Measuring Polarization Aberations in the Lab

Measuring Polarization

Aberrations in the Laboratory

THESIS

submitted in partial fulfillment of the requirements for the degree of

BACHELOR OF SCIENCE

in

ASTRONOMY ANDPHYSICS

Author : G.J.J. ‘t Hart

Student ID : 1691570

Measuring Polarization

Aberrations in the Laboratory

G.J.J. ‘t Hart

Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

June 29, 2018

Abstract

In order to improve the the polarimetric performances of VLT’s SPHERE/ZIMPOL polarimeter instrument, polarization dependent beam shifts due to reflection of a coated aluminum mirror need to be taken into

account. Therefore measurements are performed to characterize these beam shifts using a cylindrically shaped beam. In plane-of-reflection

beam shifts, as described by the Goos-Hänchen effect, and out of plane-of-reflection beam shifts, described by the Imbert-Federov effect,

have been measured. A measurement setup has been built which measures relative beam shifts of orthogonal polarizations states for

different angles of incidence. Using the measurements of the Goos-Hänchen effect a beam shift is detected in the plane-of-reflection, where no shift in out of the plane-of-reflection is detected. An effective refractive index of n =1.12 and k =2.57 has been found. Measurements

Contents

1 Introduction 7 2 Theory 11 2.1 Polarization of Light 11 2.2 Goos-Hänchen effect 13 2.3 Imbert-Federov effect 15 3 Methodology 17 3.1 Measurement Setup 17 3.2 Measurements 20

3.3 Calibration of Polarization Maintaining Optical Fiber 21

3.4 Data Reduction 23

4 Results 25

4.1 Measurement Results 25

4.2 Noise in Measurements 25

4.3 Error-Bars in Measurement Results 26

5 Discussion and Recommendations 33

5.1 Systematic Noise Analysis 33

5.2 Recommendations to Improve Measurement Accuracy 33

1

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Introduction

Up to now only young, hot, selfluminous giant exoplanets have been detected via direct imaging (Wright, 2012[1]). In order to detect old, cold exoplanets, more sensitive measurements are necessary to distinguish re-flected light from an exoplanet from the dominant parent star. In order to detect cold exoplanets in reflected visible light, a polarimeter can be used. A polarimeter measures the polarization of the observed light. Polariza-tion of light is the preference of orientaPolariza-tion of the plane of vibraPolariza-tion of the electric field. Light emitted by a star is unpolarized, i.e. the orienta-tion of the plane of vibraorienta-tion of the electric field has no preferred direc-tion. Stam et al. showed, using numerical simulations, that light reflected from Jupiter-like exoplanets is polarized due to the reflective properties of the atmosphere [2]. They suggest that sensitive polarimetry can detect these Jupiter-like exoplanets. Measurements of two orthogonal polariza-tion states have the same intensity for unpolarized starlight, where a po-larized source, such as a planet, would show up in the difference between the measurements. A Jupiter sized planet, orbiting a sun-like star at 1 AU, requires a planet/star contrast of ∼10−8 for a successful detection (Milli, (2013) [3]). Small beam shifts can thus avert cancellation of unpolarized starlight in polarimetric observations.

The high-contrast imaging polarimeter SPHERE-ZIMPOL (SPHERE: Spectro-Polarimetric High contrast Exoplanet REsearch [4], ZIMPOL: Zurich IMaging POLarimeter [5]) is installed at the Very Large Telescope (VLT), which aims for sensitive polarimetric measurements of star-systems to de-tect cold giant exoplanets. SPHERE is a complex instrument consisting of a Common Path Instrument (CPI) and three instrument arms, of which ZIMPOL is one. Currently ZIMPOL’s point source sensitivity is limited

Figure 1.1: Measured beams shift in ZIMPOL instrument (Schmid et al. (in

prep.)). The difference in horizontal en vertical polarization is shown here. Neg-ative and positive signals are visible around the center showing the beam shift.

8 Introduction

Figure 1.2: SPHERE/ZIMPOL‘s optical path. In green mirrors and

half-wave-plates are shown, which contribute to polarization aberrations. The derotator, indicated in red, contains three mirror at large angles of incident, which has the largest beam shift contribution [7].

by polarization-dependent beam shifts (Roelfsema et al., 2011 [6]). Figure 1.1 is an image taken with ZIMPOL showing the shape of the aberrations. Figure 1.2 is a schematic image of the CPI and ZIMPOL. The beam shifts are produced by the mirrors, half-wave plates (indicated in green and red, respectively) and the FLC. The largest beam shift contributions are from fold mirrors at large incident angles. The derotator in figure 1.2 has three mirrors at large angels of incidence, which accounts for the largest beam shifts. Figure 1.1 shows the beam shift as measured by ZIMPOL.

Beam shifts due to mirrors are polarization dependent, i.e. the mag-nitude and direction of the shift is different for different polarizations of light. Due to this difference in shift the centers of the the point spread functions (PSFs) for different polarizations are not at the same place after reflection. Therefore unpolarized light will not cancel-out. Polarization dependent beam shifts, due to mirrors, in the plane of reflection are called the Goos-Hänchen effect and beam shifts out of the plane of reflection the Imbert-Federov effect (see section 2.2 and 2.3).

Beam shifts in the SPHERE-ZIMPOL instrument have to be completely understood in order to increase ZIMPOL’s polarimetric sensitivity in order to detect cold giant exoplanets. This project aims to characterize beam 8

9

shifts due to reflection from an aluminum coated mirror, by means of measurements in the laboratory. Various projects have been done in or-der to measure beam shifts due to mirrors (e.g. Aiello (2009) [8]). For-mer research measured beam shifts for Gaussian shaped beams. This projects uses cylindrical shaped beams, as encountered in astronomical instruments. Furthermore, most researches measured the beam shifts out of focus or for collimated beams. In astronomical instruments the detector is always in focus. The measurements in this projects are therefore done in focus. The goals of this project are:

• Verify that beam shifts are observed in reflection from a coated alu-minum mirror for a cylindrical shaped beam of light in focus.

• Measure the magnitude of the beam shifts and validate analytical formulas which describe the beam shifts.

• Determine whether spatial and/or angular beam shifts are observed in focus.

The outline of this thesis is as follows. Section 2 gives theoretical back-ground on polarization, the Goos-Hänchen effect and the Imbert-Federov effect. The Methodology of the experiments and data-reduction is pro-vided in section 3. Experimental results and their error analysis in section 4. The Discussion and Conclusion are given in, respectively, section 5 and section 6.

2

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Theory

Before going into the experiment, theoretical background is provided. Section 2.1 gives a description of the Polarization of Light and introduces Stokes parameters. In section 2.2 and section 2.3 beam shift effects de-scribed by to the Goos-Hänchen and Imbert-Federov effect are discussed, respectively.

2.1

Polarization of Light

Light can be treated as a transverse electromagnetic wave. An electro-magnetic wave consists of an oscillating electro-magnetic and electric field, both perpendicular to the propagation direction of the wave. The polarization of light is the preferred orientation of vibration of the electric field. Unpo-larized light has no preferred direction of vibration.

Linear polarized light has a fixed orientation of the plane of vibration of the electric field. Every polarization state can be described as superposi-tion of two orthogonal linear polarized components (see figure 2.1). Such a superposition can result in three different kinds of polarization. First, if the oscillations of the two waves are in phase, the resulting wave will again be linearly polarized. Second, if the oscillation of the two linearly polarized waves are 90◦ out of phase, the beam is circular polarized. At last, if the two waves differ in amplitude and phase the light is elliptical polarized. The former two are a special case of elliptical polarization.

Most light sources consist of many emitting atoms. Atoms emit for a very short time (∼10−8s [9]) a wavetrain of linearly polarized light. These wavetrains are constantly emitted with different, randomly oriented, po-larizations. If these changes are fast and in an unpredictable way, the light is called natural or unpolarized light.

In 1852 G.G. Stokes described polarization in quantities which are a function of observables of the electromagnetic wave [9]. These quantities, called stokes parameters, are the defined as follows:

I = hE2_xi + hE2_yi, (2.1) Q= hE2xi − hE2yi, (2.2)

U =2Re(ExEy), (2.3)

12 Theory

Figure 2.1: The polarizations of light. From HyperPhysics, http: //

hyperphysics. phy-astr. gsu. edu, 1 march 2018

where E is the electric field. ~E is a complex vector with an x- and y-component. I is the total intensity. Q is intensity of the light that is lin-early polarized. Q > 0 is horizontal polarized and Q < 0 vertically po-larized. When the light has no preference in one of these two directions then Q = 0. U is the intensity of the light that is linearly polarized with orientations +45◦ (U > 0) or -45◦ (U > 0) polarization. V is the intensity of the light that is circularly polarized light, with V >0 right handed and V < 0 left handed. All polarization states of electro magnetic waves can be described using the four parameters.

We now turn to reflections from a mirror. Figure 2.2 illustrates the def-inition of p- and s-polarization. p is the component of the wave that is parallel to the reflector and s the component perpendicular to it. When the axial system has its z-axis always aligned with the propagation direction of the light, pure p- or s-polarized light just has stokes I and Q unequal to 0. For s-polarization Q= I and for p-polarization Q = −I.

12

2.2 Goos-Hänchen effect 13

Figure 2.2: Convention s- and p-polarization. An incident ray propagating in

medium 0 of reflective index N0gets reflects of a metal mirror of reflective index

N1at angle θ0. The energy entering the metal is assumed rapidly absorbed, thus

the field-magnitude of the transmitted ray is much smaller than the reflected ray. Breckenridge et al. fig.4 (2015) [10]

2.2

Goos-Hänchen effect

Reflection of light does not take place precisely at the surface of a metallic mirror. In reality, the light penetrates into the substance and many layers of atoms account for the reflected wave. Fritz Goos and Hilda Lindberg-Hänchen showed experimentally that these effects result in a spatial and angular shift compared to geometrical predictions [11]. They derived the shift from the Fresnel reflection coefficients, which describe the ampli-tude of reflected waves. The penetration results in a phase shift of the reflected wave, which causes a beam shift (Hecht optics, p.137[9]). The Goos-Hänchen effect (GH-effect) describes the shift due to different phase shifts in p- and s-polarization.

A beam of light can be described as infinitely many plane waves. In [12] a Fourier transform is performed over all these plane waves of the electric field. All waves have a slight different angle of propagation, and thus a different reflective coefficient. This assumption is only valid when the beam size is much larger than the wavelength. They ignore the y-component of the electric field in their analyses, since their approach is only valid for longitudinal shifts, i.e. shifts in the plane of reflection. Each of these waves undergoes a different phase shift which they approximate with a first-order Taylor series. The result they obtain for the

displace-14 Theory

ment, d, of the reflected beam is d = − λ 2π dφ dθ     θ0 , (2.5)

where λ is the wavelength, φ the phase and θ the angle of incidence of the incoming light wave. The derivative is the phase gradient upon re-flection at incidence angle θ0. θ0is the mean incidence angle for converg-ing/diverging beams. This formula is also known as the Artmann’s for-mula [13]. The reflection coefficient phase is dependent on polarization, thus d is dependent on polarization.

Apart from a spatial shift, the GH-effect also contains an angular shift. This shift is not expected to be observed in the measurements in this project. All measurements are done in focus, thus beams coming from different an-gles would still be focused at the same point on the detector. A mathemat-ical description is therefore left out of this report. Due to the angular shift the relative beam shift is expected to decrease at large angles of incidence [8].

The reflection coefficients are dependent on angle of incidence, polar-ization and complex refractive index. The refractive index, N = n+ik, consist of a real and a complex part. Often in literature only n is meant when referring to the refractive index, which indicates the phase velocity of the wave. The complex part of the refractive index is called the extinc-tion coefficient, and indicates the amount of attenuaextinc-tion when light prop-agates through a medium. The dependence of the reflection coefficients is studied by Breckinridge et al. (2015) using a ray tracing simulation. Breck-inridge et al. interpret the beam shift on the mirror as a tilt in the pupil plane of the optical system. Via ray-tracing they calculate the tilt in the pupil plane due to the difference in refractive index at different incidence angles. Using a Fourier transform of the tilted pupil plane they calculate the shift of the point spread function (PSF) in the focal plane (Fraunhofer far-field assumption (Hecht, p. 465 [9]).

Figure 2.3 shows the reflection coefficients for a bare aluminum mirror at a wavelength of 800nm for p-polarization and s-polarization. A tangent to the curve in refractive coefficients is equal to the derivative in eq. 2.5. This gives a prediction of the expected shift at each angle of incidence. Higher order beam shift effects, which scale with higher order derivative, are neglected in eq. 2.5.

The shortcoming of the Breckinridige et al. approach is that it is only valid for converging or diverging beams, and not for collimated beams, since there has to be a phase gradient in the incoming beam. The research 14

2.3 Imbert-Federov effect 15

Figure 2.3:Reflection coefficients for the phase of an electro magnetic wave upon

reflection at incidence angle θ from 0◦to 90◦for a bare aluminum mirror at 800 nm wavelength are shown. φrsand φrpare the reflected phase for s- and p-polarized

light. The green vertical line highlights the reflection phase at 45◦. The red and blue lines show the corresponding slope of φrsand φrp at the 45◦ incident angle.

Breckinridge et al. (2015) [10]

in this report is only concerned about converging incidence beams, thus the Breckinridge approach is valid in this case.

2.3

Imbert-Federov effect

The GH-effect describes longitudinal shifts. Apart from longitudinal de-viations from geometrical optics, also transversal shift are apparent. First order transversal shifts are called the Imbert-Federov effect (IF-effect) or the spin Hall effect of light (SHEL). The IF-effect occurs in reflection from a mirror of±45◦polarized incident light. The IF-effect is a photonic equiv-alent of the Hall effect in solids. The Hall effect in solids is the production of a voltage transverse to the current and to an external magnetic field (Si-mon, [14]). In the IF-effect the spin of the particles is replaced for the spin of photons (i.e. polarization) and the electric potential gradient by the re-fractive index gradient (Hermosa, 2012 [15]). A derivation of the IF-effect is beyond the scope of the report and can be found in Bliokh and Aiello (2013) [16].

The result they obtain for the spatial IF beam shift is d = λ

2π

sin(φs−φp)

tan θ , (2.6)

wave-16 Theory

length of the incident wave, φsand φpthe phase of the s- and p-component of the incident wave and θ is the mean incident angle.

16

3

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Methodology

In order to measure polarimetric beam shifts a measurement setup is built to measure the shifting of the PSF, described by the Goos-Hänchen and Imbert-Federov effect, with high accuracy. The measurement setup is systematically shown in figure 3.1 and photos of the measurement setup are presented in figure 3.2. The main components of the measurement setup are a Laser Modulus Diode (LMD), an aluminum coated mirror and a CCD camera. The beam produced by the LMD is polarized by a linear polarizer and thereafter spatially filtered by a polarization maintaining op-tical fiber. The beam exiting the polarization maintaining opop-tical fiber is focused onto the mirror and re-imaged onto the CCD. The mirror is ro-tated, in order to measure shifts for different angles of incidence. In ad-vance of the beam shift measurements, several calibration measurements are preformed on the polarization maintaining optical fiber. In order to measure the center of the point spread functions, a Gaussian is fitted onto the point spread functions’ cores for different input polarizations. The difference in the mean centers of the orthogonal polarizations defines the beam shift. Section 3.1 provides a detailed description of the measurement setup. In section 3.2 the preformed measurements are discussed. Section 3.3 provides a calibration of the alignment and crosstalk of the polarization maintaining optical fiber. Finally, 3.4 discusses the data reduction meth-ods.

3.1

Measurement Setup

The incident light onto the mirror has to be linear polarized alternatively between two orthogonal states during measurements. We can not simply use a rotatable polarizer, since this would affect the position of the PSF due small imperfections in the polarizer or misalignment. Correction methods are cumbersome, because there is no easy way to measure the shift due to the rotation of the polarizer, without measuring beam shifting effects of the mirror. In order to deal with this problem a PMF is used.

A PMF is an optical fiber which maintains the transmitted polarization. This maintenance only works when the incidence light has its polarization axis aligned with the fast- or slow-axis of the fiber. In case of misalign-ment, cross-talk between the two orthogonal states parallel to the axes of the PMF is present. A detailed investigation of this cross-talk is presented in section 3.3.

An LMD with wavelength 635 nm is used as light source. The LMD in the setup has a stable intensity and is partially linear polarized. The

18 Methodology

light emitted by the LMD is transmitted through a neutral-density filter (ND filter in figure 3.1), in order to increase the exposure time. This re-duces seeing aberrations, because the aberrations are averaged over the exposure time. In order to ensure the polarization of the beam, it gets transmitted through a polarizer, before it is aimed at the PMF head. The light is re-emitted by the PMF tail as a diverging beam. Two lenses, with respectively focal lengths of 75 mm and 45 mm, are used to focus the light slightly in front of the mirror. A diaphragm between the first two focusing lenses cuts out a beam with a diameter of 1.5 mm. With this beam size the PSF core is large and has no visible aberrations due to misalignment. The reflected beam is re-imaged using two lenses, with focal lengths 45 mm and 400 mm, and focused onto a CCD. The re-imaging system ensures a 8.89 magnification of the PSF. The expected shifts are < 800 nm, thus a re-imaging system is needed to magnify the shift, in order to detect it.

The mirror is placed onto a rotatable platform. The re-imaging system and camera are placed on an arm which is connected to the rotatable plat-form of the mirror. The arm moves along with the mirror to measure the beam shifts at different angles of incidence ranging from 30◦ to 85◦.

In order to prevent for seeing effects tubes are placed over the entire optical path from the PMF tail to the camera. Figure 3.2 shows a picture of the measurement setup, with and without tubes.

18

3.1 Measurement Setup 19 Polarizer LASER ND filter PMF Lens (f = 75mm) Diaphragm Lens (f = 45 mm) Detector 75 mm 120 mm 40 mm Lens (f = 45 mm) Lens (f = 400 mm) 50 mm 445 mm 400 mm

Figure 3.1:Systematic image of measurement setup used to measure polarimetric

beam shifts.

(a) Measurement setup without tubes covering optical path

(b)Measurement setup with tubes cov-ering optical path

20 Methodology

3.2

Measurements

Before each measurement, the setup has to be aligned for the angle under investigation. At first the arm is placed in the angle of interest. Thereafter, the mirror is rotated until it is at the preferred angle. The PSF is very sensitive for the alignment of the re-imaging lenses on the arm; a slight change in position of the arm largely affects the position and shape of the PSF. Fine adjustments to the angle of the arm are made until the re-imaging system is best aligned. This is where the PSF shows the least geometric aberrations.

When the setup is properly aligned the tubes are placed back over the optical path of the beam. The used CCD operates at a temperature of ∼ 42oC. While the camera is warming up there are drifts present in the location of the center of the PSF, on the order of ∼ 4 pixels. Therefore, the camera has to be warmed-up, with the tubes setup, in order to prevent these drifts to corrupt the data. All measurements in this research are done after the camera has warmed-up for at least 1h30m.

The GH measurements are performed at angles form 30◦ to 85◦ with intervals of 5◦. At each angle 7 times 60 exposures are taken, alternatively for p- and s-polarization. The polarizer in figure 3.1 is rotated, by hand, 90◦ in order to produce p- and s-polarization. Possible drifts should be present and equal in both p- and s-measurements. The exposure time for each frame is 700ms. This is the maximum frame-rate of the CCD. Each measurement is performed once for each angle. The fiber head is aligned such that its fast-axis is perpendicular to the plane of the table.

The IF measurements have the same number of frames as the GH mea-surements. The fiber tail, i.e. the emitting end of the fiber, is rotated 45◦ w.r.t. the GH measurements, in order to have ±45◦ incidence light onto the mirror. Less measurements are preformed between 30◦and 60◦; mea-surements are only preformed at intervals of 10◦. From 60◦to 85◦every 5◦ the beam shift is measured. The effect is expected to be most significant at large angles.

20

3.3 Calibration of Polarization Maintaining Optical Fiber 21

Table 3.1: Measurement settings for GH and IF measurements. ‘angle polarizer’

is the angle at which the polarizers transmission axis is aligned to the slow and fast axis of the PMF head. This angle is the angle on the rotation-mount and there-fore arbitrary. ‘angle PMF tail’ is the angle at which the slow and fast axis align with p- and s-polarization for GH measurements and±45◦for IF measurements (again arbitrary). ‘Number of polarizer rotations’ is the number of measurements done for an orthogonal pair of polarization angles.

Goos-Hänchen Imbert-Federov

Angle polarizer 41◦, 131◦ 41◦, 131◦

Angle PMF tail 97◦ 142◦

Frames per polarizer angle 60 60

Exposure time 727 ms 727 ms

Number of polarizer rotations 7 5

Total number of frames 840 600

Measured angles of incidence 30◦, 35◦, 40◦, 45◦, 30◦, 40◦, 50◦, 60◦, 50◦, 55◦, 60◦, 65◦, 65◦, 70◦, 75◦, 80◦, 70◦, 75◦, 80◦, 85◦ 85◦

3.3

Calibration of Polarization Maintaining

Op-tical Fiber

Crosstalk in a PMF is mixing of polarization states of the fiber. When the polarization of the incidence light is not properly aligned with the fast- or slow axis of the fiber, interference between the two states occurs. This in-terference changes the net polarization of the light. Polarization crosstalk in PMF has mainly three causes: misalignment of the polarization of the incidence light and the PMF head fast or slow axes, external mechanical stress on the fiber and manufacturing imperfections[12].

The first polarizer and the fiber axes have to be well aligned to mini-mize the crosstalk. When the PMF is mounted, there is no way to precisely tell the orientation of the optical axes, which is indicated on the fiber head, since it is not visible after mounting. In order to find the orientation of the optical axes of the fiber head, light of various polarizations is emitted onto the fiber. The remitted light from fiber tail is transmitted through a second polarizer, as shown in figure 3.3. This second polarizer is rotated until the transmitted intensity is at a minimum. This minimum occurs when the transmission axis of the polarizer and the fast/slow axis of the fiber head are orthogonal. Iteratively the two polarizers are rotated until a absolute minimum is found. This minimum is when the first polarizer’s

22 Methodology

Figure 3.3: Schematic overview of the setup used for the calibration

measure-ments.

transmission axis and the fast/slow-axis have the same orientation. The magnitude of the crosstalk in the PMF is quantified similar to the alignment method of the PMF with the incident polarizer. At first the fiber tail’s fast/slow axis and the second polarizer’s transmission axis are both aligned horizontally. The intensity of the PSF is measured for the first polarizer at angles ranging for 122◦ to 138◦. These angles are the angles on the polarizer mount and are therefore arbitrary. This range is chosen after looking at several angles by eye. After that the second polarizer is vertically aligned, orthogonal to the orientation of the fast/slow axis of the fiber tail, and again the intensity of the PSF is measured for the same angle range of the first polarizer. The crosstalk increases for increasing misalignment of the first polarizer and the fiber head. Figure 3.4 shows the ratio between parallel and orthogonal alignment of the PMF tail and the second polarizer. The measurements show that at an optimal alignment of the transmission axis of the first polarizer and the fast/slow axis of the PMF head. The magnitude of the crosstalk is∼0.3%.

The values of the integrated counts are the sum of all pixels inside the PSF core. The counting is done in a circular aperture around a fitted PSF center. This value is then corrected for background noise, using an annu-lus around the core. The median of the counts inside the annuannu-lus mul-tiplied with the number of pixels in the aperture is subtracted from the counts of the PSF core.

The fit through the data points in figure 3.4 is

I(α) = I0cos2(α+δα) +A0, (3.1)

where I is the measured intensity, I0the maximum intensity, α is the angle of the first polarizer, δαis a phase shift due to the arbitrary alignment of the

angles on the polarizer mount and A0is an offset. A0 is added to account for the leakage due to polarizing crosstalk in the PMF. This is an altered form of Malus’ law for linear polarizers [9]. The minimum value of this fit is at α = 131◦, at which the first polarizer and the PMF head are best coupled.

22

3.4 Data Reduction 23

122 124 126 128 130 132 134 136 138

Angle first polarizer (degrees)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 leakage ratio cos2_-fit Leakage ratio

Figure 3.4: Ratio of flux leaked through PMF due to crosstalk in the fiber.

Mini-mum at θP1 =131 deg with leakage ratio 0.0033.

3.4

Data Reduction

The data consists of 1928x1448 pixel frames. At first a dark frame is sub-tracted from each frame. The dark frame is a median of 200 frames with the same settings for the setup but with the laser turned off. After the dark frame is subtracted, a Gaussian is fitted to one of the frames in order to de-termine roughly the position of the PSF on the camera. This center is used to create a square aperture which cuts out the core of the PSFs from the raw data frames. A Gaussian is fit to the windowed PSF cores. A Gaus-sian is preferred instead of an Airy Disk, since only the center core is fitted. The first Airy Ring is often blurred due to seeing effects.

Even when the camera is warmed up, drifts in the position of the PSF are still visible (see figure 3.5). These drifts are corrected for using a lin-ear fit. The drift during a measurement is linlin-ear and not dependent on p- nor s-polarization. Thus, a fit with equal slope, but different offset is preformed on the p- and s-measurements. The drift is calculated using a least-squares first order linear fit. The drift component is subtracted from the data points. An example of this method is shown in figure 3.5. The data before the linear correction is displayed in figure 3.5a and after the correction in figure 3.5b. The black lines in figure 3.5a represent the linear fit to the data point. The beam shift is calculated using the difference in offset of the two linear fits.

24 Methodology 0 100 200 300 400 500 600 700 800

time(s)

4 2 0 2 4

x center (pixels)

0 100 200 300 400 500 600 700 800

time(s)

4 2 0 2 4

y center (pixels)

P polarizedS polarized

Linear fit

(a)Data before linear correction

0 100 200 300 400 500 600 700 800

time(s)

4 2 0 2 4

x center (pixels)

0 100 200 300 400 500 600 700 800

time(s)

4 2 0 2 4

y center (pixels)

P polarizedS polarized

mean

(b)Data after linear correction

Figure 3.5:Example of the data. These plots show a Goos-Hänchen shift

measure-ment for a 75◦ incidence angle. The first two plots show the measured centers of the PSFs. The latter two have been corrected for a linear drift.

24

4

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Results

In this section the results of the measurements described in section 3 are given. The measurement results are shown in section 4.1. An analy-sis of the noise in the measurements is given in section 4.2 and an error analysis in sections 4.3.

4.1

Measurement Results

The results of the Goos-Hänchen and Imbert-Federov measurements are shown in figures 4.1 and 4.2, respectively. The measurements are pre-formed as described in section 3.2. The graphs show the relative beam shift, i.e. the difference in beam shifts for orthogonal input polarizations. The used mirror is a coated aluminum mirror, with a unknown complex refractive index, k. Recall that the refractive index is given by N =n−ik, where n is the phase velocity coefficient and k the extinction coefficient. A least-squares fit to the data point in figure 4.1 gives a complex part of the refractive index k=2.57. Only k is fitted and n is kept constant at n =1.12 according to measurements preformed by McPeak et al. (2015) [17].

The fit in figure 4.1 calculates an effective refractive index. Mcpeak et al. finds a refractive index for uncoated aluminum of n = 1.12 and k =6.62 at a wavelength of 633nm. The mirror is coated, thus it is possible that the effective refractive index for transmissions, used to compute the reflectivity, is different from the effective refractive index for beam shifts. The reflection from different layers in the coating can result in a different beam shift, and thus a different effective refractive index, compared to bare aluminum.

A fit of a refractive index for the Imbert-Federov measurements gives k =1.3, but with a very high uncertainty and is therefore assumed to be an inaccurate result. The curve in figure 4.2 is the expected Imbert-Federov shift for n=1.12 and k =2.57.

4.2

Noise in Measurements

Tubes cover the optical path of the measurement setup in order to reduce seeing effects. This approach reduces the seeing significantly, such that the seeing is not the limiting factor in the observations. Although seeing is not a limiting factor, systematic errors limit the accuracy of the measurements. In figure 3.5 are small periodic signals visible. In figure 4.3 Large periodic signals are apparent in the Imbert-Federov measurement. This seems to be

26 Results

not the only periodic noise source in the measurements, but there could be multiple periodic noise sources in super position.

A long term measurement, shown in figure 4.4, is preformed to char-acterize the systematic periodic noise in the measurements. The exposure time is 0.998s. On the right side of figure 4.4 is a power spectrum plot-ted of the signal. This is a normalized, discrete, Fourier transform∗. This spectrum is flat except for very low frequencies. Due to zooming in at low frequencies the spectrum is not smooth. The most dominant noise source has a frequency of∼0.02Hz.

The periodic vibrations differed from day to day. The Goos-Hänchen measurements were the first measurements performed with the setup, and have the least systematic errors. The Imbert-Federov measurements, which were done two weeks after the Goos-Hänchen measurements, show much more systematic errors. The source of this sudden increase in noise is unknown up to now. There where no changes in the measurement method or setup, thus the noise source is assumed to be due to an external source.

The triangular data points in figure 4.1 and figure 4.2 show signifi-cantly more noise than all circular data points. Large periodic signals are visible in this data. This increase in noise is not due to the measurements, but appeared suddenly and remained from that day forward.

4.3

Error-Bars in Measurement Results

The error-bars, in figures 4.1 and 4.2, show a 1-σ standard deviation of the mean of the measurements. This error analysis assumes Gaussian-noise which is not entirely true is our case. Figure 4.5 shows histograms of the distribution in measured PSF positions in figure 3.5. These four histograms do not show a Gaussian distribution, which is expected in a seeing limited system. The Gaussian noise assumption is therefor a low-bound estimate for the error in determination of the means of the p- and s-polarization measurements. similar results are obtained for all circular data points in figure 4.1. Figure 4.3 shows a measurement of the Imbert-Federov-effect at 80◦. Such periodic noise is in all measurements in fig-ure 4.2 and the triangular data points in figfig-ure 4.1. Figfig-ure 4.6 shows his-tograms of the distribution in the PFS centers positions of this measure-ment. These histograms differ more from Gaussian noise patterns.

∗_{The power spectrum is made using the ‘periodogram’-function in the python’s}

‘scipy’ package. https://docs.scipy.org/doc/scipy-0.13.0/reference/generated/ scipy.signal.periodogram.html, 19-06-2018

26

4.3 Error-Bars in Measurement Results 27 30 40 50 60 70 80 90

Incidence angle ( )

1.0 0.8 0.6 0.4 0.2 0.0 0.2

Be

am

sh

ift

(

m

)

Goos-Hänchen beam shift in the plain of reflection

Model: n

t

= 1.12, k

t

= 2.57

Measurements

Noisy measurements

30 40 50 60 70 80 90

Incidence angle ( )

1.0 0.8 0.6 0.4 0.2 0.0 0.2

Be

am

sh

ift

(

m

)

Goos-Hänchen beam shift perpendicular to the plain of reflection

Model: n

t

= 1.12, k

t

= 2.57

Measurements

Noisy measutements

Figure 4.1: Measured relative shift for p- and s-polarized incidence light as a

function of angle of incidence. The triangular data point contain significantly more noise than the other points. The curve is a fit of k keeping n constant. The triangular points are not used for fitting the blue line.

28 Results 30 40 50 60 70 80 90

Incidence angle ( )

1.0 0.8 0.6 0.4 0.2 0.0 0.2

Be

am

sh

ift

(

m

)

Imbert-Federov beam shift in the plain of reflection

Model: n

t

= 1.12, k

t

= 2.57

Measurements

30 40 50 60 70 80 90

Incidence angle ( )

1.0 0.8 0.6 0.4 0.2 0.0 0.2

Be

am

sh

ift

(

m

)

Imbert-Federov beam shift perpendicular to the plain of reflection

Model: n

t

= 1.12, k

t

= 2.57

Measurements

Figure 4.2: Measured relative shift for +45◦ and−45◦ polarized incidence light

with respect to the reflection plane of the mirror as a function of angle of inci-dence. The curve is the expected beam shift according to the model for the refrac-tive index fitted in 4.1. No line is fitted to the points, since the measurements are too noisy.

28

4.3 Error-Bars in Measurement Results 29 0 100 200 300 400 500

time(s)

4 2 0 2 4

x center (pixels)

0 100 200 300 400 500

time(s)

4 2 0 2 4

y center (pixels)

P polarizedS polarized

mean

Figure 4.3: Example Imbert-Federov measurements at a 80◦ angle of incidence.

Large systematic noise is apparent in these measurements. The most dominant noise source has a frequency of 0.02 Hz.

30 Results 0 200 400 600 800 1000 1200 1400

Time (s)

2 1 0 1 2

x center (pixels)

Signal

0.000 0.004 0.008 0.012 0.016

Frequency (Hz)

0 100 200 300

Density

Power spectrum

(a)x-position of PSF and corresponding power spectrum.

0 200 400 600 800 1000 1200 1400

Time (s)

2 1 0 1 2

y center (pixels)

Signal

0.000 0.004 0.008 0.012 0.016

Frequency (Hz)

0 20 40 60 80 100

Density

Power spectrum

(b)y-position of PSF and corresponding power spectrum.

Figure 4.4: LEFT: Position of PSF x- (a) and y-position(b). Every second during

25 minutes the position is measured. RIGHT: A discrete power spectrum of the time series on the left side of the figure. The sharpness of the spectrum is due to the discreteness of the calculation method.

30

4.3 Error-Bars in Measurement Results 31 4 2 0 2 4

x center (pixels)

0 10 20 30 40 50 60 70

events

p-polarization, PSF x-position 4 2 0 2 4

x center (pixels)

0 10 20 30 40 50 60 70

events

s-polarization, PSF x-position 4 2 0 2 4

y center (pixels)

0 10 20 30 40 50 60 70

events

p-polarization, PSF y-position 4 2 0 2 4

y center (pixels)

0 10 20 30 40 50 60 70

events

s-polarization, PSF y-position

Figure 4.5: Histograms of the Goos-Hänchen measurement shown in figure 3.5.

The histograms do not show a smooth Gaussian pattern. The distributions have spikes in y-position and are not symmetric.

32 Results 4 2 0 2 4

x center (pixels)

0 10 20 30 40 50

events

p-polarization, PSF x-position 4 2 0 2 4

x center (pixels)

0 10 20 30 40 50

events

s-polarization, PSF x-position 4 2 0 2 4

y center (pixels)

0 10 20 30 40 50

events

p-polarization, PSF y-position 4 2 0 2 4

y center (pixels)

0 10 20 30 40 50

events

s-polarization, PSF y-position

Figure 4.6: Histograms of the Imbert-Federov measurement shown in figure 4.3.

The histograms show a larger deviation from a Gaussian pattern than the his-tograms in figure 4.5. The distributions have more spikes and are less symmetric.

32

5

|

Discussion and

Recommenda-tions

This section provides a discussion of the measurement results of sec-tion 4. Secsec-tion 5.1 provides an analysis of the systematic noise in the mea-surement. Section 5.2 provides suggested improvements for the measure-ment setup in 3.1 and discusses possible future research.

5.1

Systematic Noise Analysis

The deviation from a random-Gaussian-noise pattern is due to systemat-ical errors in the measurements. Large periodic vibrations are visible in most of the measurements. Figure 4.4 shows noise frequencies at roughly the same frequencies as the GH and IF measurements. Still, the observed frequencies can be one or more alias frequencies of the real noise source. The sampling frequency of the measurements in figure 4.3 is 0.727s and the exposure time of the long term measurement in figure 4.4 is 0.998s.

This noise analysis approach has some flaws. The time signal is not continues, thus a power spectrum is not smooth. Due to a falling short of data points, the density of points at the low frequency range of the power spectra limits the close analysis of the specific noise frequency. A more detailed noise analysis is needed in order to improve the accuracy of the measurements.

The periodic vibrations differed from day to day. The GH measure-ments were the first measuremeasure-ments preformed with the setup, and have the least systematic errors. The IF measurements, which were done two weeks after the GH measurements, show much more systematic errors.

5.2

Recommendations to Improve Measurement

Accuracy

A method to improve the build measurement setup is a more detailed noise analysis. If the periodic noise source can be addressed, the system can be made seeing limited, which would make more accurate measure-ments possible. It is not yet clear if the periodic noise is an alias frequency of e.g. 50Hz or due to vibrations in the building.

Apart from noise correction, faster and/or longer measurements could improve the accuracy of the setup. Faster consecutive measurements

be-34 Discussion and Recommendations

tween orthogonal can minimize the effects of the periodic noise. The po-larizer in front of the PMF can be replaced for Ferroelectric Liquid Crystal modulator (FLC). An FLC acts as a half-wave plate which optic axis orien-tation can be changed by applying an electric field on the crystal. A typi-cal switching frequency of an FLC is∼60Hz. Furthermore an FLC can be used to do much longer measurements. The measurements in the research are done by rotating the polarizer by hand. This limits the duration of the measurements and the time between polarization measurements. By im-plementing an FLC and synchronizing it with a camera, faster and more accurate measurements are possible.

Aiello et al. (2009) [8] had a similar approach to measure the spatial and angular GH-effect. Aiello et al. (2009) measures the GH-effect out of focus, in order to measure both the spatial and angular beam shifts. The mea-surements of Aiello et al. are continuous between p- and s-polarization, thus the camera captures frames while the polarizations switches. If the polarizer will be switched with an FLC effectively a combination will be made of the Aiello et al. approach and the approach in this research.

Furthermore, the input variable of the measurements can easily be changed. Measurements can be preformed with different lasers with a different wavelength, to characterize the wavelength dependence of the shifts. Also a broad band source can be used, to see the effect on a PSF of light with different wavelengths. The coated aluminum mirror can be replaced with a golden or silver mirror or with a prism to investigate the polarization dependent beam shifts due to reflection from different mate-rials with different refractive indexes.

Apart from mirrors, half-wave plates and FLCs also induce polariza-tion dependent beam shifts due to birefringence [9]. Birefringence is an optical property of a material, which says that the material has a refrac-tive index dependent on polarization and propagation direction. Due to the difference in refractive index for different polarizations, small mis-alignment or manufacturing imperfections of a retarder can induce a rela-tive beam shift between orthogonal polarizations. The setup used for the calibration measurements (figure 3.3) can be adjusted in order to measure polarization dependent beam shifts due to retarders. The second polarizer must be removed and a half-wave plate or FLC can be placed just after the diaphragm. If half-wave plate or FLC is illuminated with orthogonal po-larizations for different incidence orientations, the relative beam shift as a function of the retarders orientation can be measured.

34

6

|

Conclusion

The aim of this project is to measure the characteristics of Goos-Hänchen and Imbert-Federov shifts in reflection of cylindrical shaped light beam of a coated aluminum mirror in the laboratory. In order to do so, a measure-ment setup has been build which accurately measures relative beam shifts at different angles of incidence. The complex part of the refractive index is fitted to the data of relative beam shifts. Analysis of limiting system-atic noise is preformed and showed that systemsystem-atic periodic noise limits the measurements of which the source is unknown. Future measurements have to improve on the limiting noise factor, in order to more accurately quantify polarization dependent beam shifts.

The first aim of this project is to validate that polarization dependent beam shifts occur in reflection of a beam with cylindrical shape from a coated aluminum mirror. For Goos-Hänchen shifts a clear beam shift in the x-direction (i.e. parallel to the plane of reflection) is observed, whereas no beam shift is observed in the y-direction (i.e. perpendicular to the plane of reflection)(see figure 4.1). This means that it can be concluded that the Goos-Hänchen effect is apparent in reflection of a cylindrical beam of a coated aluminum mirror. The Imbert-Federov effect measurements show on average a shift in the y-direction which is significantly larger than the shift in the y-direction of the Goos-Hänchen measurements. On the other hand, the measurements where not able to verify analytical formulas de-scribing the Imbert-Federov effect. It can be concluded that a shift in the y-direction is detected with the Imbert-Federov measurements, but with low significance. Therefore, more accurate measurements are necessary to correctly quantify the Imbert-Federov effect.

Second, the projects aims to quantify polarization dependent beam shifts. Figure 4.1 shows a fit of the refractive index according to Artmann’s formula. The fitted complex part of the refractive index is kt = 2.57 for a real part nt = 1.12. A quantization of the Imbert-federov effect was not possible due to systematic errors.

At last, a duality of spatial and angular shifts is investigated. Expected was to only see a spatial shift, since an angular shift would not effect the position of the PSF in focus. Aiello et al. 2009 [8] showed that due to an angular shift that the relative shift would decrease for angles larger than ∼ 80◦. The measurement at large angles where very noisy, thus it is not possible to deduce a conclusion about the absence of an angular shift in the data.

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