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Renia Diamantopoulou MSc Thesis Contents Contents 1 Introduction Introduction to polarisation Theory of polarisation Polarimetry/Stokes
Renia Diamantopoulou MSc Thesis Contents Contents 1 Introduction Introduction to polarisation Theory of polarisation Polarimetry/Stokes formalism Mueller Matrices How to measure a polarised signal Modulation & Demodulation process Polarimetric Efficiencies Ellipsometry/Mueller Matrix polarimetry Ellipsometry Design of The Poly-Polarimeter Scientific Requirements Set-up Design Calculation of Photons for Normalised Stokes parameters Modulation Polychromatic Modulation Modulation Scheme for FLCs Model Development( M&m s Code) Poly-Polarimeter s Modulator code Characterisation of FLCs (Ferroelectric Liquid Crystals) General behaviour of FLCs Retardance Characterisation Polarised Fringes Experimental Results Experimental Process Modulation Process Comparison with Model Outlook And Recommendations 34 2 Chap1:Theoretical Approach 1 Introduction The purpose of the current thesis is to present the design and performance analysis of a versatile ellipsometer called The Poly-polarimeter. Its primary purpose is for it to be a complete Mueller matrix spectropolarimeter. This extremely versatile instrument will enable us to measure complete Mueller matrices and Stokes vectors with high polarimetric efficiency and high accuracy (4x4 Mueller Matrix with non-diagonal elements 10 3 and 10 2 diagonal elements)at high speed ( seconds). In astronomical polarimetry it is essential to fully understand the polarimetric properties of the instrument that is used to take the measurements, in order to be able to disentangle real astronomical signals from instrumental effects. For such astronomical instruments it is highly important to also fully understand the polarimetric properties of the individual optical elements. These instruments introduce two types of polarization is the instrumental polarisation which is the polarization that is generated by the system and the other effect is the cross talk as one polarization state is transformed into another as the light passes through the instrument. These instrumental can be caused by reflection and/or refraction processes, birefringent behaviour of material and in general, any deviation of a circularly symmetric optical system (with respect to the optical axis). therefore, it is crucial to characterise the properties of both single optical elements and the complete set up of the optical system designed to perform accurate polarimetry. The Poly-polarimeter will be able to do this operation in the wavelength range of nm and has two modes: spectropolarimetry and imaging. The first mode will obtain polarimetric measurements of the sample with a spectral resolution of 2.5nm. The second one will provide polarimetric images of the sample with a spatial resolution of 30 nm (see section 2.1 for detailed description of the scientific requirements). It will also be able to analyse samples in both reflection and trnasmission. The polarisation properties of the light and the process of measuring it is explained in the current chapter. In chapter 2, the design requirements and the development of The Poly-polarimeter is discussed. The following chapters describe a comparison between the theoretical predictions, the results from a model and the actual lab measurements. The last chapter of the current thesis presents the conclusions drawn and recommendations for future improvements based on the presented results. 1.1 Introduction to polarisation Theory of polarisation Polarisation is a wave property which describes the orientation of its oscillations. It is an important physical property of electromagnetic waves which is connected with the transversality character, with respect to the direction of propagation, of the electric and magnetic field vectors. Under this respect, the phenomenon of polarisation is not restricted to electromagnetic waves, but could in principle be defined for any wave having a transverse character, such as, for instance, transverse elastic waves propagating in a solid, transverse seismic waves, waves in a guitar string, and so on. On the contrary, polarisation phenomena are obviously inexistent for longitudinal waves, such as the usual acoustic waves propagating in a gas or in a liquid. In general, polarisation is created whenever the symmetry around the propagation direction of the light is broken.this is the case for anisotropic optics such as polarisers and retarders, but this breaking 3 POLARIZATION Chap1:Theoretical Approach e theoretical ratio of power transmitted between antennas of different polarization. These ratios due to effects such as reflection, refraction, and other wave interactions, so some practical ratios of symmetry can also be due to the change of direction of the light itself in a scattering process or due to the presence of magnetic fields.[1] Table 1. Polarization Loss for Various Antenna Combinations Since we are interested in studying the light as an electromagnetic wave lets consider an electromagnetic, monochromatic plane Ratio wave of Power of angular Received frequency ω. to It Maximum is propagating Power in vacuum along a direction that we assume as the z-axis of a right-handed reference system. In a given point of space, Receive Antenna Theoretical Practical Horn Practical Spiral the electric and magnetic field vectors of the wave oscillate in the x-y plane according to the following Polarization equations: E x (t) Ratio = E 1 cos(ωt in db -φ 1 ) as, Ratio E y (t) = ERatio 2 cos(ωt in db -φ 2 ), as where RatioE 1, ERatio 2, φ 1, in and dbφ 2 are as constants. Ratio The same oscillation can also be described in terms of complex quantities by writing ertical 0 db 1 * * N/A N/A lant (45E or 135E) -3 db! * * N/A N/A orizontal - 4 db 0 E x (t) = Re(ε 1 e iωt -20 db 1/100 ), E y (t) = Re(ε 2 e iωt N/A N/A ) (1) ircular (right-hand or left-hand) -3 db! * * * * where ε 1 and ε 2 are given by orizontal 0 db 1 * * N/A N/A lant (45E or 135E) -3 db! ε 1 = E 1 e iφ 1, ε* 2 = E 2 e iφ 2 * N/A N/A (2) ircular (right-hand or left-hand) -3 db! * * * * The polarisation of an electromagnetic wave is defined as the orientation of the electric field ircular (right-hand) vector. Recall that the 0 db electric field1 vector is perpendicular * to * both the direction * of travel* and the ircular (left-hand) magnetic field vector. - 4 The db polarisation 0 is described -20 dbby the geometric 1/100 figure -10 traced db by the 1/10 electric lant (45E or 135E) field vector upon a stationary -3 db plane perpendicular! to* the direction* of propagation, * as the wave * travels through that plane. An electromagnetic wave is frequently composed of two orthogonal components e same as theoretical E x and E y as shown in Figure 1. ansmit and receive antenna polarization will give the same results. ion of an defined as the c field vector. Antenna with two ield vector is orthogonal conductors e direction of E y Y E N field vector. x cribed by the Direction of Travel y the electric tionary plane N X direction of The sum of the E field vectors determines the sense of polarization wave travels ectromagnetic sed of (or can Figure Figure1. 1: Polarization polarisation Coordinates Coordinates o orthogonal Figure 1. This may be due to the arrangement of power input leads to various points on a flat raction of active elements The geometric in an figure array, traced or by many the sum other of the reasons. electric field vectors over time is, in general, an ellipse as shown in Figure 1. Under certain conditions the ellipse may collapse into a straight line, in which case the polarisation is called linear. In the other extreme case where the two components are ure traced by the of equal sum magnitude of the electric and 90 field out ofvectors phase, the over ellipse time willis, become in general, circular as an shown ellipse in Figure as shown 2. Thus in nditions the ellipse may collapse into a straight line, in which case the polarization is called linear. eme, when the two components are of equal magnitude and 90E out of phase, the ellipse will wn in Figure 3. Thus linear and circular polarization are the two special cases of elliptical 4 Chap1:Theoretical Approach linear and circular polarisation are the two special cases of elliptical polarisation. Linear polarisation may be further classified as being vertical, horizontal, or slant. Figure 2 depicts plots of the E field vector while varying the relative amplitude and phase angle of its component parts. The composition of two orthogonal oscillations of the same frequency gives rise to an ellipse. As we are mostly interested in studying astronomical objects, we will try to analyse the background theory of processing the obtained data from satellites or telescopes. Figure 2 depicts plots of the E field vector while varying the relative amplitude and phase angle of its component parts. Ratio of E y E x 4 Wave is travelling toward viewer - Out of the paper Vertical polarization 2 1 Counter Clockwise RHCP Clockwise LHCP 1/2 0 Horizontal polarization -180E -135E -90E -45E 0E +45E +90E +135E +180E Phase angle between E Field Vectors Figure 2. Polarization as a Function of E y/e x and Phase angle Figure 2: polarisation as a Function of E y /E x and Phase angle For a linearly polarized antenna, the radiation pattern is taken both for a co-polarized and cross polarized response. The polarization quality is expressed by the ratio of these two responses. The ratio between the responses must typically be great (30 db or greater) for an application such as cross Polarimetry/Stokes formalism B/2 B polarized jamming. For general applications, the ratio 0 E 6B indicates system power loss due to polarization mismatch. For y 4B All astronomical circularly polarized sources antennas, are partially radiation patterns polarised are usually to some degree. In order to characterise the polar- 2B isation state from the astronomical source, several formalisms have been developed. Polarisation taken with a rotating linearly polarized reference antenna. The reference antenna rotates many times while taking in astronomical 6B measurements contexts around the isazimuth considered of the antenna to that be is best beingdescribed by the Stokes-Mueller E x formalism. The tested. The resulting antenna pattern is the linear polarized 4B polarisation of an astronomical source can be describedzat first approach with the four-term Stokes 2B gain with a cyclic ripple. The peak-to-peak value is the axial vector S B ratio, = (I, and Q, represents U, V).The the polarization Stokes quality formalism a circular is complete in the sense that it describes all partially 0 polarisedpolarized polarisation antenna. The states typical Figure RWR antenna 3 represents has a maximum the Stokes parameters. As far as the four parameters 3 db axial ratio within 45E of boresight. X in Stokes vector are concerned, I denotes the intensity regardless of polarisation, Q and U describe the two-dimensional state of linear polarisation, and V represents circular polarisation. For any antenna with an aperture area, as the aperture Figure 3. Circular Polarization - E Field E. Landi Degl'Innocenti: The Physics of Polarization is rotated, the viewed dimension along the axis remains constant, while the other viewed dimension decreases to zero at 90E rotation. The axial ratio of an antenna will get worse as the antenna is rotated off boresight because the field contribution from the axial component will remain fairly constant and the other orthogonal component will decrease with rotation. I Y Q u V Figure 3: Pictorial representation of the Stokes parameters as seen looking down the beam FIGURE 1. Pictorial representation of the Stokes parameters. The observer is supposed to face the radiation source. 5 where the notations are similar to those employed in the former equation and where the indices f and s stand, respectively, for the fast-axis and the slow-axis. The ideal retarder acts by introducing a supplementary phase factor, called retardance in the electric field component along the slow axis. If 5 = TT/2, the retarder is also called a quarter-wave plate, if 5 IT, it is called a half-wave plate, and so on. It can be easily Chap1:Theoretical Approach The Stokes vector is defined as : I E xe x + E ye y Q E S = = xe x EyE y U E xe y + EyE = x V i(eye x E xe y ) I 0 + I 90 = I 45 + I 45 = I RHC + I LHC I 0 I 90 I 45 I 45 I RHC I LHC, (3) where E x and E y are given by eq. (1), (2). The linear combination of Stokes parameters for a single intensity measurement after filtering for linear or circular polarisation can be given from the following equations. I = I 0 + I 90 = I 45 + I 45 = I RHC + I LHC (4) I = I 0 + I 90 = 1 2 (I + Q) (I Q) = 1 2 (I + U) (I U) = 1 2 (I + V) + 1 (I V) (5) 2 Q = I 0 I 90 = 1 2 (I + Q) 1 (I Q) (6) 2 U = I 45 I 45 = 1 2 (I + U) 1 (I U) (7) 2 V = I RHC I LHC = 1 2 (I + V) 1 (I V) (8) 2 Eq.5-7 are an example of optimal demodulation for the modulation scheme we used in our project (for more details see section 1.2 for definition of modulation/demodulation). Notice the structure of the right term of Eq(5) and Eq(6). It becomes obvious that polarimetry is a differential technique. To verify that the light is linearly polarised in the vertical direction, we should measure the intensity through a vertical polariser but also through a horizontal polariser. The polarised Stokes parameters Q, U, V are all obtained through differences in such measurements. (see i.e. Eq.(6) third part) We have witnessed many cases where the absolute values of Q, UandV are not relevant, therefore the Stokes vector is normalised by the intensity I. With this method the normalised Stokes parameters are independent of the photometric accuracy of the instrument. The most common used forms of the Stokes parameters are the fractional polarisations Q/I, U/I, V/I. By using the normalised values the computation of the degree of polarisation (P) can also be obtained as follows: Q 0 P = 2 + U 2 + V 2 1 (9) I It would also be wise to define the degree of linear polarisation (P L ) and the angle of linear polarisation (φ L ): P L = Q 2 + U 2 ; (10) I 6 Chap1:Theoretical Approach φ L = 1 2 arctanu Q (11) From the definition of Stokes parameters it is easy to conclude that I 2 Q 2 U 2 V 2 0. Note that I 0 but Q, U, V can be either positive or negative. In a homogeneous isotropic medium there is no preferable polarisation mode (except of the case of astrophysical magnetised plasmas). However, for a preferred direction there are always two orthogonal modes that can propagate through the medium without changing their polarisation form. Although the polarisation of these two modes remains unchanged, they travel at different velocities in the case of a birefringent medium (the medium has two refractive indices, one for each mode). In a linearly birefringent medium its modes have linear polarisation. Radiation that has exactly the polarisation of one of those two modes will not be changed, but for any other polarisation angle, or for circular polarisation, the polarisation form will change as the radiation passes through the medium. Circular birefringence on the other hand causes relative phase shifts between two circularly polarised general, birefringence will be elliptical. [3] Mueller Matrices What can be concluded from above is that the Stokes parameters denote the flow of radiant energy in specific vibrations of the electromagnetic field and all four are expressed in the same units. When radiation passes through a medium the state of polarisation may change. The interaction of the Stokes parameters with matter can be described by introducing the 4x4 Mueller matrix with a general structure given by: I I Q I U I V I I Q Q Q U Q V Q M = I U Q U U U V U I V Q V U V V V Thus, the incoming Stokes vector is connected with the incoming vector through the equation[4][5]: S out = M S in, (12) The first element in every Mueller matrix M[1,1] represents the transmission of unpolarised light. In the cases where we use normalised Stokes parameters then the Mueller Matrix can be normalised with the first element M[1,1]. In a complicated optical system which consists of n elements, each of the n elements is represented by a different Mueller matrix. The total Mueller Matrix of the whole system is given by the equation: S out = M n M n 1...M 2 M 1 = M total S in, (13) We have to be extremely careful with the order of the multiplication of the matrices since the elements 4x4 Mueller Matrices are multiplicative but not commutative. In a 3-D coordinate system formed by [Q/I, U/I, V/I], the normalised Stokes vector for fully polarised light is located on a sphere with radius 1. For almost all conceivable astronomical objects, Mueller matrix is the most general description of the processing of polarised radiation. In case where we are not absolutely certain about all the elements in the Matrices, the safest path to follow is to examine the behaviour of all optical elements contributing to the polarisation and then decide whether we should neglect them or try to measure 7 Chap1:Theoretical Approach them. Some Mueller matrices used in applications express the modification of the Stokes parameters due to the properties of the medium they travel in. They are normally functions of positions within the medium. Another area of application using Mueller matrix is the scattering of sunlight within the planetary atmospheres. 1.2 How to measure a polarised signal A recent field of growing interest is the detection of exoplanets. Since the signal of exoplanets is polarised we need to figure a really accurate way to calculate the receiving signal. The very first thing one needs to do is to observe the astronomical source (i.e. exoplanets) with a polariser positioned at 0 and at 90. The intensity measured in each case is different. This process of rotating the polariser to take different intensity measures is called modulation. To complete the process one needs to take at least two measurements and this is also obvious from Eq4-Eq8. In order to receive the signal coming from the astronomical source one needs to practically subtract those two measurement we did with the polariser. This process is called demodulation. So after the modulation and demodulation process what you will result with is the signal from the object you study Modulation & Demodulation process Recalling the definition of the Stokes parameters in equation 4-8, one requires at least two photon flux measurements to retrieve I, Q, UorV. The modulation of the polarimeter describes how the various measurements yield the Stokes vector. In principle to measure Q and U one could rotate a polariser, however, this makes the measurement to be dependent on the polarisation properties of the optics after and before the analyser. Having a fixed analyser makes the measurement independent of the optics after it, in other words the transmission of the system is reduced. Some times it is necessary to introduce additional polarisation optics acting as modulators. Their job is to convert the polarised signal into the polarisation that analyser actually analyses. Four measurements with a half wave then yield (I±U)/2 and (I±Q)/2). A half wave plate (HWP) which shifts the polarization direction of linearly polarised light (e introduces a phase difference of radians between perpendicular axes) It is of high importance that the read-out of the detector is synchronised to thew modulation sequencing. Such a modulation scheme can be generalised with the use of liquid crystals. [1] A single measurement of the polarimetric signal is not enough to determine all four Stokes parameters of the incoming radiation. In order to change the polarimetric signal so as to measure all four Stokes parameters one has to change the orientation of the optical axis or insert a rotation to the retarder. Since we are able to measure only intensities, we are restricted to detecting linear combinations of the
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