# User:Tyler Wynkoop/Notebook/Optics Lab/Polarization Study Data Record

## Contents

## Polarization Study Data Record

The first two weeks were simply learning to understand the equipment. Polarization is a complicated concept. We used a Helium Neon-Laser for all experiments with an initial linear polarization of unknown orientation.

**Measuring the Polarization Shift in a Half-Wave Plate**

Starting with a linearly polarized beam of unknown angle, we measure the angles Starting Point, minimum transmission

- Half wave plate at 180
- Linear Polarizer ~59.5

Second Point, minimum transmission

- Half wave plate at 190
- Linear Polarizer ~79.5

Note: We ran out of measuring ticks on the linear polarizer so we are resetting. We measure the places of zero transmission of the linear polarizer (LP) with respect to the half-wave plate (HWP). Starting Point

- HWP at 16
- Linear Polarizer at -90

Second Point

- HWP at 20
- LP at ~-81.5

Third Point

- HWP at 30
- LP at ~-60.5

Fourth Point

- HWP at 40
- LP at ~-40.5

Fifth Point

- HWP at 50
- LP at ~-19.5

Sixth Point

- HWP at 60
- LP at ~+.5

Seventh Point

- HWP at 70
- LP at ~+19

Eighth Point

- HWP at 80
- LP at ~+39.5

Ninth Point

- HWP at 90
- LP at ~+58.5

Tenth Point

- HWP at 100
- LP at ~+77.5

**Quarter-Wave Plate**

We know that that ~+18 on our LP yields minimum transmission from the laser, thus the LP is orthogonal to the polarization of the beam at ~+20.

The lab manual instructs us to set the laser to a half S half P orientation. We set the LP to a minimum (~+18) and set an HWP in between laser and the LP. The HWP we set so that it had no effect on the intensity of the beam (i.e. still at minimum), therefore the HWP must be aligned with the beam and orthogonal to the LP. Experimentally, this was at 320, arbitrarily chosen out of the eight possible points. To achieve half S half P orientation, we must rotate the polarization of the beam by 45 degrees. Since we determined above that the polarization shift of the HWP is 2θ, we must rotate the HWP by 22.5 degrees in either direction, we subtracted yielding 297.5. We experimentally verified this by rotating the LP by 45 degrees, yielding a maximum, the expected result.

Next we inserted a Quarter Wave Plate (QWP) in the beam to achieve circular polarization. To do this, we rotated the QWP 45 degrees from the polarization vector of the beam. To verify whether the beam is circularly polarized, we rotated the LP all the way around, yielding little or no variation in the intensity of the beam. The lab manual instructs us to use a green laser (as oposed to our red laser) and do the same test, but we have non available. According to our results, the green laser would be totally cancelled out at some point because the wavelengths are different, i.e. the HWP and the QWP would not work properly.

We also proved that transmitting a beam through a QWP twice is the same as passing it through a HWP once. We put a mirror in our setup to reflect the beam back through the QWP. Using an LP we found orientations in which the LP transmitted the beam one way, but not the other.

## The Fresnel Rhomb

The first part of the Fresnel Rhomb experiment is understanding the concept behind the divice. The device is a block of glass designed to create two total internal reflections.

The lab manual asks what the index of refraction would be on a perfect Fresnel rhomb, with angle of incidence,θ=π/2 and parallelogram angle α=π/4.

The formula for the index of refraction as a function of these angles is taken from an optics text. However, this formula has proven extremely difficult to wield. Either the correct formula eludes me or the existing formula must be graphically analyzed using software, as it has proven impossible to solve.

**Determining "n" experimentally**

The next step is to experimentally determine the index of refraction by prescribing the incident angle and measuring the rhomb angle. I can adjust the angle of incident in order to produce a full quarter element. To test the whether how closely the rhomb mimics a quarter wave plate, I place a linear polarizer in the beam, after it has passed through the rhomb. If the beam is completely circularly polarized, there should be no attenuation through the linear polarizer.

After consulting with Professor Deils, it seems that there is a better way to find when the beam is circularly polarized. This this set up, a linear beam goes through a half-wave plate to control the direction of polarization. The beam then enters a LP set for full transmission. The beam then enters the rhomb. After the rhomb, the beam is reflected back on itself via a mirror, back through the rhomb and hits the LP. Because the rhomb acts as a QWP, two passes through it produces a HWP effect, causing the beam to be linearly polarized once more, but the polarization vector is offset by 90 degrees. When the beam strikes the LP again, there should be no transmission on the return trip. Adjusting the angle of the rhomb should produce this total extinction when the angle is perfect. This proved to be quite difficult experimentally, since the laser reflected off the LP and the rhomb several times, producing many errant beams. Secondly, in order to distinguish between the reflected beam and the initial beam, the mirror must be set to reflect the beam at a slightly different angle. This means that the return trip for the beam was not perfectly the same, and perfect quarter wave phase shift is impossible to observe. Therefore, a minimum is sought.

Our rhomb angle is found to be 59 degrees. To find this we set the beam to ninety degree incidence with the input face, recorded the angle on the mount, then found when the beam was perfectly parallel to a long face. The difference of these two (31 degrees) is the complement of the rhomb angle.

**The Brewster's angle method**

The Brewster's angle method used the laser, incident on a rhomb face at the Brewster's Angle. In short, Brewster's angle is the angle at which vertically polarized components of the beam are transmitted, and horizontal components are reflected.

The goal here is to orient the linearly polarized beam completely vertical, and strike the rhomb at the Brewster's angle. There should be no reflection under these circumstances.

The first step is to orient the laser for complete vertical polarizaition. Until now, the beam was unknown, and, more importantly, it was not necessary for it to be known. To make it known, I used an LP of known orientation, and spun the laser in its carriage until I got zero transmission. With the laser now aligned vertically, I placed the rhomb in the beam. I adjusted the angle of incident until I got minimum reflection. Further tuning, by rotating the laser in its carriage and adjusting the incident angle produced near extinction. This was as close as I could produce in the lab for the Brewster's angle. I then aligned the rhomb face normal to the beam. The difference of these two is the Brewster's angle. The Brewster's angle is directly related to the index of refraction.

This method is known to be inaccurate though, as the Brewster's angle is produced strictly by "eying" a minimum after reflection and requires a perfectly linearly polarized beam. Theoretically, the minimum should be total extinction as well.

Minimum reflection: 286 degrees Parallel: 230 degrees

Brewster's angle: 56 degrees Index of refraction is tan θ = n

This yields an index of refraction of 1.483.

**Minimum deviation method**

Another method to determine the index of refraction is using the minimum deviation. When a beam passes through a triangular prism, (or in this case, the acute corner of the rhomb) as the angle of incidence changes, the final angle of the beam (after it exits the prism) will change as well, but in a hyperbolic manner. There is a minimum angle at which the resultant beam will not pass below. This is known as minimum deviation and this angle is related to the index of refraction of the prism.

The procedure here, is to pass the beam through one of the acute

Minimum deviation mount angle: 343 normal mount angle: 37 minimum deviation angle: 50 degrees.

via the formula provided by the optics text: n=sin(θ)/sin(A/2) where A is the rhomb angle. this yields an index of refraction n=1.556

## Depolarization Upon Reflection

5 degrees at a time

As close to normal as equipment allows

- incident angle; min angle; min intensity; max angle; max intensity
- 6;97;4.5mV;0;23.5mV
- 8;90;4.5mV;4;41.6mV
- 13;90;4.6mV;0;447mV
- 18;90;4.5mV;3;471mV
- 23;90;4.5mV;3;460mV
- 28;90;4.5mV;359;430mV
- 33;90;4.5mV;0;450mV
- 38;90;4.7mV;2;460mV
- 43;90;5.0mV;0;460mV
- 48;90;5.4mV;0;465mV
- 53;90;6.4mV;0;420mV
- 58;90;8.5mV;0;465mV
- 63;90;11.7mV;0;453mV
- 68;90;11.5mV;0;101mV
- 73;90;19.0mV;0;133mV
- 78;90;110mV;0;400mV
- 80;90;150;0;332mV

## Depolarization upon reflection, take 2

The above data did not settle well with me, so I returned to the lab and tried again. The set up is the same, but with Dr. Diels advice, I looked again for a solution. The goal of this experiment is to try to determine the angle of a linearly polarized incident beam at which the polarity switches.

I found that at incidence angle 77 degrees on the mirror, the minor axis of the elliptical polarization is maximized, and is at +102 degrees from the horizontal, from the laser's point of view. The maximum voltage (intensity) was 1.994 V at +12 degrees, and the minimum was .539 V at +102 degrees.

The next step is to plot the the ratio of the vertical and horizontal intensities P and the phase angle difference Δ as a function of the incidence angle. To do this, I need to prescribe the magnitude of the angle of incidence, and measure the magnitude of the parallel and perpendicular components, as well as the polarization angle of the minimum. For simplicity's sake, note here that my polarizer measures from the vertical. Therefore, the polarization angle of minimum intensity, as measured by my equipment corresponds to the angle of maximum intensity measured from the horizontal that is used in the equations.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P=sqrt(A_per/A_par)}**
and
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle tan(2/beta)=(2*A_per*A_par/(A_par^2-A_per^2)}**
where
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle I=sqrt(A)}**

**Data:**
Note: 155 is as high of an angle as the equipment allows

- angle;I_per;I_par;beta
- 1.) 155;.615;2.015;19
- 2.) 154;.583;2.003;12
- 3.) 152;.512;1.937;11
- 4.) 145;.518;1.987;6
- 5.) 135;.337;1.870;-7
- 6.) 125;.235;1.860;-12
- 7.) 115;.144;1.904;-15
- .) 105;.133;1.850;-16