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Lab on Faraday Rotation Faraday Rotation Tatsuya Akiba∗ Truman State University Department of Physics (Dated: March 6, 2019) This paper reports the results of analyzing the Faraday rotation effect of SF-59 glass subject to a magnetic field generated by current through a solenoid. We verify Malus' Law to determine the orientation of the first polarizer. We set the nominal angle at an optimal φ = 45◦ to make relative intensity measurements at various magnetic field strengths. The relative intensity is measured using a photo diode and a simple circuit that allows for voltage readout. The exact magnetic field strength generated by a current through a solenoid of finite length is calculated and used to compute magnetic field strengths at various currents. A plot of the magnetic field strength integrated over the length of the optical material against the Faraday rotation angle is produced and a linear fit is obtained. The experimental value of the Verdet constant was obtained from the slope of the linear fit, and compared to two theoretical values: one provided by the manufacturer and one predicted by the dispersion relation. I. INTRODUCTION of known polarization state through an optical material in a magnetic field to obtain an experimental value of Faraday rotation is a magneto-optic effect that causes the Verdet constant. The magnetic field is generated by the polarization vector of light to rotate as it passes a solenoid, and we control the magnetic field strength by through an optical material in an external magnetic field controlling the current passing through the solenoid. The [1]. First discovered by Michael Faraday in 1845, it was laser passes through an initial polarizer to set the light the first experimental evidence linking light to magnetic to a particular linear polarization, then goes through the phenomena [2]. As the magnetic field is applied, the ma- optical material in the magnetic field, and passes through terial used for the experiment exhibits circular birefrin- a second polarizer before the relative intensity is read gence, the splitting of the material's refractive index into using a photodiode. distinct values for right and left circularly polarized light [3]. Since right and left circularly polarized light form a II. PROCEDURE basis for polarization states, the initial linearly polarized state can be expressed as a linear combination of the II.1. Verifying Malus' Law right and left circularly polarized states. Now, as the light passes through the optical material, the two circu- Before we carry out actual measurements to quantify larly polarized states propagate at different rates due to the Faraday effect and obtain an experimental value of the difference in refractive indices, causing one circular the Verdet constant, we must identify the initial polar- polarization state to pick up a phase shift relative to the ization state of the ingoing monochromatic light. We do other [3]. As we translate our superposition of circular this by rotating the second polarizer through 360◦ and polarization states after it has passed through the mate- verifying Malus' Law: rial back to linear polarization, we notice that the polar- 2 ization angle has rotated by a certain angle θ with respect I = I0 cos φ, (2) to our initial polarization due to the relative phase shift. The equation that governs the Faraday rotation effect where I0 is the ingoing light intensity, I is the outgoing is: light intensity, and φ is the angle between the ingoing π∆nL polarization state and the polarization angle of the po- θ = VBL = ; (1) larizer [4]. λ where V is the Verdet constant which depends on the optical material used, B is the magnetic field strength, II.2. Verdet Constant and the Optical Material: L is the length of the material, and ∆N is the difference SF-59 in refractive indices for right and left circularly polarized light, and λ is the wavelength of light passing through The Verdet constant is a material-dependent and the material [3]. wavelength-dependent constant that describes the In this experiment, we will pass monochromatic light strength of the Faraday rotation. It follows the dispersion relation given by: π b ∗ [email protected] V = a + 2 2 : (3) λ λ − λ0 2 [3] the efficiency factor depends on the photodiode and the The particular optical material used for this experi- laser orientation, but since these are unchanged through- ment is SF-59 glass, for which we know the constants a, out our experiment, we can treat the efficiency factor b, and λ0 as follows: as a constant. Since it is generally more convenient to perform signal readout for voltage output rather than −9 −1 a = 2843:20 × 10 T ; current output, we set up a simple circuit schematically shown in Figure1. To avoid reverse bias, we keep the re- b = 112:5192 × 10−20 m2=T, and verse bias voltage to well below 300mV as recommended by the manufacturer, and switching between three re- sistors allows us to accommodate this restriction. The λ0 = 175:3 nm three different resistor settings are: 1kΩ, 3kΩ, and 10kΩ. The photocurrent and the voltage output are related by According to our dispersion relation, at λ = 650nm we Ohm's Law [6]: should have: V ≈ 27:6rad=Tm: Vout = IpR: (5) However, the manufacturer claims that the optical mate- From Equations4 and5, we obtain: rial has a Verdet constant of 23 rad/Tm at a wavelength of 650nm [5]. We will compare our experimental value to both of these theoretical values to evaluate the accuracy Vout = RI (6) of our experiment. Thus, in terms of relative intensities, we have: II.3. Measuring Intensity with the Photodiode V I out = ; (7) V0 I0 where I0 is the maximum possible intensity of incident light on the diode when there is no magnetic field and the second polarizer is aligned with the first polarizer, and V0 is the corresponding voltage output. The efficiency factor and the resistance R cancel out, provided that we use the same resistor for the data set. Therefore, we can measure the relative intensities by measuring relative voltage outputs. I p II.4. The Magnetic Field and Calculating the Vout Faraday Rotation We can calculate the magnetic field strength as a func- tion of the position along the axis of the solenoid of finite length by applying the Biot-Savart Law. Let us set up the coordinate system such that the x-axis is aligned with FIG. 1. This is a schematic diagram of a circuit used to mea- the central axis of the solenoid and the origin is located sure incident light intensity using a photodiode. The switch at exactly half-way through the solenoid. Then, allows the use of three different resistors to adjust the ampli- µ NI Z L=2 dx0 fication of the photocurrent to the voltage output. B = 0 R2 x 0 2 2 3=2 2 −L=2 [(x − x ) + R ] A photodiode is an electronic device the converts inci- 0 1 L L dent light intensity into a proportional current signal [3]. µ0NI x + 2 x − 2 = @q − q A : There is some efficiency factor, a proportionality con- 2 (x + L )2 + R2 (x − L )2 + R2 stant, that relates the intensity of light incident and the 2 2 output photocurrent, so we may write: (8) [7] Ip = I; (4) For our system, we know that the length of the solenoid L is 150mm and the radius R is 17.5mm [3]. We can make where Ip is the output photocurrent, is the efficiency a further simplification since the manufacturer claims factor, and I is the incident light intensity. In general, that the field strength can be calculated by 11.1mT/A 3 at the center of the solenoid [5]. From our equation, at III. RESULTS x = 0, we have: III.1. Malus' Law 0 1 ◦ ◦ µ0NI L The second polarizer was rotated through 360 at 10 B0 = @q A increments. The second polarizer's polarization angle in 2 L2 2 4 + R degrees is plotted against the output voltage in Figure ◦ 0q 2 1 2. In addition to the 10 -incremented measurements, we L + R2 µ0NI 4 directly measured the maxima and minima to obtain the =) = B (9) 2 @ L A 0 following results: q ◦ 0 L2 2 1 Vmax;1 = 0:1519V, when θmax;1 = (65 ± 2:5) ; µ NI 4 + R =) 0 = (11:1mT=A)I 2 @ L A ◦ Vmax;2 = 0:1518V, when θmax;2 = (244 ± 2:5) ; Hence, we obtain: ◦ Vmin;1 = 0:0002V, when θmin;1 = (335 ± 2:5) ; and q 0 L2 2 1 4 + R ◦ B =(11:1mT=A) Vmin;2 = 0:0002V, when θmin;2 = (156 ± 2:5) : x @ L A We will use these angle measurements that give max- 0 1 (10) L L imum and minimum voltage output to determine the x + 2 x − 2 @q − q A I nominal angle φ. Specifically, we will say that for our L 2 2 L 2 2 (x + 2 ) + R (x − 2 ) + R experimental purposes, φ = 0 when the second polarizer ◦ is set to approximately θmax;1 = 65 . Our first equality in Equation1 assumes a uniform magnetic field strength along the material, but for a cur- rent through a solenoid of finite length, this is not the 0.18 case as we have shown. For a solenoid along the x-axis 0.16 and a magnetic field strength that depends on the posi- tion along the axis, we can infer from Equation1 that: 0.14 0.12 0.1 dθ = VBdx Z 0.08 (11) V Voltage, Output =) θ = V Bdx; 0.06 ` 0.04 where the integral is taken over the region where the 0.02 optical material lies [5].
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