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This article describes the several ways OpticStudio models polarization-based optical phenomena. The purpose of this article is to examine the strengths and proper applications of these features when modeling polarization-based optics. The features discussed include the Polarization Pupil Map, Jones Matrix, Birefringence, surface coatings and more. These are important for practical applications such as waveplates and isolators.
Authored By Kyle Hawkins and Nicholas Herringer
Introduction
Polarization effects are utilized in a myriad of optical systems over a wide array of fields. OpticStudio provides the means to model any of these applications by allowing users to specify an input polarization for the light coming into the system, alongside settings in Sequential mode that allow surfaces to interact with it. The three ways to model polarization-dependent media are through Jones Matrix surfaces, Surface Coatings, and Birefringent In/Out surfaces.
It is important to know the purpose of the polarizer you are trying to model in order to select the appropriate surface type. This article will describe setting up a polarized system as well as the pros and cons of each of the above surfaces/surface settings.
Generating a polarized light source
All of the polarization analyses in OpticStudio require an input polarization state, usually given as a Jones vector (Jx, Jy) with optional starting phases in X and Y. Input polarization states are entered into OpticStudio in two ways. The first is by explicitly defining the initial polarization state (Jx, Jy, and X/Y Phase) in individual analysis settings, such as with the Polarization Ray Trace and Polarization Pupil Map.
The second method is to apply the polarization state in the ‘Polarization’ section of the System Explorer. The state entered here will be considered in the calculations for any applicable analysis windows by checking the “Use Polarization” setting (or similar) in the individual analysis settings.
The “Method” (or reference state) defined in the System Explorer is required to convert the 2D Jones Matrix inputs (Jx, Jy) into 3D electric field components (Ex, Ey, Ez). While it is tempting to think of Jx as corresponding to the amount of energy in the S-polarization state and Jy as corresponding to the amount of energy in the P-polarization state, such definitions are ambiguous without a plane of incidence (see this forum post for more information). That is, a ray launched from a source and travelling in free space along ray vector K needs a reference to accurately define the polarization state vectors, S and P. In OpticStudio, the following references are available:
X Axis Reference: The P-vector is determined from K cross X, and S = P cross K (Default);
Y Axis Reference: The S-vector is determined from Y cross K, and P = K cross S;
Z Axis Reference: The S-vector is determined from K cross Z, and P = K cross S.
Note: S, P and K (direction of propagation) are always orthogonal. These are represented by the red, blue, and green vectors, respectively, in the table below.
This approach allows users more versatility when generating input polarizations. Default surfaces have a surface normal in the z-direction, so the classic S and P definitions would limit the user to generating the Z-Reference input polarizations.
Polarization Pupil Map
The Polarization Pupil Map is arguably the most useful tool in OpticStudio to quickly view the polarization state of an optical beam at a given surface. That said, it does have some nuances which must be fully understood to accurately interpret its results.
In general, OpticStudio performs time-independent calculations (i.e. it presents a snapshot in time of a steady-state system). However, this is not the case with the Polarization Pupil Map, which plots the endpoint of the electric field vector (Ex, Ey) on a cartesian graph as time moves through one period at the defined surface. This is because it is the variation of phase with time that determines the orientation of the polarization ellipse. For almost all systems, it is not important whether OpticStudio looks forward in time or backward in time for this calculation, because it assumes that the system is at steady state. By default, the calculation looks forward in time (i.e. at the phase of the ray that “will hit” the specified surface if the system were to advance forward in time by some small amount).
Additionally, users should note that when defining the initial phase shift between Jx and Jy, a positive value for the “X-Phase“ or “Y-Phase” will result in Jx lagging spatially with respect to Jy, and vice versa. For example, defining X-Phase = 90 degrees and Y-Phase = 0 degrees will cause the Ex-component of the electric field to lag 90 degrees behind the Ey component (see below).
Assuming that (Jx, Jy) = (0.707,0.707), the resulting polarization ellipse will be clockwise, circularly polarized, as seen by the Polarization Pupil Map.
Types of polarization-dependent media
In OpticStudio there are several ways to manipulate the polarization state of a given beam. Some of these methods involve the introduction of polarization-dependent surfaces and materials. Here we will present three such methods and describe their general applications within OpticStudio.
Jones Matrix
The Jones matrix surface is an idealized construct designed for light at normal incidence. It is a two-by-two matrix that modifies a Jones vector (which describes the electric field) according to
$$
\begin{bmatrix}E_x \\E_y \\\end{bmatrix} = \begin{bmatrix}A & B \\ C & D\\\end{bmatrix} \begin{bmatrix}E_x \\E_y \\\end{bmatrix}
$$
where A, B, C, D, Ex, and Ey are all complex numbers (see "How to use the Jones Matrix surface" for more information). The matrix can be used to explain 3-D phenomena with a 2-D vector because light propagation is assumed to be along the Z-axis. Thus, the electric field lives in the X-Y plane. If rays are indeed aligned with the Z-axis of the system, this surface can provide an ideal simulation for relative phase change between S and P states, as well as transmission of S and P states.
OpticStudio will allow users to apply the Jones Matrix surface to oblique angles of incidence, but the results in this configuration will represent, by definition, an approximation. The calculation does not consider the effect of the matrix for electric field components along z, the splitting of beams into ordinary and extraordinary components (if modeling birefringent materials), or Fresnel coefficients.
Jones matrices that describe retarders should not be used with oblique incident angles. To accurately calculate off-axis relative phase change, the Birefringent In and Birefringent Out surfaces should be used.
Jones matrices that describe polarizers can offer a decent approximation off-axis. The surface will allow electric fields to transmit in the z-direction and behave as they would on axis for the X- and Y-electric field components. The resulting component of the electric field parallel to K is then subtracted so that the electric field remains perpendicular to K. To create a polarizing surface that acts on Ez components, coatings should be used.
Surface coatings
OpticStudio allows users to define both real and idealized thin film optical coatings and apply those coatings to any surface in an optical design. OpticStudio also contains a large catalog of predefined coatings which includes a wide range of commonly used coatings. More information on defining and applying coatings can be found at "How to define metal materials in OpticStudio." Although coatings can be used for a variety of applications, this discussion will focus on how they affect the polarization state of light rays.
When discussing coatings in this context, one must consider the fact that the amplitude and polarization state of the electric field is described by a vector:
$$\vec{E} = \begin{bmatrix}E_x \\E_y \\Ez\\\end{bmatrix}$$
where Ex, Ey, Ez, are complex valued. The electric field vector, \(\vec{E}\), must be orthogonal to the propagation of the ray vector. At a boundary between two media, the transmittance, reflectance, and phase of the electric field is different for the S- and P-components of the field. The S-component of the field is the projection of \(\vec{E}\) that lies along the axis orthogonal to the plane of incidence, while the P-component lies within the plane of incidence. The plane of incidence contains both the ray propagation vector and the surface normal vector at the intercept point. Note: with this definition of the S- and P-components, the distinction between them becomes ambiguous when the ray propagates normal to the surface.
From this, we can see that when polarized light is incident upon a surface, the S- and P-polarization states are defined relative to that surface. If there is a coating applied to that surface, the portion of light transmitted may change significantly depending on the reference Method defined in the System Explorer.
Take, for example, a point object some finite distance in front of a planar surface with a P-state passing coating; light from this point is defined as having the initial polarization state Jx=0, Jy=1. Using an X or Y Axis Reference, the amount of S- and P-polarized light transmitted through (or blocked by) through the system varies significantly over the surface. This is because the input polarizations Jx and Jy remain parallel to the global X- and Y-Axes, respectively, at all incident points on the surface.
When using a Z Axis Reference, however, the Jx and Jy states change as the ray vector rotates around the global Z-Axis, so there is no blocked polarization state.
Thus, when using coatings, one must take care in properly relating the coating definitions with the input polarization reference method.
In applying the concepts above, users may use the Ideal2 and Table Coating formats to individually specify the real and imaginary amplitude transmission and reflection coefficients for both S- and P-polarized light. These coating formats work very well for modeling ideal polarizers. Additionally, the CODA optimization operand may be used to optimize coatings for specific polarization values.
Birefringent In/Birefringent Out
Birefringent materials operate differently than Jones Matrices or coatings inside of OpticStudio. To define a birefringent component in Sequential Mode, the user must define two surfaces in the Lens Data Editor, a Birefringent In surface and a Birefringent Out surface. Within the physical space bounded by these surfaces, OpticStudio requires two materials, one to model the ordinary index of refraction and one to model the extraordinary index of refraction of the birefringent media. To do this, OpticStudio uses the material index defined for the Birefringent In surface as the ordinary index. It then appends “-E” to the material name and searches through the currently loaded Materials Catalog for that name; the material with that name is used for the extraordinary index.
With this approach to defining the birefringent media, the Birefringent In/Out surfaces allow users to calculate Fresnel coefficients and absorption to give a much more accurate intensity transmission calculation, as compared to the Jones Matrix surface. That is, users can optionally trace the ordinary or extraordinary beam independently or trace one while accounting for the phase rotation due to the other. This is governed by the Mode Flag, and it allows users greater flexibility in how they model birefringent effects based off the angular deviation between the ordinary and extraordinary beams for a given system. More information on uses for the Mode Flag can be found on the forum at "What does selecting a mode flag on a "Birefringent In" surface do?."
The only limitation of Birefringent In/Out surfaces with regard to modeling birefringence is that they do not consider the effects of ray splitting. To account for ray splitting, the system should be converted to Non-Sequential Mode.
Applications of polarization-dependent surfaces
This section presents brief examples of how to define a birefringent retarder and an optical isolator in OpticStudio. For a more in-depth design example on birefringent polarizers, see "How to design birefringent polarizers."
Practical retarders
Optical retarders (or waveplates) are optical components which deliberately alter the polarization of incident light from one state to another. This example describes how to construct an effective zero-order retarder with a λ/4 phase change, also known as a quarter-wave plate, which will convert linearly polarized light to circularly polarized light. It utilizes the birefringent material quartz and a HeNe laser (632.8 nm).
Generally, the retardance of the plate is given by:
$$\Gamma + 2 \pi m = \frac{2 \pi \Delta n}{\lambda} d$$
where ∆n is the difference in refractive index between ordinary and extraordinary modes, λ is the wavelength of light, d is the length of the crystal, and Γ is the retardance in radians. The variable m is a natural number which represents the order of the wave plate. With this definition, the relative phase change is not affected by the order due to the 2π periodic nature of light. That said, higher-order wave plates are physically thicker than lower order-wave plates, they are more vulnerable to thermal expansion, have amplified error in retardance for off-axis beams, and have amplified error for retardance if wavelength differs from the designed value.
In practice, few true zero-order wave plates are produced because the required crystal widths are too thin and delicate for manufacturing. Instead, effective zero-order wave plates are made from two thicker uniaxial crystals (usually of the same material) with crossed crystal axes. These are not as effective as a true zero-order wave plates, but as they are more easily manufactured, they offer a good compromise between performance and manufacturability.
To build such a component in OpticStudio, the Lens Data Editor should appear something like the following.
Note that this Lens Data Editor defines a 10th-order quarter-wave plate (purple), followed by a 10th-order, zero relative phase change plate (green). The combination gives an effective 0-order quarter-wave plate. The thicknesses of the two birefringent crystals are calculated as follows.
$$d_{s1} = \left(\frac{\pi}{2} + (10 \times 2 \pi)\right) \frac{\lambda}{2 \pi \Delta n} = 0.1713 mm$$
$$d_{s3} = \left(10 \times 2 \pi\right) \frac{\lambda}{2 \pi \Delta n} = 0.6998 mm$$
As mentioned above, OpticStudio only traces one set of rays at a time, but the Birefringent In/Out Surfaces allow users to account for both the ordinary and extraordinary rays. In this system, setting the Mode Flags to 2 or 3 provides very accurate models for the output state of the system, because quartz is not a strongly birefringent material, so the angular deviation between the ordinary and extraordinary beams is small. Also, the propagation distance within the crystal is relatively small, so the beams will almost completely overlap at the defined image plane. Using Mode 2 and a 45-degree linearly polarized input beam, the output is perfectly circularly polarized light on-axis. This matches exactly the expected results with a true zero-order wave plate (see below).
However, as the incident angle of the beam increases, the effective zero-order plate begins to add more retardance than the true zero-order plate, resulting in elliptically polarized and ultimately nearly linearly polarized light. At 31.5 degrees, the effective zero-order plate essentially functions as a half-wave plate rather than a quarter-wave plate.
It is also interesting to analyze how these systems behave when only accounting for the ordinary or extraordinary beams. An easy way to compare results for each case is to define a Multi-Configuration Editor as follows. Here, the PRAM operand is applied to the Mode Flag on the Birefringent In surfaces. For Configuration 3, the Modes are set to zero (ordinary beam) for each; this is marked “O-O” in the comment line. For Configuration 4, they are set to 0 (ordinary) and 1 (extraordinary) for Surfaces 1 and 3, respectively (marked as “O-E” in the comment line), and so on.
Isolators
Optical Isolators are components which allow the transmission of light in only one direction. Such components typically invoke magneto-optical phenomena, such as the Faraday Effect. Although OpticStudio does not currently have any surfaces which model such magneto-optical effects, it can mimic the behavior of on-axis optical isolators via the Jones Matrix Surface.
The optical materials inside of isolators affect incident beams differently depending on the direction of propagation. That is, for a linearly polarized beam traveling in a given direction, the material will rotate the beam by some angle α; when traveling in the reverse direction, the material will rotate the beam by -\(\alpha\). The rotation angle, α, in radians is defined by
$$\alpha = \nu B d$$
where \(\nu\) is the Verdet constant (proportionality constant of rotation in radians per tesla-meter), B is the magnetic flux density applied to the magneto-optical media (teslas) and d is the length of the medium (in meters).
In OpticStudio, the rotation angle can be defined via a Jones Matrix surface with:
$$
\begin{bmatrix}A & B \\C & D \\\end{bmatrix} = \begin{bmatrix}\cos{\alpha} & -\sin{\alpha} \\ \sin{\alpha} & \cos{\alpha}\\\end{bmatrix}
$$
However, this assumes that there is no electric field propagation in the z-direction. This will also not calculate the effect of the media itself on the z-component or the extra rotation due to the extra off-axis propagation distance.