Deviations from Malus Law

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Pearl Leonard
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1 From: Steve Scott, Jinseok Ko, Howard Yuh To: MSE Enthusiasts Re: MSE Memo #18a: Linear Polarizers and Flat Glass Plates Date: January 16, 2004 This memo discusses three issues: 1. When we measure the light intensity through a linear polarizer that is rotated through 360 o, we see small deviations from the Malus law I(θ) = I min + (I max I min ) cos 2 θ. What causes these deviations? 2. If linearly polarized light passes through a flat glass plate that does not have an antireflection coating, the light s polarization direction will change slightly if the incidence angle is not normal to the glass surface. How big is this effect? 3. What is the effect of passing light through photoelastic modulators (pems) at nonnormal incidence? Deviations from Malus Law As described in Memo 16, to determine the transmission axis of a polarizer,a horizontal laser beam will be directed at a vertical glass plate with an angle of incidence equal to the Brewster angle, so that the reflected light is 100% polarized vertically ( E in the vertical direction). The reflected light is passed at normal incidence through the linear polarizer mounted on a rotating stage. The rotating stage will rotate the polarizer until the transmitted light intensity is maximized, which identifies the TA of the polarizer. There are several potential imperfections in the experimental setup and in the polarizer itself which could compromise this procedure. These are: Warped Polarizer Surface At normal incidence, each dielectric/air interface will reflect about 4% of the incident light (unless the surface is coated with an anti-reflective film). At near-normal incidence, the transmitted light fraction varies albeit weakly as we will see - with the angle-of-incidence. A warped polarizer surface presents a varying angle of incidence to the incident light beam as it is rotated, which will generate a spurious variation in the transmitted light fraction. For this reason, we plan to sandwich the polarizer film between two glass plates, to minimize the potential warping and the associated spurious light tranmission. As we show below, it appears that the effect of a warped surface is negligible unless the polarizer surface is badly warped - a local change in the surface normal of 5 o is required to reduce the polarizer s transmission by 0.1%. The actual surface irregularities seem to be of order 1 o (as measured by observing the motion of a laser beam that is reflected off the surface). So it isn t clear whether we need to take this precaution. 1

2 From Born and Wolf (7th Edition, p. 44) the reflection and transmission coefficients for light polarized perpendicular or parallel to the plane of incidence are: R = sin2 (θ i θ t ) sin 2 (θ i + θ t ) R = tan2 (θ i θ t ) tan 2 (θ i + θ t ) T = sin 2θ i sin 2θ t sin 2 (θ i + θ t ) sin 2θ i sin 2θ t T = sin 2 (θ i + θ t ) cos 2 (θ i + θ t ) (1) and so the net reflection and transmission coefficients for light polarized at an arbitrary angle α i to the plane of incidence are: R = R cos 2 α i + R sin 2 α i T = T cos 2 α i + T sin 2 α i (2) In the usual experimental situation, there are two interfaces - at the first light goes from air into glass (or plastic) and at the second the light goes from glass back into air. At each, we can apply Snell s rule n i sin θ i = n t sin θ t to obtain the angle of the light in the transmiitted region given the angle of the incident light. It is then straightforward in idl to compute the fraction of light which is transmitted through a dielectric medium as a function of its dielectric constant n, the angle of incidence θ i, and the polarization angle α i. The results are shown in Figs. 1 and 2. The important result is that the transmission factor varies quite weakly with angle-ofincidence, for example the tranmission factor changes by only as the angle of incidence moves from purely normal to 1 o off-normal. An angle of about 5.5 o is required to change the transmission by 0.1%, and an angle of about 17 o is required to change the transmission by 1%. The magnitude of the effect of angle-of-incidence on transmission is the almost the same for light polarized parallel or perpendicular to the plane of incidence up to about 20 o off-normal incidence, although the sign is different. Improper Polarizer Orientation The polarizer film should be oriented perpendicular to the incident laser beam, but in practice the actual angle-of-incidence will be slightly greater than zero. As shown in Fig. 3, if light polarized at an angle θ is incident on a surface (which could be, for example, the first surface of a polarimeter to measure the light s polarization) at some non-zero incidence angle β, in the referece frame of that surface the light will have a slightly greater polarization angle θ given by ( ) tan θ tan θ = tan 1 (3) cos β The results are shown in Figs. 4 and 5. We should be able to orient the polarizer film with an angle of incidence less than 1 degree, and the incident light will be vertically 2

3 Figure 1: Transmission factor for polarized light parallel (red) and perpendicular (blue) to the plane of incidence, as a function of incidence angle. Figure 2: Fractional change in the transmission coefficient between normal incidence and a given angle of incidence for light polarized parallel (red) and perpendicular (blue) to the plane of incidence. 3

4 E y = E sin θ E E y = E y θ E x = E cos θ E y β E x = E y cos β θ E x E y E x tan θ = = θ = tan θ tan [ ] -1 cos β tan θ cos β Figure 3: Geometry of polarized light incident upon a tilted surface that leads to Eq. 3. polarized with an accuracy of 0.1 o (determined by the accuracy of the digital level), so the associated error is very small, less than degrees. So we expect that our inability to align the rotating stage exactly normal to the incident light does not represent a significant error in determining the transmission axis of the linear polarizer. But note that the effect may not be so insignificant for the actual linear polarizer inside the mse diagnostic. Ray tracing shows that some of the light incident on the polarizer has an angle of incidence of 10 o on some channels. I think this could lead to offsets of several tenths of a degree. These offsets would be channel dependent, since the light rays for different channels have different angles of incidence at the polarizer. Dirt on the Polarizer Surface Dirt on the polarizer surface will scatter or absorb some of the incident light. We estimated previously that to obtain an accuracy of 0.1 o through a series of transmitted intensity measurements, the statistical error on each measurement needs to be better than about 0.1%. So dirt on the polarizer surface becomes an issue if it scatters more than about 0.1% of the incident light or more precisely, the uniformity of the light scattering by dirt needs to be less than 0.1%. According to varsik/dust/dustmicro.html, on a clean optical surface there are about 200 dust particles per square millimeter with a size distribution microns, which collectively obscure 0.98% of the optical surface. After 20 hours of exposure (exposure to what sort of enviroment wasn t described on the web 4

5 Figure 4: Difference between the actual polarization angle and the apparent polarization angle in the reference frame of a tilted surface. The various curves represent different tilt angles (in degrees) of the surface, where zero corresponds to a surface that is normal to the incident light.. Figure 5: Same as Fig. 4, but expanding the data near zero 5

6 site), the fraction of surface occupied by the dust increased to 3.3%. This is a surprising amount of dust for a clean surface. On an circular area 3mm in diameter (the size of the laser beam), there would be about 1400 dust particles. If the dust were distributed statistically, then we would expect a random variation of 1400/1400 3% in the total area covered by the dust over an ensemble of 3mmdiameter circles. Assuming that each dust particle scatters 100% of the light incident upon it, the variation in the transmitted light intensity amongst randomly selected 3mm-diameter beam images would be = which is just a little smaller than the factor of 10 3 or so that would account for the measured anomalies in the measured intensity versus angle. A filter surface that was slightly dirty, or a non-uniform distribution of dust would be enough to account for our measurements. So we should ensure that the polarizer surface is clean prior to use. Also, we can expand the beam with a beam expander from 3mm to 3cm, which increases the total number of dust particles by a factor of 100, and thereby decreases the variability by a factor of 10. Nonuniform Orientation of Transmission Axis Polarizing films are typically produced by stretching a polymer in a certain direction. So presumably the uniformity of the transmission axis is related to the uniformity of the stretching operation. We should look into this issue further - it isn t unreasonable to expect variations of as much as a few tenths of a degree across large polarizing films. Corning quotes a variation of the direction of transmission axis of ±0.5 o over a distance of 3.4 cm for its Polarcor absorptive glass polarizer. Nonuniform Light Absorption Across Polarizer Surface A perfect polarizer would absorb 50% of incident unpolarized light, i.e. it would pass all of one polarization and stop all of the orthogonal polarization. But our polarizing film absorbs about 70the incident light. If the absorption coefficient is not uniform across the polarizer surface (as might be the case if the polarizer thickness were not uniform), the transmitted intensity would obviously vary correspondingly. Rotation of Polarized Light By a Flat Glass Plate It is well known that light reflected from the surface of a dielectric, such as glass, is partially polarized. The reflected light becomes fully polarized at the Brewester angle. Correspondingly, unless the angle of incidence is normal, the transmitted light is also partially polarized. This will cause the polarization angle of linearly polarized light to rotate by a small angle when it passes through a glass plate at non-normal incidence. The transmission coefficients for S- and P- polarized light incident on a dielectric are given by T = ( nt cos θ t n i cos θ i ) [ ] 2 2 sin θt cos θ i sin(θ i + θ t ) 6

7 T = ( ) [ ] 2 nt cos θ t 2 sin θ t cos θ i (4) n i cos θ i sin(θ i + θ t ) cos(θ i θ t ) Upon entry to the glass, n i = 1.0 and n t 1.5 and applying Snell s rule we get θ t = arcsin(sin(θ i )/n t ). Upon exit from the glass, one must apply Eq. 4 again to compute the tranmission coefficient at the second interface. Thus the net transmission is the product of the tranmission coefficients at the first and second air-glass interfaces. I wrote a small idl procedure to evaluate Eq. 4 at the two interfaces. The result is illustrated in Fig. 6 as R T, the ratio of the total perpendicular transmission coefficient to the total parallel transmission coefficient. The S- and P- polarizations are transmitted equally at normal incidence, but at off-normal incidence, the ratio of the S- transmission coefficient to the P- transission coefficient differs by a few percent. At the mse pems and polarizer, the angles of incidence range from normal to about 15 degrees off-normal. Figure 6: Relative transmission of S versus P polarized light through a dielectric with n = 1.5 with air on both sides as a function of the angle-of-incidence. If linearly polarized light is incident on a glass plate, one of the electric field components 7

8 is reduced by a factor R T relative to the other, and so the polarization angle changes from θ to arctan(tan θ/ R T ). The net effect is illustrated in Fig. 7 and amounts to a few tenths of a degree change in the polarization angle for angles of incidence within 20 o of normal. Figure 7: Rotation of polarization angle, in degrees, imposed by the different transmission coefficients for S- and P- polarized light for a simple glass plate with index of refraction = 1.5 at near-normal incidence. Each curve represents the change in polarization angle for the specified angle-of-incidence. Effect of non-normal incidence at PEM The following was taken from The phase shift imposed by a retarder is φ = 2πd(n e n o ) (5) λ where n e is the index of refraction along the optic axis and n o is the index of refraction in the plane perpendicular to the optic axis. When the direction of the light is not perpendicular to the optic axis, the ordinary component still lies in the plane perpendicular to the optic axis and has an index of refraction n o. But the index of refraction for the extraordinary component becomes n e n o n eff = n 2 e cos 2 θ + n 2 o sin 2 θ [ n e 1 n ] e n o cos 2 θ (6) n o 8

9 where θ is the angle between the optic axis and the direction of propagation in the material. The latter approximation applies for the usual limit where n e n o n o. Let β be the angle between the propagation direction and the surface normal. The effective thickness of the material increases as 1/ cos β. It is straightforward to show that the total retardation is given by φ = (n e n o )d sin 2 θ λ cos β = (n e n o )d 1 for rotation parallel to optic axis λ cos β = (n e n o )d cos β for rotation perpendicular to optic axis (7) λ So for a zero-order retarder that imposes a phase shift of φ = π, the effective retardance for rays incident at 20 o off-normal would vary from π cos 20 o = 0.940π to π/ cos 20 o = 1.064π. The effect on the fft amplitudes enters as J 2 (A 1 ) and J 2 (A 2 ) where A 1 and A 2 are the retardances for pem-1 and pem-2 respectively. To the extent that the rays have the same angle-of-incidence at both pems, the angle effect on the retardances will be the same, and so there will be negligible effect on the inferred intensity ratio I 2ω1 /I 2ω2 and a negligible effect on the inferred pitch angle. 9

Müeller Matrix for Imperfect, Rotated Mirrors

Müeller Matrix for Imperfect, Rotated Mirrors From: Steve Scott, Howard Yuh To: MSE Enthusiasts Re: MSE Memo #0c: Proper Treatment of Effect of Mirrors Date: February 1, 004 (in progress) Memo 0c is an extension of memo 0b: it does a calculation for