Faraday Effect Physics 341 – Advanced Physics Laboratory I

Faraday Effect Physics 341 – Advanced Physics Laboratory I. Introduction The Faraday Effect is the rotation of the polarization plane of linearly polarized light when the light propagates through a medium along the direction of an applied magnetic field. Virtually any material--solid, liquid, or gas—exhibits Faraday rotation. The extent of the rotation may be described empirically by the formula = , where is the rotation angle in degrees, B is the magnetic field in Tesla, L is the sample length in meters, and V is a material-dependent∆ constant known as the Verdet constant. Your goal∆ in this experiment is to measure the Verdet constant of a sample of water. Students undertaking this experiment will need to understand the rudiments of how to carry out calculations involving light polarization. A concise primer on light polarization is presented in Section II. The theoretical prediction of the value of the Verdet constant for a given material is a complicated undertaking, involving the effect of the applied magnetic field on the electronic wavefunctions in the medium, and the subsequent interaction of the medium's electrons and the propagating light. We can however determine a connection between optical activity (the rotation of the plane of linearly polarized light) and the propagation velocities of left- and right- circular polarization states using classical optics. This connection is described in Section III. A semiclassical model explaining the Faraday effect in solids is presented in the text by Fowles listed in the references. The rotation of light is a small effect in most materials at attainable magnetic fields, so it is not easy to measure accurately. In our version of the experiment, a small oscillating magnetic field is applied using a solenoid. Although the field is relatively small, its frequency is known precisely, and a lock-in amplifier can be used to measure the resulting signal within a small bandwidth of the expected frequency. Electrical noise at other frequencies is thereby rejected, so the lock-in technique allows a very considerable enhancement of the signal-to-noise ratio. This approach is very commonly used in precision measurements of small signals. The experimental set-up and lock-in amplifier are described in Section IV. II. Understanding Light Polarization A. Polarization and Jones Vectors Light is a transverse wave, with the electric and magnetic field vectors perpendicular to each other and to the direction of propagation , where the wavevector has magnitude and points in the direction of propagation. In fact, the vectors , , and form a 2 �⃗ �⃗ right-handed triple (like the vectors , , and ). We usually describe the polarization �⃗ �⃗ �⃗ ̂ ̂ � state by specifying the vector. We will consider the case of light propagating in the z- direction, with the E field in the form of a harmonic traveling wave. The most general polarization state is therefore�⃗ cos( + ) = = ( ) cos( + ) 0 − 0 − where Re denotes�⃗ taking the real part of the complex quantity. As a shorthand, we � 0 � �� 0 � � suppress writing Re and the common − exponential factor ( ), and the resulting − abbreviation = is known as the Jones vector. We can further simplify the 0 notation by realizing�⃗ that the overall phase of the electric field is not usually of interest, � 0 � so letting = , we can write = − 0 up to an overall phase constant (i.e. up�⃗ to a� multiplicative� factor of ). 0 As a specific example of an important polarization state, consider left -circularly polarized (LCP) light, whose Jones vector is given by . Remember that the exponential phase factor ( ) and Re have been omitted1 for brevity; restoring them, we obtain the actual electric field− components as � � ( ) 1 ( ) cos( ) = = / ( ) = . − sin( ) − Note that the above implies that + = 1, so that the tip of the electric − field vector � � �� 2 − �� moves on a circular path about the2 origin.2 If we fix z (say z = 0 for− simplicity), − the electric field components are given by cos = , sin so that the electric field vector rotates in a counterclockwise circle when viewed against � � � � the direction of propagation. A similar analysis indicates that right circularly polarized (RCP) light is represented by the Jones vector . Often it is convenient to normalize Jones vectors;1 that is, to multiply by a constant so the result has unit amplitude. The amplitude of� −the� electric field vector is given by = = . 2 1 2 ∗ 1 The vector is the normalized Jones� �⃗vector� �⃗for∙ � ⃗LCP light; hence, is the 1 1 Jones vector for LCP light with amplitude . √2 � � √2 � � 0 0 B. Polarizers, Rotators, and Retarders: Jones Matrices Several optical components are used to alter the polarization state of light. A polarizer transmits the component of the electric field parallel to its axis and eliminates the other component. A rotator rotates the polarization state of linearly polarized light. A retarder adds a phase shift (retardation) to the component of the electric field parallel to its slow axis, relative to the component of electric field parallel to its fast axis. The propagation of light through a series of optical components is conveniently handled using 2 × 2 Jones matrices. If the matrix describing an optical component is given by and the incident Jones vector is , the transmitted electric field has the Jones vector � � �� = . + Note that the components a, b, c, d of the Jones matrix and the components Ex, Ey of the � � �� + � Jones vector may be complex numbers . A series of optical elements can be handled by successive multiplication by the appropriate Jones matrices, e.g. 2 2 1 1 � � � � �� if the light first passes through the 2component2 1 with1 Jones matrix and then 1 1 � � passes through the component with Jones matrix . 1 1 2 2 We can determine the elements of a Jones matrix� by considering� the effect of an optical element on two orthogonal polarization states2 (basis2 vectors). As an example, let's find the Jones matrix for a linear polarizer with its axis along the x-direction. We set = since the x-component of the electric field is passed, and 1 1 � � �0� = �0� since the y-component is blocked. Evaluating the products, 0 0 � � �1� = �0� = so a = 1 and c = 0. Also = = so b = d = 0. 1 1 0 0 Thus the desired0 Jones0 matrix representing a linear polarizer 1 with its axis 0along the x- � � � � 1 � 0� � � � � � � � � � � direction is . Similar reasoning can be used to obtain the Jones matrix for a linear 0 0 0 0 polarizer oriented along the y-axis, which is , and that of a linear polarizer with its � � 0 1 1 1 axis oriented at 45° to the positive x and y axes,� which� is . 1 1 1 The Jones matrix for a phase retarder can be found by applying2 similar reasoning. We � � have seen that LCP light has Jones vector . Two Jones vectors , are 1 1 orthogonal if = 0, where the asterisk denotes complex conjugation.1 2 Hence LCP √2 �� ⃗ �⃗ ∗ and RCP polarizations1 2 are orthogonal since = 0. By requiring that a �⃗ ∙ �⃗ ∗ left circular polarizer (also known as a quarter1-wave1 plate)1 1 passes LCP polarization and √2 √2 − blocks the orthogonal RCP polarization, one can� show� ∙ � that� �the� Jones matrix for a left 1 circular polarizer is . 1 1 The Jones matrix 2for a rotator− can be found using similar considerations. Say = � � represents a rotation by θ about the z-axis so that = and 1 cos � =� . Then we can find the components of the Jones�0� matrix�sin �M as before, and 0 −sin cos sin the result is = . �1� � cos �sin cos − � � III. Optical Activity Optically active materials rotate the plane of polarization of linearly polarized light. Such materials therefore act as rotators. This property stems from the fact that the propagation velocities of LCP and RCP light differ in optically active materials. To understand the connection of optical rotation and propagation velocity, let us take the refractive indices of LCP and RCP polarizations to be and . The propagation = = velocities will therefore be and ; the wavelengths in the medium will be = and = where λ is the wavelength in vacuum; and the wave vectors will be = = and = = . Suppose the linearly polarized light incident on 2 2 2 2 the optically active medium has Jones vector . We can express this as a linear combination of LCP and RCP light, as 1 0 = �+� . 1 1 Remember that we have suppressed 1writing 1the common1 factor ( ). After �0� 2 �� 2 �−� propagating a distance L through the medium, the each polarization state,− traveling at its own velocity, acquires a phase factor: + 1 1 1 1 1 ( ) / 1 ( ) / 1 ( ) / =2 �� 2 �−� + 2 1 + 2 1 −1 2 − − 2 = ( ) / �� + � � � 2 − + 2 ( + )/2 − = ( ) / �� ( − −)/2 + 2 cos = ( ) / � − � −sin − where we have used the abbreviation + 2 = ( ) /2. Discarding the overall phase � � cos factor, we see that the Jones vector after− transmission through the medium is , − sin so the light is still linearly polarized but the polarization direction has rotated by angle θ. � � Note that if > 0, the polarization vector has rotated counterclockwise looking− out of the origin along the positive z-axis (along the direction of propagation)—such a medium, where > , is called levorotatory. If instead < , the sense of rotation in opposite and the medium is called dextrorotatory. IV. Experimental Apparatus A. Experimental Set-Up A schematic of the experiment is shown in Figure 1. Polarized light from a helium- neon laser is incident on polarizer P. The orientation of P with respect to the laser polarization can be adjusted so that the laser beam has a convenient intensity—large enough to measure but not so large as to saturate the photodiode detector. The light is transmitted through a sample chamber S containing distilled water.

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