This article shows how to model and design real monochromatic and achromatic waveplates in OpticStudio. It will demonstrate how to work with birefringent materials, build a merit function to calculate retardance, and use the Universal Plot to visualize the retardance vs the waveplate thickness.

**Authored By Takashi Matsumoto**

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# Birefringent material and waveplate

Most popular waveplates use the birefringent properties of materials. Birefringent means that the material has an index of refraction that depends on the polarization and direction of propagation of light. Various types of birefringent materials exist, but uniaxial crystal type materials are used for wavelength plates. This type of material has two fixed refractive index axes that are perpendicular to each other and one of it is crystal optic axis. Generally, a wave consists of two polarization components. These components will be governed by different effective refractive indices.

The 1^{st} index is parallel to the direction of the crystal optic axis and the 2^{nd} index is orthogonal to the 1st axis.

Figure 1. Image of birefringent material and path of rays

(https://www.microscopyu.com/techniques/polarized-light/principles-of-birefringence )

These 2 directions are called “fast” and “slow” axis and the refractive index values are called “ordinary” index and “extraordinary” index. The fast axis has a low index and the phase velocity of light is faster than the orthogonal direction of the axis.

Generally, the complete polarized light can be regarded as consisting of two polarization components. These components will be governed by different effective refractive indices. Because of the material and polarization property, the incident polarized light is separated into a fast or a slow axis when propagating through the material.

To make a waveplate, the birefringent material is cut into a plate, with the orientation of the cut chosen so that the crystal optic axis is parallel to the surfaces of the plate.

For example, let’s consider a vertically linear polarized ray that hits a waveplate at an angle of 45 degrees compared to the fast axis. The light travels through the waveplate and the polarization components are split into a “fast” and a “slow” axis. These 2 components accumulate phase at different rates. The difference of phase between the rays is called “retardance” as in Figure 2.

This is the fundamental theory of birefringent waveplate.

Figure 2. Image of polarization in a birefringent half waveplate.

(https://en.wikipedia.org/wiki/Waveplate#Quarter-wave_plate)

# Calculate monochromatic quarter waveplate

Understanding the previous theory is necessary to design monochromatic waveplates.

For example, a quarter waveplate will introduce a quarter wavelength between the two polarization components of light. To design a quarter waveplate, the thickness t of the parallel plate can be computed using the equation below.

\(\left ( m+\frac{1}{4}\right )\ast \lambda = t \ast \left ( n_{e}\left ( \lambda \right ) -n_{o}\left ( \lambda \right )\right )\)

Where:

- m is the order of the waveplate
- λ is the wavelength
- t is thickness of the birefringent parallel plate
- n
_{e}and n_{o}are the extraordinary and ordinary refractive indexes.

OpticStudio contains general birefringent materials in the “Birefringent” catalog. To use that material catalog, select the right catalog in the “Material Catalog” tab in the System Explorer as in Figure 3.

Figure 3: The “Material Catalogs” tab in the System Explorer

For now, let’s use the QUARTZ material. This material means generally crystallized silica and is called a “crystal”. The “Fast” axis direction is defined in the “QUARTZ” material and the “Slow” axis direction is defined in the “QUARTZ-E” material.

The index of refraction can be checked in the Dispersion Diagram in OpticStudio.

Figure 4. Dispersion Diagram in OpticStudio

This feature provides a list and a graph of the index of refraction vs wavelength as shown in Figure 5.

Figure 5. “Dispersion Diagram” for “QUARTZ” and “QUARTZ-E”

Set Glass 1 to “QUARTZ” and Glass 2 to “QUARTZ - E” to check the values as shown in Figure 5.

Here n_{o} = 1.5487281 and n_{e} = 1.5579932 at a wavelength of 0.5 µm.

The minimum thickness of a quartz plate can be calculated using the previous equation and the thickness is t=13.491µm.

# Modeling monochromatic quarter waveplate

Now let’s model this waveplate in OpticStudio. A sample file called “Monochromatic-wave-plate.ZAR” can be found in the attachments.

The material is “QUARTZ” and the crystal optic axis is along the X direction. The direction of propagation is along the Z-axis and the wavelength is 0.5 µm in this file.

First, let’s set the polarization of the incident rays. In the “Polarization” tab of the “System Explorer”, Uncheck “Unpolarized” and set the polarization for a perfect right-handed circularly polarized incident ray as shown in Figure 6.

Figure 6. Polarization setting in the System Explorer

“X-Phase” is the phase angle in degrees of the Jones Vector. It indicates the phase delay of electric wave front of the X-direction. If an observer turns around and watches incoming rays, the observer first sees the electrical amplitude along the Y direction and then along the X direction.

This polarization is named “right-handed circular polarization” because the energy of the electric field (electrical amplitude) turns clockwise for the observer. Note that to represent perfect circular polarization in OpticStudio, the values of “J_{x}” and “J_{y}” must be the same and have a 90 degrees phase difference

A birefringent material can be modeled using the “Birefringent In” and “Birefringent Out” surfaces in OpticStudio. The “Birefringent In” surface has many important parameters, namely the X, Y, Z-cosines and the Mode parameter.

- The X, Y, Z-cosines parameters: define the crystal optic axis. Usual waveplates have flat plane surfaces which are parallel to the crystal optic axis. However general optical elements have not such orthogonality and these parameters exist to address such situation.
- Mode parameter: define the way of calculation and draft rays.
- When this value is 0 or 1, OpticStudio traces the ordinary or the extraordinary ray only.
- When this value is 2 or 3, OpticStudio traces the ordinary or extraordinary ray and the polarizations of the rays are calculated as a sum of the polarizations.

These Modes are used when the ordinary and extraordinary beam form a single beam, which is the case in a waveplate.

Figure 7. Explanation of mode 2 and 3 for "Biregringent In" in help file

Let’s set the Mode to 2 or 3.

Now the incident ray has a right-handed circularly polarization, and the polarization is defined so that the phase of the X-direction is delayed by 90 degrees. A monochromatic waveplate can be designed with the thinnest condition with the fast axis parallel to the X direction. To do that, Insert 2 surfaces after the 1^{st} surface and set the 2^{nd} surface as “Birefringent In” and the 3rd surface as “Birefringent out”.

- Set the X, Y, Z-cosines parameters to “1,0,0”.
- Set Mode to 2.
- Set the material to “QUARTZ” and enter a thickness of “0.013491” for the 2
^{nd}surface.

This comes from the previous calculation. - The thickness of the 1
^{st}and 3^{rd}surfaces can be set to 0.1 for visualization.

Most of all the other settings are default settings. The aperture type is Entrance Pupil Diameter with an Aperture Value of 0.1. Wavelength 1 is “0.5”. Figure 8. shows the Lens Data Editor, the 3D Layout, and the Polarization Pupil Map in these conditions.

Figure 8. Sample model in OpticStudio

According to the Polarization Pupil Map, the right-handed circularly polarization is changed into a linear polarization. This can be confirmed using the CODA operand in the merit function.

A description of the CODA operand can be found in the help files. CODA can calculate the retardance of rays, the phase difference.

Figure 9. Explanation of CODA operand in help file

Figure 10 shows the value of the merit function and the result is roughly 0 as expected.

Just a comment here. The thickness of the waveplate could have been optimized instead of being computed. To do that, change the weight of the CODA operand to 1 and set the thickness of surface 2 as a variable.

Another interesting result is to observe how the retardance changes with the thickness of the 2nd surface using the Universal Plot 1D.

Figure 10. Result of “Merit function” and “Universal Plot”

In Figure 10, the CODA operand returns values between -π and π.

Next, let's consider how to calculate the generated optical path difference by the waveplate. OpticStudio has an OPTH operand to calculate the optical path. OPTH calculates the optical path of surfaces. However, each surface needs to have an isotropic/uniform material.

So, in this case, the Multi-Configuration Editor can be used to change “Birefringent In” and “Birefringent Out” surfaces into “Standard” surfaces and the materials for these surfaces. In the sample file, the material for config 2 is “QUARTZ” and for config 3 “QUARTZ-E”. As there are no multiconfiguration operands to change the surface type, the “IGNR” multiconfiguration operand is used to ignore the original surfaces.

Figure 11. shows the Multi-Configuration Editor, the Universal Plot and the Merit Function

In the Multi-Configuration Editor, Configuration 1 is the waveplate setting, Configuration 2 calculates the phase with the ordinary index, and Configuration 3 calculates the phase with the extraordinary index.

In the Merit Function Editor, the CONF operand changes configurations, and the OPTH operand computes the phase. Let's read the merit function:

- The values of row 6 and line 9 are optical paths in ordinary and extraordinary index in lens unit (in this case mm)
- The values of row 7 and 10 are wavenumbers
- The value of 12 is the difference between values of row 7 and row 10
- The value of row 17 is the radian of the value of row 12

Finally:

To calculate the retardance, set the weight of row 3 to 1.0 and the weight of row 17 to 0.0 in the Merit Function.

To calculate the difference of optical path in waveplate, set the weight of row 3 to 0.0 and the weight of row 17 to 1.0 in the Merit Function.

# Modeling achromatic quarter waveplate

Now, let’s model an achromatic waveplate for a broadband light source. A sample file called “Achromatic-wave-plate.ZAR” can be found in the attachments.

An achromatic waveplate can be considered in the same way as achromatic lenses, i.e., two or more waveplates of different materials that combined will cancel chromatic dispersion. For example, this website https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=854 lists “Quartz” and “Magnesium Fluoride” as materials for achromatic waveplates. OpticStudio contains “QUARTZ” and “MgF2” in the Birefringent catalog.

The model will contain two sets of "Birefringent In" and "Birefringent Out" surfaces using these materials. The crystal (optic) axes of the two sets of birefringent materials are orthogonal. For example, if the crystal (optic) axis of the front waveplate is the X-direction, the crystal (optic) axis of the rear waveplate will be in the Y- direction. This is to make efficient use of the color dispersion effect.

Here the sample file models an achromatic quarter waveplate from 0.5 to 0.7µm. As in the previous model, light of the incident rays has a right handed circular polarization as shown in Figure 12.

Figure 12. Model of Achromatic-waveplate

The system layout is as in Figure 12. Compared to Figure 8, there is one more set of "Birefringent In" and "Birefringent Out" surfaces. The optical path length of light propagating through this series of waveplates can be calculated in the X- and Y-directions as below:

\( X\left ( \lambda \right ) : n_{1o} \left ( \lambda \right )t_{1} + n_{2e}\left ( \lambda \right )t_{2} \)

\( Y\left ( \lambda \right ) : n_{1e} \left ( \lambda \right )t_{1} + n_{2o}\left ( \lambda \right )t_{2} \)

Where:

- n
_{1o}and n_{2o}are ordinary indexes in the 1^{st}and 2^{nd}wave-plates - n
_{1e}and n_{2e}are extraordinary indexes in the 1^{st}and 2^{nd}waveplates - t
_{1}and t_{2}are the thicknesses of the 1^{st}and 2^{nd}waveplates - λ is the wavelength

Since the refractive index is a function of the wavelength in these equations, the thickness of the waveplate with the lowest color dispersion can be calculated through optimization.

The “CODA” operand can be used to calculate the retardance at different wavelength. The merit function for designing this system is equal to the square sum of the retardances.

To control the thickness of the waveplates, the TTHI operand can be used.

Optimizing this merit function means minimizing this formula.

\( \sum_{i=0}^{n}\left ( X \left ( \lambda _{n} \right ) + 0.25 \lambda _{n} -Y \left ( \lambda _{n} \right ) \right )^{2} \)

To find a good solution, the “Hammer Current Optimization” is needed as it will avoid local minimum. The thickness of surfaces 2 and 4 are set as variables. After optimizing, the result can be seen as shown in Figure 13.

Figure 13. After Optimization result

According to Figure 13, the merit function is close to zero and the Polarization Pupil Map shows that the circular polarization is changed into a linear polarization.

Now the relationship between the thicknesses of the 2 waveplates and the retardance can be checked using the Universal Plot 2D.

To do this, change the weight to 0 for the TTHI operand because that operand doesn’t relate to retardance. The result is shown in Figure 14.

Figure 14. Universal Plot of the merit function

According to the plot, the retardance appears to be the smallest when the thickness difference is constant.

This suggests that the difference between the two waveplates, rather than the overall thickness of the waveplates, is important for achromatic waveplates. In Figure 15, the scale was changed to show more clearly the optimal range of thickness.

Figure 15. Universal Plot of the merit function max value is 0.3

# Summary

This article shows how to model and design real waveplates in OpticStudio. Once the waveplate is designed, the performance can be evaluated using merit functions in the “Universal plot”.

# References

**Polarization: **Investigating OpticStudio’s polarization features: https://support.zemax.com/hc/en-us/articles/1500005486441**Universal Plot: **Tutorial 5: Parameter visualization: https://support.zemax.com/hc/en-us/articles/1500005578542**Multiconfiguration: **How to model an adaptive optical system: https://support.zemax.com/hc/en-us/articles/1500005488241**Coordinate Break: **How to work in global coordinates in a sequential optical system: https://support.zemax.com/hc/en-us/articles/1500005487781**Birefringent: **How to design birefringent polarizers: https://support.zemax.com/hc/en-us/articles/1500005486841

What does selecting a mode flag on a "Birefringent In" surface do?: https://community.zemax.com/got-a-question-7/what-does-selecting-a-mode-flag-on-a-birefringent-in-surface-do-655

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