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This to certify that the thesis titled “UNDERSTANDING THE ORBITAL ANGULAR

MOMENTUM OF LIGHT” submitted by Bankim Chandra Das, to the Indian Institute

Technology, Madras for the partial fulfillment of the requirement for the degree of Master of

Science in Physics, is bonafide work under my supervision. The contents of the thesis in full

or partial have not submitted to any other Institute or university for the award of any degree

or diploma.

Date: Dr. C. Vijayan

Project guide

Professor

Department of Physics

IIT Madras

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Contents

Acknowledgement 4

Chapter 1: Orbital Angular Momentum: An Overview

1.1 Introduction 5

1.2 The Angular Momentum of light 7

1.2.1 Spin Angular Momentum (SAM) 7

1.2.2 Orbital Angular Momentum 9

1.3 The Orbital Angular Momentum Phenomena 11

References 14

Chapter 2: Energy Flow in a paraxial vortex beam: the spin and the orbital flow

2.1 Introduction 15

2.2 Spin and Orbital energy Flows 16

2.3 The Transverse energy flow 17

2.3.1 Gaussian Beam 19

2.3.2 Laguerre Gaussian (LG) Beam 20

References 24

Chapter 3: Higher order Gaussian beams

3.1 Introduction 25

3.2 The paraxial wave equation formulation 25

3.3 Solution of the paraxial wave equation 26

3.3.1 Fundamental Gaussian beam 26

3.3.2 Hermite-Gaussian beam 29

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3.3.2.1Intensity distribution 30

3.3.3 Laguerre- Gaussian Beam 32

3.3.3.1 Superposition of the LG beams 35

References 35

Chapter 4: Generating Laguerre – Gaussian (LG) beams

4.1 Introduction 36

4.2 Generation the LG beam by using spiral phase plate 36

4.3 Generation the LG beam by using the fork diffraction grating 37

4.4 Generation the LG beam by using the π/2 mode converters 40

References 46

Chapter 5: Experiments with Laguerre- Gaussian Beams

5.1 Introduction 47

5.2 Double Slit diffraction with LG beams 47

5.3 Demonstration by simulation 51

References 52

Summary and Conclusion 54

Appendix

“Mathematica” code for phase plot 55

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Acknowledgements

First and foremost I would like to thank my project guide Prof. C. Vijayan for his support

that made this work possible. I am very much thankful to my guide that he chose a topic of

my interest where the theory, simulation and experiments all were there. He has always

supported me and urged me to pursue research works and also has given me many

opportunities for improving the quality of my work.

I am grateful to Prof. M. V. Satyanarayana, who helped me to build up the ideas. In the

experimental part it would not have been possible to do the experiments in this short time

without the help of Mr. U. Somasundaram and I thank him wholeheartedly. When we have

started the work we did not have a clear idea about which set up to use to start with but at the

end of this journey we were able to generate the beam experimentally in an economic and

easy manner and to do some simple experiments which gave us some interesting results.

I gratefully acknowledge the opportunity provided by Dr. Nirmal K Viswanathan of

University of Hyderabad to visit the beam optics lab of the University, as a part of my project

work. In that short visit to the lab I got some idea about the experiment and the current

research works in this field.

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Chapter 1

Orbital Angular Momentum: An Overview

1.1 Introduction

It is well known that the polarized light has spin angular momentum (SAM) but the

light can carry orbital angular momentum also (OAM). But this property was less understood

before 90’s and some recent studies show that this is a very useful property and a very

emerging field for research.

If the light is circularly polarized then it carries a spin angular momentum ±ħ per photon.

And if the light has an azimuthal phase dependence of exp[ἰ𝑙φ] where 𝑙 can be any integer

the light can carry an Orbital angular momentum of 𝑙ħ per photon.

From the ratio of the angular momentum to the energy both spin and orbital momentum can

be determined easily.

If an ideal light beam is circularly polarized and travels as a plane wave then the spin angular

momentum is given by Jz = Nħ and the total energy associated with this is W= Nħω, Where

N is the no. of photons.

So, the ration of SAM to energy is,

𝐽𝑧

𝑊=

𝑁ħ

𝑁ħ𝜔=

1

𝜔 (1.1)

In fact a more general derivation can be done. If the light is elliptically polarized instead of

circularly polarized then it has a spin−1 ≤ 𝜎 ≤ 1. ( -1 is for right circularly polarized and +1

is for left circularly polarized and 0 for linear polarized light.)

In that case the ratio of SAM to energy is

𝐽𝑧

𝑊=

𝜎

𝜔 (1.2)

Now If the light beam has l dependent azimuthal phase exp[ἰ𝑙φ] such that it has an electric

field u(ρ, φ, z)= u0(ρ, φ, z)e-ikzei𝑙φ then,

𝐽𝑧

𝑊=

𝑙

𝜔 (1.3)

If the beam is has both polarized and azimuthal phase dependence then the ratio becomes

𝐽𝑧

𝑊=

𝑙±𝜎

𝜔 (1.4)

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Here the σħ comes due the spin angular momentum and 𝑙ħ comes due to the orbital angular

momentum.

In deriving the above relation no assumption has been made of the form of the amplitude

distribution u(ρ, φ, z) or u(x, y, z). So If σ=0 that is if the light beam is linearly polarized

there the orbital angular momentum will be present due to the spatial variation of the and the

OAM will be dependent on 𝑙.

The relationship between angular momentum and the linear momentum is �� = 𝑟 × �� . Where

𝑟 is the distance from the reference point and �� is the linear momentum �� = m �� . Now it can

be shown that the momentum of an electro-magnetic field is related to the poynting

vector �� = 휀0( 𝐸 × �� ). So the angular momentum is �� = 𝑟 × 휀0( 𝐸 × �� ). Any angular

momentum component in the propagation direction let in z direction requires transverse

components of linear momentum e.g x-y Plane. So, any angular momentum in Z direction

requires a component of electric and/or magnetic field in the z direction. So for a plane wave

which has only transverse electric and magnetic component that is poynting vector is parallel

to the direction of the propagation can’t carry any angular momentum along Z direction.

But if the light is circularly polarized having plane wave front then it can carry an angular

momentum (spin) of ±ħ depending upon the handedness of the beam. This is because due to

the finite aperture (edge effects) there exists an axial component of the E-M field. A detailed

treatment of this edge effect always returns a finite angular momentum of ±ħ [1].

The origin of the Orbital angular momentum of light can easily be understood. If the beam

has a azimuthal phase dependence exp[ἰ𝑙φ] in the transverse plane of the beam where 𝑙 can

be any integer and φ is the azimuthal phase in the plane such beam has a helical phase fronts

(Figure 1.1).

Figure 1.1

Shows Helical wave front for l=1

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where 𝑙 determines the no. of intertwined helix. This type of beam has axial components of

electromagnetic field or equivalently the pointing vector has azimuthal component. So, they

carry an orbital angular momentum along the beam axis.

1.2 The Angular Momentum Of light

1.2.1 Spin Angular Momentum (SAM):

It is an intrinsic property of light and comes due to the polarization of light (the

vectorial property of light) and the finite aperture effect.

We will derive the SAM for circularly polarized light and then generalize it for arbitrary

polarization states.

We know that electromagnetic wave carry energy associated with pointing vector which is

given by,

𝑆 = 휀0( 𝐸 × �� ) (1.5)

The linear momentum �� is proportional to the 𝑆 . and the angular momentum is related to

�� = 𝑟 × �� .

Let us assuming a vector potential 𝐴 ,

𝐴 = �� 𝑢(𝑥, 𝑦, 𝑧)𝑒ἰ𝑘𝑧 (1.6)

Here �� = 𝛼 �� + 𝛽�� , is the unit vector of arbitrary polarization states. α, β is in general

complex number. U(x, y, z ) is the spatial amplitude distribution.

Now �� = 𝛻 × 𝐴 , if we assume the Lorentz gauge [2]

�� = [(𝜕𝑢

𝜕𝑧+ ἰ𝑘𝑢) ( 𝛼 �� − 𝛽��) + (𝛽

𝜕𝑢

𝜕𝑥− 𝛼

𝜕𝑢

𝜕𝑦) 𝑧 ] 𝑒𝑥𝑝[ἰ𝑘𝑧] (1.7)

�� = ἰ𝜔[𝛼𝑢�� + 𝛽𝑢�� − ἰ

𝑘(𝛼

𝜕𝑢

𝜕𝑥+ 𝛽

𝜕𝑢

𝜕𝑦)]𝑒𝑥𝑝[ἰ𝑘𝑧] (1.8)

Taking the time average of the poynting vector 𝐸 × �� , we found that the linear momentum

𝑝 =𝜀0

2(�� ∗ × �� + �� × �� ∗ ) (1.9)

Putting the value of equation (1.7 & 1.8) and after simplifying we get

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𝑝 =ἰ𝜔𝜀0

2( 𝑢�� 𝑢∗ − 𝑢∗�� 𝑢) − 2ἰ𝑘|𝑢|2𝑧 + (𝛼𝛽∗ − 𝛽𝛼∗)(�� |𝑢|2 × 𝑧 ) (1.10)

The first term in the equation leads to the component of 𝜑 direction. And the(𝛼𝛽∗ − 𝛽𝛼∗) is

the general polarization term and leads to the spin angular momentum. Now if the amplitude

of the electric field is in the form of u(x, y, z) = 𝑢0𝑒−(𝑥2+ 𝑦2) then the first term goes to zero.

The last term in which we are interested in this derivation leads to a spin σ in 𝒛 direction as,

it is in 𝜑 direction here and cross product �� gives an angular momentum in 𝒛 .

If α=1, β=1 which is linearly polarized light 𝛼𝛽∗ − 𝛽𝛼∗ = 0, so, for linearly polarized light

no angular momentum is there.

If α=1, β=±ἰ, which is circularly polarized light, Then 𝛼𝛽∗ − 𝛽𝛼∗ = ±2ἰ. There for σ can

take ±1 depending upon the handedness of the polarization.

Now if we consider the cylindrical co-ordinate system for the amplitude of the field it is 𝝋

independent.

�� |𝑢|2 × 𝑧 = 𝜕

𝜕𝑟|𝑢|2𝜑 (1.11)

The angular momentum density is the 𝒛 direction will be given by

𝑗𝑧 = (𝑟 × 휀0( 𝐸 × �� )𝑧 = 휀0𝑟( 𝐸 × �� )𝜑 = 1

2휀0𝜔𝜎𝑟

𝜕

𝜕𝑟|𝑢|2 (1.12)

This shows that the density of the SAM is proportional to the handedness of the polarization.

Now, finally if we take the ratio of the total angular momentum to the total energy

integrating across the beam cross section we get

𝐽𝑧

𝑊=

ʃ ʃ𝑟𝑑𝑟𝑑𝜑(𝑟 ×𝜀0( 𝐸 ×�� )𝑧

𝑐 ʃ ʃ𝑟𝑑𝑟𝑑𝜑( 𝐸 ×�� )=

𝜎

𝜔 (1.13)

This is same as stated earlier. Now in the semi-classical way the total energy of the photon

W= Nħω. So angular momentum per photon is

𝐽𝑧

𝑁= 𝜎ħ (1.15)

So, the angular momentum is proportional to the σ = (𝛼𝛽∗ − 𝛽𝛼∗) = { ±1, 0}. the value of σ

will dependent on the polarization state. For Circular polarization σ= ±1, for linear

polarization σ= 0, for elliptical polarization σ will lie between -1 to +1. So in general −1 ≤

𝜎 ≤ +1.

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There for the Spin angular momentum of light is either – ħ or +ħ for right circularly

polarization and left circularly polarized light respectively.

1.2.2 Orbital Angular Momentum (OAM):

In the previous section we have derived the SAM per photon assuming the wave front

is a plane wave. That is there is no spatial variation of phase along the transverse direction

but instead of this let there is azimuthal phase dependence in the amplitude of the field.

Let the vector potential is

𝐴 = �� 𝑢(𝑟, 𝜑, 𝑧)𝑒ἰ𝑘𝑧

Where 𝑢(𝑟, 𝜑, 𝑧) = 𝑢(𝑟, 𝑧)0𝑒−𝑟2

𝑒𝑥𝑝[−ἰ 𝑙 𝜑]𝐿𝐺𝑝𝑙 (r)1 , it is a beam of Laguerre- Gaussian

type.

For simplicity of our calculation we will deal with p=0, i.e. 𝐿𝐺0𝑙 (r). We know that 𝐿𝐺0

𝑙 (r) =1.

So the amplitude of the field becomes

𝑢(𝑟, 𝜑, 𝑧) = 𝑢0(𝑟, 𝑧)𝑒−𝑟2

𝑒𝑥𝑝[−ἰ 𝑙 𝜑] (1.16)

The Index “𝑙” can be any integer which is the charge or helicity of the beam.

Substituting the value of “u” in equation (1.10, 1.12) we can calculate the momentum and

total angular momentum of the light beam. After simplifying we get 𝑧 and 𝜑 components of

the momentum.

𝑝𝜑 = 𝜔𝜀0𝑙

𝑟|𝑢|2 + 2ἰ𝜔휀0𝜎|𝑢|2 (1.17)

𝑝𝑧 = 𝑐𝑘휀0|𝑢|2 (1.18)

And the orbital angular momentum density along the z axis is,

𝑗𝑧 = 𝜔휀0𝑙|𝑢|2 + 2ἰ𝜔휀0𝜎𝑟|𝑢|2

Integrating over the beam cross section, It is possible to find the total angular momentum and

the energy ratio.

1 𝐿𝐺𝑝

𝑙 (x) is associated Laguerre polynomial. Which is defined as 𝐿𝐺𝑝𝑙 (x ) =

𝑒𝑥𝑥−𝑙

𝑝!

𝑑𝑝

𝑑𝑥𝑝 (𝑒−𝑥𝑥𝑛+𝑘).

The generating function for this is g(x, z) = 𝑔(𝑥, 𝑧) = exp (−

𝑥𝑧

1−𝑧)

(1−𝑧)𝑘+1 .

𝐿𝐺0𝑙 (x )= 1.

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𝐽𝑧

𝑊=

ʃ ʃ𝑟𝑑𝑟𝑑𝜑(𝑟 ×𝜀0( 𝐸 ×�� )𝑧

𝑐 ʃ ʃ𝑟𝑑𝑟𝑑𝜑( 𝐸 ×�� )=

𝜎+𝑙

𝜔 (1.19)

Again from the above away the total angular momentum per photon will be

𝐽𝑧

𝑁= (𝜎 + 𝑙)ħ (1.20)

Where σ is the spin angular momentum and 𝑙 is the orbital angular momentum. Here we can

see the total angular momentum is not limited to the value ±ħ. But for this type of beam

𝐿𝐺0𝑙 (r) this can take any value because 𝑙 can be any integer value 𝑙 = {0, ±1, ±2, ±3 …}

The total angular momentum is dependent on the polarization sate but the orbital angular

momentum is dependent only on the value of 𝑙. Earlier we saw that the total angular

momentum is ±ħ. But here the extra 𝑙ħ term comes due to the azimuthal phase dependence

of the beam that is it comes due to the term 𝑒𝑥𝑝[−ἰ 𝑙 𝜑] in the electric field.

Figure 1.2 Simulated phase variation of the LG beam for P=0, l=1 in the transverse plane.

In the figure 2 the phase variation is shown. For l=1 the total azimuthal phase change is 2π

for a particular z = constant plane.

For l=2 total phase change will be 4π in a transverse plane of the helical beam.

0

2π

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1.3 The Orbital Angular Momentum Phenomena:

Simple analogy and comparison between spin states and the orbital states in different

situations are useful to study their property but the properties of most of the higher orbital

angular momentum are still unknown. First we need to distinguish the structures of the light

beam and also its properties when it is converted to a helical (vortex) beam, e.g. LG beam.

LG beam is not only the beam which has orbital angular momentum (OAM) but also the

Bessel Beam [3], Mathieu beam [4]; Ince-Gaussian beam [5] can also carry the OAM. In all

the cases the interference with the plane wave gives the spiral interference pattern.

Nowadays there are so many ways for generation of LG beams. But first it was achieved by

using cylindrical lenses which acts like a mode converter ( 𝜋

2 converter). Although the details

are very interesting, it will be discussed later chapters.

Another approach of generation is diffractive element prior to the same azimuthal variation

in the diffraction grating one can generate the LG beam of the same helicity. The gratings

look like forks. The beam produced with these grating the first order diffraction spot will be

a annular intensity cross section which is due to the term 𝑒𝑥𝑝[−ἰ 𝑙 𝜑] phase term. Indeed

the similar beams were studied earlier but none of those works recognized the property of

angular momentum. Now days the similar diffractive elements are produced by computer

generated holograms and recently by the Spatial light modulators [foot note].

If we use spiral phase plate we can still generate the helical beams. If the spiral plate has an

optical thickness 𝑡 =𝜆𝑙𝜑

2𝜋, where 𝜑 is the azimuthal phase angle. When a plane wave is

transmitted through a spiral plate the wave is converted to a helical phase characterized by an

azimuthal phase structure 𝑒𝑥𝑝[−ἰ 𝑙 𝜑] . But such spiral plates are hard to manufacture but the

conversion frequency is very good. The output will be purely a vortex beam. For a linear

momentum ħk, this gives an azimuthal component 𝑙ħ

𝑘 and it gives an orbital angular

momentum component 𝑙ħ per photon [6].

The processes for generation of the helical beam described above are not unique, there are

other ways also. The interference between the two plane waves gives sinusoidal fringe

patterns but three or more plane [7] waves leads to a point of destructive interference around

which the phase advances or retards by 2π. Also it can be shown that is you allow interfering

of two Gaussian beams with a little angle between them the output will be a vortex type

beam.[8] In these vortex beam there is a complete darkness in the center of the beam and this

a point of perfect phase singularity. This singular point neither carries angular momentum

nor the energy. The phase singularity maps out lines with complete darkness in the space.

But about the point of singularity there lies an intensity distribution.

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In order to generate pure LG mode the cylindrical lens converter method is a conventional

approach. When light goes through cylindrical lens mode converter it acts mathematically

equivalent to polarized light through quarter wave plate. Indeed the two states orbital angular

momentum can be represented on the Poincare Sphere. The details will be described later.

Similarly the jones matrix formulation which describes the propagation of the polarized light

in an optical system can also be implemented for the orbital angular momentum of light. In

this case the corresponding matrices will be (N + 1) × (N +1) instead of 2×2 matrix as in the

case of polarization states where N is the order of the beam. [9] There are also joint matrices

for the light which has Polarization states and carry orbital angular momentum. Alternatively

we can apply the jones matrices for spin states first and then for the orbital angular

momentum separately this is analogous to the separation of spin part and space part in the

wave function.

First experimental demonstration of Orbital Angular Momentum was in optical tweezers. It

uses the gradient forces of a tightly focused beam of light to trap the micro particles

surrounding in a fluid. If the radius of the micro particle is r, its mass is proportional to r3 and

its rotational moment of inertia is proportional to r5. So, If the size of the particle is decreased

then it is easier to move rather easier to rotate. In the figure[3] it is shown how a particle is

trapped with circularly polarized light (left) and with LG beam with a particular charge 𝑙.

When the beam is passed through the liquid the after absorbing the light and the angular

momentum the micro particles starts to rotate at a particular frequency. This is a direct

experimental demonstration of transferring the angular momentum of light to mater.

Figure 3 Optical tweezers. Left side trapped with circularly polarized light, Right side with

LG beam with a charge 𝑙. [This figure is taken from reference no. I]

13 | P a g e

Such transfer of the angular momentum and the torque applied to the particles basically

converts the tweezers to the optical spanners. The interesting thing happens when the beam

carries both OAM and SAM. If the beam is circularly polarized and charge of the beam is

𝑙=1, Then the total angular momentum will be either ħ+ ħ =2ħ, or ħ- ħ = 0, depending upon

the handedness of the beam. If the handedness of angular momentum are same then the total

angular momentum will be additive and the particle will start to rotate with twice frequency.

But now if a 𝜆2⁄ plate is introduced then the handedness of polarization state will be

reversed or if a π converter is introduced then the charge will be –𝑙 and the total angular

momentum will be zero. In this case the rotation of the particle is ceased but the particle will

still be trapped [10].The ability to stop the particle in this way by using the Laguerre

Gaussian beam is a direct evidence of transferring the SAM and OAM to the matter.

It should be noted that the Spin angular momentum is intrinsic property as it is independent

of the choice of the axis but the orbital angular momentum is dependent on the choice of the

axis. Though if the transverse momentum is zero then the σ and 𝑙 is independent of the

choice of axes, they are invariant under the shift of axis. So, the OAM can be said to be a

quasi-intrinsic property. But for the beam which has a transverse momentum in that case 𝑙 is

extrinsic.

For spin angular momentum and polarized no spatial and temporal coherence is needed. We

can polarize the light by a normal bulb light. But for the helical beam the situation is quite

different. For perfect vortex beam the spatial coherence is needed as it is a helical wave front

but there is no restriction in the temporal coherence. Now if we send the incoherent light

through the center of a spiral plate this creates a vortex in the far filed which are incoherent

with respect to each other such that at center the average energy and intensity is not zero at

the center. The azimuthal energy flow and the momentum is proportional to the radius but at

large radius it is similar to the perfect helical beam that is the energy flow is inversely

decaying function of the radius. So, whether the light beams giving zero intensity or non-zero

intensity on the axis is an indication of the spatial coherence of the beam. Degrading the

spatial coherence destroys the fidelity of the zero intensity and the phase singularity on the

axis of the beam. If the beam is not perfectly spatially coherent, when it is transmitted or

diffracted we will get an average angular momentum. This kind of vortex beam is similar to

Rankine Vortex [11].

14 | P a g e

Reference:

1. J.W. Simmons and M. J.

Guttmann, States,Waves, and

Photons: a Modern Introduction

to Light (Addison-Wesley, 1970)

2. Optical vortices, Henry Sztul

3. McGloin, D. and Dholakia, K.

(2005) Bessel beam: diffraction

in a new light, contemporary.

Physics, 46, 15–28.

4. Gutierrez-Vega, J.C., Iturbe-

Castillo, M.D., and Chavez-

Cerda, S. (2000) Alternative

formulation for invariant optical

fields: Mathieu beams. Opt.

Lett., 25, 1493–1495

5. Bandres, M.A. and Gutierrez-

Vega, J.C. (2004) Ince-

Gaussian beams. Opt. Lett., 29,

144–146.

6. Turnbull, G.A., Robertson,

D.A.,Smith, G.M., Allen, L., and

Padgett,M.J. (1996) Generation

of free-space Laguerre-

Gaussian modes at millimetre-

wave frequencies by use of a

spiral phaseplate. Opt.

Commun., 127, 183–188

7. O’Holleran, K., Padgett, M.J.,

and Dennis, M.R. (2006)

Topology of optical vortex lines

formed by the interference of

three, four, and five plane

waves. Opt. Express, 14, 3039–

3044.

8. Pravin Vaity, A. Aadhi, and R.

P. Singh (2013). Formation of

optical vortices through

superposition of thwo Gaussian

beams. (050.4865)

9. Allen, L., Courtial, J., and

Padgett, M.J. (1999) Matrix

formulation for the propagation

of light beams with orbital and

spin angular momenta. Phys.

Rev. E, 60, 7497–7503.

10. N. Simpson, K. Dholakia, L.

Allen, and M. Padgett,

“Mechanical equivalence of

spin and orbital angular

momentum of light: an optical

spanner,” Opt. Lett. 22, 52–54

(1997)}

11. G. A. Swartzlander and R. I.

Hernandez-Aranda, “Optical

Rankine vortex and anomalous

circulation of light,” Phys. Rev.

Lett. 99, 163901 (2007). 59. G.

A. Swartzlander, “Peering into

darkness with a vortex spatial

filter,” Opt. Lett. 26, 497–499

(2001)

Book references:

I. Twisted Photons: Jaun. P.

torres, Lluies torner

II. Optical Vortices: Henry Sztul

III. Review paper “Optical Angular

momentum: origin, behavior,

applications”. Alison m. Yao,

Miles j. Padgett

15 | P a g e

Chapter2

The Energy Flow of paraxial vortex beam: the spin and the orbital

flow

2.1 Introduction

Rotational properties of the light are the growing interest in the current research in optics.

Generally the properties are related to the beam circulatory energy flow of the transverse

direction of propagation. The internal energy flow of a light beam can be separated into the

spin flow density and orbital flow density associated with the polarization state and spatial

inhomogeneity. Such kind of energy flow is represented in the field of so called “Optical

Vortices” which is an interesting concept in the new field called “singular Optics”.

The rotational properties are generally related to the mechanical momentum and it is related

to the 𝑝 =𝑆

𝐶2. The angular momentum can be transmitted to the other objects like micro

particle, Micro machines, and cold atoms. Two types are flow associated with the transverse

energy flow. One is intrinsic spin flow which is related to beam polarization and it depends

on the vector rotations of the every point on the beam cross section. The orbital flow is

dependent on the spatial inhomogeneity of the beam (the screw dislocation of the phase

singularity of the optical vortices).2

In this chapter we mainly focus on the two types of the beams. First, the normal Gaussian

beam and then the one vortex type beam LG beam with zero radial index for the simplicity of

the calculations. For the Gaussian beam it is shown that there is no orbital flow as there is no

phase dependence on the transverse plane and for the LG beam there exists both the spin and

orbital flow.

2 Twisted photons: Applications of Light with Orbital Angular Momentum.

Edited by Juan P. Torres and Lluis Torner 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40907-5

16 | P a g e

2.2 Spin and Orbital Energy Flows:

In the previous chapter we discussed about the spin to orbital angular momentum

conversion. This is the main concern here. The transformation comes because of the strong

spatial transverse inhomogeneity. In the paraxial case the contribution of spin and orbital

angular momentum can be separated perfectly but for the transformation such as strong

focus, passing the beam through small aperture they are related to the deviation from the

paraxial condition, the separation of the spin and orbital angular momentum is impossible.

But still the contribution of polarization and the contribution due to the inhomogeneity can

be separated, namely the density of the energy flow can be written as

𝑆 = 𝑆𝐶 + 𝑆𝑂

(2.1)

Where the 𝑆𝐶 , 𝑆𝑂 are the so-called spin and orbital flow density [1, 2]. Where the density can

be decomposed to

𝑆𝑐 =

𝐶

16𝜋𝐾𝐼𝑚[�� × (𝐸∗ × �� )], 𝑆𝑜

=𝐶

8𝜋𝐾𝐼𝑚[(𝐸∗ . (�� )�� )] (2.2)

Where C is the velocity of light and k is the wave vector. �� is the electric field. (𝐸∗ . (�� )�� ) is

the invariant berry notation [1].of the differential operator which defined as in the Cartesian

co-ordinates

[(𝐸∗ . (�� )�� )]𝑗 = 𝐸𝑥∗ 𝜕𝐸𝑥

𝜕𝑗+ 𝐸𝑦

∗ 𝜕𝐸𝑦

𝜕𝑗+ 𝐸𝑧

∗ 𝜕𝐸𝑧

𝜕𝑗 (2.3)

Where j = x,y,z. In reference to the eq. 2.1, 2.2 the electromagnetic angular momentum w.r.t

a reference point 𝑅0 can be written as

�� =1

𝑐2 𝐼𝑚[∫[(�� − 𝑅0 ) × 𝑆 ]𝑑3𝑟 = 𝐿𝑐

+ 𝐿𝑜 (2.4)

Now we can again decomposed the spin and angular part from the equation which can be

shown as

𝐿𝑐 =

1

8𝜋𝜔𝐼𝑚[∫(𝐸∗ × �� ) 𝑑3𝑟 (2.5)

𝐿𝑜 =

1

8𝜋𝜔𝐼𝑚[∫(�� − 𝑅0

) × (𝐸∗ . (�� )�� )𝑑3𝑟] (2.6)

The integration is performed over the whole space and the condition for the electric field ��

will rapidly go to zero as R → ∞.

17 | P a g e

It can be seen from equation 2.5 Lc denotes the vector nature of the light in contrast to the Lo,

Which is position independent. In the case of the paraxial beam propagating along the z

direction the Lc coincide with the usual definition of the spin angular momentum. [3]

Hence it can be referred as non-paraxial angular momentum. But for Lo we can see that this

is dependent on the reference point. It can be related to the non- paraxial beam. When the

beam is tightly focused the beam has initially a well-defined paraxial spin and orbital angular

momentum. But after the focusing the momentum are redistributed to non-paraxial AMs (eq.

2.4) to the focused beam. This generally defined as the transformation or conversion of spin

to orbital angular momentum.

Experimentally this can be achieved. As in the previous chapter we discussed about the

optical tweezers. The paraxial and non-paraxial version can be distinguished from the motion

of the trapped particle. Under the spin angular momentum the particle will rotate with its

own axis regardless the position from the beam axis. But while for the orbital angular

momentum the particle will start to rotate w.r.t the beam axis [4].

This is direct evidence to the spin-orbital conversion when the beam is tightly focused. The

trapped particle is of the order of micrometer. If the intensity is high enough the size of the

trapped particle will increase.

2.3 The Transverse energy flow:

Let a paraxial beam is propagating along “z” direction.

The electric filed �� can be written as [2, 5]

�� = 𝐸⊥ + 𝑒��𝐸𝑧 = exp(ἰ𝑘𝑧) [𝑢 +

ἰ

𝑘𝑒𝑧�� . 𝑢 ] (2.7)

Where u= u(x,y,z) is the complex amplitude of the electric field and it is slowly varying

envelop (eq. 2.7). 𝑒�� is the beam’s longitudinal direction. We can write on the basis of the

circular polarization as

��𝜎 = 1

√2 (��𝑥 + ἰ𝜎��𝑦) (2.8)

Here ��𝑥, ��𝑦 are the unit vectors of the transverse direction. σ = ±1, is the handedness of the

polarization of the beam.

From the above we can write the amplitude of the electric filed vector as,

�� = ��+𝑢+ + ��−𝑢− (2.9)

18 | P a g e

𝑢𝜎 = 𝑢𝜎(𝑥, 𝑦, 𝑧) is the scalar amplitude of the of the complex field depending upon the

polarization state (σ). The partial intensity and the phase can be written as using the Gaussian

units

𝐼𝜎 = 𝑐

8𝜋|𝑢𝜎(𝑥, 𝑦, 𝑧)|2 (2.10)

And

𝜑𝜎 = 1

2ἰln

𝑢𝜎

𝑢𝜎∗ (2.11)

Now if we substitute this, the spin and orbital flow density (eq. 2.2) reduces to the following

form

𝑆𝑐 =

1

2𝑘��𝑧 × �� (𝐼−1 − 𝐼+1) =

1

2𝑘𝑟𝑜𝑡[��𝑧(𝐼+1 − 𝐼−1)] =

1

2𝑘𝑟𝑜𝑡[��𝑧𝑠3] (2.12)

[2] Where 𝑠3 is the fourth Stock’s parameter3 which characterize the degree of the circular

polarization [6]. The energy flow occurs in the beam due to the inhomogeneous 𝑠3.

The situation become easier is the beam has circular intensity profile and the circular

polarization. In this case the stoke parameter 𝑠3 equals σ𝐼𝜎. And if we consider the polar

frame due to the circuital symmetry then,

𝑟 = √𝑥2 + 𝑦2 , 𝜑 = 𝑡𝑎𝑛−1[𝑦

𝑥] (2.13)

The corresponding spin flow density of the eq. - 2.12 becomes

𝑆𝑐 = −

𝜎

2𝑘[−𝑒��

1

𝑟

𝜕

𝜕𝜑+ 𝑒��

𝜕

𝜕𝑟]𝐼𝜎 (2.14)

Where the 𝑒𝑟, 𝑒𝜑 are the usual unit vectors in polar co-ordinates. In comparison with the

above eq.- 2.14 we can write the orbital flow of eq. – 2.2 as following[2].

𝑆0 =

1

𝑘𝐼𝜎�� 𝜑𝜎 =

1

𝑘 𝐼𝜎[𝑒��

𝜕

𝜕𝑟+ 𝑒��

1

𝑟

𝜕

𝜕𝜑]𝜑𝜎 (2.15)

3 Stocks parameters are

𝑆0 = 𝐼 𝑆1 = 𝑝𝐼 cos(2𝜒) cos (2𝜓) 𝑆2 = 𝑝𝐼 cos(2𝜒) sin (2𝜓)

𝑆3 = 𝑝𝐼 sin(2𝜒) Where 𝐼, 𝑝, 𝜓, 𝜒 are the spherical co-ordinates of the three dimensional Cartesian vectors (𝑆0, 𝑆1, 𝑆2, 𝑆3). 𝐼 is the total intensity of the beam. 𝑝 is the degree of polarization. This can be represented in the poincar’e sphere also.

19 | P a g e

We can notice that there is a great similarity in between the two equations 2.14, 2.15. Both

spin and orbital density depend on the beam spatial inhomogeneity and the polarization state.

They are directly related to the radial and azimuthal derivatives of intensity and the phase

profile of the beam. However Orbital flow density is originated mainly due to the

inhomogeneity in the transverse phase distribution. From the eq. 2.15 it can be seen it will

also depend on the transverse intensity but this only modifies the intensity of the orbital flow

but the variation depends on the transverse phase profile. In the case of the spin flow it is

only dependent on the transverse amplitude inhomogeneity and polarization state but not in

the phase distribution. However there exists a difference between the interrelation of the spin

Sc, orbital So and the derivatives of the intensity, phase. From the eq. 2.15 S0 is directly

related to the phase gradient and always directed along the phase gradient. But the spin Sc is

orthogonal to the intensity gradient (𝑟𝑜𝑡[��𝑧𝑠3]). Though it seems like there is a difference

between them but for trapping of micro particles both acts equivalently. Now we will discuss

some example so that idea of the flow density will be clear.

2.3.1 Gaussian Beam:

For the paraxial Gaussian beam the electric field becomes

�� = 𝐸0 exp [−

𝑥2+𝑦2

𝜔(𝑧)2]𝑒ἰ(𝑘𝑧−𝜔𝑡)𝑒

(ἰ𝑘(𝑥2+𝑦2))

2𝑅(𝑧) exp [𝑖𝜓(𝑧)] (2.16)

Here all the quantities are expressed in usual notation. 𝜓(𝑧) is the Gouy phase term.

Now if we choose the cylindrical polar co- ordinate the eq. 2.10 intensity distribution and

2.11 phase distribution become (for a fixed z plane)

𝐼𝜎 = 𝐼𝜎0exp [−𝑟2

𝜔2], (2.17)

𝜑𝜎 = 0 (2.18)

As there is no azimuthal phase dependence 𝜑𝜎 becomes zero. So the wave front become flat,

No orbital flow is there, 𝑆𝑜 = 0.

The spin flow is determined by the radial derivative of eq. 2.14

𝑆𝑐 = −

𝜎

2𝑘[−𝑒��

1

𝑟

𝜕

𝜕𝜑+ 𝑒��

𝜕

𝜕𝑟]𝐼𝜎

𝑆𝑐 = −

𝜎

2𝑘 𝜕𝐼𝜎

𝜕𝑟 𝑒�� (2.19)

Substituting eq. 2.17 in eq. 2.19 the spin flow of the normal Gaussian beam becomes,

𝑆𝑐 = 𝜎𝑒��

𝑟

𝑘𝜔2 𝐼𝜎0 exp [−𝑟2

𝜔2] (2.20)

20 | P a g e

(a) (b)

Figure (2.1)

(a) Shows the simulated spin flow density

in the transverse plane of the beam for

σ=1 (b) shows the radial profiles of

intensity and the spin flow.

From figure (2.1a) it is clear that the energy flows azimuthally. The length of the arrows

shows the relative density of the flow. This is plotted for the left circularly polarized light

and there is no change of direction in the energy flow. From figure (2.1b) the intensity is

maximum at the center and decrease radially but the flow density has a maximum at 𝑟 =𝜔

√2.

And then gradually decreases as clear from both pictures.

2.3.2 Laguerre Gaussian (LG) Beam:

A slightly complicated situation comes when we consider the LG beams. This

type of beam has orbital helicity. We will consider both the spin (polarized) and the orbital

charge. The screw dislocation of the wave front produces the vortices [3,7]. Restricting

ourselves as mentioned earlier we will consider zero radial index (p=0).In this case eq. 2.10,

2.11 becomes,

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

r

EnergyFlo

wdensi

ty

Radial Profiles

𝑆𝑐

𝐼

21 | P a g e

𝐼𝜎 =1

𝑙!𝐼𝜎0(

𝑟

𝜔)2|𝑙|exp [−

𝑟2

𝜔2] (2.21)

𝜑𝜎 = 𝑙𝜑 (2.22)

The normalization constant restricts that for all 𝑙 the total power will be the same. Now the

flow equation 2.14 becomes

𝑆𝑐 = −𝜎𝑒��

1

𝑙!

1

𝑘𝜔 (

𝑟

𝜔)2|𝑙|−1 𝐼𝜎0{ |𝑙| − (

𝑟

𝜔)2} exp [−

𝑟2

𝜔2] (2.23)

Using eq. 2.15 the orbital flow becomes

𝑆0 = 𝑒��

1

𝑙!

1

𝑘𝜔 (

𝑟

𝜔)2|𝑙|−1 𝐼𝜎0 𝑙 exp [−

𝑟2

𝜔2] (2.24)

We can see from the above equations there exist a simple relation between them.

𝑆𝑐 =

𝜎

𝑙 ( |𝑙| − [

𝑟

𝜔]2 ) 𝑆0

(2.25)

The derived dependencies are plotted using Mathematica in the figure (2.2 a-d). At any 𝑙 the

corresponding transverse energy flow goes to zero and also when r → ∞ the energy also goes

to zero. In the intermediate region the spin and orbital flow both poses a maxima ant then

gradually decreases.

The orbital flow |𝑆0| eq. 2.24 has a maximum at

𝑟

𝜔= √

2|𝑙|−1

2, (|𝑙| > 0) (2.26)

Whereas the spin flow |𝑆𝑐| has two extremes at,

(𝑟

𝜔)2 = |𝑙| +

1

4 ± √

16|𝑙| + 1

4

This corresponds to the maxima of |𝜕

𝜕𝑟𝐼𝜎| on inner and outer radius of the donut type of rings.

22 | P a g e

(a) (b)

(c) (d)

Figure 2.2

Radial profiles of the normalized intensity

(I), Spin flow density (Sc), Orbital flow

density (So), Total Transverse energy flow

(S) in the units of 𝑟/𝜔, for circularly

polarized beam with zero radial index

(p=0). (a) σ =1,𝑙=0. (b) σ = 1,𝑙 = 1. (c) σ

= 1, 𝑙 =2. (d) σ =-1, 𝑙 = 1

.

From the graph we can see when 𝑙=0, the extremum disappears and we get the spin flow

maximum at 𝑟 =𝜔

√2.

As we can see from the graphs of figure 2.2 the magnitude of the spin flow and orbital flows

are in the same order. As the orbital flow is related to the mechanical momentum it can drive

or rotate the micro particle [4]. This can be done by then spin also. This can be taken into the

account to the experimental fact of spin to orbital angular momentum conversion [8].

The total orbital flow becomes in terms of the orbital momentum,

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

r

Ener

gyFlo

wdensi

tyRadial Profiles

0.0 0.5 1.0 1.5 2.0 2.5 3.00.2

0.1

0.0

0.1

0.2

0.3

0.4

r b

Energ

yFlo

wdensi

ty

Radial Profiles

0.0 0.5 1.0 1.5 2.0 2.5 3.00.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

r b

EnergyFlo

wdensi

ty

Radial Profiles

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.0

0.2

0.4

0.6

r b

EnergyFlo

wdensi

ty

Radial Profiles

𝑆𝑐

𝐼

𝑆𝑐

𝐼

𝐼 𝐼

𝑆𝑐 𝑆𝑐

𝑠 𝑠

𝑆0 𝑆0

𝑆0

𝑠 𝑠

23 | P a g e

𝑆 = 𝑆𝑐 + 𝑆0 = (1 − 𝜎|𝑙|

𝑙+ 𝜎

1

𝑙 𝑟2

𝜔2) 𝑆 0 (2.27)

In the figure (2.2) it is marked as S. From the graph it is clear that the spin and orbital flow

can support or suppress themselves depending upon the handedness. In the region 𝑟

ω <𝑙, the

orbital flow dominates but for other regions like at the periphery of the rings the spin flow

dominates.

(a) (b) (c)

Figure 2.3

Maps of the Orbital flow S0 , Spin flow Sc,

Total energy flow (S) on a transverse

plane of a right circular polarized beam

with σ= -1, 𝑙 =1. For orbital flow there is

no change in handedness. At every point

polarization direction is same. b & c

circular streamline shows the energy flows

vanishes for corresponding flow

components.

At the near field region 𝑟

𝜔≪ 1 an interesting situation comes. From the equation 2.25 the

magnitude of the spin and orbital density are almost equal. Now if the handedness of

both 𝑆𝑐 , 𝑆0 are same then the total flow of energy eq. 2.27 tends to zero (fig. 2.2b). Formally

the handedness of spin flow is determined by not only the σ but also the factor 𝜕

𝜕𝑟𝐼𝜎

(eq. 2.19).

On the contradiction to the other graphs if the polarization handedness is opposite to the

orbital rotation then the total flow will be added up and give a maximum flow of rotational

24 | P a g e

energy available for the flow of circularly polarized LG beam with a given 𝑙 as clear from the

figure 2.2d curve S.

The flow diagrams are represented in the figure 2.3 which agrees the curves of the figure

2.2d. As we can see the orbital moment has no change of handedness. It flow only one

direction. The relative length of arrows shows the strength of the flow density. It starts from

zero and attain a maximum value then again decreases with no change of flow direction.

Whereas the spin and the total flow density has a change of handedness. It is clear from the

figure (2.3b-c).One region has positive flow and the other region has the reverse flow. The

regions of the opposite flow are indicated by a stream line (black circle) in the figures. The

stream line corresponds to the zero energy flow. It is clear that the total energy flow decrease

rapidly after the streamline circle. So if we are able to decrease the width of the total flow of

the curve 2.2d then we can confine the energy in a small ring. That is we can do the

confinement of the energy.

Here we have shown only few combinations of the σ and 𝑙. But there can be so many

combinations. Even we can do the calculation with fractional spin (with elliptically polarized

light).

References:

1. Berry, M. (2009) Optical currents.

J. Opt. A: Pure Appl. Opt., 11,

094001 (12 pp)

2. Bekshaev, A.Ya. and Soskin, M.S.

(2007) Transverse energy flows in

vectorial fields of paraxial beams

with singularities. Opt. Commun.,

271, 332–348.

3. Soskin, M.S. and Vasnetsov, M.V.

(2001) Singular optics. Prog. Opt.,

42, 219–276.

4. Simpson, N.B., Dholakia, K., Allen,

L., and Padgett, M.J. (1997)

Mechanical equivalence of spin and

orbital angular momentum of light:

an optical spanner. Opt. Lett., 22,

52–54.

5. Bekshaev, A. and Soskin, M. (2007)

Transverse energy flows in

vectorial fields of paraxial light

beams. Proc. SPIE, 6729, 67290G

6. Shurkliff, W.A. (1962) Polarized

Light, Harvard University Press,

Cambridge, MA

7. Allen, L., Padgett, M.J., and

Babiker, M. (1999) Orbital angular

momentum of light. Prog. Opt., 39,

291–372

8. Nieminen, T.A., Stilgoe, A.B.,

Heckenberg, N.R., and Rubinsztein-

Dunlop, H. (2008) Angular

momentum of a strongly focused

Gaussian beam. J. Opt. A: Pure

Appl. Opt., 10, 115005.

25 | P a g e

Chapter 3

Higher order Gaussian beams

3.1 Introduction:

We are familiar with the normal Gaussian beams. But there are other beams also.

These are the higher order Gaussian beams. Hermite Gaussian beam [1], Lagurrere Gaussian

beam [2], Bessel Beam [3], Mathieu beam [4]; Ince-Gaussian beam [5] are some examples of

the higher order Gaussian beams. Here we will discuss only the normal Gaussian beam then

the Hermite- Gaussian beam and lastly the Lagurrere Gaussian (LG) beam. In the

experimental part we use the Hermite-Gaussian (HG) beam to generate LG beam. For this

we need to discuss about the HG beams. We will show how the solution of the paraxial wave

equation changes if we impose some constraints to the amplitude of the wave.

3.2 The paraxial wave equation formulation

We know the Maxwell’s equations. In a source free regions the equation becomes

�� . �� = 0 (3.1)

�� . �� = 0 (3.2)

�� × �� = −𝜕��

𝜕𝑡 (3.3)

�� × �� = 휀0𝜇 0𝜕��

𝜕𝑡 (3.4)

If we take the 2nd derivative of the curl equations the wave equation becomes,

𝛻2�� − 1

𝑐2

𝜕2��

𝜕𝑡2 (3.5)

Now let an arbitrary electric filed has a form 𝐸 (𝑥, 𝑦, 𝑧; 𝑡) = 𝐴 (𝑥, 𝑦, 𝑧)𝑒ἰ𝜔𝑡 . If we substitute it

in the eq. 3.5 the wave equation becomes

(𝛻2 + 𝑘2)𝐴 (𝑥, 𝑦, 𝑧) = 0 (3.6)

This is also known as the Helmoltz equation. Here the �� is wave

26 | P a g e

Now we assume that the general form of the 𝐴 is �� (𝑥, 𝑦, 𝑧)exp (−ἰ𝑘𝑧). Here �� is the shape

of the wave front and the exp(−ἰ𝑘𝑧) denotes the wave is propagating along the z direction.

Now if we substitute this in eq 3.6 we will get

(𝜕2𝑢

𝜕𝑥2 + 𝜕2𝑢

𝜕𝑦2 ) + 𝜕2𝑢

𝜕𝑧2 − 2ἰ𝑘 𝜕𝑢

𝜕𝑧= 0 (3.7)

Now if we consider the paraxial approximation that is the envelop of the amplitude is a

slowly varying function along the direction of propagation that is,

𝜕2𝑢

𝜕𝑧2 ≪ 𝜕𝑢

𝜕𝑧,

𝜕2𝑢

𝜕𝑧2 ≪ 𝜕2𝑢

𝜕𝑥2, 𝜕2𝑢

𝜕𝑧2 ≪ 𝜕2𝑢

𝜕𝑦2 (3.8)

We can then neglect the 𝜕2𝑢

𝜕𝑧2 term. Then the eq 3.7 becomes

(𝜕2𝑢

𝜕𝑥2+

𝜕2𝑢

𝜕𝑦2 ) − 2ἰ𝑘

𝜕𝑢

𝜕𝑧= 0

Or 𝛻𝜏2𝑢 − 2ἰ𝑘

𝜕𝑢

𝜕𝑧= 0 (3.9)

This is known as the paraxial wave equation. 𝛻𝜏2 is known as the transverse operator. The

solution of the equation depends on the chosen co-ordinate system. If we choose the

Cartesian co-ordinate system (x, y, z) we will get the Hermite Gaussian beam, if we choose

the cylindrical polar co- ordinate (r, φ, z) the solution would be the Laguerre- Gaussian beam

and if we choose the elliptical co-ordinate (η, ζ, z) the results will be Ince Gaussian beam. In

the following section we will discuss about the solution of HG and LG beams.

3.3 Solution of the paraxial wave equation

3.3.1 Fundamental Gaussian beam

The simple solution of the paraxial wave equation is the fundamental Gaussian beam.

The solution is

𝑢(𝑥, 𝑦, 𝑧) = 𝑢0

𝜔(𝑧)𝑒𝑥𝑝 (−

𝑥2+𝑦2

𝜔(𝑧)2) 𝑒𝑥𝑝 (−𝑖𝑘

𝑥2+𝑦2

2𝑅(𝑧)2− 𝑖 𝜓(𝑧)) (3.10)

𝑢(r, φ, z) = 𝑢0

𝜔(𝑧)𝑒𝑥𝑝 (−

𝑟2

𝜔(𝑧)2) 𝑒𝑥𝑝 (−𝑖𝑘

𝑟2

2𝑅(𝑧)2− 𝑖 𝜓(𝑧)) (3.11)

27 | P a g e

The 1st equation is expressed in Cartesian co-ordinate and the 2nd equation is in cylindrical

polar coordinate. This is the fundamental Gaussian beam and it is defined as TEM00. Figure

3.1 depicts the beam profile and the intensity distribution of a fundamental Gaussian beam.

(a) (b) (c)

Figure 3.1

Intensity distribution of the

fundamental Gaussian beam

TEM00. (a) shows the simulated

intensity distribution in a plane,

(b) corresponding 3D false color

simulated Image, (c) 2d view of

Intensity variation along transverse

direction(x direction).

Here all the symbols are in usual notations.

𝜔(𝑧) = 𝜔0√(1 + (𝑧

𝑧0)2) (3.12)

𝑅(𝑧) = 𝑧 [1 + (𝑧0

𝑧)2] (3.13)

𝜓(𝑧) = 𝑡𝑎𝑛−1 (𝑧

𝑧0) (3.14)

𝜔0 = √𝜆𝑧0

𝜋 (3.15)

ω0 is the beam waist at z = 0 plane. Z0 is the Rayleigh length. ω (z) is the beam waist of the

beam at z plane. R (z) is the radius of curvature at z along the propagation direction. 𝜓(𝑧) is

10 5 5 10x

0.2

0.4

0.6

0.8

1.0I

28 | P a g e

the Gouy Phase which varies from –π/2 to π/2. It increase -π/4 to π/4 almost linearly when |z|

< z0. After the Rayleigh length it varies very slowly. It is become slowly varying function for

|z| > z0. [figure 3.2]

(a) (b)

Figure 3.2

(a) Gouy phase dependence along the z direction, (b) variation of radius of

curvature along the z axis.

Now it is clear that the Gouy function is a slowly varying function for large distance along

the propagation direction. So we can neglect the term for simplicity of the calculation. The

radius of curvature is also tending to infinite as we go large distance along the propagation

direction. So, the wave front becomes the plane wave front for the fundamental Gaussian

beam. The phase part becomes z = constant if we neglect this two terms. So it represents a

plane wave front. [Figure 3.3]. But in general these are the parabolic wave fronts.

Figure 3.3

Phase diagram for the fundamental Gaussian

beam. It is constant for all points of a particular

plane.

10 5 5 10z

1.5

1.0

0.5

0.5

1.0

1.5

10 5 5 10z

20

10

10

20

R z

29 | P a g e

3.3.2 Hermite-Gaussian beam:

The fundamental Gaussian beam is not only the solution of the

paraxial Helmholtz equation. Some solutions are non-Gaussian type and they have but the

share the same parabolic wave fronts of the Gaussian beams. Now we consider a wave front

whose complex envelope is a modulated function of the Gaussian wave front.

𝑢(𝑥, 𝑦, 𝑧) = 𝑋 (√2𝑥

𝜔(𝑧))𝑌 (

√2𝑦

𝜔(𝑧)) exp [𝑖 𝑍(𝑧)]𝑢𝑔(𝑥, 𝑦, 𝑧) (3.16)

Here ug (x, y, z) is the Gaussian envelope. 𝑋 (√2𝑥

𝜔(𝑧)), 𝑌 (

√2𝑦

𝜔(𝑧)) are the real function. If we

solve the paraxial Helmholtz equation the solution becomes the Hermite- Gaussian beams

[1]. The final complex amplitude of the beam including the propagation factor becomes

𝐴𝑚,𝑛(𝑥, 𝑦, 𝑧) = 𝑢𝑚,𝑛 [𝜔0

𝜔(𝑧)]𝐻𝑚 (

√2𝑥

𝜔(𝑧))𝐻𝑛 (

√2𝑦

𝜔(𝑧)) 𝑒𝑥𝑝 (−

𝑥2 + 𝑦2

𝜔(𝑧)2)

× 𝑒𝑥𝑝 (−𝑖𝑘𝑧 − 𝑖𝑘𝑥2+𝑦2

2𝑅(𝑧)2− 𝑖(𝑚 + 𝑛 + 1) 𝜓(𝑧)) (3.17)

Here 𝐻𝑚(𝑥) is the Hermite polynomial4 of order m. m can be any integer. m = 0, 1, 2,3 …

𝐺𝑚(𝑥) = 𝐻𝑚(𝑥)𝑒𝑥𝑝 [−𝑥2

2] is known as Hermite- Gauss function [1]

(a) (b) (c) (d)

Figure 3.4

Simulated Field amplitude of HG modes (a) 𝐺0, (b) 𝐺1, (c) 𝐺2, (d) 𝐺3

4 𝐻𝑚(𝑥) is defined as 𝐻𝑚(𝑥) = (−1)𝑚𝑒𝑥2 𝑑𝑛

𝑑𝑥𝑛 𝑒−𝑥2. 𝐻0(𝑥) = 1, 𝐻1(𝑥) = 2𝑥, 𝐻2(𝑥) = 4𝑥2 − 2

10 5 5 10x

0.2

0.4

0.6

0.8

1.0G0

10 5 5 10x

4

2

2

4

G1

4 2 2 4x

2

1

1

2

G2

4 2 2 4x

4

2

2

4

G3

30 | P a g e

3.3.2.1 Intensity distribution

The intensity of Hermite Gaussian beam is 𝐼𝑚,𝑛 = | 𝐴𝑚,𝑛|2.

𝐼𝑚,𝑛(𝑥, 𝑦, 𝑧) = 𝑢𝑚,𝑛2 [

𝜔0

𝜔(𝑧)]2 𝐻𝑚

2 (√2𝑥

𝜔(𝑧))𝐻𝑛

2 (√2𝑦

𝜔(𝑧)) 𝑒𝑥𝑝 (−

2(𝑥2+𝑦2)

𝜔(𝑧)2) (3.18)

Figure 3.5 illustrates the normalized intensity distribution of some HG modes. The width of

the beam is higher for higher value of m, n. It is clear from the pictures.

Figure 3.5

Simulated intensity distribution on

the transverse plane of some HG

beam.

The orders (m, n) are indicated on

the top of the pictures.

As it is clear from the picture the no. of vertical lobes will be n + 1, and for horizontal lobes

it will be m + 1. The order of the modes is defined as N = n + m. The beam waist increases

as a function w (z). These beams have the rectangular symmetry except the HG00. It has the

circular symmetry. It is basically fundamentally Gaussian mode. As H0(x) = 1. So, it reduces

to the fundamental mode. Figure 3.6 shows 3D view of HG11 mode.

31 | P a g e

(a) (b)

Figure 3.6

Intensity distribution of HG11 mode. (a) Transvers intensity distribution. (b)

False color image; 3D view of the corresponding mode.

We have generated some of the higher order HG modes experimentally. The details of the

generation process will be discussed in the chapter 4. In the figure 3.7 some the

experimentally generated beam is shown.

(a) (b) (c) (d)

Figure 3.7

Experimentally generated HG modes . (a) HG01, (b) HG10, (c) HG20, (d) HG02.

32 | P a g e

3.3.3 Laguerre- Gaussian Beam:

Hermite- Gaussian beams form a complete set of solutions of the paraxial

Helmholtz equation. Any mode can be written as the superposition of these modes. However

these are not the only solution of the paraxial Helmholtz equation. If we choose the

cylindrical polar co- ordinate (r, φ, z) the solution would be Laguerre-Gaussian (LG) mode.

This type of beam has cylindrical symmetry. After solving the Helmholtz equation the

complex amplitude of the LG beam becomes [1, 2, 6]

𝐴𝑝,𝑙(𝑟, φ, z) = 𝑢𝑝,𝑙 [𝜔0

𝜔(𝑧)] [

𝑟

𝜔(𝑧)]|𝑙|

𝑒𝑥𝑝 [−𝑟2

𝜔2(𝑧)] 𝐿𝑝

|𝑙| (2𝑟2

𝜔2(𝑧)) 𝑒𝑥𝑝[−𝑖𝑙𝜑]

× 𝑒𝑥𝑝 (−𝑖𝑘𝑧 − 𝑖𝑘𝑟2

2𝑅(𝑧)2− 𝑖(2𝑝 + |𝑙| + 1) 𝜓(𝑧)) (3.19)

Here 𝐿𝑝|𝑙|(𝑥) is the associated Lagurre polynomial. The symbols are in usual notations which

are defined in eq 3.12- 3.4. These beams are known as 𝐿𝐺𝑝𝑙 beams [Fig 3.8]. These beams

Figure 3.8

Simulated intensity (1st row) and

phase distributions (2nd row) on

the transverse plane of some LG

beam. The orders (p, 𝑙) are

indicated on the top of the picture

33 | P a g e

are circularly symmetric. The intensity is proportional [𝑟

𝜔(𝑧)]|𝑙|

𝑒𝑥𝑝 [−𝑟2

𝜔2(𝑧)] 𝐿𝑝

|𝑙| (2𝑟2

𝜔2(𝑧)).

So if | 𝑙 | ≠ 0, then at r = 0, there is a zero intensity at the center of the beam and the beam

has annular intensity patterns. These beams are the vortex beams. If the p index is zero then

𝐿0|𝑙|(𝑥) = 1. The radius of the𝐿𝐺0

|𝑙|(𝑥) vortex is proportional to |𝑙|1/2. It is clear

Figure 3.9

Experimentally generated LG01

beam. (a) Transverse intensity

distribution. (b) False color image; 3D

view of the corresponding

simulated beam. (c) Variation of

intensity along horizontal direction

(x axis).

from the pictures [ fig. 3.8]. The “p” index denotes the no. of rings in the beam will be

p + 1.” 𝑙” represents the helicity of the beam. We have generated experimentally 𝐿𝐺01 .Figure

[3.9]. Because of the term 𝑒𝑥𝑝[−𝑖𝑙𝜑] in the amplitude of the electric field there is azimuthal

phase dependence in the transverse plane. The Gouy phase term is multiplied by (2p + |𝑙| +

1). So the effect in phase is little bit bigger than the HG and fundamental Gaussian mode.

The phase term has some interesting properties. The phase term is

𝑙𝜑 + 𝑘𝑧 + 𝑘𝑟2

2𝑅(𝑧)2+ (2𝑝 + |𝑙| + 1) 𝜓(𝑧) (3.20)

Now if we assume that the radius of curvature of the beam is infinite and the Gouy phase

change is negligible for larger distances then the last two term of the equation becomes

slowly varying functions. They can be neglected. So the wave front of the LG beam becomes

10 5 0 5 10x

0.0020.0040.0060.0080.0100.0120.014

I

34 | P a g e

𝑙𝜑 + 𝑘𝑧 = 𝑐𝑜𝑛𝑠𝑡 (3.21)

This is an equation of helix. For a particular z value there is azimuthal phase dependence on

the transverse plane. The phase dependence is simulated for some of the LG mode in the

figure 3.8. As the azimuthal phase dependence is 𝑒𝑥𝑝[−𝑖𝑙𝜑], For 𝑙 = 0 the wave will be

normal Gaussian beam with plane wave front. = 1 the phase will rotate 0 to 2π. For = 2 the

phase will change 0 to 2π twice and so on. Equation 3.21 shows the wave front is a helical

surface having 𝑙 intertwined helix Figure [3.10].

(a) (b)

(c) (d)

Figure 3.10

Simulated helical wave fronts of the LG beams. (a) 𝑙 = 0, (b) 𝑙 = 1, (c) 𝑙 = 2, (d) 𝑙 = 3.

The order of the beam is defined as 𝑁 = 2𝑝 + |𝑙|. p can be any integer and 𝑙 can take any

value between -∞ to +∞. The index 𝑙 is related to the orbital angular momentum of light as

discussed in the previous chapters.

35 | P a g e

3.3.3.1 Superposition of the LG beams:

If the electric field of one mode is 𝐿𝐺𝑝′𝑙′ and the other

mode has field 𝐿𝐺𝑝𝑙 , then the intensity of the resultant beam would be

𝐼 = [𝐿𝐺𝑝′𝑙′ + 𝐿𝐺𝑝

𝑙 ][𝐿𝐺𝑝′𝑙′ + 𝐿𝐺𝑝

𝑙 ]∗

In the figure 3.11 some of the intensity patterns of the superposed beam is shown. From the

figure it is clear the number of radial cuts in a ring will be |𝑙 − 𝑙′| and the number of rings

will be p+ 1 as earlier.

Figure 3.11

Normalized simulated intensity plots of some superposition of LG beams. LG05 + LG01, LG05

+ LG0-5, and LG24 + LG2-4 (left to right).

References:

1. Fundamental of photonics : B.

Saleh

2. Lasers, A. E. Siegman.

3. McGloin, D. and Dholakia, K.

(2005) Bessel beam: diffraction in

a new light, contemporary.

Physics, 46, 15–28.

4. Gutierrez-Vega, J.C., Iturbe-Castillo, M.D., and Chavez-Cerda,

S. (2000) Alternative formulation for invariant optical fields: Mathieu beams. Opt. Lett., 25, 1493–1495

5. Bandres, M.A. and Gutierrez-Vega, J.C. (2004) Ince-Gaussian beams. Opt. Lett., 29, 144–146.

6. Gaussian beam in the optics course: E. J. Galvez.

0

1

36 | P a g e

Chapter 4

Generating the Laguerre – Gaussian (LG) beams

4.1 Introduction:

There are so many ways we can generate the LG beams which already mentioned in

the 1st chapter .Here we will see it in details. They can be generated directly from the laser

cavity[1] or by using spiral plates [2], or by using fork diffraction [3] or using Spatial Light

Modulators and also by using mode converters[4]. The mode converts are basically two

cylindrical lenses In which if we pass the Hermite Gaussian beam the output will be vortex

beam (LG beam).

We have used the mode converters to produce the LG beam experimentally. We have dealt

with only the HG01 beam, as it can be easily generated. And used it to produce the LG beam

with 𝑙= 1.

To produce the higher order Gaussian beams (HG) we have cascade two grating separated by

π/2 phase difference. And then the higher order modes will come. And then it is to be send

through the cylindrical lenses in order to get the LG beams. We will see the details shortly.

4.2 Generation the LG beam by using spiral phase plate:

This is the old technique and perhaps the easiest way to understand how the helical

beam can be generated using a helical phased surface. A simulation (using Mathematica) is

shown in figure 4.1 to show this. The optical thickness of the spiral plate increases

azimuthally according to 𝑙𝜆𝜃

2𝜋(𝑛−1). Here n is the refractive index, λ is the wave length of the

light, 𝑙 is the topological charge or the helicity of the beam and θ is the azimuthal angle. It

seems like it is easier to understand but to make it is not so simple. It requires extreme

precision in the pitch of the helical surface.

Spiral plate demonstrate easily how the OAM transferred from the optical system to the light

and why the beam should carry the orbital angular momentum. If a plane wave is incident of

the plane surface of the spiral plate the beam will refract in the azimuthal direction. Thus the

linear momentum acquires an azimuthal component w.r.t the radius vector when it

propagates, which results an angular momentum along the beam propagation direction.

At radius r the azimuthal angle becomes 𝑙𝜆

2𝜋(𝑛−1). Now from the snell’s law the angular

deviation of the transmitted ray (𝑛−1)𝑙𝜆

2𝜋(𝑛−1)𝑟=

𝑙𝜆

2𝜋𝑟. Now we know that the liner momentum is

ħk. So if we multiply this the angular momentum becomes 𝑙 ħ per photon.

37 | P a g e

(b)

(a)

Figure 4.1

(a) Spiral plate (b) generation of the helical wave with the spiral plate. In this case

𝑙=0 transformed to 𝑙 = 3.

4.3 Generation the LG beam by using the fork diffraction grating:

An alternative way of generating the beam is use of diffractive optical element. A

helical phase profile (exp[ἰ𝑙φ]) can convert a Gaussian beam to helical beam whose wave

front has 𝑙 fold corkscrew (fig. 4.2). But in the practical situation the phase profile is added to

a linear phase ramp. The sum is expressed as modulo of 2π in order to get a fork type grating.

The form of the fork grating can be calculated mathematically. If we superpose a Plane wave

front of TEM00 and the LG0𝑙 we will get the interference pattern which will be a fork type

fringe. Let 𝐸1 be the electric field of TEM00 and 𝐸2

be the electric field of LG0 . Then the

intensity of the interference is

𝐼 = (𝐸1 + 𝐸2

)(𝐸1 + 𝐸2

)∗ (4.1)

= |𝐸1 |2 + |𝐸2

|2 + (𝐸1 𝐸2

∗ + 𝐸2 𝐸1

∗ ) (4.2) The last term of the above eq. is the main interest. Let

𝐸1 = 𝐸00

exp [ἰ(𝜔𝑡 + 𝑘𝑥 𝑠𝑖𝑛𝛽) , 𝐸2 = 𝐸0𝑙

exp [ἰ(𝜔𝑡 + 𝑙𝜑) (4.3)

s

z

r

θ

38 | P a g e

Fig 4.2

Simulated helical phase profile exp (ἰ𝑙φ)

converts a plane wave to a helical wave

corresponding to 𝑙 fold helicity. Here the

phase profile is for 𝑙 =3. The output has

three intertwined helix.

We assume that the plane wave has a wave vector of magnitude k = (k sinβ, 0, k cos β)

whereas the LG beam has a wave vector k= (0, 0, k). The LG beam has a azimuthal phase

dependence. So the phase dependence assumed to be 𝑙φ. The interference comes due to the phase difference between the two waves. The phase difference δ becomes

𝛿 = 𝑙𝜑 − 𝑥𝑘 𝑠𝑖𝑛𝛽 (4.4)

The condition for the binary grating 𝛿 = 𝑛𝜋.

So the our expression for obtaining the binary fork grating is (fig. 4.4)

𝑛𝜋 = 𝑙 𝑡𝑎𝑛−1 (𝑦

𝑥) − 𝑥𝑘 𝑠𝑖𝑛𝛽 (4.5)

In order to get the blazed diffraction grating we have to use modulo function. If we use

blazed diffraction grating instead of the binary grating the energy loss will be less and the

intensity of the first order diffraction will be higher. If the binary grating function is

𝑓(𝑥, 𝑦) = 𝑙 𝑡𝑎𝑛−1 (𝑦

𝑥) − 𝑥𝑘 𝑠𝑖𝑛𝛽 (4.6)

The expression for the blazed grating becomes,

𝑔(𝑥, 𝑦) = exp [−ἰ 𝑚𝑜𝑑(𝑓(𝑥, 𝑦), 2𝜋] (4.7)

39 | P a g e

+

(a) (b) (c)

Figure 4.3

A linear phase ramp is added to a phase

diagram of helical phase of 𝑙 = 1,

resulting a fork diffraction grating. This is

a blazed diffraction grating. (a) phase

diagram, (b) linear phase ramp, (c) fork

diffraction. [Simulated using Mathemtica]

Here basically we are putting the function in an envelope which has a period of 2π. The mod

() is the modulo function. In the figure 4.4 we have simulated the binary grating and as well

as the blazed grating for 𝑙 = 1.

(a) (b)

Figure 4.4

The computer generated for grating. (a) Binary grating for 𝑙=1. (b) Blazed grating for 𝑙=1.

1

0

=

40 | P a g e

4.4 Generation the LG beam by using the π/2 mode converters:

The specific design of cylindrical lenses would transform between HG to LG mode.

If we send the HG beam through cylindrical lenses at 450 it can be decomposed in to a set of

HG modes and if the same set of HG mode can be rephrased through the mode converters the

LG beam will come. The rephrasing occurs due to the different Gouy phase shift for different

orders. We can describe the transformation by the jones matrix formulation of the mode

converters [5, 6]. The cylindrical mode converters have two forms; one is π/2 converter and the other one in π

converter. The π/2 converter transformed any HG mode of order m, n, oriented at 450 w.r.t

the cylindrical lenses into the LG beam with mode indices 𝑙 = m - n and p = min (m, n). The

π converter transforms the hleicity or handedness of the beam. It is equivalent to the dove

prism. Mathematically the π/2, π converters [Figure 4.5] is equivalent to the λ/4, λ/2 wave

plates respectively.

(a) (b)

Figure 4.5

(a) π/2 converter (b) π converter

[Figures generated using Mathematica]

For this method first we have to generate the higher order HG beams. We have used a grating

which is printed in a plastic transparent sheet with high dpi printer. The grating basically

made of two parallel grating shifted by π/2 phase or λ/2 distance w.r.t each other (figure 4.6).

√2𝑓

2𝑓

41 | P a g e

(a) (b)

Figure 4.6 The gratings for generating HG beams.

(a) The binary grating.

(b) Corresponding blazed grating

Now if we send normal Gaussian beam the output will be HG beams. We have used the He-

Ne laser of wave length 632.8 nm. Figure 4.7 shows our experimental set up. We got all the

higher orders with one zero index (HG00 HG01, HG02, HG03).Figure[4.8]

Figure 4.7 experimental set up for generating HG beam, LG beam and double slit diffraction

Grating for HG mode Mode converter

Diffraction grating CCD

Laser

42 | P a g e

Figure 4.8

Higher order Gaussian modes. From left to right

𝐻𝐺01, 𝐻𝐺10, 𝐻𝐺20, 𝐻𝐺02. The 1st row shows simulated beam profile. 2nd row shows corresponding

experimentally generated beam profiles. 3rd row shows simulated 2d view of the beam profile. 4th row

shows corresponding simulated 2D intensity distributions.

4 2 0 2 40.00.20.40.60.81.01.21.4

y

Inte

nsi

ty

4 2 0 2 40.0

0.5

1.0

1.5

x

Inte

nsi

ty

4 2 0 2 40

2

4

6

8

y

Inte

nsi

ty

4 2 0 2 40

2

4

6

8

x

Inte

nsi

ty

43 | P a g e

If we send the beam through rotator and rotate the beam trough beam at 450 It can be shown

that this beam will have set of HG modes [7]. By the jones matrix formulation for the Higher

modes the mode rotation matrix for order N=15 would be

𝑅(𝜑) = [cos (𝜑) sin (𝜑)−sin (𝜑) cos (𝜑)

] (4.1)

If we send the HG01 beam at the output will be HG01 at 450. But it has both components HG01,

HG10.

The input column vector would be [10]. The output will be 𝑅(𝜑) × 𝐻𝐺01.

1

√2[

1 1−1 1

] × [10] =

1

√2[11] (4.2)

So the output will have both HG modes. Now if we rephrased it through the cylindrical

lenses then the output will be a LG mode of order N = 1. 𝑙 =1. P= 0. The matrix for the π/2

converter analogous to the quarter wave plate

𝐶(𝜋/2) = [1 00 −ἰ

] (4.3)

The output is

𝐶(𝜋/2) × 𝑅(𝜑) × 𝐻𝐺01

[1 00 −ἰ

] ×1

√2[

1 1−1 1

] × [10] =

1

√2[1−ἰ

] (4.4)

From the output we can say if the two HG modes added with a π/2 phase difference we will

get the LG beam as output beam. Experimentally we got exactly the same beam (fig 4.9). It

can be notice that if the input beam is not 450 w.r.t the cylindrical lenses the output will not

be a perfect vortex beam, because different order will get different Gouy phase shift. The

dependence of the rotation of the beam w.r.t the lenses axis can be studied easily. In our

experiment we placed the grating at 450 instead of using the beam rotator. If we use the

rotator the final out intensity is very less. So we rotate the grating directly 450 and it is easier

to do also.

5 The order of the mode is define as N = m + n = 2p + |𝑙|. Where m, n are the indices of the HG beam and p, 𝑙 are

the indices of the LG beam. The order N is always constant for a transformation. The column vector will have N+1 no. of elements and the order of each matrix (mode rotation matrix, 𝐶(𝜋/2), 𝐶(𝜋) ,mode filter) will be (N+1)× (N+1).

44 | P a g e

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4.9

Combining the Hermite- Gaussian modes

in order to get the LG modes. Figure

shows the simulated and experimental

transverse intensity distribution.

In the figure - 4.9 the upper row shows HG mode at 450 can be decomposed to a set of HG01,

HG10. If it is rephrased then we will get LG mode (in the lower row).

1

√2𝐻𝐺10 −

ἰ

√2𝐻𝐺01 = 𝐿𝐺1

0 (4.5)

In the figure 4.4 a, e correspond to HG01, (b), (f) corresponds to HG10, (c) corresponds to

HG01 at 450. These entire pictures are simulated pictures. (d) Corresponding experimentally

generated HG01 beam. (g) is the simulated LG10 beam. (h) is the corresponding

experimentally generated beam.

This process can be generalized further in order to get the higher orders LG beams. For this

we need to generate the higher HG beams. If we use a crossed parallel grating we will get the

HG11, HG12, HG22, HG21 etc. Then if we superpose these we will get LG beams of higher 𝑙.

1

√2

1

√2 =

1

√2

−ἰ

√2

=

45 | P a g e

For an example if we send the HG20 beam at 450 to the cylindrical lenses then it can be

decomposed HG20, HG11 and HG02. If we rephrased it we will get the LG20 beam (figure

4.10).

In the matrix representation the output is

𝐶(𝜋/2) × 𝑅(𝜑) × 𝐻𝐺20 (4.6)

In this case the order of the matrix will be N+ 1 = m+ n + 1= 3

[1 0 00 ἰ 00 0 −1

] ×

[

1

2

1

√2

1

2

−1

√20

1

√21

2−

1

√2

1

2 ]

× [100] =

[

1

2ἰ

√2

−1

2]

(4.7)

So we can conclude that the output will be

1

2𝐻𝐺20 +

ἰ

√2𝐻𝐺11 −

1

2𝐻𝐺02 = 𝐿𝐺2

0 (4.8)

(a) (b) (c) (d)

Figure 4.10

The computer generated results for generating the LG02 beam. (a) HG2,0 (b) HG1,1 (c) HG0,2

(d) LG20

1

2 +

ἰ

√2 −

1

2 =

46 | P a g e

References:

1. M.V. Berry, “Paraxial beams of

spinning light,” In M.S. Soskin

and M.V. Vasnetsov, editors,

Singular Optics,v. 3487, p. 6-

11. SPIE (1998).

2. V. V. Kotlyar, A. A. Almazov, S.

N. Khonina, V. A. Soifer, H.

Elfstrom, and J. Turunen,

”Generation of phase

singularity through diffracting

a plane or Gaussian beam by

a spiral phase plate,” J. Opt.

Soc. Am. A 22, 849-861

(2005).

3. N.R. Heckenberg, R. McDuff,

C.P. Smith, and A.G. White,

“Generation ofoptical phase

singularities by computer

generated holograms,” Opt.

Lett. 17(1992).

4. M. Padgett, J. Arlt, N. Simpson,

and L. Allen, “An experiment

to observe the intensity and

phase structure of

LaguerreGaussian laser

modes,” Am. J.Phys. 64, 77-

82 (1996)

5. M. W. Beijersbergen, L. Allen,

H. van der Veen, and J. P.

Woerdman, “Astigmatic laser

mode converters and transfer

of orbital angular momentum,”

Opt. Commun. 96, 123–132

(1993)

6. Allen, L., Courtial, J., and

Padgett, M.J. (1999) Matrix

formulation for the

propagation of light beams

with orbital and spin angular

momenta. Phys. Rev. E, 60,

7497–7503.

7. Allen, L., Courtial, J., and

Padgett, M.J. (1999) Matrix

formulation for the

propagation of light beams

with orbital and spin angular

momenta. Phys. Rev. E, 60,

7497–7503

Book references:

1. Optical Vortices: Henry Sztul

2. Gaussian beams: E. J.Galvez

3. Gaussian beam in the optics course: E. J.Galvez

4. Review paper “Optical Angular momentum: origin, behavior, applications”. Alison m. Yao, Miles j. Padgett

47 | P a g e

Chapter 5

Experiments with Laguerre- Gaussian Beam

5.1 Introduction:

Till now we are familiar with the interference pattern, diffraction geometry of

the plane wave with the fundamental Gaussian beams. But here we tried to show the

diffraction effects when the beam has a helical cross section which has azimuthal phase

dependence. That is, we have done the diffraction experiment with the LG beams. It gives

some interesting results. Here we basically focused on diffraction from the double slit and

the circular aperture. These simple experiments with LG beam lead to a better understanding

and interpretation of OAM in the interference pattern. Lastly we will discuss a simulated

experiment that how we can show the LG beam has OAM with the simulations.

5.2 Double Slit diffraction with LG beams:

We are familiar with the Young’s double slit experiment with the plane wave. Unlike

the plane wave if the beam has some phase dependence in the transverse plane then the

diffraction pattern will be more interesting. The LG beams have a phase variation. The

electric field amplitude of the beam is proportional to

𝐴𝑝,𝑙(𝑟, φ, z) ∝ [𝑟

𝜔(𝑧)]|𝑙|

𝑒𝑥𝑝 [−𝑟2

𝜔2(𝑧)] 𝐿𝑝

|𝑙| (2𝑟2

𝜔2(𝑧)) 𝑒𝑥𝑝[−𝑖𝑙𝜑] (5.1)

If the radial index p is zero. 𝐿0|𝑙|(𝑟) = 1. So the amplitude becomes proportional to

[𝑟]|𝑙|𝑒𝑥𝑝[−𝑟2]𝑒𝑥𝑝[−𝑖𝑙𝜑] (5.2)

This beam has a phase singularity and zero intensity at the center of the beam. Now if we use

a double slit and place the dark position on the opaque space then a diffraction pattern will

apprear. The experimental setup is shown in the figure 5.1. We have used He-Ne laser of

wavelength λ = 632.8nm and the mode is LG01. A double slit with opaque space a = 0.2 mm

and slit width d = 0.1 mm is used. In the young’s double slit experiment the wave front is

plane wave front. So there is no initial path difference between the two interfering waves on

the slits. The path difference comes when the waves propagate and interfere on the screen.

48 | P a g e

The standard derivation shows that the intensity will vary with the optical phase difference δ

with the following relation.

𝐼 ∝ 𝐶𝑜𝑠2(𝛿/2) (5.3)

∝ 𝐶𝑜𝑠2 (𝜋𝑑𝑥

𝜆𝐷) (5.4)

Here d is the slit width and D is the distance between the screen and the diffracting slit. We

assume that the propagation direction is z and the intensity variation is in the horizontal x

direction. The optical phase difference 𝛿 = 𝜋𝑑𝑥

𝜆𝐷 is derived from the path difference between

the interfering waves.

Figure 5.1

Experimental set up for generating the LG beam and diffraction patterns.

But, for the LG beams the situation is different. As it has azimuthal phase dependence on the

transvers plane there is an initial phase dependence of the interfering waves.[figure 5.2] So

the LG beam will show the interference due to two reasons. One is due to the optical path

length of the interfering waves and the other one is due to the initial phase difference of the

waves of the two slits as there is azimuthal phase dependence. In figure 5.2 we have shown

the phase variation of the LG01 mode when it passes through the slits. Let the first slit (right)

Grating for HG mode Mode converter

Diffraction grating CCD

49 | P a g e

has the phase variation 𝜑1(𝑦) and the 2nd slit (left) has phase variation of 𝜑2(𝑦). So the

additional phase variation along the y axis is

𝛻𝜑(𝑦) = 𝜑2(𝑦) − 𝜑1(𝑦) (5.5)

The optical path difference is same when the LG beam falls on the double slit but due to the

azimuthal phase dependence there is an optical phase difference which is given by equation

5.5. The phase variation along the y direction is shown in the figure 5.2. So in the intensity

pattern the additional phase 𝛻𝜑(𝑦) should be added in the argument of the cosine function.

So the intensity distribution on the screen is proportional to the term by the analogy with the

plane wave interference [1]

𝐼 ∝ 𝐶𝑜𝑠2 (𝜋𝑑𝑥

𝜆𝐷+ 𝛻𝜑(𝑦) ) (5.6)

Now we have to calculate 𝛻𝜑(𝑦) which is nothing but proportional to tan−1 𝑦

𝑥. Now for a

fixed x (as the slits are vertical) it will vary only with y. so intensity proportional to

𝐼 ∝ 𝐶𝑜𝑠2 (𝜋𝑑𝑥

𝜆𝐷+

1

2tan−1 𝑦 ) (5.7)

(a) (b) (c) (d)

Figure5.2

(a) Simulated phase variation

of the LG01 beam on the

plane of the diffraction

grating. (b) Phase

variation along left slit 𝜑2(𝑦),

(c) phase variation along

right slit 𝜑1(𝑦),

(d)Legends.

0

2 π

50 | P a g e

In figure 5.3 we have shown the simulated interference pattern and the experimental

diffraction pattern for LG01 beam. Both patterns look like the same. In the experimental

(a) (b) (c)

(d)

Figure 5.3

(a) Simulated double

diffraction pattern for LG01,

(b) experimentally obtained

fringes, false color CCD

image (2D view)

(c) 3D view of the

experimental fringes,

(d) Experimentally observed

intensity variation along the

x direction.

figure, there is some distortion in the fringes. This is due to the distortion on the LG beam we

have used and the distance between the screen and the diffraction grating. In our

experimental set up due to lack of distance between the screen and the diffraction grating the

1

0

I

n

t

e

n

s

i

t

y

X position

51 | P a g e

fringes are not proper. Actually we generated the beam by using the transformation of the

HG beams so it needs some distance to focus the beam using the cylindrical lens converter.

But the simulated patterns shows that the diffraction pattern is write. The intensity

distribution is clear from the 3D view [figure 5.3c]. In figure 5.3d the intensity distribution

along the horizontal (x) axis is shown (experimental) this is similar to the normal intensity

distribution. Intensity distribution along x direction is similar to the intensity distribution

with the normal plane wave front because the initial phase variation of the incident beam

varies vertically (y direction). This is clear from the fringe patterns. We have used the beam

with 𝑙 = +1, but If we use the 𝑙 = −1 the fringe will be shifted in the opposite direction.

But we didn’t have the π converter so we can’t change the helicity of the beam.

The LG beam is generally called the twisted beam but it is difficult to understand the light.

This simple double silt diffraction experiment shows the twist nature of the light. Due to the

azimuthal phase variation we got the twisting effect in the far filed. If we have the LG beams

with the opposite helicity we can demonstrate the twisting nature more clearly because if we

have 𝑙 = +1 the twist will be in left side (due to left helicity) and for 𝑙 = −1 the twist will

be in the right side (due to left helicity).

5.3 Demonstration by simulation

The OAM present in the LG beams is demonstrated by the following simulation:

The LG beam is dependent on 𝑟, 𝛷, 𝑧 it has azimuthal phase dependence.

𝐴𝑝,𝑙(𝑟, φ, z) ∝ [𝑟

𝜔(𝑧)]|𝑙|

𝑒𝑥𝑝 [−𝑟2

𝜔2(𝑧)] 𝐿𝑝

|𝑙| (2𝑟2

𝜔2(𝑧)) 𝑒𝑥𝑝[−𝑖𝑙𝜑] exp [−𝑖𝑘𝑧]

So if we move the screen along the z- direction the phase structure and the intensity

distribution will start to rotate. This can be shown by the animation by continuously

changing the z- variable. The code is given appendix.

It is easier to show the phase structure rotation than to show the intensity rotation because

intensity is completely symmetric with respect to origin {r=0}. Here some of the snap shots

are given for different “z distance”. And the distance is written on the top of the plots. The

distance is varied from 0 to 6.27.

The result of the simulation is shown in Fig. 5.4 If we carefully see, the white marked radial

line is rotated from zero to 2π.

52 | P a g e

As the white line rotates we can conclude that the Poynting vector rotates along the

direction of propagation. And it is a measure of angular momentum (the mathematical

derivation can also be shown). So the beam has angular momentum.

Here “l =1” is chosen. So the phase is rotated only 2 π for a given distance. If we

Chose arbitrary “l” it will rotate (2 π ∗ 𝑙) times for that given distance. This can be

shown by the Mathematica code given here.

We have chosen only LG beam without any polarization. If the light is circularly

polarized spin angular momentum will come into action. Then depending upon the

total momentum l +σ the structure will rotate according to that manner.

If we consider the rotation of intensity structure we have placed the beam

asymmetrically to show the variations.

Figure 5.4

Phase rotation along the propagation direction.

Reference

Optical Vortices: Henry Sztul

2π

0

53 | P a g e

Summary and Conclusions

The main objective of this project work is to study and understand the concept of

orbital angular momentum of light i.e. the twisting properties of the photons. Photons have

not only the translation motion but it can also have rotation attributes. In order to understand

these properties we have to choose a vortex beam which has such properties. Beams of

Laguerre - Gaussian (LG) cross section are examples of such beams. We mainly studied the

properties of the LG modes. Our work involved going through recent relevant literature,

simulation of light beams of appropriate beam profiles and lastly experimental verification of

some parts of the theoretical simulations.

In Chapter, 1 an overview of the angular momentum of light is provided. In order to do that,

we have discussed the spin and angular properties of the light. The spin angular momentum

arises due to the polarization state of the beam. This is an intrinsic property. It can take the

values between -1 to +1. The angular momentum for the spin is 𝜎ħ per photon. The orbital

angular momentum arises due to the spatial inhomogeneity of the medium. The orbital

angular momentum can be 𝑙ħ per photon where 𝑙 can be any integer value between −∞ to

+∞.

In Chapter 2, we mainly focused on the energy flow in the vortex beam. The radial

dependence on the transverse plane of spin and orbital flow density is discussed. Some

energy flow diagrams are simulated to show the flow direction and how the magnitude of

energy varies if the beam has only spin property and also if it has both the spin and angular

momentum properties.

In Chapter 3, the higher order Gaussian beam solution are described. The fundamental

Gaussian mode, Hermite Gaussian mode and Laguerre Gaussian modes are studied in detail.

The paraxial Helmholtz equation can have different modes if we solve it in different co-

ordinate systems. For examples, if we solve it in rectangular Cartesian coordinate the

solution will be Hermite-Gaussian modes. If it is solved in cylindrical polar coordinate the

Laguerre-Gaussian mode will be the solution. All the modes are complete and orthogonal.

In Chapter 4, generation of the vortex beam is discussed. Some of the generation processes

were elaborated in this context. The use of spiral phase plate, the fork diffraction grating and

the mode converters in experimentally generating vortex beams is described. We have

generated the 𝐿𝐺01 beam using the cylindrical mode converter. Also we generated the

Hermite Gaussian modes. The theoretical transformation theory was also elaborated and it is

also experimentally verified. The mode converters are mathematically equivalent to the

retardation plates (quarter and half wave plate).

54 | P a g e

In Chapter 5, some experiments are presented. The classic Young’s double slit experiment is

performed with an LG beams and the experimental result was also verified theoretically and

by the simulation. Also a demonstration was designed by the simulation to show that the

vortex beam has the angular momentum of light and energy flows helically the Poynting

vector propagates.

The twisting property of light is an interesting topic in the current research areas. This

is a quite new field and this was experimentally observed in the 1990’s. Many of the

properties of the twisted photons are still unknown. The aim of this project is to understand

the properties of the twisted photons. There are many applications of this vortex beam. In

optical communication, bio imaging, optical trapping, driving the micro machines the vortex

beam is used. In fiber optics communication the beam is used. The great advantage of this

beam is that it can carry the orbital angular momentum of any integer values. Recently a

beam was generated which had a topological charge ( 𝑙) of 256. Such beams enable the

storage and coding of a large amount of data. Recent studies also demonstrated that the OAM

beam can transfer a data of 2.5 terabits per second6.

Here in this project some basics properties of light beams with angular momentum are

discussed. Some theoretical simulations and some simple experiments are done. The main

conclusion of the work is that vortex beams can indeed be simulated using simple software

and implemented in the laboratory with simple optical set ups. A variety of beams with

optical angular momentum can be generated by an appropriate configuration of usual optical

elements which help to envisage the practical implementation of the upcoming applications.

Further, performing basic experiments in Optics such as interference leads to very interesting

results.

6 http://en.wikipedia.org/wiki/Orbital_angular_momentum_of_light

55 | P a g e

Appendix

“Mathematica” code for phase plot

LGp[p_, l_] = 𝑟^Abs[𝑙]Exp[𝐼𝑙𝑡]Exp[𝐼𝑘]LaguerreL[𝑝, 𝑙, 𝑟^2]

LGc[p,l−] ≔ TransformedField[Cylindrical → Cartesian, LGp[𝑝, 𝑙], {𝑟, 𝑡, 𝑘} →

{𝑥, 𝑦, 𝑧}]

Manipulate [DensityPlot [Arg [𝑒ⅈ𝑧+ⅈ𝑙ArcTan[𝑥,𝑦](𝑥2 + 𝑦2)Abs[𝑙]

2 LaguerreL[𝑝, 𝑙, 𝑥2

+ 𝑦2]] , {𝑥, −4,4}, {𝑦, −4,4},Mesh → None, PlotPoints → 150, FrameLabel

→ {X,Y}, PlotLabel

→ Style[{Phase Diagram,P=, 𝑝,L=, 𝑙}, Bold, 20], ColorFunction

→ Hue, PlotLegends

→ BarLegend[{Hue, {0,2Pi}}]] , {{𝑝, 0,p}, Table[𝑖, {𝑖, 0,5,1}], ControlType

→ Setter, ControlPlacement

→ Top}, {{𝑙, 1,l}, Table[𝑖, {𝑖, 0,5,1}], ControlType

→ Setter, ControlPlacement

→ Top}, {{𝑧, 0,Distance of the screen}, 0,10,.01, ImageSize

→ Tiny, Appearance → Labeled}, TrackedSymbols

→ {𝑝, 𝑙, 𝑧}, ControlPlacement → Left, SaveDefinitions → True]