Wave Plates

Introduction

FRED has the capability to simulate polarization of rays through an optical system. Light sources can be randomly, circularly, or linearly polarized. Optical components that filter or manipulate polarization, such as birefringent wave plates and polarizers, can be accurately modeled. Some simple examples of polarization modeling in FRED can include absorptive dichroic and wiregrid polarizers, a calcite half-wave plate, and the Maltese cross phenomenon. Each of these features can be applied to more complex optical systems such as Liquid Crystal Displays (LCDs), interferometers, and polarization microscopes.

Wave plates are made from materials that have different real refractive index values for ordinary and extraordinary rays. Oriented properly, the wave plate can shift one polarization component of light with respect to another, transforming its polarization state. Quarter-wave plates change linear to circular polarization and visa-versa. Half-wave plates change x-polarized light into y-polarized light or RHC into LHC polarized light.

Starting with the FRED system in the X-Polarizer example, a wave plate element is added after the x-polarizer (Figure 1). There are two methods to model a wave plate. The simplest method is to assign a ½ wave plate coating a surface. Under the Coatings category of the FRED document, the user can right-click and select Create a New Coating…. Under the drop-down menu, “Polarizer/Waveplate Coating (Jones matrix)” can be selected. For this example, the “1/2 wave +45 Fast Axis” is chosen for the coating type. This ensures that the wave plate crystal axis is rotated at 45 degrees with respect to x-polarized incident light.

Figure 1. Randomly polarized light is filtered through an x-polarizer. The remaining light passes through a +45° ½ wave plate (yellow), which transforms x-polarized light into y-polarized light.

A more accurate approach to model a wave plate is assign a custom birefringent material to a rod element. Under the Material category of the FRED document, the user can right-click and select Create a New Material…. Under the drop-down menu, “Sampled Birefringent and/or Optically Active Material” can be selected. For this example the crystal axis is orientated to +45° (0.707,0.707,0), and the following material properties are defined (based on a calcite crystal): Wavelength=0.59 µm, no=1.658, ne=1.486, ko=0, ke=0.

To function as a ½ wave plate, the length of the rod must be chosen such that the ordinary and extraordinary polarization components are displaced by a net value of ½λ:

Where L=rod length, λ is light wavelength in system units, K=an integer, and no and ne are ordinary and extraordinary components of the birefringent index of refraction. A raytrace through this volume birefringent material will split each ray into ordinary and extraordinary components. As a result, the Polarization Spot Diagram will display each separate component (Figure 2).

Figure 2. Polarization Spot Diagram after x-polarized light passes through a calcite ½ wave plate. Ordinary and extraordinary components of the polarization are plotted as separate rays.

To ensure that light is indeed y-polarized, the Coherent Vector Wave Field is displayed at the detector surface. From the right-click menu the “Show X Component of Field” can be selected. Then, right-clicking again “Show Statistics” can be chosen to observe the integral of energy in the x-polarization component. Comparing the X component with the Y component confirms that nearly all incident energy is in the y-polarization component.

The thickness of the wave plate determines the fraction of x- and y-polarized light that reaches the detector. To illustrate this, the rod wave plate is replaced with a 3° wedge of calcite. X- and y-components of the coherent field are shown in Figure 3.

Figure 3. X- and y-components of the coherent vector field upon detector after x-polarized light passes through a wedge calcite crystal with +45° optical axis. The waveplate thickness varies along the y-dimension, and thus acts as a ½ waveplate at periodic locations along the wedge.