COMSOL 6.3 - Gaussian Beam Propagation Through an Optical Prism

Gaussian Beam Propagation Through an Optical Prism

Optical prisms are transparent optical elements with polished surfaces that are designed to disperse or refract light. As light hits a prism boundary at an oblique angle of incidence, the ray path is deviated from its original direction after passing through the prism boundaries. The amount of deviation depends on the refractive indices of the prism and the surrounding medium. For instance, when light propagates from a low index medium to a prism with high refractive index, as it passes the interface between the two media, the transmitted light moves toward the normal of the interface.

Triangular prisms are the most familiar type of optical prism. Their geometry is simple, consisting of triangular base and rectangular sides. The most widely used prism materials include glass, fluorite, and acrylic. To make optical prisms transparent to light and minimize reflection, anti–reflection coatings are applied on the prism boundaries. Optical prisms are used in a wide variety of practical applications such as spectroscopy design, binoculars, cameras, fiber optics communication, periscopes, microscopes, among many others.

Figure 1: Schematic of a right–angled triangular prism indicating the refraction of light.

Model Definition

This model demonstrates the refraction of light through an optical prism. Figure 1 shows the schematic. A right–angled triangular optical prism with acute angles αp and 90° − αp is surrounded by air. The prism boundaries are coated with anti-reflection coatings. A light ray with wavevector ki is incident at an angle αi. As light propagates from a low refractive index material (air) to a high index material (prism), the transmitted light moves closer to the boundary normal where angle of transmission is αtp. This ray with wavevector ktp hits the second prism boundary at an angle αip and exits in the air medium at an angle αt.

The anti-reflection (AR) coatings at the prism boundaries are defined using a Transition Boundary Condition that models a thin layer with refractive index nAR and thickness dAR. These two parameters are calculated from the propagation directions of the light and the refractive indices of the surrounding media. Note that for oblique angles of incidence, the refractive index and layer thickness of the AR coatings are different for s- and p-polarized lights.

Assuming lossless media, as shown in Figure 1, the refractive index and thickness of the AR coating layer, lying between the incident (air) medium and the prism, for s- and p-polarized incident lights can be derived from Ref. 1 as

where np is the refractive index of the prism material. Now, the refractive index and thickness of the AR coating lying between the prism and exiting media are expressed as

This model simulates the refraction of s- and p-polarized Gaussian beams passing through a right-angle triangular optical prism in a 2D geometry using the Electromagnetic Waves, Frequency Domain interface. A Matched Boundary Condition feature is used to launch the Gaussian beam and to properly absorb the reflected and refracted waves with known wave directions. A Transition Boundary Condition feature is used to model the anti–reflection coatings applied to the prism boundaries using the above equations. Wavelength Domain study steps are used to solve for the domain fields. Reflectance and transmittance are also calculated using an integration operator.

Results and Discussion

Figure 2 show the nonzero electric field component of an s-polarized Gaussian beam. The anti-reflection coatings at the prism boundaries almost eliminate all reflection at the prism boundaries.

Figure 2: Electric field component of an s-polarized Gaussian beam propagating from left to right through a right–angled triangular optical prism.

Figure 3 shows the dominant electric field component of a p-polarized Gaussian beam.

Figure 3: Dominant electric field component of a p-polarized Gaussian beam propagating from left to right through a right angle triangular optical prism.

Table 1 shows the calculated reflectance and transmittance for the s- and p-polarized Gaussian beams.

Table 1: Reflectance and transmittance.
Reflectance, s-polarized wave	Transmittance, s-polarized wave	Reflectance, p-polarized wave	Transmittance, p-polarized wave
0.0105	0.9856	0.0089	0.9893

Reference

1. M. Born and E. Worlf, Principle of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Elsevier, 2013.

Wave_Optics_Module/Optical_Scattering/gaussian_beam_propagation_optical_prism

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

Geometry 1

The geometry is very simple, consisting of a rectangle and a polygon.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Wave and Geometric Parameters

In the window, under click .

In the window for , type Wave and Geometric Parameters in the text field.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file gaussian_beam_propagation_optical_prism_parameters_geom.txt.

Anti-Reflection Coatings Parameters

In the toolbar, click

and choose > .

In the window for , type Anti-Reflection Coatings Parameters in the text field.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file gaussian_beam_propagation_optical_prism_parameters_ar.txt.

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type W.

In the text field, type H.

Locate the section. From the list, choose .

Click

Prism

In the toolbar, click

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

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


x (µm)	y (µm)
-H*tan(anglePrism)+dOffset	-H/2
5[um]	H/2
5[um]	-H/2

Click

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Right-click and choose .

In the toolbar, click

to close the window.

Materials

Prism

In the window, under right-click and choose .

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

Select Domain 2 only.

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	np	1	Refractive index

Click the

button in the toolbar.

Electromagnetic Waves, Frequency Domain (ewfd)

First, simulate the model for an s-polarized Gaussian beam.

In the window, under click .

In the window for , locate the section.

From the list, choose , as the beam possesses out-of-plane electric field component only.

Matched Boundary Condition 1

In the toolbar, click

and choose .

Select Boundaries 1 and 2 only.

In the window for , locate the section.

Click

In the dialog, type Input Boundaries in the text field.

Click .

Later, this selection will be used to define the integration operator to calculate the input power of the Gaussian beam.

In the window for , locate the section.

From the list, choose .

In the w0 text field, type w0.

In the p0 text field, type W/2.

Now, define the electric field vector of the s-polarized beam propagating along the x direction.

Specify the Eg0 vector as


1	z

Specify the ki,dir vector as


1	x
0	y

Reference Point 1

The reference point and the incident wave direction together define the optical axis of the Gaussian beam.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Specify the r0 vector as


-W/2	x

Matched Boundary Condition 2

Add a matched boundary condition to absorb the transmitted wave from the prism.

In the toolbar, click

and choose .

Select Boundaries 7 and 9 only.

In the window for , locate the section.

Click

In the dialog, type Output Boundaries in the text field.

Click .

Later, this selection will be used to define the integration operator to calculate the transmitted power of the Gaussian beam.

In the window for , locate the section.

Find the subsection. Specify the ks,dir vector as


kxScaDir	x
kyScaDir	y

Matched Boundary Condition 3

Add a matched boundary condition to absorb the reflected wave from the first prism boundary.

In the toolbar, click

and choose .

Select Boundary 3 only.

In the window for , locate the section.

Click

In the dialog, type Reflection Boundaries in the text field.

Click .

Later, this selection will be used to define the integration operator to calculate the reflected power of the beam from the prism.

In the window for , locate the section.

Find the subsection. Specify the ks,dir vector as


-cos(2*alphaInc)	x
sin(2*alphaInc)	y

Matched Boundary Condition 4

Add another matched boundary condition to the truncated prism boundary to absorb the transmitted wave inside the prism.

In the toolbar, click

and choose .

Select Boundary 4 only.

In the window for , locate the section.

Find the subsection. Specify the ks,dir vector as


cos(alphaIncPrism)	x
-sin(alphaIncPrism)	y

Transition Boundary Condition 1

Add a transition boundary condition to model the anti-reflection coating on the incident prism boundary.

In the toolbar, click

and choose .

Select Boundary 5 only.

In the window for , locate the section.

From the n list, choose . In the associated text field, type n1_s.

From the k list, choose , and keep the default value 0.

In the d text field, type d1_s.

Transition Boundary Condition 2

Add another transition boundary condition to model the anti-reflection coating on the transmission-side prism boundary.

In the toolbar, click

and choose .

Select Boundary 6 only.

In the window for , locate the section.

From the n list, choose . In the associated text field, type n2_s.

From the k list, choose , and keep the default value 0.

In the d text field, type d2_s.

In the window, right-click and choose , to simulate for the p-polarized Gaussian beam.

Electromagnetic Waves, Frequency Domain 2 (ewfd2)

In the window, right-click and choose .

In the dialog, click .

Now, set up the physics interface settings to simulate for the p-polarized Gaussian beam.

In the window for , locate the section.

From the list, choose , as the beam possesses in-plane electric field components only.

Matched Boundary Condition 1

In the window, expand the node, then click .

In the window for , locate the section.

Specify the Eg0 vector as


1	y
0	z

Transition Boundary Condition 1

In the window, click .

In the window for , locate the section.

In the n text field, type n1_p.

In the d text field, type d1_p.

Transition Boundary Condition 2

In the window, click .

In the window for , locate the section.

In the n text field, type n2_p.

In the d text field, type d2_p.

Definitions

Define the operators and variables to calculate the reflectance and transmittance.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Integration 2 (intop2)

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Integration 3 (intop3)

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Reflectance and Transmittance Calculation

In the window, right-click and choose .

In the window for , type Reflectance and Transmittance Calculation in the text field.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file gaussian_beam_propagation_optical_prism_variables.txt.

Study 1

Wavelength Domain (s-polarization)

In the window, under click .

In the window for , type Wavelength Domain (s-polarization) in the text field.

Locate the section. In the column of the table, under , clear the checkbox for .

Locate the section. In the text field, type lda0.

Wavelength Domain (p-polarization)

Right-click > and choose .

In the window for , type Wavelength Domain (p-polarization) in the text field.

Locate the section. In the column of the table, under , clear the checkbox for .

In the column of the table, under , select the checkbox for .

In the toolbar, click

Results

Global Evaluation 1

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
R_s	1	Reflectance, s-polarized wave
T_s	1	Transmittance, s-polarized wave
R_p	1	Reflectance, p-polarized wave
T_p	1	Transmittance, p-polarized wave