Alan Stanley
Wed, 11/03/2021 - 07:16
Edited Text
THE ORBITAL ANGULAR MOMENTUM OF LIGHT

by
Nicholas Paul Pellatz
A thesis submitted in partial fulfillment of the requirements
for graduation with Honors in Physics.

Whitman College
2014

Certificate of Approval
This is to certify that the accompanying thesis by Nicholas Paul Pellatz has been
accepted in partial fulfillment of the requirements for graduation with Honors in
Physics.

________________________
Mark Beck, Ph.D.

Whitman College
May 15, 2014

Contents
1 Introduction

1

2 Theory

5

2.1

Orbital Angular Momentum in Paraxial Beams . . . . . . . . . . . .

5

2.2

Generation of the LG Modes . . . . . . . . . . . . . . . . . . . . . . .

8

2.3

Interference of the LG Modes . . . . . . . . . . . . . . . . . . . . . .

11

3 Experiment

14

3.1

Table Setup for Generating and Interfering LG Modes . . . . . . . . .

14

3.2

Generating Single LG Modes . . . . . . . . . . . . . . . . . . . . . . .

16

3.3

Superpositions of LG Modes . . . . . . . . . . . . . . . . . . . . . . .

21

3.4

Sorting Even and Odd LG Modes . . . . . . . . . . . . . . . . . . . .

24

4 Conclusion

28

A Detailed Experimental Setup

30

B LabVIEW Programming

32

iii

Abstract
Here we present the results of an exploration of the Laguerre-Gaussian (LG) modes
of laser light. These modes, each characterized by an integral index l, carry orbital
angular momentum due to their helical phase fronts. We demonstrate methods of
generating the LG modes from a simple Gaussian mode using computer-generated
holograms and find that we can generate each of the LG modes with indices l = −3
through l = 3. We also predict and confirm the behavior of positive and negative LG
modes in a superposition. The intensity pattern of a superposition of an LG mode
with index l and one with index −l is a symmetric arrangement of 2l bright spots
which rotate as the relative phase between the two beams is adjusted. Finally, we
explore an interferometric method of sorting even and odd modes from a superposition
and find that we can separate an l = 0 mode and an l = 1 mode from a superposition
of the two.

iv

List of Figures
1.1

The intensity profile of a simple Gaussian mode. . . . . . . . . . . . .

1

1.2

The intensity profile of an l = 1 LG mode. . . . . . . . . . . . . . . .

2

1.3

An l = 1 forked diffraction grating. . . . . . . . . . . . . . . . . . . .

3

2.1

Simplified depiction of how a hologram works. . . . . . . . . . . . . .

9

2.2

Simulated interference of tilted LG beam and Gaussian reference beam. 11

2.3

Starting on the left and going right, l = −1,+2, and +3 holograms. .

2.4

Starting top-left and going clockwise, interference patterns for l1 − l2 =

11

2, 4, and 6 (δ = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.5

Intensity patterns for l1 − l2 = 2 with δ = π/2 and δ = π. . . . . . . .

13

3.1

Modified Sagnac interferometer used to form superpositions of LG modes. 15

3.2

In the top-left, a photograph of the first diffracted beam using an l = 1
hologram. This photograph is 300 × 300 pixels with an exposure time
of 10ms. In the bottom left, the intensity of each pixel from the image
above plotted vertically. The top-right and bottom-right images are
theoretical fits of the data on the left to Equation 3.1. . . . . . . . . .

3.3

17

In the top-left, a photograph of the first diffracted beam using an l = 2
hologram. This photograph is 360 × 360 pixels with an exposure time
of 10ms. In the bottom left, the intensity of each pixel from the image
above plotted vertically. The top-right and bottom-right images are
theoretical fits of the data on the left to Equation 3.1. . . . . . . . . .
v

19

LIST OF FIGURES
3.4

vi

On the left, a photograph of the first diffracted beam using an l = 3
hologram. This photograph is 400 × 400 pixels with an exposure time
of 10ms. On the right, a theoretical fit of the data to Equation 3.1. .

3.5

20

Starting on the left and going right, first-order diffracted beams for
l = 1, l = 2, and l = 3 holograms shown on the same scale (each
photograph had an exposure time of 10ms and was cropped to 360×360
pixels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

20

In the top row, theoretical intensity patterns for the superpositions of
LG modes labeled above. Subsequent rows show photographs of superpositions with each row having slightly more relative phase between
the interfering beams than the row above. Here, each photograph had
an exposure time of 10ms and was cropped to 320 × 320 pixels. . . .

3.7

An illustration of the way a Dove prism flips a laser beam over the axis
defined by it’s lower base. . . . . . . . . . . . . . . . . . . . . . . . .

3.8

23

24

Altered version of the interferometer in Figure 3.1 with Dove prisms
in each arm. The Dove prisms are oriented at 90◦ with respect to each
other for sorting odd and even LG modes. . . . . . . . . . . . . . . .

3.9

25

Illustration of what happens to even and odd modes in the two arms
of our sorting interferometer. . . . . . . . . . . . . . . . . . . . . . . .

26

3.10 Photographs of the two outputs of the sorting interferometer when the
input is a superposition of l = 0 and l = 1 states. Each photograph has
a slight increase in the relative phase δ from the one above it. If the
top photograph has δ = 0, then the third photograph has δ = π, and
the final photograph has δ = 2π. The photographs all have resolution
1260 × 460 pixels and were taken with an exposure time of 50 ms. . .

27

A.1 Photograph of our table setup including both the generating interferometer and the sorting interferometer. . . . . . . . . . . . . . . . . .

30

A.2 Detailed diagram of the table setup shown in Figure A.1. . . . . . . .

31

LIST OF FIGURES

vii

B.1 Block diagram for WindSettings.vi which sets parameters for one window. 32
B.2 Front panel for Open External Window.vi. . . . . . . . . . . . . . . .

33

B.3 Block diagram for Open External Window.vi. . . . . . . . . . . . . .

34

B.4 Basler program used to capture photographs with the USB camera
used in the lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Chapter 1
Introduction
Without thinking, it it is easy to make the mistake of assuming that the output
of a laser is a uniform column of light. The more complicated truth is that the
output of any laser, being comprised of electromagnetic waves, must satisfy Maxwell’s
equations. This condition on a laser beam restricts the possible light output to a
number of various modes. For instance, the output of a typical laser is in the Gaussian
mode. A laser in the Gaussian mode shining on a screen is brightest in the middle
and gets dimmer further from the center as can be seen in Figure 1.1. In fact, the
intensity profile across any diameter of the beam follows the familiar Gaussian“bell
curve” shape. Of course there are many other modes which a beam of laser light can
be in and many ways of transforming a beam in the Gaussian mode to another mode.

Figure 1.1: The intensity profile of a simple Gaussian mode.
1

CHAPTER 1. INTRODUCTION

2

In this study, we were most interested in the set of modes known as the LaguerreGaussian (LG) modes. The electric field of the LG modes is characterized by two
indices, usually p and l. Here we restrict ourselves to an investigation of the subset
of the LG modes whose p index is 0. The index l can then take on any integer value.
The intensity profile of the LG mode with l = 1 is shown in Figure 1.2. This intensity
profile, however, hides the more complicated structure of the LG modes. Recall
that the Poynting vector characterizes the direction and magnitude of energy flow
for electromagnetic waves. In a simple Gaussian beam, the Poynting vector points
straight along the beam axis. In the LG modes on the other hand, the Poynting
vector moves in a spiral about the beam axis as the beam propagates [1]. The value
of the index l characterizes the tightness of the Poynting vector’s spiral.
There are several different methods for creating a beam in an LG mode. One
method involves putting the laser beam through two π/2 mode converters. This
method, however, requires the input to be in one of the Hermite-Gaussian modes.
Each Hermite-Gaussian mode gets converted into a different LG mode [1]. The
method preferred in this study is one that uses a Spacial Light Modulator (SLM)
to get a variety of LG modes from a single Gaussian beam. An SLM has a liquid
crystal display which allows the creation of any diffraction grating needed. This technology is widely used in commercial LCD projectors [2]. On the SLM, we place a

Figure 1.2: The intensity profile of an l = 1 LG mode.

CHAPTER 1. INTRODUCTION

3

Figure 1.3: An l = 1 forked diffraction grating.
special forked diffraction grating, an example of which is illustrated in Figure 1.3.
The grating shown is an l = 1 grating, which means that the first order diffracted
beam when this grating is illuminated with a Gaussian beam will be the l = 1 LG
mode. In fact, the mth order diffracted beam will be in the l = m LG mode. The
mathematical origins of these forked gratings will be explored further in Chapter 2.
What is perhaps most surprising about the LG modes is that they carry orbital
angular momentum. A beam of light in the LG mode with index l carries orbital angular momentum equal to l~ per photon, and because l can only take on integer values
the orbital angular momentum is quantized [3]. We will motivate these assertions in
Section 2.1. The fact that l can only take on integer values follows from an exponential eilφ term in the electric field amplitude of the LG modes (here, φ is the angle in
the plane perpendicular to the beam axis). As we should expect, this term oscillates
about the beam axis with a frequency dependent on l. Of course, if φ is increased
by 2π, then the field should be the same. In other words, eilφ = eil(φ+2π) = eilφ eil2π .
But eil2π will equal 1 only if l is an integer. Thus, the orbital angular momentum is
quantized for beams with φ dependence eilφ .
The orbital angular momentum variable exhibits quantum properties very similar
to those of the spin variable for photons. It has been shown experimentally that

CHAPTER 1. INTRODUCTION

4

pairs of photons can be entangled in their orbital angular momentum [4]. There
has even been a demonstration of an indeterminacy relation between the angular
position of a photon in a beam and its orbital angular momentum [5]. What sets
orbital angular momentum apart from spin angular momentum, however, is the fact
that for photons there are only two spin states available, but infinitely many orbital
angular momentum states. The variable l can take on any integer value. We also
have good classical reasons to believe that the quantum states corresponding to the
LG modes form a complete and orthogonal basis set [3]. In other words, the orbital
angular momentum state of any photon can be written as a linear combination of LG
states and none of the LG states themselves can be written as a linear combination
of any other LG states. For this reason, we say that the LG states define an infinitedimensional Hilbert space.
The study of the quantum properties of the LG states is still relatively young
(the first demonstration of entanglement in the orbital angular momentum variable
was as recent as 2001 [4]). The infinite-dimensional nature of the LG state space
makes it a potentially very fruitful space in which to study entanglement effects and
try to understand them on a deeper level. But in order to perform entanglement
experiments, we must first have a solid understanding of how the LG modes can be
generated, superimposed, and sorted efficiently. The research presented here represents an important step towards being able to study entanglement of orbital angular
momentum states in an undergraduate laboratory.

Chapter 2
Theory
2.1

Orbital Angular Momentum in Paraxial Beams

It is instructive to first see how light beams treated classically can carry orbital angular
momentum. The work within this section will all be done within the approximation
that the transverse spatial distribution of the beam of light does not vary rapidly as
the beam propagates (known as the paraxial approximation). Let u(x, y, z) be the
spatial amplitude distribution for our beam of light and let the beam axis lie along
ẑ. We will then take our vector potential to be oriented along the x-axis:
A = u(x, y, z)ei(kz−ωt) x̂
Here, k is the wave vector and ω is the angular frequency. The magnetic field is then



i ∂u
i ∂u
B = ∇ × A = ik u −
ŷ +
ẑ ei(kz−ωt) .
k ∂z
k ∂y
Within the paraxial approximation, however, ku is much greater than
tude. In that case, the magnetic field becomes


i ∂u
B ≈ ik uŷ +
ẑ ei(kz−ωt) .
k ∂y

∂u
∂z

in magni-

(2.1)

Now, according to Maxwell’s equations, the electric field E has dependence on both
the time derivative of A and the spatial derivative of the scalar potential V . So the
5

CHAPTER 2. THEORY

6

next thing to do is choose a gauge to work within and find the scalar potential V
in that gauge. In the Lorentz gauge, the relationship between the vector and scalar
potentials is given by [6]
∇·A=−

1 ∂V
.
c2 ∂t

Plugging in our expression for A and solving for V , we find that


∂u i(kz−ωt) −1 ∂V
c2 i(kz−ωt) ∂u
∂ c2 i(kz−ωt)
.
e
= 2
=⇒ V = e
=
ue
∂x
c ∂t

∂x
∂x iω
Then, knowing A and V , the electric field is



∂A

c2
i(kz−ωt)
E = −∇V −
=− ∇
ue
+ iωuei(kz−ωt)
∂t

∂x

c2 ∂
u∇ei(kz−ωt) + ei(kz−ωt) ∇u + iωuei(kz−ωt)
=−
iω ∂x

(2.2)

I have written out the gradient of V in this way in order to make the observation that
u changes on length scales much longer than a wavelength while ei(kz−ωt) oscillates
quickly on the order of a wavelength. Therefore, the first term in brackets in Eqn.
2.2 is much larger than the second term. Throwing out the second term, we are left
with an expression for E which is similar in form to Eqn. 2.1:

i ∂u
ẑ ei(kz−ωt)
E ≈ ikc ux̂ +
k ∂x


We have now described both the magnetic and electric fields of our beam without
assuming anything about the form of u(x, y, z) other than the fact that the beam is
paraxial. A natural next step to take would be to calculate the Poynting vector for this
beam since the Poynting vector contains information about the beam’s momentum
and energy.
According to Griffiths, the Poynting vector is the energy flux density, or energy
per unit time per unit area, in an electromagnetic field [6]. It’s time-averaged form
is given by
hSi =

1
1
hE × Bi =
(E∗ × B + E × B∗ ) .
µ0
2µ0

CHAPTER 2. THEORY

7

If we then plug in our expressions for E and B, we find that




∂u∗
∂u∗
k2c 2
ikc
∗ ∂u
∗ ∂u
u
−u
x̂ + u
−u
ŷ +
|u| ẑ.
hSi =
2µ0
∂x
∂x
∂y
∂y
µ0
This equation takes a slightly simpler form if we define a gradient-like operator Ô ≡

( ∂x
x̂ +


ŷ):
∂y

hSi =

k2c
ikc
uÔu∗ − u∗ Ôu +
|u|2 ẑ.
2µ0
µ0

(2.3)

Up to this point, we have been working completely within Cartesian coordinates
even though cylindrical coordinates are the more natural choice if we want to think
about angular momentum along the z-axis. Equation 2.3 lets us transform to cylindrical coordinates by rewriting our defined operator:





1 ∂

x̂ +
ŷ =
r̂ +
φ̂ .
Ô ≡
∂x
∂y
∂r
r ∂φ
We now assume that u(x, y, z) can be written as
u(x, y, z) = u(r, φ, z) = u0 (r, z)eilφ .

(2.4)

In general, l could be any complex number. But since u(r, φ, z) is a physical quantity,
it must be single-valued. In other words, we must have
u(r, φ, z) = u(r, φ + 2π, z) =⇒ u0 (r, z)eilφ = u0 (r, z)eilφ eil2π =⇒ eil2π = 1.
This means that l must be an integer. We can then find the φ component of our
expression for the Poynting vector:




ikc
ilφ 1 ∂
∗ −ilφ
∗ −ilφ 1 ∂
ilφ
u0 e
ue
− u0 e
u0 e
hSiφ =
2µ0
r ∂φ 0
r ∂φ
lkc|u|2
=
µ0 r

(2.5)

The spatial amplitude distribution for the LG modes can, in fact, be written in the
form of Equation 2.4. This calculation will then show that any beam of light whose
azimuthal dependence is eilφ carries orbital angular momentum. What we have in
Eqn. 2.5 is the energy flux density in the φ̂ direction. To get the momentum in

CHAPTER 2. THEORY

8

that direction, we simply divide by c. The angular momentum flux density along the
z-axis then, is the cross product of r with hSiφ /c:

Lz ẑ = r ×

lkc|u|2
φ̂
µ0 r


=

lk|u|2

µ0

We now clearly see that the beam carries an amount of angular momentum along
the z axis. To get a better sense of just how much angular momentum is carried, we
can compare Lz to the energy flux density carried by the beam along its z axis:
hSiz =

k2c 2
|u|
µ0

So, the ratio of angular momentum to energy along the axis of propagation is
Lz
l
l
=
=
hSiz
ck
ω

(2.6)

In other words, the ratio of angular momentum to energy in a beam with the φdependence of Equation 2.4 is just the index l over the frequency. This result is very
suggestive since multiplying the numerator and denominator by ~ gives the familiar
form ~ω in the denominator, which is interpreted quantum mechanically as the energy
per photon in the beam. This motivates the belief that individual photons carry an
amount l~ of orbital angular momentum. The treatment up to Equation 2.6, however,
has been wholly classical, so it would be a mistake to take this as a rigorous proof of
the amount of orbital angular momentum carried by photons.

2.2

Generation of the LG Modes

In order to understand how the forked diffraction gratings described in Chapter 1 can
convert a Gaussian laser beam to a Laguerre-Gaussian one, it is first necessary to
understand a few basic principles of holography. In a general sense, a hologram is like
a photograph in that it is a way of storing optical information. Unlike a photograph,
however, a hologram stores information about both the intensity and the phase of

CHAPTER 2. THEORY

9

Figure 2.1: Simplified depiction of how a hologram works.
the field of the object. A basic diagram of how a hologram works is offered in Figure
2.1.
When the hologram is being formed, we interfere a reference beam (depicted as
a plane wave in Figure 2.1) with the light from the object we are trying to image.
The interference pattern formed by the reference wave and the object wave is what
is recorded on the holographic film. So if the reference wave is described by Eref and
the object wave by Eobj , the intensity recorded on the film is
2
2
I = |Eref + Eobj |2 = Eref
+ Eobj
+ 2Re [Eref · E∗ obj ] .

The first two terms on the right hand side of this equation correspond to the separate
intensities of the reference and object waves. The final term on the right hand side
is what contains information about the relative phase of the two beams. The way
we retrieve this information is illustrated in the right half of Figure 2.1. When the
developed holographic film is illuminated with the reference wave, an exact copy of
the object wave is output on the other side of the hologram at the angle (relative to
the reference wave) at which it was input [7]. There is also a beam output at the
opposite angle which is the complex conjugate of the object beam, and a transmitted

CHAPTER 2. THEORY

10

beam which has the same form as the reference.
What we want to do in the context of this experiment is image a mode of laser
light rather than an object. Since we don’t have the mode we are interested in to
begin with, it is easiest to generate the necessary holograms using a computer. For
reference, here is the spatial amplitude distribution of an LG mode with index l (note
that it matches the form of Equation 2.4):
√ !|l|




2r2
r2
ClLG r 2
|l|
L0
(2.7)
exp − 2
ul (r, φ, z) =
w(z) w(z)
w (z)
w2 (z)




z
r2
exp (ilφ) exp i(|l| + 1)arctan
× exp
exp (ikz) .
2
2
2(z + zR )
zR
Here, ClLG is a normalization constant, w(z) is the beam waist given by
s
z2
w(z) = w(0) 1 + 2
zR
|l|

zR = πw(0)2 /λ is the Rayleigh range, and L0 (2r2 /w2 (z)) is a generalized Laguerre
polynomial given by [3], [8]
|l|

L0 (x) =

ex d −x |l|
e x .
x|l| dx

If we simulate the interference of an l = 1 LG beam coming in at an angle (like the
object wave in Figure 2.1) with a Gaussian reference beam using Mathematica, we
see the intensity pattern in Figure 2.2. This looks much like the forked diffraction
grating in Figure 1.3 enclosed within a Gaussian intensity envelope.
As it turns out, this analysis is slightly overcomplicated for our purposes. The
important difference between a Gaussian beam and Laguerre-Gaussian beam is the
phase eilφ . At the most fundamental level, all we want to do is impart a phase eilφ
onto a beam that has a tilt in, say, the x-direction. In other words, the object beam
has a phase eilφ relative to the reference beam, and the reference beam has a phase
eikx relative to the object beam [9]. The intensity profile of this simplified hologram
is then
I(x, y) = eikx + eilφ

2

= 2 + 2cos(kx − lφ) = 2 + 2cos(kx − larctan(y/x)).

(2.8)

CHAPTER 2. THEORY

11

In this equation, |l| varies the number of forks in the pattern, the sign of l varies
whether the pattern is oriented upright or inverted, and k varies the spacing of the
fringes. A few holograms created with this intensity profile are shown in Figure 2.3.
Since (eilφ )∗ = e−ilφ = ei(−l)φ , these holograms generate an LG mode with index l
in the first diffracted order and an LG mode with index −l in the first order on the
opposite side of the central transmitted beam.

Figure 2.2: Simulated interference of tilted LG beam and Gaussian reference beam.

Figure 2.3: Starting on the left and going right, l = −1,+2, and +3 holograms.

2.3

Interference of the LG Modes

In order to predict what the interference (or superposition) of two LG modes would
look like we could plug Equation 2.7 into Mathematica and plot the results (in fact

CHAPTER 2. THEORY

12

we will do just that in Section 3.3). But Equation 2.7 is unnecessarily cumbersome
if all we want to do is gain a qualitative understanding of what is going on when two
LG modes interfere. Since Equation 2.7 matches the form of Equation 2.4, we can
simplify things by ignoring the dependence of ul (r, φ, z) on r and z and just looking
at the interference of the phase factors eilφ .
In this simplified model, the intensity pattern of a superposition of beams with
l-values l1 and l2 is
I(φ) = eil1 φ + eil2 φ eiδ

2

= 2 + 2cos ((l1 − l2 )φ − δ) .

(2.9)

Here, δ represents some relative phase between the two beams. Since the intensity
has a sinusoidal dependence on φ, we can first think about how many times I(φ) goes
to zero through one full cycle of φ. Note that I(φ) is zero when cos((l1 − l2 )φ − δ) is
−1. This occurs when
(l1 − l2 ) φdark − δ = (2n + 1) π

=⇒

φdark =

(2n + 1) π + δ
(l1 − l2 )

(2.10)

where n is an integer. In the case where (l1 − l2 ) = 1, there is one value of φdark
between 0 and 2π for fixed δ. In fact, the number of dark regions is exactly the
difference between l1 and l2 . Figure 2.4 shows example intensity patterns for three
different values of (l1 − l2 ). In the special case where l1 = l2 , the intensity is constant
in φ and varies up and down with the phase difference δ.
From Equation 2.9, we can also see that changing the relative phase δ between the
two beams has the effect of rotating the intensity pattern in the plane. This feature
is demonstrated in Figure 2.5. We’ll see in Section 3.3 that the principles discovered
here (that the difference in the index l determines the number of dark regions and
that changing δ rotates the intensity pattern) apply both in theory and practice when
we work with the LG modes in their full form.

CHAPTER 2. THEORY

13

Figure 2.4: Starting top-left and going clockwise, interference patterns for l1 − l2 =
2, 4, and 6 (δ = 0).

Figure 2.5: Intensity patterns for l1 − l2 = 2 with δ = π/2 and δ = π.

Chapter 3
Experiment
3.1

Table Setup for Generating and Interfering LG
Modes

As mentioned in Chapter 1, we used a Spacial Light Modulator (SLM) to display the
computer-generated holograms needed to transform a simple Gaussian beam into a
Laguerre-Gaussian beam. The SLM has a liquid crystal display (LCD) on it that is
set in front of a mirror. Each pixel on the LCD contains a birefringent liquid crystal.
If a voltage is applied to the pixel, the liquid crystal modifies the index of refraction
for one polarization direction by an amount proportional to the voltage. If the laser
polarization is parallel to the the axis of the crystals, this creates a phase grating
which allows the LCD to mimic whatever is on the computer monitor it is connected
to in exactly the same way as a common LCD projector. Generating the LG modes
does not necessarily require any more complicated experimental setup than a laser
incident on the SLM. Superposing two LG modes, however, does require a little more
ingenuity. In general, we would like to be able to superpose any two LG modes, but
we only have one SLM available in the lab. The table setup which solves this problem
is shown in Figure 3.1.
The interferometer in Figure 3.1 is a modified version of a common-path Sagnac
14

CHAPTER 3. EXPERIMENT

15

Figure 3.1: Modified Sagnac interferometer used to form superpositions of LG modes.
interferometer. The two beams in this interferometer travel essentially the same path
(in opposite directions), but there is some lateral separation between the two beams.
This means that the two beams both hit the SLM, but in different places. So, we can
generate two different LG modes with one SLM by placing different holograms at the
two positions where the two beams hit the SLM, aligning everything so that it is the
first order diffracted beams which recombine at the beam splitter. The component
towards the bottom of Figure 3.1 is a glass slide inserted at an angle into one of the
beams. By varying the angle of the slide, we can vary the relative phase δ between
the two beams.
It’s also worth noting that before the laser was input to the interferometer, it
went through a linear polarizer, a half-wave plate, a neutral density filter, and it was
coupled to a single mode optical fiber. The linear polarizer allowed us to vary the
intensity of the beam while the half-wave plate rotated the polarization of the beam
to achieve maximum diffraction efficiency. The neutral density filter simply cut down

CHAPTER 3. EXPERIMENT

16

the intensity of the laser so that the camera wouldn’t be saturated. Finally, the optical
fiber ensured that the beam was in a pure Gaussian mode. A complete diagram of
the table setup, including the additional interferometer described in Section 3.4, is
given in Appendix A.
The camera we used in the lab was a Basler acA 1600 USB 3.0 digital camera.
This camera has a resolution of 1628×1236 and a pixel size of 4.4µm×4.4µm. In order
to manage both the holograms on the SLM and the images from the USB camera at
the same time, it was helpful to use two computer monitors. One monitor was solely
for controlling the SLM. We ended up writing a program in LabVIEW which could
open two holograms at specified positions on the SLM monitor, move the holograms
around on the monitor, change the size of the holograms, and switch holograms for
generating different LG modes. This program is discussed briefly in Appendix B.
With this setup, we could have total control over the holograms displayed on the
SLM while taking pictures with the camera using the second monitor.

3.2

Generating Single LG Modes

The very first question we can ask once our table is completely set-up is whether we
can generate single LG modes with some degree of purity. In other words, we want
to make sure that the holograms described in Section 2.2 work as we expect them
to. In generating these single modes, we used the same experimental setup detailed
in Figure 3.1. Instead of displaying two holograms on the SLM, however, we just
displayed one so that we would have only one first-order diffracted beam incident on
the camera. Experimental results are compared with theoretical predictions for an
l = 1 LG mode in Figure 3.2.
In Figure 3.2, we can qualitatively see that the theory and experiment are very
similar. It’s likely that what we have generated is indeed an l = 1 LG mode. To get
a more quantitative measure of the agreement between theory and experiment, we

CHAPTER 3. EXPERIMENT

17

Figure 3.2: In the top-left, a photograph of the first diffracted beam using an l = 1
hologram. This photograph is 300 × 300 pixels with an exposure time of 10ms. In
the bottom left, the intensity of each pixel from the image above plotted vertically.
The top-right and bottom-right images are theoretical fits of the data on the left to
Equation 3.1.
first fit the data to a theoretical model found by taking the magnitude squared of the
amplitude distribution in Equation 2.7 (for simplicity, we evaluate at z = 0 and use
w0 ≡ w(0)):
I0
|ul (r, φ, 0)| = 2
w0
2

If we recall that r =

p

√ !2|l|

2 2 2

r 2
2r2
r
2r
|l|
exp − 2 exp
L0
2
w0
w0
zR
w02

(3.1)

x2 + y 2 , then the only fitting parameters in Equation 3.1 are

the overall amplitude I0 , the beam waist w0 , and the (x, y) position of the center of the
beam. Using Mathematica’s NonLinearModelFit function, we found the theoretical
intensity pattern plotted in the bottom-right of Figure 3.2. To then quantify the
agreement between the data and the fit, we normalize the intensities If it (x, y) and

CHAPTER 3. EXPERIMENT

18

Idata (x, y) such that
ZZ

ZZ
If it (x, y)dxdy =
A

Idata (x, y)dxdy = 1

(3.2)

A

where A is the region captured in the photograph. Since intensities are always nonnegative, it follows that
ZZ

p
If it Idata dxdy ≤ 1.

0 ≤

(3.3)

A

Because of the condition in Equation 3.2, if the integral in Equation 3.3 exactly equals
1, then If it = Idata everywhere in the region. The closer this integral is to 1, the more
completely If it and Idata overlap.
Using Mathematica again to numerically evaluate the integrals in Equations 3.2
and 3.3, we found that the overlap between If it and Idata was about 0.980 while the
fitted value of w0 was 340µm for the l = 1 mode pictured in Figure 3.2. So, we can
say with confidence that we can generate an l = 1 LG mode using the SLM in the
expected manner. The parameter w0 , however, is only the true waist size for a simple
Gaussian (l = 0) beam. To get a better measure of the width of the beam, we can
maximize Equation 3.1 with respect to r using Mathematica to set the derivative of
Equation 3.1 equal to zero and solve for r. If we call the value of r which gives the
maximum intensity rwaist , then
s
rwaist = w0 zR

zR2

|l|
− w02

(3.4)

where, again, zR = πw02 /λ (we used a laser with λ = 532nm). Note that this equation
only represents the beam waist for non-zero values of l. With this definition, rwaist
for the fit in Figure 3.2 was 240µm. Similar data using an l = 2 hologram is shown
in Figure 3.3. The calculated overlap between If it and Idata for this mode was 0.910
while the rwaist for the fit was 323µm.

CHAPTER 3. EXPERIMENT

19

Figure 3.3: In the top-left, a photograph of the first diffracted beam using an l = 2
hologram. This photograph is 360 × 360 pixels with an exposure time of 10ms. In
the bottom left, the intensity of each pixel from the image above plotted vertically.
The top-right and bottom-right images are theoretical fits of the data on the left to
Equation 3.1.
Although the overlap is strong between the data and the fitted curve for l = 2,
the photograph in the top-left of Figure 3.3 shows some asymmetry in the central
dark region. The reason for this is not entirely clear. We found that changing the
alignment of the hologram on the SLM did not clear up this problem. It may be that
there were irregularities in the input laser beam (i.e. the input beam may not have
been in a pure Gaussian mode) which were exacerbated as we increased the value of l
in the hologram displayed on the SLM. In Figure 3.4, we can see that the asymmetry
is indeed worse for higher l. In the case of an l = 3 hologram, the overlap between
the data and the fitted curve was 0.894 while rwaist for the fit was 426µm.
It’s interesting to note that in Figures 3.2 through 3.4, the first diffracted beams
appear to be getting dimmer as the order of the hologram increases. The simplest

CHAPTER 3. EXPERIMENT

20

explanation for this trend is illuminated by Figure 3.5, where photographs of the first
diffracted beams are shown on the same scale for l = 1, l = 2, and l = 3 holograms.
In Figure 3.5, it’s clear that as l increases, the waist size of the output beam increases.
Assuming that the same amount of power is channeled into the first diffracted beam
for each l, this means that the same amount of power is being spread over larger areas
for larger l. Hence, the output beam seems dimmer for larger l. It is also possible
that the power in the first diffracted beam is not the same for each l, but this would
most likely be an effect secondary to the effect of increasing waist size.

Figure 3.4: On the left, a photograph of the first diffracted beam using an l = 3
hologram. This photograph is 400 × 400 pixels with an exposure time of 10ms. On
the right, a theoretical fit of the data to Equation 3.1.

Figure 3.5: Starting on the left and going right, first-order diffracted beams for l = 1,
l = 2, and l = 3 holograms shown on the same scale (each photograph had an exposure
time of 10ms and was cropped to 360 × 360 pixels).

CHAPTER 3. EXPERIMENT

3.3

21

Superpositions of LG Modes

To create superpositions of LG modes, we used the experimental setup of Figure 3.1,
placing two holograms on the SLM instead of one. In order to align the interferometer,
we found it easiest to first place simple diffraction gratings on the SLM so that we
had Gaussian beams in the first diffracted order. This way, we would know the two
output beams were perfectly aligned when the intensity of the spot on the camera
faded in and out as we changed the relative phase (as opposed to having fringes going
across the spot in some direction). With the interferometer aligned, we then switched
the holograms on the SLM to, say, l = 1 and l = −1, and achieved the results laid
out in Figure 3.6.
In the top row of Figure 3.6, we see simulated interference patterns for the addition
of positive and negative LG modes. These were generated in Mathematica using
Equation 2.7 in its full form. Note the similarities of these patterns to the ones
illustrated in Figure 2.4. In each case, the difference between the values of l in the
two beams is equal to the number of bright spots (or dark spots). The simulated
patterns are to be compared with the photographs in the second row of Figure 3.6.
Although there are a few minor asymmetries, especially for the higher order patterns,
the shape of each simulated pattern closely matches the patterns in the photographs.
For example, the overlap between the data and a fitted curve for the addition of l = 1
and l = −1 modes was 0.955.
The other feature being illustrated in Figure 3.6 is the way that the interference
patterns change as the relative phase between the two beams is varied. Going down
any one of the columns in Figure 3.6 corresponds to making small changes in the
phase of one of the beams (by tilting the glass slide of Figure 3.1). As we expected
based on the predictions of Figure 2.5, the patterns rotate about their centers as the
phase is adjusted. This is true even for the superpositions of higher order modes.
The shape of the interference pattern, for the most part, remains unchanged as the
entire pattern rotates. We also note that, as with single LG modes, the brightness of

CHAPTER 3. EXPERIMENT

22

the interference patterns diminishes as we increase the magnitude of the l values in
the component beams because of the extra area which the higher-order patterns take
up.

CHAPTER 3. EXPERIMENT

23

Figure 3.6: In the top row, theoretical intensity patterns for the superpositions of
LG modes labeled above. Subsequent rows show photographs of superpositions with
each row having slightly more relative phase between the interfering beams than the
row above. Here, each photograph had an exposure time of 10ms and was cropped
to 320 × 320 pixels.

CHAPTER 3. EXPERIMENT

3.4

24

Sorting Even and Odd LG Modes

Beyond generating and interfering LG modes, we’d also like to be able to extract
information about the orbital angular momentum of a laser beam. This can be done
using the holograms of Section 2.2. If, for example, an l = 1 LG mode is incident
on an l = −1 hologram, the first diffracted beam will be in a simple Gaussian mode
which can be coupled to a single mode fiber. Modes besides l = 1 will not produce
a Gaussian in the first diffracted beam. This method is simple, but it can only test
for single LG modes. It cannot tell us exactly what the orbital angular momentum
of a laser beam is. We explored a more general interferometric method of measuring
orbital angular momentum laid out in Ref. [10].
The interferometric method of sorting relies on an optical device called a Dove
prism. The workings of the Dove prism are illustrated in Figure 3.7. Light incident
on the angled sides of a Dove prism is refracted and then goes through a total internal
reflection off of the lower base of the prism. That total internal reflection flips the
transverse profile of the beam over the axis which runs along the lower base.
The design of the interferometer we used for sorting odd and even modes is shown
in Figure 3.8. This interferometer is essentially the interferometer in Figure 3.1 with
Dove prisms in each arm. The reason for having two Dove prisms oriented at a 90◦
angle to each other is illustrated in Figure 3.9. In Figure 3.9, we see phase profiles for
an odd (l = 1) and an even (l = 2) LG mode. These phase profiles are something like
what you would see looking down the beam axis if your eyes could detect electric fields.

Figure 3.7: An illustration of the way a Dove prism flips a laser beam over the axis
defined by it’s lower base.

CHAPTER 3. EXPERIMENT

25

Looking closely at the phase profiles, we see that if we flip an odd mode horizontally
in one arm of the interferometer and vertically in the other, the products are π out
of phase with each other. If we do the same thing to an even mode, however, the
products are in phase with each other. Thus, even modes will interfere constructively
at one output of the interferometer and odd modes will interfere constructively at the
other. In other words, depending on δ, only even modes will exit through one port
and only odd modes will exit through the other. This is the reason for placing an
extra mirror at the second output of the interferometer and redirecting the output to
the camera.

Figure 3.8: Altered version of the interferometer in Figure 3.1 with Dove prisms in
each arm. The Dove prisms are oriented at 90◦ with respect to each other for sorting
odd and even LG modes.
Our table was set up so that the output of the interferometer creating superpositions of LG modes was the input for the sorting interferometer (see Appendix A).
The simplest superposition we could sort was a superposition of an l = 1 state with
an l = 0 state. Visually, sorting this state would also be easier since the l = 0 state
is a simple Gaussian which has no central dark spot. Some photographs of the two
sorted output beams side-by-side are shown in Figure 3.10. In the first photograph

CHAPTER 3. EXPERIMENT

26

Figure 3.9: Illustration of what happens to even and odd modes in the two arms of
our sorting interferometer.
of Figure 3.10 we can clearly see an l = 0 mode on the left and an l = 1 mode on
the right. Subsequent photographs in Figure 3.10 show how the two outputs change
as the relative phase between the two arms of the interferometer is adjusted. The
output beam on the right begins in an l = 1 state, morphs into an l = 0 state, and
then morphs back into an l = 1 state (while the left side does the opposite). This is
exactly how we should expect the sorting interferometer to work.
Although this method of sorting LG modes is not perfectly general, it is possible
to sort the even and odd modes further by putting each through additional interferometers with Dove prisms [10]. The idea is to cut the angle between the Dove prism
orientations in half with each additional interferometer and so split the superpositions
into two further categories. For example, if you have a beam of even modes from the
first interferometer and put that through an interferometer where the Dove prisms are
oriented at 45◦ with respect to each other, you can sort the even modes into modes
in which l is an even multiple of two and modes in which l is an odd multiple of two.
So, in an application where you might know exactly how many modes you will have
to sort, you can build the number of interferometers required and have a perfectly
general sorting mechanism.

CHAPTER 3. EXPERIMENT

27

Figure 3.10: Photographs of the two outputs of the sorting interferometer when the
input is a superposition of l = 0 and l = 1 states. Each photograph has a slight
increase in the relative phase δ from the one above it. If the top photograph has
δ = 0, then the third photograph has δ = π, and the final photograph has δ = 2π.
The photographs all have resolution 1260×460 pixels and were taken with an exposure
time of 50 ms.

Chapter 4
Conclusion
Here we have demonstrated experimental methods for generating Laguerre-Gaussian
modes from a simple Gaussian mode, interfering LG modes and predicting the form
of the interference pattern, and finally, sorting odd and even LG modes. We’ve found
that although the first diffracted beams from higher order holograms contain some
unexpected asymmetries, the overlap between the experimental intensity patterns
and the fitted theoretical curves is high for l = 1, 2, and 3. In fact, we’ve shown that
we can generate all of the LG modes with indices l = −3 through l = 3 (the negative
LG modes have identical intensity patterns to those shown in Figures 3.2-3.4).
Using these single LG modes, we’ve been able to form interference patterns between positive and negative modes and not only predict the shape of the intensity
patterns, but also the fact that the patterns rotate as the relative phase of the two
interfering beams is altered. The rotation of the interference patterns is evidence of
the twisting phase fronts of the LG modes, which cannot be observed from the intensity patterns of single modes alone. In sorting an l = 0 mode from an l = 1 mode,
we’ve also seen how varying the relative phase between the two modes can switch the
outputs at which odd and even modes constructively interfere.
There are two strengths of the experimental methods described here which are
worth noting. First, all of the methods are simple enough to be performed in an

28

CHAPTER 4. CONCLUSION

29

undergraduate laboratory. The most advanced piece of equipment necessary was
the Spacial Light Modulator. Second, all of the methods can, in principle, work
using single photon sources ([4],[10]). This means that the methods could be useful
in performing quantum mechanical experiments. In this work, we haven’t thought
much about the LG modes in a quantum mechanical context, yet it is their quantum
properties which motivate much of the current research being done with LG modes.
The LG modes correspond to a basis of orbital angular momentum states which
is infinite in dimension, discrete, orthogonal, and complete. It is hoped that the
methods set out here will enable future undergraduate researchers to explore quantum
phenomena in this unique basis.

Appendix A
Detailed Experimental Setup
Here we present a more complete picture of the experimental setup used in this
research. Figure A.1 is a photograph of the lab table as it was set up for sorting odd
and even LG modes. Figure A.2 is a detailed diagram of the same setup.

Figure A.1: Photograph of our table setup including both the generating interferometer and the sorting interferometer.

30

APPENDIX A. DETAILED EXPERIMENTAL SETUP

Figure A.2: Detailed diagram of the table setup shown in Figure A.1.

31

Appendix B
LabVIEW Programming
After generating the holograms of Section 2.2 in Mathematica, we wanted to be able
to display holograms on the monitor being mirrored by the SLM. We also wanted to
be able to move the holograms around on the screen, zoom in and out, and change
each hologram to different orders. Since LabVIEW provides pre-built programs made
for opening external windows and performing operations on them, it was a natural
choice for solving these problems. Figure B.1 shows the block diagram of a program
called WindSettings.vi written to take the identifying number of a previously-opened
window and set its center point, zoom factor, width, height, and screen coordinates.
There is also an option to bring the window to the front.

Figure B.1: Block diagram for WindSettings.vi which sets parameters for one window.

32

APPENDIX B. LABVIEW PROGRAMMING

33

The front panel of the main program which makes use of WindSettings.vi is shown
in Figure B.2. This program, called Open External Window.vi, opens several windows
and puts a single hologram in each. The windows are split into two groups (right
and left), within which the zoom, center, and position coordinates are all the same.
This means that only two holograms are visible on the monitor at a time because
one hologram will always be in front. To change which hologram is in front, the
user can click the buttons labeled with the order of the hologram. A portion of
the block diagram for this program is shown in Figure B.3. There we see how the
program opens several windows from image files and then sets their parameters using
WindSettings.vi.

Figure B.2: Front panel for Open External Window.vi.

APPENDIX B. LABVIEW PROGRAMMING

34

Figure B.3: Block diagram for Open External Window.vi.
Finally, in Figure B.4 we include a screen shot of the program used to take photographs with our camera. This program was provided by Basler, the makers of the
camera. Along the top panel are some controls for taking single shots, taking a continuous shot, and saving the captured image. There are many different settings available
on the left. The setting most often adjusted in this research was the exposure time.

APPENDIX B. LABVIEW PROGRAMMING

35

Figure B.4: Basler program used to capture photographs with the USB camera used
in the lab.

Bibliography
[1] Miles Padgett, Johannes Courtial, and Les Allen. “Light’s orbital angular momentum.” Physics Today 57.5 (2004): 35-40.
[2] Derek Huang, Henry Timmers, Adam Roberts, Niranjan Shivaram, Arvinder
S. Sandhu. “A low-cost spatial light modulator for use in undergraduate and
graduate optics labs.” Am. J. Phys. 80, 211 (2012).
[3] L. Allen, M. J. Padgett, and M. Babiker. “IV The orbital angular momentum of
light.” Progress in Optics 39 (1999): 291-372.
[4] Mair, Alois, et al. “Entanglement of the orbital angular momentum states of
photons.” Nature 412.6844 (2001): 313-316.
[5] Franke-Arnold, Sonja, et al. “Uncertainty principle for angular position and angular momentum.” New Journal of Physics 6.1 (2004): 103.
[6] Griffiths, David J. Introduction to Electrodynamics. 3rd ed. Upper Saddle River,
New Jersey: Prentice Hall, 1999. Print.
[7] Kenyon, Ian R. The Light Fantastic 2nd ed. New York: Oxford University Press,
2011. Print.
[8] Abramowitz, Milton and Stegun, Irene A. “Orthogonal Polynomials” Handbook
of Mathematical Functions. Washington, D.C.: National Bureau of Standards,
1964. Print.

36

BIBLIOGRAPHY

37

[9] Carpentier, Michinel, Salgueiro, and Olivieri. “Making optical vortices with
computer-generated holograms.” Am. J. Phys. 76, 916 (2008): 916-921.
[10] Leach, Padgett, Barnett, Franke-Arnold, and Courtial. “Measuring the Orbital
Angular Momentum of a Single Photon.” Phys. Rev. Lett. 88, 25 (2002).