Continuous Wave Magneto Optic Kerr

Transcription

1 Indian Institute of Science Education and Research, Kolkata Advanced Experimental Physics (PH4201) Semester 8 Laboratory Report Continuous Wave Magneto Optic Kerr Effect Submitted by: Roopam K. Gupta 11MS072 B.S - M.S Department of Physical Sciences Under the Guidance of: Dr. Chiranjib Mitra IISER Kolkata

2 Abstract In this report, three experimental setups, designed to study the magnetization characteristics of Magnetic samples by utilizing the Magneto-Optical Kerr Effect(MOKE) are constructed. Linear methods are investigated as a means of studying static magnetization properties of the sample. Data are taken for single layer thin films.

3 0.1 Introduction MOKE can be categorized by the direction of the magnetization vector with respect to the reflecting surface and the plane of incidence. Polar MOKE When the magnetization vector is perpendicular to the reflection surface and parallel to the plane of incidence, the effect is called the polar Kerr effect. To simplify the analysis, near normal incidence is usually employed when doing experiments in the polar geometry. Longitudinal MOKE In the longitudinal effect, the magnetization vector is parallel to both the reflection surface and the plane of incidence. The longitudinal setup involves light reflected at an angle from the reflection surface and not normal to it, as above in the polar MOKE case. In the same manner, linearly polarized light incident on the surface becomes elliptically polarized, with the change in polarization directly proportional to the component of magnetization that is parallel to the reflection surface and parallel to the plane of incidence. This elliptically polarized light to first-order has two perpendicular E vectors, namely the standard Fresnel amplitude coefficient of reflection r and the Kerr coefficient k. The Kerr coefficient is typically much smaller than the coefficient of reflection. Transversal MOKE When the magnetization is perpendicular to the plane of incidence and parallel to the surface it is said to be in the transverse configuration. In this case, the incident light is also not normal to the reflection surface but instead of measuring the polarity of the light after reflection, the reflectivity r is measured. This change in reflectivity is proportional to the component of magnetization that is perpendicular to the plane of incidence and parallel to the surface, as above. If the magnetization component points to the right of the incident plane, as viewed from the source, then the Kerr vector adds to the Fresnel amplitude vector and the intensity of the reflected light is r + k 2. On the other hand, if the component of magnetization component points to the left of the incident plane as viewed from the source, the Kerr vector subtracts from the Fresnel amplitude and the reflected intensity is given by r k 2. Quadratic MOKE In addition to the polar, longitudinal and transverse Kerr effect which depend linearly on the respective magnetization components, there are also higher order quadratic effects, for which the Kerr angle depends on product terms involving the polar, longitudinal and transverse magnetization components. Those effects are referred to as quadratic Kerr effect. Quadratic magneto-optic Kerr effect (QMOKE) is found strong in Heusler alloys such as Co 2 F esi and Co 2 MnGe. 1

4 In this semester I have studied the longitudinal MOKE for which the calculation are shown in the next section. In the calculations first I have shown the effect on Electric field and then I have also tried to include the signal detection of the photodiode. 0.2 Theoretical (Optical) Understanding of Longitudinal MOKE Figure 1: Schematic MOKE setup for showing the calculations Starting with Jones formalism, We can write any electric field vector as: [ ] E0p e E = iφp E 0s e iφs Here: φ p : phase in horizontal component. φ s : phase in vertical component. For a left circularly polarized light we can write, with E 0p = E 0s and φ s = φ p + π/2 [ ] E0 e E iφp L = E 0 e iφp+π/2 Normalizing the above we get: And similarly: E L = 1 [ ] 1 2 i E R = 1 [ ] 1 2 i 2

5 Now we can write the normalized Jones vectors as: [ ] [ ] 1 E h = and E 0 0 v = 1 Where E h : Horizontal component and E v : Vertical Component of the polarized state. There are some points to be kept in mind before doing the calculations: 1. Boundary Boundary conditions (according to solutions of Maxwell s equations) dictate that there has to be a continuity of electromagnetic field vectors across any interface. 2. For reflection boundary conditions dictate the time independent relation connecting incident and reflected waves. Basically, ω i = ω r. [ω i : frequency of incident wave, ω r : frequency of reflected wave.] Effect of optical element on polarized state can be represented by 2 2 matrix Jones Matrix. Lets say that the polarizer is set at θ p relative to ˆp direction. [ ] cos θp 0 JonesM atrix[p olarizer] = 0 sin θ p When incident E i wave passes through polarizer it becomes: E i = E 0 cos θ p ˆp + E 0 sin θ p ŝ When polarized light is incident upon a magnetic sample its horizontal and vertical components are modified differentially by ˆp and ŝ components of sample s magnetization. Magnetization can be written as linear superposition of ˆp and ŝ components. E r = S E i S = m 2 t S t + m 2 l Sl m t = M t ; m l = M l M s M [ s S t r t = pp rps t ] [ S l r l = pp rps l ] r t sp r t ss All the r s are called Fresnel coefficients. In the symbol r l sp, l: longitudinal effect, sp: relating reflected s-wave to incident p-wave. For ss and pp: Represent that how much of the polarized light is reflected. For sp and ps: Give rise to net rotation and elliptical polarization. r l sp r l ss 3

6 Since Magnetization is restricted on the plane. Considering the Longitudinal Kerr Effect: m 2 t + m 2 l = 1 ( ) ( ) nβ β rpp l β nβ = nβ + β rss l = β + nβ ( ) rps l = rsp l sin θβκ 2 = n 2 β (nβ + β )(β + nβ ) ( ) ( ) nβ β rpp t κ 2 sin 2θ = nβ + β 1 + n 2 (n 2 cos 2 θ 1) + sin θ ( ) β nβ rss t = β + nβ rps t = rsp t = 0 [ here β = cos θ, β = 1 sin2 θ ], n: refractive index of medium, θ: angle of incidence n 2 measured from sample normal, κ 2 = in 2 Q, Q: contains all quantum mechanical information about spin-orbit coupling. If Q goes to zero, we see no coupling effect which implies no MOKE. [ Er,p E r,s E r = (m 2 t S t + m 2 l Sl ) E i ] [ = m 2 r t pp r t ] [ ] [ ps E0 cos θ p t + m 2 r l pp r l ] [ ] ps E0 cos θ p E 0 sin θ l p E 0 sin θ p r t sp r t ss r l sp r l ss E r,p = (m 2 t r t pp + m 2 l rl pp)e 0 cos θ p + m 2 l rl pse 0 sin θ p & E r,s = m 2 l rl spe 0 cos θ p + (m 2 t r t ss + m 2 l rl ss)e 0 sin θ p Taking simplest case θ p = 90 From the above we infer the following: E r,p = m 2 l rl pse 0 E r,s = r l sse 0 1. Vertical component modified by r l ss independently of the sample s magnetization. 2. Longitudinal KE produced a horizontal polarized component proportional to square of sample s magnetization 4

7 Now analyzing at angle θ a E t = E rp cos θ a ˆp + E rs sin θ a ŝ Here we can take angle θ a such that we specifically get MOKE information, Hence taking θ a = 0 which gives E t E rp Now as we have understood how the light shall appear after the whole process, we can also write the equation for signal detection by photodiode normalized to incident intensity. I I 0 = M 4 l M 4 s 0.3 Lock In Amplifier I E t 2 I 0 E 0 2 = E rp 2 E 0 2 = m lrps l 2 sin 2 θ cos 2 θ n 4 Q 2 n 4 β 2 (nβ + β ) 2 (β + nβ ) 2 A lock-in amplifier is a type of amplifier that can extract a signal with a known carrier wave from an extremely noisy environment. Depending on the dynamic reserve of the instrument, signals up to 1 million times smaller than noise components, potentially fairly close by in frequency, can still be reliably detected. It is essentially a homodyne detector followed by low pass filter that is often adjustable in cut off frequency and filter order. Whereas traditional lock-in amplifiers use analog frequency mixers and RC filters for the demodulation, state of the art instruments have both steps implemented by fast digital signal processing for example on an FPGA. Usually sine and cosine demodulation is performed simultaneously, which is sometimes also referred to as dual phase demodulation. This allows the extraction of the in-phase and the quadrature component that can then be transferred into polar coordinates, i.e. amplitude and phase, or further processed as real and imaginary part of a complex number (e.g. for complex FFT analysis). Recovering signals at low signal-to-noise ratios requires a strong, clean reference signal the same frequency as the received signal. This is not the case in many experiments, so the instrument can recover signals buried in the noise only in a limited set of circumstances. Basic Principles Operation of a lock-in amplifier relies on the orthogonality of sinusoidal functions. Specifically, when a sinusoidal function of frequency f1 is multiplied by another sinusoidal function of frequency f2 not equal to f1 and integrated over a time much longer than the period of the two functions, the result is zero. Instead, when f1 is equal to f2 and the two functions are in phase, the average value is equal to half of the product of the amplitudes. 5

8 In essence, a lock-in amplifier takes the input signal, multiplies it by the reference signal (either provided from the internal oscillator or an external source), and integrates it over a specified time, usually on the order of milliseconds to a few seconds. The resulting signal is a DC signal, where the contribution from any signal that is not at the same frequency as the reference signal is attenuated close to zero. The out-of-phase component of the signal that has the same frequency as the reference signal is also attenuated (because sine functions are orthogonal to the cosine functions of the same frequency), making a lock-in a phase-sensitive detector. For a sine reference signal and an input waveform U in (t), the DC output signal U out (t) can be calculated for an analog lock-in amplifier by: U out (t) = 1 T t t T sin [2πf ref s + ϕ] U in (s) ds where is a phase that can be set on the lock-in (set to zero by default). If the averaging time T is large enough (e.g. much larger than the signal period) to suppress all unwanted parts like noise and the variations at twice the reference frequency, the output is U out = V sig cos θ where V sig is the signal amplitude at the reference frequency and θ is the phase difference between the signal and reference. Many applications of the lock-in only require recovering the signal amplitude rather than relative phase to the reference signal. For a simple so called single phase lockin-amplifier the phase difference is adjusted (usually manually) to zero to get the full signal. More advanced, so called two phase lock-in-amplifiers have a second detector, doing the same calculation as before, but with an additional 90 degree phase shift. Thus one has two outputs: X = V sig cos θ is called the in-phase component and Y = V sig sin θ the quadrature component. These two quantities represent the signal as a vector relative to the lock-in reference oscillator. By computing the magnitude (R) of the signal vector, the phase dependency is removed: The phase can be calculated from R = X 2 + Y 2 = V sig. tan θ = Y/X. Signal measurement in noisy environments The essential idea in signal recovery is that noise tends to be spread over a wider spectrum, often much wider than the signal. In the simplest case of white noise, even if 6

9 the root mean square of noise is 103 times as large as the signal to be recovered, if the bandwidth of the measurement instrument can be reduced by a factor much greater than 106 around the signal frequency, then the equipment can be relatively insensitive to the noise. In a typical 100 MHz bandwidth (e.g. an oscilloscope), a bandpass filter with width much narrower than 100 Hz would accomplish this. The averaging time of the lock-in-amplifier determines the bandwidth, and allows very narrow filters, less than 1 Hz if needed. However this comes at the price of a slow response to changes in the signal. In summary, even when noise and signal are indistinguishable in the time domain, if the signal has a definite frequency band and there is no large noise peak within that band, noise and signal can be separated sufficiently in the frequency domain. If the signal is either slowly varying or otherwise constant (essentially a DC signal), then 1/f noise typically overwhelms the signal. It may then be necessary to use external means to modulate the signal. For example, when detecting a small light signal against a bright background, the signal can be modulated either by a chopper wheel, acoustooptical modulator, photoelastic modulator at a large enough frequency so that 1/f noise drops off significantly, and the lock-in amplifier is referenced to the operating frequency of the modulator. In the case of an atomic force microscope, to achieve nanometer and piconewton resolution, the cantilever position is modulated at a high frequency, to which the lock-in amplifier is again referenced. When the lock-in technique is applied, care must be taken to calibrate the signal, because lock-in amplifiers generally detect only the root-mean-square signal of the operating frequency. For a sinusoidal modulation, this would introduce a factor of 2 between the lock-in amplifier output and the peak amplitude of the signal, and a different factor for non-sinusoidal modulation. In the case of nonlinear systems, higher harmonics of the modulation-frequency appear. A simple example is the light of a conventional light bulb being modulated at twice the line frequency. Some lock-in-amplifiers also allow separate measurements of these higher harmonics. Furthermore, the response width (effective bandwidth) of detected signal depends on the amplitude of the modulation. Generally, linewidth/modulation function has a monotonically increasing, non-linear behavior. 0.4 Photo Elastic Modulator The PEM is a resonant device whose precise oscillating frequency is determined by the physical properties of the optical element/transducer assembly. The electronic head, optical head and the cables that connect them make up a circuit that operates like a 7

10 crystal-controlled oscillator circuit. The PEM controller controls the amplitude of the PEM oscillations and generates a reference signal. A feedback signal from the head assembly is used by the controller to monitor the PEM oscillation amplitude and to provide timing for the generation of the reference. Principle of Operation The phenomenon of photoelasticity is the basis of operation for the PEM. If a sample of transparent solid material is stressed by compression or stretching, the material becomes birefringent, that is, different linear polarizations of light have slightly different speeds of light when passing through the material. PEM uses a rectangular shape for the modulator optical element. A fused silica bar is made to vibrate with a natural resonant frequency of about 50 khz. This vibration is sustained by a quartz piezoelectric transducer attached to the end of the bar. At the center of the optical element an oscillating birefringence occurs at a frequency of about 50 khz. The magnitude of the birefringence is controlled electronically by the PEM Controller. Retardation effects of compression and extension The effect of the modulator on a linear polarized monochromatic light wave is shown in slide 3 of the Polarization Primer. The plane of polarization is at 45 to the modulator axis before passing through the modulator. If the optical element is relaxed the light passes through with the polarization unchanged. If the optical element is compressed, the polarization component parallel to the modulator axis travels slightly faster than the vertical component. The horizontal component then leads the vertical component after light passes through the modulator. If the optical element is stretched, the horizontal component lags behind the vertical component. The phase difference between the components at any instant of time is called the retardation or retardance. The peak retardation is the amplitude of the sinusoidal retardation as a function of time. The retardation (in length units) is given by A(t) = z[nx(t) ny(t)] where z is the thickness of the optical element and nx(t) and ny(t) are the instantaneous values of refractive index along the x and y directions. Common units for retardation include distance (nanometers, microns), waves (quarter-wave, half-wave), and phase angle (radians, degrees) The PEM-100 Controller can display retardation in waves or phase angle. 8

11 Quarter wave retardation An important condition occurs when the peak retardation reaches exactly one-fourth of the wavelength of light. When this happens, the PEM acts as a quarter-wave plate. Figure 1 shows this condition at the instant retardation is at its maximum. Figure 2: Quarter Wave Retardation using PEM The polarization vector traces a right-handed spiral about the optic axis. Such light is called right circularly polarized. For an entire modulator cycle, Figure 1 shows the retardation vs. time and the polarization states at several points in time. The polarization oscillates between right circular and left circular, with linear (and elliptical) polarization states in between. Half Wave Retardation Another important condition occurs when the peak retardation reaches one-half of the wavelength of the light. When this happens, the PEM acts as a half-wave plate at the instant of maximum retardation and rotates the plane of polarization by 90. Figure 3: Half Wave Retardation using PEM At maximum retardation, the polarization states are linear, rotated by 90. The half-wave retardation condition is particularly important for calibration of the PEM. 9

12 0.5 Optical Chopper An optical chopper is a device which periodically interrupts a light beam. Three types are available: variable frequency rotating disc choppers, fixed frequency tuning fork choppers, and optical shutters. A rotating disc chopper was famously used in 1849 by Hippolyte Fizeau in the first non-astronomical measurement of the speed of light. Optical choppers, usually rotating disc mechanical shutters, are widely used in science labs in combination with lock-in amplifiers. The chopper is used to modulate the intensity of a light beam, and a lock-in amplifier is used to improve the signal-to-noise ratio. To be effective, an optical chopper should have a stable rotating speed. In cases where the 1/f noise is the main problem, one would like to select the maximum chopping frequency possible. This is limited by the motor speed and the number of slots in the rotating disc, which is in turn limited by the disc radius and the beam diameter. 0.6 Nirvana Auto Balanced Photodetector The auto-balancing technology allows elimination of background noise from dynamically changing systems, including thermal drifting and wavelength dependence, enabling you to achieve the perfect power balance between reference and signal beams. Circuit uses a low frequency feedback loop to maintain automatic DC balance b/w signal and reference arms. 1. Circuit behaves as a variable gain beam splitter. This cancels the common mode laser noise with greater than 50dB rejection. 2. Loop bandwidth can be adjusted depending on the application Working The photoreceiver operates in three distinct modes: 1. Signal mode. 2. Balanced mode. 3. Auto-balanced mode. The output of the photodetector (A) can be expressed as A = ((I S g) I R ) Rf. Here, I S is the signal photodiode current, I R is the reference photodiode current, R f is the value of the feedback resistor, and g is the current-splitting ratio, which describes how much of the reference current comes from the subtraction node (I s ub) and how much comes from ground. In signal mode, g = 0 and no reference photocurrent comes from the 10

13 subtraction node. Here, the output A is simply an amplied version of the signal current. In balanced mode, g = 1, and all the reference photo current comes from the subtraction node. In this mode, A = (I S I R ) R f, the photo detector behaves as an ordinary balanced photoreceiver, where laser noise is canceled if the DC photo currents are equal. In auto-balanced mode, g is electronically controlled by a low-frequency feedback loop to maintain equal DC photocurrents cancelling laser noise regardless of the photocurrent. How to Use For auto balanced mode, Ideally reference photodiode should receive more that is twice as much compared to signal photo diode, P ref /P sig = 2. Loop Bandwidth knob controls the gain-compensation cutoff frequency which determines the speed of the auto-balancing effect. Cutoff frequency should be set as high as possible, but below the modulation frequency of interest to obtain best voice cancellation. ( f c = P ) sig 1 P sig R P ref 101 LB Here, P sig and P ref in mw. R: Photo diode responsivity in A/W. LB: Loop bandwidth knob setting. I sig = P sig R Modulation frequency must be above gain compensation cutoff otherwise noise cancellation will be degraded, because the gain compensation cutoff frequency has a single pole, the modulation frequency should be significantly above the cutoff frequency so it does not affect noise cancellation performance. Common reasons for lack of noise suppression: 1. Laser Polarization: Unexpected differential signal due to differential polarization sensitive optics. 2. Frequency Modulation. 3. Spatial modulation: Due to optics setup. 4. Polarization wiggle: Acousto-optic modulation create small rotation in the polarization of modulated light. We shall define a new Quantity CMRR : Common mode rejection ratio. ( ) VcommonMode CMRR = 20 log 10 V autobal 11

14 V commonmode : Detector output voltage proportional to P commonmode (The laser power present on the reference and signal photodiode at the given frequency). V autobal : Output voltage at the same frequency in auto-balanced mode. 0.7 Experimental Setup In this section I shall describe all the experimental setups that were made during the semester. There were total of three setups: Setup 1 Figure 4: MOKE setup with PEM The working of PEM is described in the above section. The settings were: 1. 50KHz frequency was set(pem). 2. λ/4 retardation was set(pem). 3. The polarizer was set at 45 with respect to the laser polarization. 4. The analyzer was set at 45 with respect to the laser polarization. 5. The magnetic field was varied from tesla to 0.02 tesla. Result 12

15 Figure 5: Graph between Magnetization and Magnetic Field Setup 2 Figure 6: MOKE setup with Chopper The working of Optical Chopper is described in the above section. The settings were: 1. 50KHz frequency was set. 2. The polarizer was set at zero degree and the analyzer was rotated from 0 to The magnetic field was kept constant. Result 13

16 Figure 7: Graph between Captured Intensity and Analyzer Angle Setup 3 Figure 8: CW Pump Probe setup with PEM and Chopper The working of the PEM, Optical Chopper and Auto Balanced PhotoDetector have been explained before. The above setup is a CW Pump probe with PEM and Optical chopper placed. The settings would be: 1. P1 is at 45 with respect to the laser beam. 14

17 2. P2 is at 45 with respect to the laser beam. 3. P3 is at zero degree with respect to the laser beam. 0.8 Summary In short following studies were done: 1. Understanding of MOKE using Jones Matrix formalism. 2. Working of Lock In Amplifier. 3. Working of Photo Elastic Modulator. 4. Working of Optical Chopper. 5. Three optical setups were made throughout the semester and the respective studies have been shown before. 15

18 Bibliography [1] Wikipedia [2] Phase sensitive detection: the lock-in amplifier by Dr. G. Bradley Armen, The University of Tennessee [3] LASER PHYSICS by PETER W. MILONNI, JOSEPH H. EBERLY 16