This blog will develop the basic theoretical understanding necessary to begin using Polarization features and design Polarization components in Zemax.
The Polarization of light plays a critical role in much of our modern technology, and it plays a very important role in research experiments in quantum mechanics, which challenges our assumptions about the nature of reality.
Electromagnetic Waves
Electromagnetic waves are derived from the Maxwell Equations for the unifying theory of electricity and magnetism, that when space is empty produce wave equations that are not independent, and travel at a speed, c, which is the same in all reference systems. The first equation, Gauss's Law for the Electric Field is proportional to the Charge Density. It says that a static charge will produce an electric field. The second equation, Gauss's Law for the Magnetic Field, which says there are no magnetic dipoles (a strange imbalance in nature that we are still searching for) and a north pole must come with a south pole to form a close loop. The third equation, Faraday's Law, says that a changing magnetic field depends on the curl of the electric field. Faraday noticed that if you have a loop of wire with no power source, you can induce a current in the wire if you push a magnet through it. The fourth equation, Ampere-Maxwell's Law, J is the current density and it says that a changing electric field depends on the curl of the magnetic field (a nice symmetry with Faraday's Law). Essentially you can create a magnetic field capable of moving a compass needle with an alternating or changing electric field.
These equations can be combined to create couple electromagnetic waves. If you take the time derivative of the Ampere-Maxwell's Equation, and substitute Faraday's Law into Ampere's for the time derivative of the magnetic field, you will isolate the electric field (or the magnetic field). By the quick use of a vector identity the Ampere's equation will simplify and another substitute of Gauss's Law for the Electric Field in free space (to cancel terms) you arrive at two wave Equations, one describing the time dependent electric field and one describing the time dependent magnetic field. However, because of Ampere's Law the equations are not independent of each other.
Polarization
If you think about it, at the time before Maxwell, electricity and magnetism were two completely different physical phenomena. We just showed that together they describe light as a wave. However, light is more complicated than that, being that it sometimes behaves as a particle, photon. The derivation of electromagnetic waves above is important for polarization because it isolates mathematically what light is. A changing electric field induces a changing magnetic field, and a changing magnetic field induces a changing electric field. In empty space they can propagate, voila light.
If we just look at the oscillations of the electric field or magnetic field, and describe their oscillations with a vector changing in space and time, we can describe the polarization. There are two methods used for visualizing the state of polarization: observing a three-dimensional snapshot in time of the electric vector, and observing the time evolution of the electric vector at a fixed location on the z-axis over an xy-plane. The direction of propagation, electric and magnetic fields are always orthogonal, which makes for some interesting states of polarization.
Figure 1.1 Plane-Wave traveling along the z-axis.
It is standard practice to only look at the electric field when describing wave equations. In the next section we will describe the properties of polarization using an ideal plane wave at a single wavelength for simplicity. In the image above, U is the electric vector in the plane of wavefronts.
Polarization Unit Vector
All states of polarization can be described as elliptical. Lets look at a solution to the coupled wave equations above (technically this is a solution for the homogenous Helmholtz equation), for a Plane Wave at a single wavelength traveling along the propagation vector k, which for simplicity, is the z-axis. We can then observe the properties by looking at the xy-plane.
In general, the polarization state is defined by the complex three dimensional unit vector, a. For our geometry, traveling along the z-axis the polarization vector is described as:
and 
are real scalar constants and 
is the phase difference between x and y components of the field. In general the leading constant is complex. These three terms contain everything we need to correctly analyze and model our polarization states in Zemax.
Polarization in Zemax
Sequential Mode: In sequential mode navigate to the System Explorer (usually on the left). Expand the Polarization tab. Uncheck the Unpolarized box. The Jones Matrix fields become present as seen below.
Figure 1.2 Snip of the Zemax OpticStudio System Explorer, focused on the Polarization tab.
Figure 1.3 Polarization vector related to Zemax OpticStudio notation
Non-Sequential Mode: In this mode the System Explorer does not contain the information to control the Polarization input. First, go to the Non-Sequential Component Editor, insert a source. Open the Object Properties for this Object. Click the 3rd option, Sources. Unclick Random Polarization. Now we have the Jones inputs necessary to control the Polarization for this specific source, see below.
Figure 1.4 Zemax snip of the Non-Sequential Component Editor, focused on the Object Properties and Polarization inputs within the sources tab.
States of Polarization
For this section we will use the Polarization Pupil Map feature in Sequential Mode to visualize the various states of polarization.
Linear Polarization: is defined when the phase difference between X-Phase and Y-Phase is 0.
We can plot this on the xy-plane for various orientations of Linear Polarization. The Electric field is confined to oscillate along this line, hence the name. Any combination of a_x and a_y besides the trivial solution of 0,0 will always define a linear polarization state for phi = 0.
Figure 1.5 Visualization of Linear Polarization states.
Figure 1.6 Example of Zemax OpticStudio Polarization Pupil Map for Linear Polarization state.
Left-hand circular (LHC) Polarization: is defined when the phase difference between X-Phase and Y-Phase is pi/2 and a_x = a_y = 1/sqrt(2).
We can visualize and trace the electric field at various point in time. It can be shown the as time is allowed to progress the Electric field will trace what looks like a unit circle if looking directly in front of the light. The second image to the right shows the progression of time and location with the z-axis.
Figure 1.7 Visualization of Left Hand Circular Polarization state.
Figure 1.8 Example of Zemax OpticStudio Polarization Pupil Map for Left Hand Circular Polarization state.
Right-hand circular (RHC) Polarization: is defined when the phase difference between X-Phase and Y-Phase is -pi/2 and a_x = a_y = 1/sqrt(2).
The trace for Right-hand circular is exactly the same as Left-hand circular (surprise!), however as time increases the Electric field traces the opposite way.
Figure 1.9 Visualization of Right Hand Circular Polarization state.
Figure 1.10 Example of Zemax OpticStudio Polarization Pupil Map for Right Hand Circular Polarization state.
Elliptical Polarization: is the most general state of plane-wave polarization. It just so happens that the above states Linear, and Circular Polarizations are just special cases of elliptical polarization.
Figure 1.11 Visualization of Elliptical Polarization state.
Figure 1.12 Example of Zemax OpticStudio Polarization Pupil Map for Elliptical Polarization state.
Summary
Zemax uses Jones Matrix inputs to control Polarization. In order to properly model and understand the Polarization Pupil Map, it is necessary to understand where these inputs come from. We partially derived the wave equation from the Maxwell Equations to understand what Polarization is. The Electric and Magnetic Field oscillations. If we map this to a 2D plane we can visualize them and name them. Then we looked at the equation for a Plane-Wave traveling along the z-axis, which is typical for Sequential Mode, and broke out the equation to show the complex Polarization unit vector and related this directly to Zemax. Finally, we showed the various states of Polarization and how all can be considered elliptical.
Figure 1.13 Illustration of various polarization states for different phase differences.
Image and notation credit to Prof. Tom D. Milster, University of Arizona
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