# Magneto-optic Modulators and Sensors

This is a continuation from the previous tutorial - optical isolators and circulators.

Polarization and amplitude modulators that are based on the Faraday effect and are driven by currents or magnetic fields can be easily realized. In comparison to the electro-optic polarization and amplitude modulators discussed in previous tutorials, these devices have similar functions but quite different characteristics.

The mechanism responsible for magneto-optic polarization modulators is circular birefringence, whereas that for electro-optic polarization modulators is linear birefringence.

If the input optical wave is linearly polarized, the output of an ideal magneto-optic polarization modulator is linearly polarized, but hat of an electro-optic polarization modulator is elliptically polarized in general and is linearly polarized only when the applied voltage is equal to an integral multiple of the half-wave voltage.

The basic structure of both magneto-optic and electro-optic amplitude modulators consists of a polarization modulator and a polarizer-analyzer pair.

The basic configuration of a magneto-optic amplitude modulator is simply that of the polarization-dependent optical isolator shown in figure 7-7(a) [refer to the optical isolators and circulators tutorial], except that $$\theta_\text{F}$$ for a modulator can have any value and the polarizer at the output is now referred to as the analyzer.

Usually, there is absorption loss in the Faraday rotator, as well as in the polarizer and the analyzer. If the Faraday rotator has negligible magnetic circular dichroism, the transmitted optical wave remains linearly polarized, though attenuated.

The intensity transmittance of the modulator is then given by

$\tag{7-45}T=\frac{I_\text{out}}{I_\text{in}}=T_0\text{e}^{-\alpha{l}}\cos^2(\theta_\text{F}-\theta_\text{p})=\frac{T_0}{2}\text{e}^{-\alpha{l}}[1+\cos2(\theta_\text{F}-\theta_\text{p})]$

where $$\alpha$$ and $$l$$ are the absorption coefficient and the length, respectively, of the Faraday rotator and $$T_0$$ accounts for losses in the polarizer, the analyzer, and other components such as the nonmagnetic substrate supporting a magnetic film. If the input optical wave is linearly polarized, $$0\lt{T_0}\le1$$. If it is unpolarized, $$0\lt{T_0}\le1/2$$.

If the absolute value of $$\theta_\text{F}$$ is small, $$\theta_\text{p}$$ is chosen to be $$45^\circ$$ for a linear response. Then $$T$$ varies linearly with $$\theta_\text{F}$$:

$\tag{7-46}T=\frac{T_0}{2}\text{e}^{-\alpha{l}}(1+\sin2\theta_\text{F})\approx{T_0}\text{e}^{-\alpha{l}}\left(\frac{1}{2}+\theta_\text{F}\right)$

In this case, the transmittance $$T$$ has the highest sensitivity in response to variations in the value of $$\theta_\text{F}$$ around the point $$\theta_\text{F}=0$$. Because $$\theta_\text{F}$$ of a paramagnetic or diamagnetic Faraday rotator is linearly proportional to the magnetic field, and thus is also linearly proportional to the modulating current, a linear response that has a high sensitivity over a large dynamic range can be obtained for a modulator using such a Faraday rotator.

In certain applications, however, a value of $$\theta_\text{p}$$ different form $$45^\circ$$ is chosen for objectives other than a linear response.

To measure the value of the Faraday rotation angle $$\theta_\text{F}$$ independently of the fluctuations in the input optical intensity and the absolute calibration of the detection system, a dual-quadrature polarimetric configuration as shown in figure 7-13 below can be employed.

In this configuration, a polarizing beam splitter, such as a Glan prism, is used to divide the output beam into two orthogonal linearly polarized beams detected by two differential photodetectors of matched responsivity.

The output readings from the two photodetectors are taken to compute a normalized difference signal

$\tag{7-47}\text{DS}=\frac{P_1-P_2}{P_1+P_2}$

where $$P_1$$ and $$P_2$$ are the optical powers of the two beams detected by the differential photodetectors.

By properly orienting the principal axis of the polarizing beam splitter with respect to that of the input polarizer for $$\theta_\text{p}=45^\circ$$ so that $$\text{DS}=0$$ when $$\theta_\text{F}=0$$, the difference signal has the following dependence on the Faraday rotation angle:

$\tag{7-48}\text{DS}=\sin2\theta_\text{F}$

Current and magnetic field sensors

A magneto-optic amplitude modulator can be used as a current or magnetic field sensor. For this kind of application, a linear response is desired. Therefore, the absolute value of $$\theta_\text{F}$$ is kept small within the range of operation, and the analyzer is carefully oriented at $$\theta_\text{p}=45^\circ$$ with respect to the polarizer so that (7-46) is valid. Paramagnetic or diamagnetic materials, such as silica glass, terbium-doped glasses, TGG, Bi12SiO20 (BSO), and Bi12GeO20 (BGP), are used.

There are two different types of current sensors, namely, linked and unlinked. In a linked sensor, the conductor carrying the current to be measured is fully enclosed by the magneto-optic medium. In an unlinked device, the magneto-optic medium does not fully enclose the conductor.

Figure 7-14 below shows two examples of the linked type.

In figure 7-14(a), the Faraday rotator is made of a monolithic magneto-optic material, such as a single piece of silica glass. The conductor passes through the central opening of this medium. A linearly polarized wave is guided by total internal reflection at the properly shaped corners to travel closely along the magnetic field line encircling the conductor. To multiply the Faraday rotation angle, sophisticated optical design for guiding the optical wave to encircle multiple turns around the conductor can be implemented over this basic structure.

Alternatively, an optical fiber wound around the conductor, shown in figure 7-14(b), can be used. For a linked current sensor, the Faraday rotation angle given in (7-25) [refer to the Faraday effect tutorial] has to be modified because both the magnetic field and the optical path loop around the current.

Using Ampere's law, we have

$\tag{7-49}\theta_\text{F}=V\oint{\pmb{H}_0\cdot\text{d}\mathbf{l}}=VNi$

where $$N$$ is the number of turns for which the optical path encircles the current $$i$$, and $$V$$ is the Verdet constant.

Example 7-4

A fiber-optic current sensor of linked type as shown in figure 7-14(b) consists of 20 turns of silica fiber wound around the conductor. The detection scheme has the dual-quadrature polarimetric configuration shown in figure 7-13 with a polarized He-Ne laser at $$\lambda$$ = 632.8 nm used as the light source. The sensor has a dynamic range from 1 A to 1 kA. What is the smallest Faraday rotation angle the sensor is required to measure? What is the largest linearity error in the measurement?

At 632.8 nm wavelength, the Verdet constant of silica fiber is $$V=3.93\times10^{-6}\text{ rad A}^{-1}$$ from table 7-1 [refer to the Faraday effect tutorial]. To obtain a current reading of 1 A at the lower end of its dynamic range, the sensor, with $$N=20$$, is required to be capable of measuring a Faraday rotation angle as small as

$\theta_\text{F}=3.939\times10^{-6}\times20\times1\text{ rad}=78.6\text{ μrad}$

Linearity error of the measurement occurs because of the difference between the signal, $$\text{DS}=\sin2\theta_\text{F}$$, that is obtained from the reading of the sensor according to (7-48) and the response, $$2\theta_\text{F}$$, that is directly proportional to the current. It increases as the absolute value of $$\theta_\text{F}$$ increases from zero toward $$\pi/2$$. Therefore, the largest linearity error occurs at the upper end of the dynamic range at $$i=1\text{ kA}$$ for which $$\theta_\text{F}=78.6$$ mrad. It is found as

$\text{Linearity error}=1-\frac{\sin2\theta_\text{F}}{2\theta_\text{F}}=1-\frac{\sin(2\times0.0786)}{2\times0.0786}\approx0.41\%$

Unlinked magneto-optic current sensors can also take a variety of different structures. Figure 7-15(a) shows a simple structure, where a more sophisticated structure is shown in figure 7-15(b).

In the figure 7-15(b) structure, the Faraday rotator is placed in the gap of a magnetic core that serves the purpose of a flux concentrator to enhance the sensitivity of the device by concentrating the magnetic flux through the rotator.

Figure 7-15(c) shows yet another structure consisting of a Faraday rotator that is looped around by a current-carrying conductor in the form of a solenoid; the relation in (7-49) also applies to this structure with $$N$$ being the number of turns of the conducting wire in the solenoid.

One can also take advantage of the nonreciprocal nature of the Faraday effect to multiply the total Faraday rotation angle in an unlinked device, thus further enhancing the sensitivity of the device, by properly applying total-reflection coatings on the rotator surfaces for the optical wave to have multiple internal passes in the rotator, as also shown in figure 7-15(b).

There are advantages and disadvantages for both linked and unlinked types of devices. As can be seen from (7-49), a linked device measures the current directly and is virtually immune to interference from external stray magnetic fields.

An unlinked device does not measure the current directly, but measures the magnetic field induced by the current. It requires careful calibration for a correct reading of the current because it is susceptible to external interference and spatial variations of the magnetic field strength.

One major problem of linked devices, however, is the maintenance of the correct polarization direction of the linearly polarized optical wave looping around in the rotator.

All changes in the polarization direction have to be caused solely by the Faraday effect. All other effects, such as improper internal reflection in a monolithic rotator and linear birefringence in a fiber caused by bending stress, that lead to polarization changes have to be eliminated in order to obtain a correct reading of the current being measured.

The bandwidth of a magneto-optic current sensor of either type is ultimately limited by the optical transit time through the sensing element. Therefore, the device is capable of sensing AC currents at very high frequencies even when relatively long fibers are used.

The light source used does not have to be polarized because it is polarized by the polarizer at the input end of the device before entering the Faraday rotator.

From the above discussions, it is clear that any unlinked current sensor can also be used as a magnetic field sensor.

Spatial light modulators

A magneto-optic spatial light modulator consists of one- or two-dimensional spatial array of independently addressable Faraday rotators placed between a polarizer and an analyzer.

A single-crystal magnetic thin film, commonly a bismuth-substituted iron garnet film of a high specific Faraday rotation coefficient, is grown on a lattice-matched, transparent, nonmagnetic garnet substrate, typically Gd3Ga5O12 (GGG) or its derivatives such as one doped with Ga, Mg, and Zr. This substrate allows a large amount of Bi to be incorporated into the iron garnet film for a large specific Faraday rotation.

The magnetic film is structured into a one- or two-dimensional array of isolated mesas using microprocessing technology. Each mesa defines a pixel (picture element) of the spatial light modulator.

It is required that the film has a sufficiently large uniaxial magnetic anisotropy with a positive anisotropy constant in the direction normal to the surface, thus ensuring that the magnetization always points either up or down normal to the film surface.

The magnetization state of a pixel is controlled by two orthogonally running conductors that intersect at one corner of the mesa, as shown in figure 7-16 below.

Switching of the magnetization state is accomplished by changing the magnetization direction.

A uniform magnetic field stronger than the saturation field of the film material can be externally applied to the entire array to switch all of the pixels in the array to a given state, thus refreshing the array by erasing any existing pattern.

The array can then be configured into any desired pattern by switching the magnetization state of selected pixels using the magnetic field generated by the currents flowing through the matrix conductors.

The combined magnetic field generated by the currents flowing through the two conductors intersecting at a selected mesa is sufficient to initiate the switching, but not that generated by the current through either conductor alone. Thus, the entire array of pixels can be electrically addressed using orthogonally crossing matrix drive conductors.

In switching the magnetization state of a pixel, the magnetic field triggers movement of the magnetic domain wall across the mesa, as also illustrated in figure 7-16.

If the magnetic field exceeds the saturation field and lasts long enough, the domain wall can sweep across the entire mesa, resulting in complete switching of the magnetization direction.

If, instead, the currents generating the magnetic field are terminated at the moment when the domain wall reaches the bottom of the film but has not swept across the mesa, the mesa will be nucleated, containing equal areas magnetized in opposite directions.

Consequently, there are three different magnetization states for a pixel: two uniformly magnetized states and the nucleated state.

When a pixel is in one of the two uniformly magnetized states, a linearly polarized optical wave transmitted by the pixel experiences a Faraday rotation angle of either $$\rho_\text{F}l$$ or $$-\rho_\text{F}l$$, where $$\rho_\text{F}$$ is the specific Faraday rotation and $$l$$ is the thickness of the film.

When it is in the nucleated state, the Faraday rotation for the optical wave transmitted by the pixel averages out to be zero.

A spatial light modulator can be used either in binary operation, by switching between the two uniformly magnetized states of opposite magnetization direction, or in ternary operation, by switching among all of the three different magnetization states.

The basic configuration of a transmission-mode magneto-optic spatial light modulator in binary operation is illustrated in figure 7-17 below.

The input light, which can be either polarized or unpolarized to begin with, is polarized by the polarizer. Using (7-45), the transmittance for the two uniformly magnetized states can be expressed as

$\tag{7-50}T=T_0\text{e}^{-\alpha{l}}\cos^2(\pm\rho_\text{F}l-\theta_\text{p})$

In order for the device to have the highest possible contrast ratio, the surfaces of the Faraday rotator are antireflection coated to eliminate reflections that can introduce improper polarization changes to the transmitted light.

In addition, the value of $$\theta_\text{p}$$ has to be chosen such that $$T_\text{OFF}=0$$ in the OFF state. One choice is $$\theta_\text{p}=90^\circ-\rho_\text{F}l$$ for the magnetization state yielding a Faraday rotation angle of $$-\rho_\text{F}l$$ to represent the OFF state. Another choice is $$\theta_\text{p}=90^\circ+\rho_\text{p}l$$ for the magnetization state corresponding to a Faraday rotation angle of $$\rho_\text{F}l$$ to represent the OFF state. In either case, the transmittance in the ON state is

$\tag{7-51}T_\text{ON}=T_0\text{e}^{-\alpha{l}}\sin^2(2\rho_\text{F}l)$

Clearly, the optimum thickness of the magnetic film that yields the highest transmittance in the ON state depends on the values of both $$\alpha$$ and $$\rho_\text{F}$$. In practice, $$T_\text{OFF}$$ is never exactly zero because of residual reflections from the rotator surfaces and the circular dichroism in the film. Ignoring the effect of circular dichroism, the OFF-state transmittance is given by

$\tag{7-52}T_\text{OFF}=T_0R_1R_2\text{e}^{-3\alpha{l}}\sin^2(2\rho_\text{F}l)$

where $$R_1$$ and $$R_2$$ are the reflectivities of the two surfaces of the Faraday rotator consisting of the magnetic film on a substrate.

The contrast ratio of the device is given by

$\tag{7-53}\text{Contrast ratio}=\frac{T_\text{ON}}{T_\text{OFF}}=\frac{\text{e}^{2\alpha{l}}}{R_1R_2}$

Clearly, to maximize the contrast ratio, the residual reflections have to be minimized with high-quality antireflection coating while maximizing the ON-state transmittance.

Example 7-5

A Bi:YIG film of 10 μm thickness on GGG substrate is used for a transmission-mode magneto-optic spatial light modulator operated at 632.8 nm wavelength. At this wavelength, the film has an absorption coefficient of $$\alpha=0.108\text{ μm}^{-1}$$ and a specific Faraday rotation of $$\rho_\text{F}=1.68\times10^{-2}\text{ rad μm}^{-1}$$. The sample is antireflection coated on both surfaces with $$T_0=0.9$$ caused only by the absorption in the polarizer and analyzer. What is the ON-state transmittance of the modulator? If the film thickness is increased to 16 μm, what is the ON-state transmittance?

For $$l$$ = 10 μm, we find with the given parameters that

$T_\text{ON}=0.9\times\text{e}^{-0.108\times10}\times\sin^2(2\times1.68\times10^{-2}\times10)=3.3\%$

For $$l$$ = 16 μm, the transmittance is increased to

$T_\text{ON}=0.9\times\text{e}^{-0.108\times16}\times\sin^2(2\times1.68\times10^{-2}\times16)=4.2\%$

Further increase in the film thickness does not increase the ON-state transmittance but results in a decrease in the ON-state transmittance.

If the analyzer is oriented such that $$\theta_\text{p}=90^\circ$$, equation (7-50) yields the same transmittance, which is proportional to $$\sin^2(\rho_\text{F}l)$$, for the two uniformly magnetized states. These two states can still be distinguished because their transmitted fields have a $$\pi$$ phase difference.

Consequently, this arrangement leads to the binary phase-only mode of operation.  In this configuration, the transmittance of a pixel in the nucleated state is zero. A ternary phase-only mode of operation consisting of the +1, 0, and -1 states is possible if the nucleated state is also included in the operation.

Using a reflection-mode spatial light modulator, the magnetic film can be halved in thickness while maintaining the same contrast ratio as that of a transmission-mode device.

In a reflection-mode device, the back surface of the Faraday rotator is made totally reflective while the front surface is antireflection coated. Light entering the rotator from the front surface passes through the film twice before leaving the film, also from the front surface.

Because of the nonreciprocal characteristic of the Faraday effect, the Faraday rotation is cumulative for both passes. The linear absorption is also cumulative. Therefore, if the film of a reflection-mode device is half as thick as that of a transmission-mode device, the reflectance of the reflection-mode device in any particular state is the same as the transmittance of the transmission-mode device in the corresponding state.

The magneto-optic spatial light modulator has several unique features that enable it to find many useful applications, such as parallel optical signal processing, optical pattern recognition, image coding, and reconfigurable optical interconnects.

On the one hand, the device is electrically addressable and has a high frame rate because the magnetization state of a pixel can be switched in a time as short as 1 ns. In practical applications, typical current pulses used for the switching are on the order of 100 ns.

On the other hand, the device has the nonvolatility to hold a pattern for a long time because the pixels do not spontaneously demagnetize without the externally applied magnetic field or the controlling current pulses.

A very high contrast ratio can be obtained by optimizing the film thickness and by eliminating the absorption and stray reflections of the optical components as much as possible.

The size of a pixel is typically on the order of 10 μm x 10 μm to 100 μm x 100 μm. A very large number of pixels can be incorporated in a device for a high image resolution.

The next part continues with the magneto-optic recording tutorial