

i ei φ2 −e−i φ22i E1

i eiφ2 +e−i

φ2

2 E1

(3.58)

which once again gives is of Euler’s form and can be rewritten as:(EA

EB

)=

(i sin(φ2 )HG01

i cos(φ2 )HG01

)(3.59)

As it can be seen, if some input E1 was given of the form HG10 only then therespective outputs would be given by Eq. (3.56) for EA and EB and if E1 wasgiven of the form HG01 only then the respective outputs for EA and EB wouldbe given by Eq. (3.59). The input mode E1 can be controlled to be simply oneor the other state but it is very straightforward to represent any input mode forE1 that can be written in the form Eq. (3.30) In this case, the odd and evencomponents can be added and then written in the form Eq. (3.47)Which can bemathematically expanded by using the respective equations above as:

E1 = cos(θ12

)× Eq. (3.56) + eiφ1 sin(θ12

)× Eq. (3.59) (3.60)

then the overall equation becomes(EA

EB

)=

(cos(φ2 ) cos( θ12 )HG10 + i sin(φ2 )eiφ1 sin( θ12 )HG01

i cos(φ2 )eiφ1 sin( θ12 )HG01 − sin(φ2 ) cos( θ12 )HG10

)(3.61)

3.10 2 Beam Input

Now that the specific but easier case of one beam input has been determined,the equation for the input of two beams can be resolved as well. Picking upfrom Eq. (3.53):(

EA

EB

)=

1

2

(eiφ2 − e−i

φ2 Πx i(ei

φ2 + e−i

φ2 Πx)

i(eiφ2 + e−i

φ2 Πx) −(ei

φ2 − e−i

φ2 Πx)

)(E1

E2

)(3.62)

Looking at this equation it is immediately possible to break up the compo-nents with the Πx from the components without them to get(

EA

EB

)=

1

2

((eiφ2 iei

φ2

ieiφ2 −ei

φ2

)+

(−e−i

φ2 ie−i

φ2

ie−iφ2 e−i

φ2

)Πx

)(E1

E2

)


Now at this point it is important to write out the beams E1 and E2 in theirexpanded form(

EA

EB

)=

1

2

((eiφ2 iei

φ2

ieiφ2 −ei

φ2

)+

(−e−i

φ2 ie−i

φ2

ie−iφ2 e−i

φ2

)Πx

)(cos θ12 HG10 + eiφ1 sin θ1

2 HG01

cos θ22 HG10 + eiφ2 sin θ22 HG01

)

and then by using the properties of matrix multiplication, the beam can berewritten as to get(

EA

EB

)=

1

2

(eiφ2 iei

φ2

ieiφ2 −ei

φ2


2 HG01


)(3.63)

+1

2

(−e−i

φ2 ie−i

φ2

ie−iφ2 e−i

φ2

)(cos θ12 ΠxHG10 + eiφ1 sin θ1

2 ΠxHG01

cos θ22 ΠxHG10 + eiφ2 sin θ22 ΠxHG01

)

A moment should be taken to look at the second element with the Πx op-erator multiplying each element. By definition this parity operator directlyaffect the x component only of the transverse Electric field function. Since anytransverse field can be resolved into a linear combination of HG10 and HG01.Further, by definition the mode HG10 will exhibit the parity in x while theHG01 mode does not. Mathematically this means that the HG10Πx → −HG10

and HG01Πx → HG01 and thus our matrix becomes

(EA

EB

)=

1

2

(eiφ2 iei

φ2

ieiφ2 −ei

φ2


2 HG01


)(3.64)

+1

2

(−e−i

φ2 ie−i

φ2

ie−iφ2 e−i

φ2

)(− cos θ12 HG10 + eiφ1 sin θ1

2 HG01

− cos θ22 HG10 + eiφ2 sin θ22 HG01

)

Chapter 4

Mathematical Theory

4.1 The Unitary Matrix

Looking at the models derived for 1 beam input and the resulting simu-lations, the intensity of the beams were taken by integrating the electric fieldfunctions in mathematica. The input beams and output beams had an overallmagnitude equal to 1 after normalization. In the 1 beam input case it seemsthat the matrix preserves Isometry. It was hypothesized that the matrix fromEq. (3.53) is Unitary. A unitary matrix is a square n × n complex matrix Uthat satisfies the condition:

U†U = UU† = In (4.1)

where In is the identity matrix in n dimensions and U† is the Hermitian adjointof U To prove that the equation is indeed Unitary, we restate it from Eq. (3.53)for convenience:

1

2

(eiφ2 − e−i

φ2 Πx i(ei

φ2 + e−i

φ2 Πx)

i(eiφ2 + e−i

φ2 Πx) −(ei

φ2 − e−i

φ2 Πx)

)(4.2)

In this form it is not clear how to go about finding the complex conjugateof the matrix, but by using the fact that

eiϕ = cosϕ+ i sinϕ = A+Bi (4.3)

where A represents the real component and B represents the imaginary compo-nent the matrix can be simplified. First, let ϕ = φ

2 so that Eq. (4.2) becomes

1

2

(eiϕ − e−iϕΠx i(eiϕ + e−iϕΠx)

i(eiϕ + e−iϕΠx) −(eiϕ − e−iϕΠx)

)(4.4)

but we know that eiϕ = cosϕ+ i sinϕ so our equation now becomes

1

2

(cosϕ+ i sinϕ− [(cosϕ− i sinϕ)Πx] i(cosϕ+ i sinϕ+ [(cosϕ− i sinϕ)Πx])

i(cosϕ+ i sinϕ+ [(cosϕ− i sinϕ)Πx]) −(cosϕ+ i sinϕ− [(cosϕ− i sinϕ)Πx])

)(4.5)

29

30 CHAPTER 4. MATHEMATICAL THEORY

which we expand to get

1

2

(cosϕ− cosϕΠx + i sinϕ+ i sinϕΠx i(cosϕ+ cosϕΠx + i sinϕ− i sinϕΠx)

i(cosϕ+ cosϕΠx + i sinϕ− i sinϕΠx) −(cosϕ− cosϕΠx + i sinϕ+ i sinϕΠx)

)(4.6)

which we then factorize to get

1

2

((1− Πx) cosϕ+ (1 + Πx)i sinϕ i((1 + Πx) cosϕ+ (1− Πx)i sinϕ)

i((1 + Πx) cosϕ+ (1− Πx)i sinϕ) −((1− Πx) cosϕ+ (1− Πx)i sinϕ)

)(4.7)

Expanding the elements being multiplied by i we get:

1

2

((1− Πx) cosϕ+ (1 + Πx)i sinϕ (1 + Πx)i cosϕ− (1− Πx) sinϕ)

1 + Πx)i cosϕ− (1− Πx) sinϕ) −((1− Πx) cosϕ+ (1− Πx)i sinϕ)

)(4.8)

but once again we notice that this is simply of the form of Eq. (4.3) so then wecan rewrite our matrix as

1

2

(a+ bi c+ dic+ di −a− bi

)(4.9)

where

a = (1− Πx) cosϕ

b = (1 + Πx) sinϕ

c = −(1− Πx) sinϕ

d = (1 + Πx) cosϕ (4.10)

We call this equation U and hence find U† as well such that

U =1

2


)(4.11)

and

U† =1

2

(a− bi c− dic− di −a+ bi

)(4.12)

Now we can multiply our matrices from Eq. (4.11) × Eq. (4.12) as shown

UU† =1

4


)(a− bi c− dic− di −a+ bi

)(4.13)

and we can expand the matrices to get the matrix((a+ bi)(a− bi) + (c+ di)(c− di) (a+ bi)(c− di) + (c+ di)(−a+ bi)

(c+ di)(a− bi) + (−a− bi)(c− di) (c+ di)(c− di) + (−a− bi)(−a+ bi)

)(4.14)

4.1. THE UNITARY MATRIX 31

which simplifies to

UU† =1

4

(a2 + b2 + c2 + d2 2bci− 2adi−(2bci− 2adi) a2 + b2 + c2 + d2

)(4.15)

now we must take a moment to consider the individual elements, consideringthe first element of the matrix we have a2 +b2 +c2 +d2. Substituting Eq. (4.10)and rearranging we get

(1− Πx)2 cos2 ϕ+ (1− Πx) sin2 ϕ+ (1 + Πx)2 sin2 ϕ+ (1 + Πx)2 cos2 ϕ (4.16)

and since we know that sin2 ϕ+ cos2 ϕ = 1 we get

(1− Πx)2 + (1 + Πx)2 (4.17)

which we expand to get

1− 2Π2x + Π2

x + 1 + 2Π2x + Π2

x (4.18)

but we know that Π2x = 1 so

1− 2Π2x + 1 + 1 + 2Π2

x + 1 = 4 (4.19)

We now consider the second element 2bci − 2adi and again SubstitutingEq. (4.10) and using Π2

x = 1 we get

2((1 + Πx) sinϕ)(−(1− Πx) sinϕ)− 2((1− Πx) cosϕ)((1 + Πx) cosϕ)

= 2(sin2 ϕ)((1 + Πx)(1− Πx))− 2(cos2 ϕ)((1 + Πx)(1− Πx))

= 2(sin2 ϕ)(1− Π2x)− 2(cos2 ϕ)(1− Π2

x)

= 2(sin2 ϕ)(1− 1)− 2(cos2 ϕ)(1− 1)

= 0 (4.20)

Now that the values of each element of the matrix has been determined, weget

UU† =1

4

(a2 + b2 + c2 + d2 2bci− 2adi−(2bci− 2adi) a2 + b2 + c2 + d2

)=

1

4

(4 00 4

)=

(1 00 1

)= I2 (4.21)

which is indeed the identity matrix in two dimensions. Hence the general matrixEq. (3.53) is Unitary.

32 CHAPTER 4. MATHEMATICAL THEORY

Chapter 5

Simulating theMathematical Model

5.1 1 Beam Input

FILLER

33

34 CHAPTER 5. SIMULATING THE MATHEMATICAL MODEL

Chapter 6

Experiment

6.1 Apparatus

6.1.1 Iris

Filler

6.2 Procedure

6.2.1 Setting up the interferometer

6.2.2 Borrowing beam from 2D Parity Sorter

35

36 CHAPTER 6. EXPERIMENT

Chapter 7

Results and Analysis

7.1 HeNe Laser

Once the interferometer was aligned, it became important to start under-standing the beams that were output at each port. The initial output at bothports was a simple Gaussian (HG00 mode). To better understand this HG00

consider that it is the simplest HG mode and the fundamental mode that is seenwhen looking at a normal beam of laser light. Since the n subscript of this modeis 0 it is even and it is hypothesized that if only an HG00 mode were passedthrough the a perfectly calibrated interferometer, then at some ideal phase dif-ference between the beams propagating within it, the HG00 mode would be seencoming out of only 1 port. This would be the instance where the interferometeris acting like a sorter and hence the even mode would be sorted out of 1 portand the odd out of the other as shown in FIG 7.1 . In the actual experiment,these theoretical conditions were not always held. As the glass plate was tilted,the intensities of the Gaussian beams seem to visibly change until the brightnessof the beams had essentially swapped between the two output ports. While oneport dimmed significantly as the phase was changed by using the glass plate,it never truly became completely dark because of“leakage” of the mode in theinterferometer. Regardless, this was still a confirmation that sorting was indeedoccuring within the interferometer. To minimize this, the polarizer was addedand set to 90◦ (so that the beam was polarized for better interference in theinterferometer). This produced the output shown in Fig.7.2

The next step was to try and optimize the alignment of the beams that wereoverlapping in the interferometer. This could be quantified by achieving thedimmest possible output at the odd port of the interferometer output. Duringthe alignment process the polarizer was initially not a part of the apparatus.The beam was simply passed through two irises before entering into the inter-ferometer. Since the polarizer consists of some material that has a differentoptical density from air, any beam passing it would be refracted and potentiallyskewed. Thus it was worth the effort to realign the laser to compensate for

37

38 CHAPTER 7. RESULTS AND ANALYSIS

Figure 7.1: Sorting of the HG00 mode. When this mode is input to the interfer-ometer and phase difference in the interferometer φ = 0 then the beam wouldcome out of port B. Conversely if the phase difference was φ = π then the beamwould come out of port A. For any other phase difference, the HG00 would beoutput of both ports with varying intensity.

any slight misalignment that introducing the polarizer would have caused in theinterferometer. In doing this, the output of the interferometer is shown in Fig.7.3:

where the most interesting consequence of the changes was that there becamea visible HG10 mode. This was modeled in section To better understand what isbeing seen in Fig. 7.3 we consider that the He-Ne laser is not simply producingpurely a HG00. What is actually being produced in the laser beam includes theHG00, HG01 and HG10 modes.

The table shows the results of the intensities observed for the input andoutput beams where the values indicated for each field below were the result of

Table 7.1: Measurements of the voltages in the interferometer. Values observed

Port Avg Voltage (mV)Input 1382 (± 13)Port A 374 (± 13)Port B 668 (± 13)

the average of 5 consecutive readings. From the table the following efficiencyvalues were calculated: From the table it can be seen that there is about a25% loss in the total input power. In an ideal situation the total input powerwould be the same as the total output power. Due to the conservation of energy,

7.1. HENE LASER 39

(a) Port A (b) Port B

Figure 7.2: Output at both ports for polarized input beam

Table 7.2: Efficiency values of the Interferometer.

Output Efficiency Sorting Efficiency0.75 (± 0.01) 0.21 (± 0.01)

this is not possible in practical setting because some of the light energy is lotseverytime the beam is reflected, refracted and generally as the beam propagatesthrough space. The closer the reading for output efficiency is to 1 the better.On the contrary, the sorting efficiency should ideally be 0.

Following this a polarizing beam splitter was then used to get better polar-ized light into the interferometer. This yielded the following results:

Table 7.3: Measurements of the voltages in the interferometer with PBS.

Port Avg Voltage (mV)Input 550 (± 13)Port A 328 (± 13)Port B 148 (± 13)

and the following efficiency valuesFrom these values it can be understood that


(a) Port A (b) Port B

Figure 7.3: Output at both ports for polarized and realigned input beam

Table 7.4: Efficiency values of the Interferometer with the PBS in place.

Output Efficiency Sorting Efficiency0.87 (± 0.01) 0.33 (± 0.01)

7.2. 2D SORTER OUTPUT 41

7.2 2D Sorter Output

A beam from the 2D Intererometer was also borrowed. The


Chapter 8

Conclusions

8.1 Stuff

This is dummy text.

43

44 CHAPTER 8. CONCLUSIONS

Appendix A

Extra Stuff

This is dummy text.

45

46 APPENDIX A. EXTRA STUFF

Appendix B

Extra Stuffing

This is dummy text.

47

48 APPENDIX B. EXTRA STUFFING

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