When light propagates in a magneto-optical material, its polarization rotates. The rotation angle is called the Faraday rotation angle. The Faraday rotation angle is linearly proportional to the magnetic field, which is applied along light propagation direction. In a ferromagnetic material and a super paramagnetic material, the Faraday rotation angle is linearly proportional to the magnetization of sample along light propagation direction.

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There are 3 methods of measurement of the Faraday rotation angle:

1) Crossed- polarizers

precession: poor
sensitivity:poor
implementation: simple
distribution by optical anisotropy: very sensitive (it is bad)

2) Balance detectors

precession: moderate
sensitivity:moderate
implementation:moderate
distribution by optical anisotropy: very sensitive (it is bad)

3) 2x2 switch

precession: high
sensitivity:high
implementation:moderate/difficult
distribution by optical anisotropy: not sensitive (it is very good!!!)

Which method should I use for measurements of Faraday rotation angle?

1) Use the setup of crossed polarizers only when you are using a camera and you measure 2D distribution of faraday rotation angles. Do not use it with a photo detector. The sensitivity of this method is very poor! It can measure the Faraday rotation angle about 1 degree and larger.

2) Use the setup of balanced detectors when the angle of the Faraday rotation in your sample is greater than 10 mdeg and your sample is not anisotropic. The setup of this method is very simple.

3) Use the setup of a 2x2 optical switch when (a) you are testing a fiber-optic component (b) the sample is anisotropic or it have an optical activity (c) the angle of Faraday rotation is smaller than 10 mdeg; (d) the light absorption in the sample is high.

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Disturbing effect: Polarization rotation due to the optical anisotropy and the optical activity

The polarization of light can be rotated in an anisotropic material and a material with optical activity. The optical anisotropy and the optical activity severely disturb the measurements of the Faraday rotation angle.

The influence of the optical anisotropy and the optical activity on the measurements of the Faraday rotation angle can be significantly reduce by measuring the Faraday rotation angle for two opposite directions of the applied magnetic field and subtracting data from each other. Since the Faraday rotation angle has opposite polarities for opposite direction of magnetic field and the anisotropy and the optical activity are not influenced by the polarity of magnetic field, this method is working very well for a transparent MO material. However, for an absorptive materials this method does not exclude the significant contribution of the optical anisotropy and the optical activity on measured Faraday rotation angle. The influence the optical anisotropy and the optical activity can be fully excluded using measurements with a 2x2 switch, because it measure only a non-reciprocal response.

When it is possible, measure the Faraday rotation for two opposite directions of the magnetic field and subtract data from each other!!!

The measurements with a 2x2 optical does not require the reversing of the magnetic field. However, if it is possible it is better to do it.

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Set-up of crossed -polarizers

This method was invented by Faraday about 250 years ago. Still now it is the most-used method to measure the magneto-optical (MO) effect. The major merit of this method is its simplicity. It can be used with a camera or a microscope to see 2D MO image. The major demerit of this method is its very poor sensitivity. Only relatively large rotation angles can be measured.

Figure 1 shows the conventional setup for measurements of Faraday rotation. Light is passing  from the light source to the detector through the sample, which is placed between crossed polarizer and analyzer. When polarization of light is not rotated by the sample, light is blocked by the analyzer and it can reach the detector. When the polarization is rotated, a small amount of light can pass the analyzer and it can be detected.

merits:

- simplicity;
-simple implementation;
-It does not need high dynamic range of detector;
-It can be used with a camera;

 

demerits:

-very bad sensitivity
-very bad precision;
-sensitive to optical anisotropy;
- only a relatively-large rotation angle can be measured;
-it can not distinguish between positive and negative rotation angles;

 

 

 

 

The demerit of this setup of the crossed-polarizers is a bad sensitivity, because of the following reason. Figure 2 shows the transmission of light through a polarizer and an analyzer as function of angle between them. The angle of 90 degrees corresponds to the crossed polarizer and analyzer. For this angle the 1st derivation of the transmission with respect to the angle equals zero.  This means that the change transmission as function of a rotational angle is small and it is proportional to square of rotation angle and the second derivation. This is reason of a bad sensitivity of the conventional setup for the Faraday rotation measurements. The sensitivity of the conventional setup usually does not exceed 1 mdeg and often it is larger than 10-50 mdeg.

 

Why this method does not need a high dynamic range of a detector? Why a camera can be used?

It is because most of light is blocked by the crossed polarizers. Only a tiny amount of light reaches detector. In contrast, in the methods of balanced detectors and 2x2 switch a substantial amount of light reach the detector and the substantial detector current is weakly modulated due MO response.

 

 

 

 

 

 

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Set-up of balanced detectors

It is hard to trace who was first to introduce this method. It would credit following British scientists (A. P. Heberle, J. J. Baumberg, and K. Kohler, Phys. Rev. Lett. 75, 259 (1995)). At first, this method was used in order to distinguish a positive and negative Faraday rotations. Only after a while, its high sensitivity has been recognized.

 

merits:

- moderate sensitivity;
-it can distinguish between positive and negative rotation angles;;
-small angles can be measured;
-;

 

demerits:

-It needs a high dynamic range of detectors
-implementation is complex;
-It is difficult to use it with a camera;
- sensitive to optical anisotropy;

 

 

 

Note: Often the intensity of light source or magnetic field is modulated. Then, the detectors are connected to differential inputs of a lock-in amplifier.

Note: Output of a photo detector is current. Input of a lock-in amplifier is voltage. Therefore, only amplified detectors can be used with a lock-in amplifier.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Can setup of balanced detectors be utilized with a camera?

In principle, it is possible. In this case, two CCD matrixes should be used with a polarization-splitter cube. In a commercial camera, a splitter cube with 2 or 3 CCD matrixes are often used to record a color information. With some modifications such camera could be converted into a camera of balanced detectors. I do not company, which makes such camera. It would be greater improvement for the Kerr microscopy to have such camera.

 

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Setup for for measurements of Faraday rotation, which utilizes 2x2 switch

Zayets et al, Patent appl., 2016-126905, 2016; in Japanese; in English

 

merits:

- very high sensitivity;
-it can distinguish between positive and negative rotation angles;;
-very small angles can be measured;
-it can be used with a detector with a narrow dynamic range;
-it is not sensitive to optical anisotropy;

 

demerits:

-implementation is complex;it needs fiber/free-space coupling.
-implementation is complex;
-It is difficult to use it with a camera;
- ;;

Fig.5 the setup for measurements of the Faraday rotation with a 2x2 optical switch. It consists of an source of light, a light detector, a 2x2 optical switch,, a pulse generator and a lock-in amplifier. The 2x2 switch is switching between two opposite directions of light propagation. The pulse generator generates a sequence of electrical pulses, which switch the 2x2 switcher between its two states. Also, the pulses are used as the reference signal for the lock-in amplifier. The angle between axes of the polarizers is 45 degrees. The amplitude of the modulation is proportional to the value of the Faraday rotation of the sample. Therefore, it is detected with a high- sensitivity by the lock-in amplifier.

Reason why the output signal of the detector is modulated proportionally to the value of the Faraday rotation of the sample:

Since the Faraday effect is a non-reciprocal effect, the polarization rotates in opposite directions for opposite propagation directions of light. For example, if in case of the propagation in the up-direction the Faraday rotation is positive, the polarization of light at the polarizer will be larger than 45 degrees and the detection signal will be smaller.  In the case of the propagation in the down-direction the polarization of light at the polarizer will be smaller than 45 degrees and the detection signal will be larger. Measuring the difference of detected light between two opposite propagation direction by the lock-in technique, the Faraday rotation angle can be evaluated with a high precision and a high sensitivity. The angle of 45 degrees between polarizers corresponds to the highest sensitivity for measurement of the Faraday rotation angle, because the derivation of the transmission between polarizers (See Fig.4) with respect to the angle between polarizers is the largest in this case.

 

Why sensitivity of the setup with a 2x2 switch is much better than the sensitivity of the setup with balanced detectors?

(1) Use of lock-in technique;

(2) Modulation of magneto-optical constants of a sample at a high speed;

(3) Modulation of non-reciprocal properties of a sample at a high speed;

(4) Suppression of influence of optical anisotropy of sample and its intrinsic optical activity on magneto-optical response.

Achieving a high sensitivity with a detectors of a low dynamic range

 

Note:

High-sensitivity photo detectors have a relatively narrow dynamic range. Therefore, they are very sensitive and they can detect light with a high precision and a low noise level, but they can detect light only of a small intensity.

Photo multipliers: high sensitivity, moderate dynamic range

CCD camera: very high sensitivity, very narrow dynamic range

In case when intensity of light at the detector is small, a detector of a smaller dynamitic range can be used and a noise level is smaller.

 

In contrast to the setup of balanced detector, where the angle between polarizers should be exactly 45 degrees, the setup of 2x2 optical switch operates at any angle between polarizers.

The setup with a 2x2 optical switch has a very high sensitivity, but still may use a photo detector of a narrow dynamic range. It might be possible even use a CCD camera

 

 

 

 

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2x2 optical switch

There are many 2x2 switches on market. All, I know, have a fiber for input and output. I do not know any bulk-type 2x2 optical switch. I believe it is possible to make a bulk-type 2x2 optical switch using mirrors. A bulk-type 2x2 optical switch could be very useful in a measurement setup with a camera.

For MO and non-linear measurement, the critical parameter for a 2x2 switch is the cross-talk between input 1 and input 2. It should be minimized as much as possible!!!

The optical properties of a sample is measured for two opposite directions of light propagation. Since the direction of light is changed by a 2x2 optical switch without any change or distortion of the optical alignment,  a tiny difference of optical properties for opposite propagation directions can be measured with a high precision.

 

 

 

 

 

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Other setups

 

There is another possible setup for measurement of Faraday rotation angle, where a continuously- rotating polarizer is used. On my experience it does not provide sensitivity better than the setup with crossed polarizers. However, I have not tested all possible setups.