# Polarization Optics

Course Overview
This laboratory course will introduce the student to the polarization properties of light and unique interactions with optical components.  A theoretical understanding of concepts will be reviewed to gather the concepts necessary to study their effects.  Using these learning techniques a more complete understanding and ability to apply polarization concepts to real-world systems will be conveyed.
Course Duration

Approximately 3 hours (self-paced)

Learning Objectives
By the end of this pre-lab assignment, students should be able to:
• Understand what polarized light is
• Use polarized light correctly in optical simulations
• Utilize polarization effects at optical interfaces
• Incorporate polarization sensitive optics
• Review polarization applications
Course Outline - Introduction to Polarization
Polarization is a fundamental property of light and electromagnetic waves, describing the orientation of their electric field vectors. The electric field vectors can be broken down into two orthogonal states.  These states can oscillate in or out of phase to create linear, circular, and elliptical polarization.  Each has unique applications and can be generally attributed to specific sources.  Naturally occurring sources are normally linearly polarized or unpolarized.  An unpolarized source is simply light that propagates with a random polarization that changes over time and some ratio of the two polarization states.  Circular and elliptical polarization states are generally unnatural and are created artificially using wave plates and retarders.    Additionally, polarization analysis techniques, such as polarimetry, facilitate advancements in fields ranging from astronomy to telecommunications by providing insights into the properties of received light and enabling precise measurement and manipulation. Overall, polarization serves as a cornerstone in understanding and utilizing light and electromagnetic waves across numerous scientific and engineering disciplines.

## Polarization States

Polarization can be described by two orthogonal oscillating electric fields in a 2D plane.  If the two states oscillate in phase, the resulting polarization is linear.  Figure 1 demonstrates an in-phase linearly polarized light wave.
If the electric field vectors happen to start oscillating out of phase, say ¼ of a wavelength, then circular polarization will occur. This VIDEO is a good depiction of different polarization states and should be viewed.
In reviewing the video, one can see that the electric field vectors can take on infinite phase differences with respect to each other, but only three states can be produced: linear, circular, or elliptical polarization.
Laser sources tend to be linearly polarized due to the gain media used to create the lasing action. LEDs are unpolarized as there is no mechanism to create a preferred polarization state output. Knowing the polarization state of a light source is an important parameter in optical systems. For instance, attenuating light can be achieved using neutral density filters or polarizers. Using a polarizer as an attenuator with an LED would cut out around 50% of the light throughput, whereas a neutral density filter is polarization insensitive. If the system is to produce light for an imaging application, then a polarizer may reduce the output to an undesirable level.
Using optical components, such as windows or beam splitters, at angles other than normal incidence could produce physical effects such as Brewster’s angle. This is the angle at which the reflected light is completely polarized. For laser beam scanning applications this increases the non-uniformity of the laser beam scan area and reduces the peak power under critical requirements.
Experiment 1

## Polarization States

The polarization state of light will affect how it interacts at interfaces with optical surfaces. Some optical substrates have inherent polarization properties such that the light will behave differently when different polarization states are incident on them. Light sources with linear polarization will act homogeneously at an interface while unpolarized, elliptical, and circular polarization will exhibit unique exploitable properties.
Polarizers are optics that interact with specific polarization states under rotation and angular offsets. Some polarizers are designed for direct incidence, i.e. for light that is incident normal to the surface, and others for specific incident angles, such as 45 degrees. Polarizers can either absorb or reflect the rejected polarization states.
In a polarizer, the polarization state that is transmitted is based on its polarization axis. Typically, this is indicated by the manufacturer as a line or arrow on the side of the polarizer, but it can easily be experimentally obtained by rotating the polarizer until the maximum amount of light has passed.
The formula to determine how much light has passed when light is incident on a polarizer is given by

1. I_{transmitted} = I_0 cos^2(\theta), this is known as Malus’s Law.

Where theta is the angle between the polarization axis of the polarizer and the polarization direction of the incident light, I_0 is the incident light source power, and I_{transmitted} is the transmitted power.

It is necessary to see that an unpolarized source will transmit 50% of its power through a polarizer at any rotation angle.  Additionally, a linearly polarized source will have full light transmission and rejection as the polarizer is rotated in 90-degree increments.  Two polarizers with orthogonal polarization axes will block any source, either polarized or unpolarized.
Exercise 1: Polarized Light Sources
• Click on the analysis portal and select run analysis
• Click on the analysis detector to bring it into its own window
Notice on the bottom of the window there are four options to view the power for each polarization state; none, x, y, and z
• Select polarization state “None”, x, and y to see how much power is present for each
We are not interested in the z component as it is not present in these polarization states
• Record these values in the table below
The “None” option includes power for all polarization states
Notice in the filled-out table from the exercise that the power in each polarization component depends on the polarization type. Power was split evenly between the two states for all types except linearly polarized light. However, there will always be some light of the opposite polarization for linearly polarized light for real-world sources, and this is determined by the polarization ratio of the light source itself. Not to be confused with the extinction ratio of the polarizer, which measures the “purity” of the polarized light output after transmission.
Extra information

The original discovery of polarization of light is attributed to the Dutch scientist Christiaan Huygens in the 17th century. In 1690, Huygens proposed that light consists of waves that propagate through a medium, much like waves in water. He observed that when light passes through certain materials or is reflected at certain angles, its oscillations become aligned in a specific direction.

Huygens’ work laid the foundation for understanding polarization, although the term “polarization” itself was not coined until later. His insights were instrumental in explaining various optical phenomena, including the behavior of light as it interacts with different materials, such as crystals and lenses.

When polarization is referenced to an optical component’s optical axis, the states are referred to as “S” and “P” states. These stand for perpendicular (S) and parallel (P). The “S” stands for “senkrecht” which is German, and translates to “perpendicular” in English.

Otherwise, if polarization is being referenced in a general way we use the terms “horizontal” and “vertical” with some surface (such as a breadboard) being the reference.

Exercise 2: Polarizer Optics
• Click on the analysis portal and select run analysis
There is a polarizer inserted into the optical path to change the amount of light that makes it to the detector.  The extinction ratio is 60 dB, so very little light from the opposite polarization will make it through to the detector when aligned with the polarization axis of the light source.
• Change the “Z” angular component of the polarizer in the simulation file to the positions specified in the table below. This simply rotates the polarizer
• Record the resulting power for each polarization state by changing the angle and running the analysis
• Keep the file open for the next exercise
If you graph the power over angle from the table then you should generate a cosine curve.  Notice if you continue to rotate the polarizer 360 degrees there are two minima and two maxima.  This should be self-evident since the transmitted power is proportional to the cosine squared of the angle.
Note that the power on the detector will always be less than the total input power because of the extinction ratio of the polarizer and Fresnel reflections. Finally, if the polarizer is rotated to the minimum transmission angle for linearly polarized light, all light will be blocked except for the amount passed, which is dictated by the extinction ratio.
Extra information
The discovery of polarizers is a culmination of centuries of scientific inquiry and innovation. Beginning with Christiaan Huygens’ wave theory of light in the 17th century, which laid the groundwork for understanding light’s behavior, subsequent contributions such as Malus’s Law further elucidated the principles of polarization.
Edwin H. Land’s invention of Polaroid material in the 20th century marked a significant milestone, providing a practical means to manipulate polarized light. By selectively absorbing light waves oscillating in one direction while allowing others to pass through, Polaroid material became the foundation for modern polarizers.
Today, polarizers play indispensable roles in various fields, from photography and display technologies to scientific research, offering precise control over the polarization of light and enabling countless technological advancements.
Exercise 3:  Multiple Polarizers
• With the previous simulation file still open, click on the polarizer in the 3D layout and duplicate it
• Rename the new one “Polarizer 2”
• Position it at [0,100,75]
• Change the rotation angle to the values in the table below, run analysis, and record the power
Notice that there are locations of minimum and maximum transmission for both a single polarizer and dual polarizers
• After the table is filled out, add a third polarizer in the same manner as the second from the above step
• Change polarizer 2 and 3 angle to z = 90
Polarizer 1 should be z = 0 degrees
• Run analysis and observe the power on the detector for both “x” and “y” polarization states
We now have “crossed” polarizers, between polarizer 1 and 2/3, and should not see any transmission to the detector
• Now change polarizer 3 angle to z = 0
• Run analysis and observe the power on the detector for both “x” and “y” polarization states
We still have crossed polarizers, between polarizer 1 and 2, and should not observe any power still on the detector
• Now change polarizer 2 angle to z = 80 and polarizer 3 to z = 90
• Run analysis and observe the power on the detector for both “x” and “y” polarization states
We now have light on the detector even though polarizers 1 and 3 are crossed.  This is contradictory.
For either a linearly polarized source and a single polarizer, or an unpolarized source and a dual polarizer setup, the results are intuitive given Malus’s law. However, as was determined at the end of exercise 3, when we add a third polarizer (or an infinite amount of them) in between crossed polarizers we will always get power through the system.

The effect we are seeing is that the polarization state is no longer referenced to the original light source’s polarization axis, but to the previous polarizer’s polarization axis.
When the second polarizer was rotated to 80 degrees, we then generated not a 90-degree angular difference between the incident light and the polarization axis, but a 10-degree angular difference. Therefore, some light will pass through the second and third polarizers as the light polarization and the third polarizer polarization axis are no longer 90-degrees in angular difference.

In reality, there are always two orthogonal electric field vectors that create the total polarization even after transmission through a polarizer. See the video in the beginning of this virtual lab for a refresher on this.
Note that the electric field vectors don’t convey the ratio of energy in each state, only the sum polarization. The magnitude shown in the final step (after polarizer 3) in Figure 3 below is only for illustrative purposes of the power that transmits through the third polarizer.
Extra information
The origin of polarizers dates back to the early 19th century, marked by the systematic investigations of scientists like Étienne-Louis Malus, who discovered light polarization through reflection in 1808. Further advancements came with the work of Jean-Baptiste Biot and Félix Savart on double refraction in crystals, and Augustin-Jean Fresnel’s development of the theory of light as a transverse wave. William Nicol’s invention of the Nicol prism in 1828 significantly improved the efficiency of polarizing light, leading to its widespread use in various optical instruments. Later, Émile Baudot’s enhancements and Edwin H. Land’s inventions, particularly the Polaroid polarizers in the 20th century, revolutionized the field, impacting photography, sunglasses, and LCD technology. Throughout history, the collaborative efforts of these scientists and inventors have driven the evolution of polarizers, making them indispensable in modern technology and everyday life.
Experiment 2

## Polarization Sensitive Surfaces and Phenomena

Optical substrates themselves can have properties that interact with polarization states in different ways. A characteristic known as bi-refringence will affect polarization states differently and is commonly used to split light into its constituent states. Bi-refringent substrates have the effect of different indices of refraction for each polarization state. Optical surfaces can be coated to selectively choose polarization properties in various interesting ways. As polarizers are manufactured specifically to reject light perpendicular to its optical axis, other types of coatings can be created to have a more limited effect.
In addition to substrate and coating specific effects, there are physical phenomena that can be utilized to exploit light polarization. Brewster’s angle is a physical phenomenon where, at a certain angle, the reflected light is completely polarized.
2. \theta_B=\frac{n_2}{n_1}, This is Brewster’s angle.
Where {n_x} is the index of refraction for medium x.
Exercise 4:  Brewster’s Angle
• Calculate Brewster’s angle for the various wavelengths and indices of refraction in the table below using equation 2
• Using the material tool, find substrates that are close to the index of refraction from the table above.
• Fill in the table below for the substrate used, index of refraction of the medium, and reflected and transmitted power at the specified angles
The offset angles are displaced from Brewster’s angle, e.g. if Brewster’s angle is 56 degrees and the angle is +1, then the angle of the Brewster’s window would be 57 degrees
• Now change the light source polarization to 90 degrees corresponding to a vertically polarized linear state and fill out the same table with the new polarization state
Notice that this polarization state does not exhibit the same behavior as the other state
The values in the table will be accurate for the simulation software ideal setup.  In a real-world optical system, there will always be some opposite polarization in the reflected beam, even when the source is incident at Brewster’s angle with the correct polarization.  As you move away from Brewster’s angle, the reflected light will reduce and become unpolarized again, as can be seen in the analysis.  As noted above, Brewster’s angle is only applicable for one polarization state.
Extra information
Brewster’s Angle, named after the Scottish physicist Sir David Brewster, refers to the angle of incidence at which light undergoing reflection from a surface becomes completely polarized. This phenomenon occurs when the angle of incidence is such that the reflected and refracted rays are perpendicular to each other. This concept finds applications in various optical devices and technologies, including polarizing filters, anti-glare coatings, and optical instruments.
Exercise 5:  Beam Sampler
• As the incident polarization angle changes, the reflected power will change.  Change the polarization angle of the light source using the light source settings
• Click on the analysis portal and select run analysis
• Fill in the table below
Extra information
The beam sampler works on the principle of Fresnel reflections which differ depending on the incident polarization. These components generally reflect 1-10% of the light incident on them. For optical setups, either the light source or the beam sampler itself can be rotated to generate the desired reflection. These are good optical components to use if the reflected energy needs to be changed for different sources. They are commonly used for sensing and power measurements during dynamic optical simulations.
Experiment 3

## Polarization Applications

Creating polarized light can be done using polarizers for linear polarization, or by polarization and retarders, or wave plates, to create more complicated states such as circular and elliptical polarization.
The phase shift from a bi-refringent substrate can be calculated using equation 3 below.
2. \Delta\theta=\frac{2\pi{d}(n_1-n_2)}{\lambda}, This is Brewster’s angle.
Where d is the thickness of the substrate, {n_x} is the index of refraction along the slow and fast axis, and \lambda is the design wavelength.
Exercise 6:  Circular Polarized Light
• Using equation 3, calculate the parameters needed to create a quarter waveplate to circularly polarize a linear light source at 660 nm
Use the user defined object that is in the 3D layout
• Once the quarter waveplate is designed and implemented, click on the analysis portal and run the analysis
• Click on the detector window and select the map view
• If the quarter waveplate was implemented correctly, the map view will show circular polarization over the spatial intensity field The polarization state will not be perfect, but should look somewhat elliptical and close to circularTry 1.301365 and 1.31 for the two indices and 12.7 mm for the thickness if other values are not working
Extra information
The origins of optical waveplates can be traced back to the 19th century, marked by the understanding of light’s wave nature and polarization phenomena. Initially spurred by scientists like Young, Huygens, Malus, and Fresnel, who laid the groundwork for the study of light as a wave, researchers began exploring ways to deliberately manipulate its polarization states. This led to the development of birefringent materials like quartz and calcite, which exhibit different refractive indices for light polarized in different directions, forming the basis for waveplate designs. As technology progressed, engineers refined these designs, leading to the creation of quarter-wave and half-wave plates with precise control over polarization states. Modern advancements in materials science and manufacturing techniques have further enhanced waveplate capabilities, making them indispensable tools across a myriad of applications in optics, telecommunications, and spectroscopy.
Analysis and Conclusion
Polarization is an important aspect in optical system design and system performance.  Not only can it be important in selecting components to take advantage of their specific properties, but they can also be a detriment if not properly understood.  This course has reviewed the fundamental properties of polarization and the interaction with various interfaces and optical components.  Polarization is a fundamental aspect of light and is critical to understand when analyzing and utilizing light sources.
Assessment
NA
Resources
The student requires an account in the 3DOptix system.
Synopsys
In this course, students study the fundamental principles of polarization optics. The curriculum focuses on the core principles of polarization light-matter interactions, crucial phenomena that dictate how light oscillates along its propagation and at the interface between different optical media. Through engaging experiments and exercises, students acquire a profound comprehension of these concepts and their applications.

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