# How to model a beam splitter in Sequential Mode

This article explains how to create a beam splitter cube in Sequential Mode. One of the biggest challenges for modeling such a system is that multiple ray paths cannot be simultaneously traced in Sequential Mode. Thus, multiple configurations are needed to trace rays along both the transmitted and reflected paths within the beam splitter. This article also discusses how to calculate the total power in both transmitted and reflected beams, accounting for polarization effects and thin-film coatings.

Authored By Nam-Hyong Kim

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## Introduction

Beam splitters can be modeled either in Sequential Mode or Non-Sequential Mode in OpticStudio.

In Non-Sequential Mode, rays can split into transmitted and reflected rays at an object interface. The ability to trace multiple transmitted and reflected ray paths simultaneously, along with the ability to easily define off-axis geometries without the use of Coordinate Breaks, are two of the core benefits of Non-Sequential Mode.

In Sequential Mode, multiple ray paths cannot be traced simultaneously. In order to trace both a transmitted and reflected ray path from a surface, each path must be modeled in a separate configuration. In this article, we will explore how to model such an interface using a practical 50/50 beam splitter design example.

## Defining the system and splitter surface

To demonstrate how to model Sequential Mode systems that require the tracing of multiple transmitted and reflected ray paths, we will construct the following polarization-independent 50/50 beam splitter cube.

The cube is made out of MgF2 coated N-BK7 glass. The 50/50 coating is ideal, being independent of polarization, incident angle and wavelength. The reflected rays, shown in green, reflect from the bottom mirror before reaching the top image surface. We will calculate the correct intensities at both image surfaces, accounting for N-BK7 bulk absorption, Fresnel losses from thin-film coated surfaces, and 50/50 splitting from ideal coatings.

Before getting started with the example, you should know how to specify system and surface properties in OpticStudio. If you do not, refer to the articles, "How to design a singlet lens" and "How to tilt and decenter a sequential optical component," before continuing.

To begin the design, Make the following adjustments in the System Explorer.

• Define the system aperture under Aperture, set Aperture Type: Entrance Pupil Diameter and Aperture Value: 15.0.
• Specify a single, on-axis field point by setting Fields...Field 1...X: 0 and Fields...Field 1...Y: 0.
• Specify a single wavelength at Wavelengths...Wavelength 1... Wavelength (µm): 0.550.
• Under Units set Lens Units: Millimeters.

Then enter surfaces in the Lens Data Editor as shown below.

In the Lens Data Editor toolbar, use Tilt/Decenter Elements to apply Tilt X: -45 to Surface 3.

Open the 3D Layout. In the Settings, specify Number of Rays: 5 and Ray Pattern: Y Fan to draw only rays along the Y-axis.

## Setting a rectangular aperture

By default, the system aperture in Sequential Mode is circular and all surfaces have circular apertures. To make the beam splitter cube, we must apply rectangular apertures its surfaces. Here, we will place 10 mm x 10 mm rectangular apertures on Surfaces 2 and 6. To apply the correct aperture on the tilted splitter surface (Surface 4), we must apply a rectangular aperture of 10 mm x sqrt (2*10). Apply these values to the appropriate surfaces via Surface Properties...Aperture...Aperture Type, ...X-Half Width, and ...Y-Half Width

Update the 3D Layout.

To remove the vignetted marginal rays from the layout, check the Delete Vignetted box in the 3D Layout settings. Update the 3D Layout again.

## Applying the splitter coating

We will now place coatings on both the outer and inner surfaces of our prism. On the exterior surfaces (Surfaces 2 and 6), set the Coating parameter to AR; similarly, for the interior splitter surface, set the Coating to I.50. The AR coating represents a quarter-wave thick MgF2 anti-reflection coating, and the I.50 coating represents an ideal 50/50 transmission/reflection coating. These increase transmission at the front and rear of the prism, and evenly divide the beam at the interior surface, respectively.

## Analyzing the transmitted intensity

We now have the straight (refracted) path of the beam splitter modeled. You can specify any amount of transmission by defining additional ideal coatings in the coating file. You can also create a non-ideal coating either by specifying coating layer thicknesses and material type or the transmission properties of the coating as a function of wavelength and incident angle. For more detailed information about how to define coatings in OpticStudio, please refer to the section of the Help Files entitled “Defining Coatings.".

The effect of thin-film coatings can only be accounted for when considering the polarization effects in the calculation or analysis, even if the coating is ideal. The total transmission at the image plane can be evaluated by any polarization-enabled analyses/calculations in OpticStudio.  We will use the Polarization Ray Trace to calculate the total chief-ray transmission at the image plane.

Open the Polarization Ray Trace (Analysis...Polarization...Polarization Ray Trace) and specify the following settings.

The total transmission is reported at the bottom of the window.

The Polarization Ray Trace is accounting for all loss mechanisms: AR-coated N-BK7 surfaces, 50/50 splitting and N-BK7 bulk absorption at the whatever wavelength the ray is traced at, and at whatever angle it makes to the surfaces.

## Modeling the reflected beam path with multiple configurations

We will now model the reflected path using multiple configurations. Open the Multi-Configuration Editor from Editors...Multi-Configuration and insert a configuration using Insert Configuration (or by pressing <Ctrl+Shift+Ins> on the keyboard).

Insert a PRAM operand in the Multi-Configuration Editor and specify the Tilt About X (Parameter 3) for Coordinate Break Surface 5.

Place a Pickup Solve on the second configuration with a Scale Factor of -1.

Because Configuration 2 will model the reflected path in the beam splitter, we need to change Surface 4's Material from N-BK7 to MIRROR. Insert to the Multi-configuration Editor a GLSS operand for Surface 4 and specify a value of MIRROR for Configuration 2. Switch the Lens Data Editor to Configuration 2 by pressing <Ctr+A> on the keyboard. The Lens Data Editor should display “Config 2/2” in the title bar.

Enter the following settings for the 3D Layout to display all configurations.

Notice how the reflected rays (in green) propagate in the wrong direction (upward). This is due to improper thickness sign convention after the mirror in Configuration 2, causing "virtual" ray propagation. Thicknesses corresponding to real propagation always change sign after a mirror. After an even number of mirrors (including zero mirrors), thicknesses are positive for real propagations and negative for virtual propagations. After an odd number of mirrors, thicknesses are negative for real propagations and positive for virtual propagations. This sign convention is independent of the number of mirrors, or the presence of Coordinate Breaks. This fundamental convention cannot be circumvented through the use of coordinate rotations of 180 degrees. Therefore, we need to change the thicknesses of Surface 5 and Surface 6 to -20 mm in Configuration 2. To do this, insert a THIC operand for Surfaces 5 and 6 and place Pickup Solves with Scale Factors of -1 for the second configuration.

For more information on virtual propagation, refer to the Help Files entry entitled “Virtual propagation,” located in the “Conventions and Definitions” section.

After updating the 3D Layout, notice that rays now propagate as expected.

## Defining the second pass beam path

Now, we will model the second-pass path, where the second beam path is retro-reflected through the beam splitter. In the second pass through the beam splitter, the rays must encounter the same geometry as on the first pass. Because rays must travel sequentially from one surface to the next, this means we must re-define the beam splitter cube for this second pass, so that the rays can interact with it again. Insert a new Surface 7 and place a Pickup Solve with a Scale Factor of -1 on its Thickness from the Surface 6 Thickness. Apply a Material of MIRROR to this surface.

Next, we need to redefine the splitter cube for the beam’s second pass through it. Insert 3 surfaces after Surface 7 in the Lens Data Editor. The Material for the first two will be N-BK7, and there will be no Material for the third. Using Tilt/Decenter Elements, apply a Tilt X of 45 degrees to the second N-BK7 surface (Surface 9). Note that the material type for the diagonal is N-BK7 and not Mirror, since after reflecting from the bottom mirror (surface 7) we need to trace the transmitted/refracted rays that reaches the top image surface.

Set the 3D Layout to display the current configuration (Configuration 2) and update the 3D Layout.

The layout will look strange for now, but that is simply because we have not yet applied surface apertures to the second pass surfaces. We will do this in the next section.

## Cleaning up the final design

To set rectangular apertures on Surfaces 8 and 12, navigate to Surface Properties...Aperture and select Pickup From: Surface 2; for Surface 10, select Pickup From: Surface 4.

Set the Surface 8, 10, and 12 Coating parameters to AR, I.50, and AR, respectively.

After updating the 3D Layout, we see that we've properly defined the second pass for the beam splitter!

However, although we have the correct setup for Configuration 2, we do not for Configuration 1, because Surfaces 7-12 are also present in that configuration. To see this, display all configurations in the 3D Layout.

To correct this, we can insert one IGNM operand into the Multi-Configuration Editor. This operand allows us to ignore a range of surfaces in a given configuration. Insert one IGNM operand with First Surface: 7 and Last Surface: 12. Notice that the 3D Layout now looks great!

## Analyzing the beam path intensities

In order to analyze the beam intensities at the two output image planes, we can open a Polarization Ray Trace for each configuration. To do this, press Clone in the first Polarization Ray Trace window toolbar.

Then choose a different configuration for each window.

You can now see the total transmission for both configurations. As expected, Configuration 2 has just under half the transmitted intensity of Configuration 1, because its beam must pass through the prism twice.

You can also modify the system to be able to pivot the cube about its center. The modified file is included as "Rotating BS.zmx."

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