This is a continuation from the previous tutorial - magneto-optic effects.

The Faraday effect is a phenomenon based on the propagation and transmission of an optical wave through a material with the presence of a magnetic field.

For the convenience of a general discussion, we consider the $$\boldsymbol{\epsilon}$$ tensor in the presence of a magnetic field or a magnetization of the form given by (7-16) [refer to the magneto-optic effects tutorial].

When there is an applied magnetic field but no spontaneous magnetization, we identify the tensor elements, $$\xi$$, $$n_\perp$$, and $$n_\parallel$$, with the corresponding elements in (7-15) [refer to the magneto-optic effects tutorial] as $$\xi=fH_{0z}$$, $$n_\perp^2=n_o^2+c_\perp{H_{0z}}^2$$, and $$n_\parallel^2=n_e^2+c_\parallel{H}_{0z}^2$$.

When there is a spontaneous magnetization, we follow the definitions of $$\xi(M_{0z})$$, $$n_\perp^2(M_{0z}^2)$$, and $$n_\parallel^2(M_{0z}^2)$$ in (7-16) because the effect is then completely determined by the magnetization regardless of whether there is an applied magnetic field or not.

The eigenvalues and the corresponding eigenvectors of $$\boldsymbol{\epsilon}$$ can be found by diagonalizing $$\boldsymbol{\epsilon}$$ through the normal procedure. By so doing, we find that the eigenvalues are

$\tag{7-17}\epsilon_+=\epsilon_0(n_\perp^2-\xi),\qquad\epsilon_-=\epsilon_0(n_\perp^2+\xi),\qquad\epsilon_z=\epsilon_0n_\parallel^2$

and the eigenvectors are, correspondingly,

$\tag{7-18}\hat{e}_+=\frac{1}{\sqrt2}(\hat{x}+\text{i}\hat{y}),\qquad\hat{e}_-=\frac{1}{\sqrt2}(\hat{x}-\text{i}\hat{y}),\qquad\hat{z}$

The complex eigenvectors, $$\hat{e}_+$$ and $$\hat{e}_-$$, respectively, are the left- and right-circularly polarized unit vectors defined in the polarization of light tutorial. These two complex unit vectors appear as eigenvectors because the $$\boldsymbol{\epsilon}$$ tensor in the presence of a magnetic field or a magnetization is not symmetric.

The eigenvalues are all real because $$\boldsymbol{\epsilon}$$ is Hermitian.

It is clearly not possible to attach the meaning of the principal axes in real space to these complex eigenvectors. Nonetheless, these eigenvectors still define the principal normal modes of polarization for proper decomposition of the electric field components of an optical wave that propagates in the medium:

$\tag{7-19}D_+=\epsilon_+E_+,\qquad{D_-=\epsilon_-E_-},\qquad{D_z=\epsilon_zE_z}$

Therefore, $$\epsilon_+/\epsilon_0$$, $$\epsilon_-/\epsilon_0$$, and $$\epsilon_z/\epsilon_0$$ are the principal dielectric constants for the three normal modes. They define the following three principal indices of refraction:

$\tag{7-20}n_+=\sqrt{n_\perp^2-\xi}\approx{n_\perp-\frac{\xi}{2n_\perp}},\qquad{n_-=\sqrt{n_\perp^2+\xi}\approx{n_\perp+\frac{\xi}{2n_\perp}},\qquad{n_z=n_\parallel}}$

The propagation constants for the normal modes are given by

$\tag{7-21}k^+=\frac{n_+\omega}{c},\qquad{k^-=\frac{n_-\omega}{c}},\qquad{k^z}=\frac{n_z\omega}{c}$

When an optical wave propagates along the $$z$$ axis, in either the positive $$z$$ or the negative $$z$$ direction, the normal modes are the circularly polarized modes $$\hat{e}_+$$ and $$\hat{e}_-$$, with propagation constants $$k^+$$ and $$k^-$$, respectively.

As mentioned in the polarization of light tutorial, if the wave propagates in the positive $$z$$ direction, $$\hat{e}_+$$ is the left-circular polarization and $$\hat{e}_-$$ is the right-circular polarization. If the wave propagates in the negative $$z$$ direction, $$\hat{e}_+$$ becomes the right-circular polarization while $$\hat{e}_-$$ becomes the left-circular polarization.

In either situation, however, $$n_+$$ and $$k^+$$ defined above remain with $$\hat{e}_+$$, and $$n_-$$ and $$k^-$$ still belong to $$\hat{e}_-$$.

With a fixed $$z$$ direction, the wavevectors for the two circularly polarized normal modes are $$\mathbf{k}^+=k^+\hat{z}$$ and $$\mathbf{k}^-=k^-\hat{z}$$, respectively, for forward propagation in the positive $$z$$ direction and are $$\mathbf{k}^+=-k^+\hat{z}$$ and $$\mathbf{k}^-=-k^-\hat{z}$$, respectively, for backward propagation in the negative $$z$$ direction.

Furthermore, we see from (7-20) that the values of $$n_+$$ and $$n_-$$, thus also those of $$k^+$$ and $$k^-$$, do not depend on the wave propagation direction. Instead, they depend only on the direction of $$\pmb{H}_0$$, or that of $$\pmb{M}_0$$ if an internal magnetization exists.

If an optical wave is initially circularly polarized, either left or right, it is in one of the normal modes. It propagates with a single propagation constant belonging to the circular polarization and maintains the same polarization state throughout its path in the medium. If it is reflected to propagate in the opposite direction, its handedness changes, but not its unit vector or its propagation constant.

If an optical wave is initially linear or elliptically polarized, its field is a superposition of the two circularly polarized normal modes. This field then decomposes into two circularly polarized orthogonal components that propagate with different propagation constants, $$k^+$$ and $$k^-$$. This phenomenon is called circular birefringence. It is know as magnetic circular birefringence because it is caused by the magneto-optic effect.

A case of special interest is the propagation of a linearly polarized optical wave in such a medium. Assume, without loss of generality, that the wave is initially linearly polarized in the $$x$$ direction at an arbitrary initial position $$z$$ = 0:

$\tag{7-22}\mathbf{E}(0,t)=\hat{x}\mathcal{E}\text{e}^{-\text{i}\omega{t}}=\frac{\mathcal{E}}{\sqrt2}(\hat{e}_++\hat{e}_-)\text{e}^{-\text{i}\omega{t}}$

with $$\mathcal{E}_+=\mathcal{E}_-=\mathcal{E}/\sqrt2$$. Both circularly polarized components propagate as normal modes with their respective propagation constants. When the wave propagates a distance $$l$$ in the positive $$z$$ direction, we have

\tag{7-23}\begin{align}\mathbf{E}(l,t)&=\hat{e}_+\mathcal{E}_+\exp[\text{i}\mathbf{k}^+\cdot\hat{z}(l-0)-\text{i}\omega{t}]+\hat{e}_-\mathcal{E}_-\exp[\text{i}\mathbf{k}^-\cdot\hat{z}(l-0)-\text{i}\omega{t}]\\&=\hat{e}_+\mathcal{E}_+\exp(\text{i}k^+l-\text{i}\omega{t})+\hat{e}_-\mathcal{E}_-\exp(\text{i}k^-l-\text{i}\omega{t})\\&=\frac{\mathcal{E}}{2}\left[\hat{x}\left(\text{e}^{\text{i}k^+l}+\text{e}^{\text{i}k^-l}\right)+\text{i}\hat{y}\left(\text{e}^{\text{i}k^+l}-\text{e}^{\text{i}k^-l}\right)\right]\text{e}^{-\text{i}\omega{t}}\\&=\mathcal{E}\left(\hat{x}\cos\frac{k^--k^+}{2}l+\hat{y}\sin\frac{k^--k^+}{2}l\right)\exp\left(\text{i}\frac{k^++k^-}{2}l-\text{i}\omega{t}\right)\end{align}

The optical field clearly remains linearly polarized because its $$x$$ and $$y$$ components are in phase, but its plane of polarization is rotated by an angle of

$\tag{7-24}\theta_\text{F}=\tan^{-1}\frac{\mathcal{E}_y}{\mathcal{E}_x}=\frac{k^--k^+}{2}l=\frac{\pi}{\lambda}(n_--n_+)l\approx\frac{\pi\xi}{\lambda{n_\perp}}l$

This magnetically induced rotation of the plane of polarization of a linearly polarized optical wave is called Faraday rotation, and this phenomenon is known as Faraday effect.

It can be shown that the plane of polarization rotates by the same amount in the same sense if the wave propagates in the negative $$z$$ direction for the same distance $$l$$. Therefore, the sense of Faraday rotation is independent of the direction of wave propagation.

A positive value for $$\theta$$ corresponds to a rotation from the positive $$x$$ axis to the positive $$y$$ axis, which is counterclockwise rotation when viewed facing against the direction of propagation. A device that provides the function of the Faraday rotation is called a Faraday rotator.

In a paramagnetic or diamagnetic material, which has no internal magnetization, the Faraday rotation for a linearly polarized wave propagating over a distance $$l$$ is linearly proportional to the externally applied magnetic field. The Faraday rotation angle in this case is generally expressed as

$\tag{7-25}\theta_\text{F}=VH_{0z}l$

where

$\tag{7-26}V=\frac{\omega{f}}{2cn_\perp}=\frac{\pi{f}}{\lambda{n_\perp}}$

is the Verdet constant (measured in radians per ampere).

In the literature, the Verdet constant is often quoted in Gaussian units (minutes per oersted per centimeter). The conversion between Gaussian and SI units is $$1\text{ min Oe}^{-1}\text{ cm}^{-1}=2.094\times10^{-2}\text{ deg A}^{-1}=3.655\times10^{-4}\text{ rad A}^{-1}$$. The Verdet constant defined in terms of (7-25) and given in radians per ampere or degrees per ampere is convenient when the magnetic field is generated by a current.

In many practical situations, however, the magnetic field is provided by a permanent magnet. Then, the Faraday rotation angle is often written in terms of the magnetic induction as $$\theta_\text{F}=VB_{0z}l$$, and the unit of the Verdet constant is correspondingly quoted as degrees per gauss per centimeter in the Gaussian system or radians per tesla per meter in the SI system. The conversion between Gaussian and SI units is $$1^{\circ}\text{ G}^{-1}\text{ cm}^{-1}={10^6}^{\circ}\text{ T}^{-1}\text{ m}^{-1}=1.745\times10^4\text{ rad T}^{-1}\text{ m}^{-1}$$.

The conversion of units for the Verdet constant from one defined in terms of $$B_{0z}$$ to one defined in terms of $$H_{0z}$$ is $$1^{\circ}\text{ G}^{-1}\text{ cm}^{-1}\rightarrow1^{\circ}\text{ Oe}^{-1}\text{ cm}^{-1}$$ for Gaussian units and $$1\text{ rad T}^{-1}\text{ m}^{-1}\rightarrow4\pi\times10^{-7}\text{ rad A}^{-1}$$ for SI units.

The Verdet constant has positive values for diamagnetic materials and negative values for paramagnetic materials. The Verdet constants of some materials of interest are listed in Table 7-1 below.

The Verdet constant of a given material is a function of both optical wavelength and temperature. In the optical spectral region, its absolute value usually increases when the optical wavelength or the temperature decreases.

In a ferromagnetic or ferrimagnetic material, which has an internal magnetization, $$\xi$$ is determined by the magnetization rather than by the applied magnetic field. The total Faraday rotation angle for an optical wave traveling over a distance $$l$$ through such a material is simply

$\tag{7-27}\theta_\text{F}=\rho_\text{F}\frac{M_{0z}}{M_s}l$

where $$M_{0z}$$ is the existing magnetization in the material and $$M_s$$ is the saturation magnetization of the material.

The Faraday rotation can be small if the material is not sufficiently magnetized; it is maximized only when the material is fully magnetized to reach its saturation magnetization.

The Faraday rotation is then characterized by the following specific Faraday rotation, or rotatory power:

$\tag{7-28}\rho_\text{F}=\frac{\omega\xi(M_s)}{2cn_\perp}=\frac{\pi\xi(M_s)}{\lambda{n_\perp}}$

which is the amount of rotation per unit length traversed by the optical wave in the material at saturation magnetization. It has the unit of radians per meter, but is often quoted in the unit of degrees per centimeter. The conversion between them is $$1^{\circ}\text{ cm}^{-1}=1.745\text{ rad m}^{-1}$$. The specific Faraday rotation can have either positive or negative values.

Many metallic ferromagnetic materials, such as Fe, Co, and Ni, have very large values of specific Faraday rotation, but they also have very large absorption coefficients and, consequently, are not very useful in many device applications that require optical transmission.

Therefore, a figure of merit for these materials is $$\rho_\text{F}/\alpha$$, often quoted in degrees per decibel, which measures the amount of Faraday rotation in a medium for a certain amount of attenuation.

The specific Faraday rotations of some ferromagnetic and ferrimagnetic materials, together with their absorption coefficients and figures of merit, are listed in Table 7-2 below.

Both the specific Faraday rotation and the absorption coefficient of a material are highly dependent on the optical wavelength and the temperature. Similar to the Verdet constant of a paramagnetic or diamagnetic material, the specific Faraday rotation of a ferromagnetic or ferrimagnetic material normally increases when the temperature or the wavelength decreases.

Significant variations can occur in either direction, however, near the optical frequencies corresponding to ferromagnetic resonances in such materials, sometimes even changing the sign of the specific Faraday rotation in a given material at certain resonance frequencies.

The Faraday effect is nonreciprocal. It has the characteristic that the sense of the Faraday rotation in a particular material is independent of the direction of wave propagation but is determined only by the direction of the external magnetic field, or that of the magnetization if the material is ferromagnetic or ferrimagnetic.

The expression for $$\theta_\text{F}$$ in (7-25) holds true for propagation in either the parallel or the antiparallel direction with respect to $$\pmb{H}_0$$, and that for $$\rho_\text{F}$$ in (7-28) is also valid for propagation in either direction with respect to $$\pmb{M}_0$$.

The amount of the Faraday rotation is doubled, rather than canceled, when an optical wave passing through a magneto-optic material is reflected to retrace its original path in the opposite direction back to the starting point. This phenomenon is a direct consequence of the fact discussed above that the propagation constant associated with each circularly polarized eigenvector is independent of the wave propagation direction and, therefore, is not changed by reflection.

The Faraday rotation is positive when the value of $$\theta_\text{F}$$, or that of $$\rho_\text{F}$$, is positive, meaning that the rotation is counterclockwise when viewed in the direction against that of $$\pmb{H}_0$$, or that of $$\pmb{M}_0$$ when an internal magnetization exists.

Therefore, the sense of positive Faraday rotation is the same as the electric current that generates $$\pmb{H}_0$$ or, in the case of ferromagnets and ferrimagnets, the current that can be conceptually associated with $$\pmb{M}_0$$.

Using the right-hand rule, the axial vector corresponding to a positive Faraday rotation points in the same direction as that of the $$\pmb{H}_0$$ or $$\pmb{M}_0$$ causing the Faraday effect. For negative Faraday rotation, the sense of rotation is opposite to that for positive Faraday rotation. Figure 7-1 below summarizes these concepts.

The Faraday rotation in a diamagnetic material is positive because its Verdet constant is positive, whereas that in a paramagnetic material is negative because its Verdet constant is negative. The Faraday rotation in ferromagnets and ferrimagnets can be either positive or negative.

Example 7-1

A Faraday rotator consists of a TGG crystal in a magnetic field that has a flux density of $$B_0=0.5\text{ T}$$ along the longitudinal axis of the crystal. If a Faraday rotation angle of 45° is desired for a linearly polarized optical beam at 800 nm wavelength traveling through the crystal, what should the crystal length be? In which sense is the polarization of the wave rotated? If the beam is reflected back at the output end of the crystal, what is the polarization direction of the reflected wave at the input end?

From Table 7-1, $$V=-65\text{ rad T}^{-1}\text{ m}^{-1}$$ at 800 nm wavelength for TGG. Because the value of $$V$$ is negative but that of $$B_0$$ is positive in this problem, the Faraday rotation angle is negative. These sense of rotation for a negative Faraday angle is clockwise when viewed facing against the direction of wave propagation. The desired Faraday rotation angle is $$\theta_\text{F}=-45^{\circ}=-\pi/4\text{ rad}$$. Therefore, the required length of the crystal is

$l=\frac{\theta_{F}}{VB_0}=\frac{-\pi/4}{-65\times0.5}\text{ m}=0.024\text{ m}=24\text{ mm}$

The total Faraday rotation angle of the reflected wave is double that of the single-pass rotation angle. Thus, the reflected wave returning to the input end is linearly polarized at 90° with respect to the polarization direction of the incident wave.

Besides attenuating the transmission of an optical wave, the optical absorption of a material has some very interesting consequences on the Faraday effect. In the presence of absorption losses, both $$n_+$$ and $$n_-$$ become complex. When the imaginary parts of $$n_+$$ and $$n_-$$ do not have equal values, the two circularly polarized normal modes experience different degrees of attenuation. This phenomenon is called circular dichroism, as distinct from the linear dichroism between two linearly polarized modes.

In the presence of circular dichroism, a linearly polarized wave undergoing a Faraday rotation does not remain linearly polarized but becomes elliptically polarized with a Faraday rotation angle, $$\theta_\text{F}$$, and a Faraday ellipticity, $$\epsilon_\text{F}$$, given by

$\tag{7-29}\theta_\text{F}=\text{Re}\left[\frac{\pi}{\lambda}(n_--n_+)l\right]\approx\text{Re}\left[\frac{\pi\xi}{\lambda{n_\perp}}l\right]$

and

$\tag{7-30}\epsilon_\text{F}=\tan^{-1}\tanh\text{Im}\left[\frac{\pi}{\lambda}(n_--n_+)l\right]\approx\tan^{-1}\tanh\text{Im}\left[\frac{\pi\xi}{\lambda{n_\perp}}l\right]$

The absorption that is directly related to the magneto-optic effect makes $$\xi$$ a complex quantity with an imaginary part, $$\xi''$$. If the background material is relatively lossless so that $$n_\perp'\gg|n_\perp''|$$, the circular dichroism is solely contributed by $$\xi''$$ and is known as magnetic circular dichroism.

A material in which the normal modes of optical wave propagation are circularly polarized is referred to as being optically active or optically gyroscopic. Such a material exhibits circular birefringence. Certain nonmagnetic materials, such as quartz and sugar solutions, possess natural optical activity in the absence of a magnetic field or a magnetization. In analogy, the existence of a magnetically induced circular birefringence in an otherwise optically nonactive material is sometimes called artificial, or induced, optical activity.

The similarities between these two phenomena are that both have circularly polarized normal modes and both can cause circular birefringence and circular dichroism. The plane of polarization of a linearly polarized wave can also be rotated while propagating through a naturally optically active medium in a way similar to Faraday rotation.

The fundamental difference between them is that natural optical activity is reciprocal, as mentioned in the polarization of light tutorial, whereas magnetically induced optical activity is nonreciprocal.

In the simplest case, natural optical activity can be described by an $$\boldsymbol{\epsilon}$$ tensor in the form of (7-16) [refer to the magneto-optic effects tutorial] but with $$\xi=\gamma\hat{k}\cdot\hat{z}$$, where $$\gamma$$ is a characteristic constant of the medium. Because of this dependence on wavevector, the values of $$k^+$$ and $$k^-$$ associated with $$\hat{e}_+$$ and $$\hat{e}_-$$ are exchanged when the propagation direction is reversed.

This characteristic, in contrast to the discussions immediately following (7-21) above, manifests the fundamental difference between natural circular birefringence and magnetic circular birefringence.

As a result, when a linearly polarized optical wave traverses a naturally optically active medium twice along the same path but in opposite directions, the angle of rotation of its polarization gained in the forward pass is exactly canceled by that obtained in the backward pass, thus returning the wave back to its exact original polarization direction.

Whereas magnetically induced optical activity exists in all materials, natural optical activity cannot exist in centrosymmetric materials. In an otherwise centrosymmetric medium, such as a liquid, the addition of molecules, such as sugar molecules, that cause optical activity breaks the centrosymmetry of the system.

The next part continues with the magneto-optic Kerr effect tutorial