Orbital Angular Momentum Beam

It is well known that light has energy. The photon has an energy of hν, where h is the Planck constant and ν is the frequency. It is also known that the photon, even though it is a massless particle, possesses a linear momentum. That is, it exerts a force in its propagation direction. This knowledge has led to ideas of spacecraft propulsion based on the radiation pressure from sunlight hitting large mirrors — so called solar sails.

Light also possesses angular momentum in two different forms — spin angular momentum and orbital angular momentum (OAM). Spin angular momentum is associated with the light field’s polarization. For circular polarization, the electric and magnetic field vectors rotate as the wave propagates. Orbital angular momentum, on the other hand, is related to the wave’s spatial field distribution and in particular the spiraling (helical) phase distribution. The realization that light also possesses OAM arrived not too long ago. The first paper was published as late as 1992 (see Ref. 1). An introduction to the field of OAM waves and the development since 1992 is found in Ref. 2.

This phase distribution produces a Gaussian donut beam (c.f. Figure 1).

Figure 1: The electric field norm, showing the ring structure of the beam.

As shown in Figure 2, the phase rotates around the optical axis as the beam propagates.

Figure 2: Phase distribution at five different locations along the direction of propagation. As shown, the phase twists as the beam propagates.

The resulting beam, also called a vortex beam or a helical beam, has a spiraling phase distribution as shown in Figure 3 below.

Figure 3: The spiraling phase variation of the vortex (OAM) beam.

One way of creating a beam with OAM is to send a Gaussian beam through a phase plate with a spiraling thickness. Thereby, the transmitted beam will get a spiraling phase front, resulting in a vortex beam.

Since vortex beams having different twists, don’t interfere with each other, the use of these waves for communication has become a very active research topic. For more information, see Ref. 3.

Model Definition

This model simulates a Laguerre–Gaussian beam with the Electromagnetic Waves, Beam Envelopes interface, using the unidirectional wave formulation. The input beam is a focused Gaussian beam with a spiral phase distribution.

For a wave propagating in the x-direction and with a transverse radial coordinate r, the Laguerre–Gauss field distribution is given by

(1)

where the spot radius varies as the wave propagates as

(2)

and the wave front curvature evolves as

Here x0 = πn0w02/λ is the Rayleigh range, k is the wave number,

is a generalized Laguerre polynomial with radial mode number m and angular mode number n. The phase function is now

(3)

This model simulates a beam with radial mode number 0 and angular mode number 1. This beam will have one radial node, located on the optical axis. The generalized Laguerre polynomial for this mode is

(4)

Comparing Equation 1with the field distribution for the lowest order Gaussian beam (for example see the Application Library model Self-Focusing), makes it possible to identify that

(5)

Thus, in the model, the expression

(6)

will be used as the Gaussian beam input amplitude.

References

1. L. Allen and others, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Physical Review A, vol.–45, no.–11, pp.–8185–8189, 1992.

2. S.M. Barnett, M. Babiker, and M.J. Padgett, “Optical orbital angular momentum,” Philosophical Transactions of the Royal Society A, vol. 375, https://doi.org/10.1098/rsta.2015.0444, 2017.

3. A.E. Willner, “Twisted light could dramatically boost data rates,” IEEE Spectrum, vol. 53, no. 8, pp. 34–39, 2016.

Wave_Optics_Module/Beam_Propagation/orbital_angular_momentum

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
lda0	1[um]	1E-6 m	Wavelength
w0	10*lda0	1E-5 m	Spot radius
zR	pi*w0^2/lda0	3.1416E-4 m	Rayleigh range
m	1	1	Azimuthal mode number
E0	1[V/m]	1 V/m	Electric field amplitude

Geometry 1

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 3*w0.

In the text field, type 3*lda0.

Locate the section. From the list, choose .

Click

Definitions

This view is very compressed in the propagation direction. The following instructions change the Camera’s View scale to make the domain expand in the x direction.

In the window, expand the node.

Camera

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

In the text field, type 50.

Click

Click the

button in the toolbar.

Materials

Air

In the window, under right-click and choose .

In the window for , type Air in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	1	1	Refractive index
Refractive index, imaginary part	ki_iso ; kiii = ki_iso, kiij = 0	0	1	Refractive index

Definitions

Now, add a function and some variables that will be used for defining the input beam.

Spot Radius

In the toolbar, click

and choose .

In the window for , type Spot Radius in the text field.

In the text field, type w.

Locate the section. In the text field, type w0*sqrt(1+(x/zR)^2).

Locate the section. In the text field, type m.

In the text field, type m.

Beam Variables

In the window, right-click and choose .

In the window for , type Beam Variables in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
rho	sqrt(y^2+z^2)	m	Transverse radial coordinate
phi	atan2(y,z)	rad	Transverse azimutal coordinate
EG0	E0(rhosqrt(2)/w(x))^mexp(jmphi)exp(jmatan(x/zR))	V/m	Gaussian beam amplitude

Plot Variables

The following instructions add two variables that later will be used for visualizing the field’s spiraling phase.

Right-click and choose .

In the window for , type Plot Variables in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
argEy	arg(ewbe.Ey)	rad	Complex argument of the y-component of the electric field
argWindow	if(argEy<-pi/2,-2pi,if(argEy>pi/2,2pi,argEy))		Complex argument in the range [-pi/2,pi/2]