MOKE detection techniques

There is several ways how to detect MOKE signal. We will briefly discuss some of those methods shortly, but before we would like to comment one thing that all of those MOKE

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detection techniques have in common. The Kerr angles are not measured absolutely, i.e.

we do not directly measure Kerr rotation and/or ellipticity of incidents- orp- polarized wave upon reflection from sample that is magnetized into desired direction. Instead, we measure change in polarization state of light with change of M direction (in the LinMOKE case we usually just invert M direction).

There are two main reasons for this. (i) It is technically very hard to find absolute zero of MO signal. With majority of the setups (or at least with all the setups known to us) some offset is present, that depend on current setup alignment and sample alignment.

The offset also often slowly drift in time. (ii) There could be other effects that cause change in the polarization state of light upon reflection, like natural anisotropy of the crystal (not the case of cubic crystals), strain etc. Thus, this differential measurement for two (or more)M directions can efficiently filter out those effects, as they are notM direction dependent. Although with measurements of hysteresis loops Kerr angles are measured directly, we are interested here in the shape of the loops and not in the exact value of the Kerr angles, where those loops usually posses considerable offset. In the case when we need to obtain exact value of Kerr angles from the loops, we must centre them around zero, which is in principle adequate to differential measurements whenM is reversed.

Besides, there is one more thing that almost all MOKE measurement methods have in common.

Measurement of Kerr ellipticity

All the methods described below are only able to directly detect change in the rotation of the plane of polarization, but are not able to directly detect ellipticity of the polar-ization. This should be common to all detection techniques that employ polarizers and are based on measurement of intensity change at the detector. Thus, to measure Kerr ellipticity, quarter-wave retarder (i.e. compensator δ = ^π₂, see Tab.2.1) is employed in the optical pathway (in our case right behind the sample). Such a compensator will literary swap the ellipticity angle for the rotation angle and vice versa. Thus, with the compensator present, the setup is still only detecting changes in the rotation of the plane of polarization, but this rotation angle is now proportional to the ellipticity of polarized light that is reflected from the sample, but didn’t pass through compensator yet. Hence the Kerr ellipticity is actually being measured.

In Jones formalism quarter-wave compensator writes:

C=e^iδ2

⎡

⎣

1 0

0 e^−iδ

⎤

⎦

δ=^π₂

= e^iπ4

⎡

⎣

1 0

0 −i

⎤

⎦. (3.1.1)

As discussed in previous chapter in Sec. 2.4, (i) in the Jones formalism we can discard the prefactors (e^iπ4 in this case) and (ii) we can join two or more Jones matrixes into one effective Jones matrix. We will do so for the reflection matrix of the sample and the Jones matrix of the compensator.

C^(π2)R=

⎡

⎣

rss rsp

−ir_ps −ir_pp

⎤

⎦. (3.1.2)

If we will now yield Kerr angles from this effective reflection matrix, we will have Φ^C(

π 2) s =irps

r_ss =−ϵ_s+iθ_s, Φ^C(

π 2) p =irsp

r_pp =−ϵ_p+iθ_p, (3.1.3) where we see that rotation have been swapped for ellipticity and vice versa.

Nearly crossed polarizers

The simplest method for MOKE detection utilize so-called nearly crossed polarizers.

One polarizer is in the incident beam pathway and the second polarizer (which we call analyzer) is in the reflected beam pathway. With crossed polarizers no light intensity get pass the analyzer to the detecetor. With magnetized sample the Kerr rotation is induced upon reflection and thus some of the light intensity will pass to the detector. Note that the rotation of analyzer and Kerr rotation of reflected beam, both have the very same effect on the intensity at the detector. This is being used to calibrate the setup and obtain volt-degree characterization for the measurement of Kerr angles. Because with completely crossed polarizer and analyzer the Kerr angle sign would be indistinguishable (from zero intensity we can go only up), the analyzer is slightly turn away from the crossed position (ca. 5^◦ from crossed position), and therefore ”nearly crossed polarizers”.

Then, the intensity drop or rise provide information on the sign, while absolute change in intensity give us information on the Kerr effect strength.

Intensity differential measurement – use of Wollaston prism

Another technique for MOKE detection utilize Wollaston prism. Wollaston prism is a polarizer that split incoming light into two beams of linear polarizations orthogonal to

each other, where both beams are deviated from the original path by an angle. How much intensity goes into each of the branch depends on how is the Wollaston prism balanced with respect to incoming polarization. If Wollaston prism will be balanced precisely at 0 degrees, the incoming s-polarized light will be split into two beams with linear polarization rotated by ±45^◦. Then, if we will turn Wollaston prism by 45^◦, one of the beam will be fully extinct and all the intensity will be in the other. Again, as in the previous case, rotation of Wollaston prism and rotation of polarization are relative to each other. Each of the beam is then detected by its own detector (usually some photodiode). With balanced Wollaston prism intensities on both diodes are the same.

When Kerr effect is introduced through sample magnetization (or rather reversion of M), intensityI1 at one diode will rise whereas intensity I2 at the other diode will fall.

From the differenceI₁−I₂ we can calculate the Kerr angles. This technique is employed with the Vector MOKE setup based at Bielefeld University as will be discussed later in the text.

Polarization modulation technique

This modulation techniques employ modulation of polarization state of light. This means that the polarization state of light will harmonically oscillate in time, producing also harmonic changes of the intensity at the detector. The signal is then being measured through lock-in amplifier. This provide us with much better signal-to-noise ratio, same as in the case of intensity modulation technique (where chopper is usually used), but furthermore filter out also effects as e.g. depolarization of the sample.

To modulate the ellipticity of polarization, photoelastic modulator (PEM) is used. The device act as a compensator with time-varying phase shift. Fused silica block inside the PEM is periodically compressed by a piezocrystal at frequency ω being usually 50 kHz. By the amplitude of compression, the amplitude of phase shift is given. To modulate rotation of polarized light, Faraday cell with capacitor (together forming LC oscillator) is used. Faraday cell consist of a coil with fused silica block inside. Magnetic field produced by the coil will induce Faraday rotation to the light passing through the fused silica. The higher the field the higher the Faraday rotation. These are the most conventional technique for modulation, but with rapid advance of technology there will be surely other approaches as is for example nematic liquid crystal modulator [105].

The measurement principle with modulation technique require more detailed treatment by Jones formalism to obtain Intensity dependencies of I_1ω and I_2ω being intensities at first and second harmonic of modulation frequencies measured by lock-in amplifier, respectively. With this technique it is actually possible to measure ellipticity without use

of compensator, as the signal ofI1ω is usually proportional to Kerr ellipticity andI2ω to Kerr rotation, yet the principle from point of physics is similar to the use of compensator (as PEM is compensator with time-varying phase shift). However measurement of Kerr ellipticity throughI_1ω does not have to be a rule. With our Spectroscopy MOKE setup based at Technical University of Ostrava, that employ modulation technique through PEM, the Kerr ellipticity is still measured through use of compensator and detection of I_2ω, as calibration ofI_1ω is problematic in our case.

There is multiple way how to obtain calibration, i.e. volt-degree characterization ofI2ω, of this technique. Similar calibration as in the case of nearly crossed polarizers could be done (i.e. by small analyzer rotation). Some setups may also employ null technique, that is described below, to measure Kerr effect.

Null technique

This technique is usually combined with the modulation technique using Faraday cell, although in principle could be used on its own. Another Faraday cell, so-called null cell, is employed within the setup, and is used to compensate Kerr effect from sample through feedback loop. Calibration of the null cell will provide us with ampere-degree charac-terization, i.e. we can calculate the Kerr angles from the current needed to compensate the Kerr effect from the sample.