# Magneto-optic effects

This is a continuation from the previous tutorial - traveling-wave modulators.

Magneto-optic materials have unique physical properties that offer the opportunity of constructing devices with many special functions not possible from other photonic devices. The most significant of these properties are that the linear magneto-optic effect can produce circular birefringence and that, unlike other optical effects in dielectric media, it is nonreciprocal. All practical magneto-optic devices exploit one or both of these two properties. Important applications of these devices include polarization control, optical isolation, optical modulation, and magneto-optic recording. The basic principles of magneto-optic effects, as well as the functions of various magneto-optic devices based on these effects, are considered in the following series of tutorials.

Because magneto-optic effects are intimately connected to the magnetic properties of materials, we first briefly summarize the fundamental magnetic properties of materials. To a certain degree there is a parallelism between the electric and the magnetic properties of materials, but this parallelism is not complete. We shall pay attention to the similarities and differences between these properties in order to gain an appreciation of the uniqueness of magneto-optic devices.

Following the similarity between (1) and (2) in the optical fields and Maxwell's equations tutorial, a ** magnetic susceptibility tensor**, \(\boldsymbol{\chi}_\text{m}\), analogous to the electric susceptibility tensor \(\boldsymbol{\chi}\) can be defined to describe the magnetization induced by a magnetic field. In parallel with (54) and (55) in the harmonic fields tutorial, we have, for magnetic fields,

\[\tag{7-1}\mathbf{M}(\mathbf{k},\omega)=\boldsymbol{\chi}_\text{m}(\mathbf{k},\omega)\cdot\mathbf{H}(\mathbf{k},\omega)\]

and

\[\tag{7-2}\mathbf{B}(\mathbf{k},\omega)=\mu_0\mathbf{H}(\mathbf{k},\omega)+\mu_0\mathbf{M}(\mathbf{k},\omega)=\boldsymbol{\mu}(\mathbf{k},\omega)\cdot\mathbf{H}(\mathbf{k},\omega)\]

In general, magnetic permeability

\[\tag{7-3}\boldsymbol{\mu}=\mu_0(1+\boldsymbol{\chi}_\text{m})\]

is a tensor for an anisotropic material.

As mentioned in the optical fields and Maxwell's equations tutorial, in any material there is no physical basis to specify a magnetization induced by the magnetic component of an electromagnetic field at an optical frequency. Because the electric and magnetic components in an electromagnetic field are mutually coupled, it is not physically meaningful to define an optically induced magnetization that is separate from the optically induced electric polarization. Consequently, at an optical frequency \(\omega\), \(\boldsymbol{\chi}_\text{m}(\omega)=0\) and \(\boldsymbol{\mu}(\omega)=\mu_0\) in all materials, including anisotropic ones.

In magnetostatics, however, a nonvanishing magnetization in response to an externally applied magnetic field can appear in a material. In this situation, a nonvanishing magnetic susceptibility \(\boldsymbol{\chi}_\text{m}\) is meaningful and \(\boldsymbol{\mu}\) clearly differs from \(\mu_0\).

Both \(\boldsymbol{\chi}_\text{m}\) and \(\boldsymbol{\mu}\) are ** symmetric tensors**. They can be diagonalized with real, orthogonal principal axes in a manner similar to the diagonalization of the symmetric \(\boldsymbol{\chi}\) and \(\boldsymbol{\epsilon}\) tensors of a nonmagnetic dielectric material that shows no optical activity. A fundamental difference between the electric and the magnetic properties of materials is that

**, as mentioned in the propagation in an anisotropic medium tutorial,**

*while the principal dielectric susceptibilities are always positive***.**

*the principal magnetic susceptibilities of lossless magnetic materials can be either positive or negative*A material is referred to as ** paramagnetic** if its principal magnetic susceptibilities are positive and as

**if they are negative. While the dielectric susceptibilities of ordinary materials are typically on the order of \(1-10\), the magnetic susceptibilities of paramagnetic and diamagnetic materials are extremely small, typically on the order of \(\pm10^{-5}\).**

*diamagnetic*In a diamagnetic material there are no intrinsic magnetic dipole moments. The negative magnetization in such a material results from the magnetic dipole moments induced by an external magnetic field, which are always aligned in opposition to the inducing field.

In contrast, a paramagnetic material consists of atoms or ions that have intrinsic magnetic dipole moments. Above a certain temperature that is characteristic of a particular paramagnetic material, these magnetic dipoles are randomly oriented in thermal equilibrium. In the presence of an externally applied magnetic field, these dipoles tend to align in the direction of the magnetic field and overshadow all diamagnetic effects in the material, resulting in a net positive magnetization. A few crystals are paramagnetic along one principal axis but diamagnetic along another, however.

In some paramagnetic solids, the intrinsic magnetic dipole moments can become orderly oriented in the absence of an external magnetic field due to their mutual interactions if the temperature is reduced below a certain critical value. Such solids are called ** magnetically ordered**. A magnetically ordered solid whose intrinsic magnetic dipoles tend to line up in the same direction has a

**and is called a**

*spontaneous magnetization***material, or a**

*ferromagnetic***. Examples of ferromagnets are Fe, Co, Ni, Gd, Dy, and the alloy MnBi.**

*ferromagnet*The intrinsic magnetic dipoles in a magnetically ordered solid can also assume alternate antiparallel directions. This ordering can still result in a net spontaneous magnetization if the alternating dipoles are different and their moments do not cancel. Material with this kind of property are called ** ferrimagnetic** materials, or

**. The most important ferrimagnetic materials are the rare-earth iron garnets, particularly Y**

*ferrimagnets*_{3}Fe

_{5}O

_{12}(YIG), and the rare-earth transition-metal (RE-TM) alloys, such as GdFe, GdCo, TbFe, GdTbFe, TbFeCo, etc.

If antiparallel alignment of the alternating dipoles in a solid results in complete cancellation of their magnetic moments, there is no net spontaneous magnetization though the solid is still magnetically ordered. Such a solid is called an ** antiferromagnetic** material, or an

**. Examples of antiferromagnets are Cr, FeO, CoO, NiO, FeF**

*antiferromagnet*_{2}, CoF

_{2}, CoCO

_{3}, and many garnets, such as Ca

_{3}Cr

_{2}Ge

_{3}O

_{12}, Ca

_{3}Mn

_{2}Ge

_{3}O

_{12}, and Mn

_{3}Al

_{2}Si

_{3}O

_{12}.

The ** critical temperature**, \(T_\text{c}\), is called the

**for a ferromagnetic or ferrimagnetic material and the**

*Curie temperature***for an antiferromagnetic material. Above \(T_\text{c}\), these materials are paramagnetic with small magnetic susceptibilities. In a ferromagnetic or ferrimagnetic material the value of the magnetic susceptibility diverges at \(T_\text{c}\) as the spontaneous magnetization appears. In an antiferromagnetic material, however, no spontaneous magnetization occurs. The magnetic susceptibility of an antiferromagnet merely reaches a finite maximum value at a temperature slightly above \(T_\text{c}\).**

*Néel temperature*Macroscopic ** magnetic domains** having different magnetization orientations normally exist in a ferromagnetic or ferrimagnetic material below its Curie temperature. Consequently, the material appears to be only weakly magnetized or completely unmagnetized. By applying an external magnetic field, the domains that are oriented along the field grow at the expense of the adversely oriented ones, thus increasing the total net magnetization of the material. This process is reversible in weak magnetic fields but shows the characteristics of a hysteresis in strong fields.

The largest magnetization reached is the ** saturation magnetization, \(M_\text{s}\)**, beyond which the magnetization does not increase further with increasing magnetic field.

Properties analogous to ferromagnetism also exist in some dielectric materials called ** ferroelectrics**. In a ferroelectric material, a spontaneous polarization appears below its Curie temperature. Examples of ferroelectric crystals are KDP, BaTiO

_{3}, LiNbO

_{3}, and KNbO

_{3}.

Because \(\boldsymbol{\chi}_\text{m}=0\) and \(\boldsymbol{\mu}=\mu_0\) at optical frequencies, the response of a material, irrespective of whether it is magnetic or nonmagnetic, to an optical field at a frequency \(\omega\) is fully described by its electric susceptibility \(\boldsymbol{\chi}(\omega)\) and, equivalently, by its electric permittivity \(\boldsymbol{\epsilon}(\omega)\). A material does respond to a static magnetic field, \(\pmb{H}_0\), however. Its optical properties can be changed by its response to \(\pmb{H}_0\), resulting in various magneto-optic effects. The electric susceptibility and electric permittivity at an optical frequency \(\omega\) thus become a function of \(\pmb{H}_0\):

\[\tag{7-4}\mathbf{P}(\omega,\pmb{H}_0)=\epsilon_0\boldsymbol{\chi}(\omega,\pmb{H}_0)\cdot\mathbf{E}(\omega)=\epsilon_0\boldsymbol{\chi}(\omega)\cdot\mathbf{E}(\omega)+\epsilon_0\Delta\boldsymbol{\chi}(\omega,\pmb{H}_0)\cdot\mathbf{E}(\omega)\]

and

\[\tag{7-5}\mathbf{D}(\omega,\pmb{H}_0)=\boldsymbol{\epsilon}(\omega,\pmb{H}_0)\cdot\mathbf{E}(\omega)=\boldsymbol{\epsilon}(\omega)\cdot\mathbf{E}(\omega)+\Delta\boldsymbol{\epsilon}(\omega,\pmb{H}_0)\cdot\mathbf{E}(\omega)\]

where \(\boldsymbol{\chi}(\omega)=\boldsymbol{\chi}(\omega,\pmb{H}_0=0)\) and \(\boldsymbol{\epsilon}(\omega)=\boldsymbol{\epsilon}(\omega,\pmb{H}_0=0)\) represent the intrinsic properties of the medium in the absence of the magnetic field.

In the case of a ferromagnetic or ferrimagnetic material, in which a static magnetization \(\pmb{M}_0\) exists, the properties of the medium at an optical frequency are dependent on \(\pmb{M}_0\). Then, instead of (7-4) and (7-5), we have

\[\tag{7-6}\mathbf{P}(\omega,\pmb{M}_0)=\epsilon_0\boldsymbol{\chi}(\omega,\pmb{M}_0)\cdot\mathbf{E}(\omega)=\epsilon_0\boldsymbol{\chi}(\omega)\cdot\mathbf{E}(\omega)+\epsilon_0\Delta\boldsymbol{\chi}(\omega,\pmb{M}_0)\cdot\mathbf{E}(\omega)\]

and

\[\tag{7-7}\mathbf{D}(\omega,\pmb{M}_0)=\boldsymbol{\epsilon}(\omega,\pmb{M}_0)\cdot\mathbf{E}(\omega)=\boldsymbol{\epsilon}(\omega)\cdot\mathbf{E}(\omega)+\Delta\boldsymbol{\epsilon}(\omega,\pmb{M}_0)\cdot\mathbf{E}(\omega)\]

While \(\boldsymbol{\chi}\) and \(\boldsymbol{\epsilon}\) are changed in the presence of \(\pmb{H}_0\) or \(\pmb{M}_0\), the magnetic permeability of the material at an optical frequency remains the constant \(\mu_0\), and the relation between \(\mathbf{B}(\omega)\) and \(\mathbf{H}(\omega)\) remains independent of \(\pmb{H}_0\) or \(\pmb{M}_0\):

\[\tag{7-8}\mathbf{B}(\omega)=\mu_0\mathbf{H}(\omega)\]

Therefore, magneto-optic effects are completely characterized by \(\boldsymbol{\epsilon}(\omega,\pmb{H}_0)\), if no internal magnetization is present, or by \(\boldsymbol{\epsilon}(\omega,\pmb{M}_0)\), if an internal magnetization is present. In general, these effects are weak perturbations to the optical properties of the material. The *first-order*, *or linear*, *magneto-optic effect* is characterized by a linear dependence of \(\boldsymbol{\epsilon}\) on \(\pmb{H}_0\) or \(\pmb{M}_0\), and the *second-order*, *or quadratic*, *magneto-optic effect* results from a quadratic dependence of \(\boldsymbol{\epsilon}\) on \(\pmb{H}_0\) or \(\pmb{M}_0\). Note that like electro-optic effects, both first- and second-order magneto-optic effects are nonlinear optical effects.

The general description of magneto-optic effects in terms of \(\boldsymbol{\epsilon}(\omega,\pmb{H}_0)\) or \(\boldsymbol{\epsilon}(\omega,\pmb{M}_0)\) is analogous to the general description of electro-optic effects in terms of \(\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\). The classification of first- and second-order magneto-optic effects is also analogous to that of first- and second-order electro-optic effects.

However, there are many important fundamental differences between magneto-optic and electro-optic effects. These differences originate from basic distinctions in the electric and magnetic characteristics of materials and are mostly tied to the fact that electric and magnetic fields follow different rules of transformation under space inversion and time reversal, as described in the optical fields and Maxwell's equations tutorial.

The major differences and their implications are summarized below.

1. **Space-inversion symmetry**. Materials with space-inversion symmetry are centrosymmetric. In such materials, no spontaneous electric polarization can exist, and the first-order electro-optic effect also vanishes. However, ** neither a spontaneous magnetization nor the first-order magneto-optic effect is forbidden in a centrosymmetric material**. This difference is due to the fact that under space inversion, the polar vectors \(\mathbf{P}\) and \(\mathbf{E}\) change sign, but the axial vectors \(\mathbf{M}\) and \(\mathbf{H}\) do not. Therefore, amorphous solids can be ferromagnetic or ferrimagnetic but cannot be ferroelectric. The first-order magneto-optic effect appears in gases and liquids, as well as in amorphous solids and nonpolar cubic crystals, where the Pockels effect does not exist.

2. **Time-reversal symmetry**. Materials with time-reversal symmetry are lossless and reciprocal. A lossless dielectric material not subject to an external magnetic field possesses time-reversal symmetry because it is reciprocal. Time-reversal symmetry is lost in a dielectric material when it has a loss or gain, or when an external magnetic filed is applied to it. A magnetically ordered material, regardless of whether it is lossless or not, is nonreciprocal and thus does not possess time-reversal symmetry.

3. **Reciprocity**. In an optical system that has time-reversal symmetry, an optical signal can be run backward in time without changing the reality of the physics. This is not possible if the medium involved is nonreciprocal or it has a loss or gain. However, there is a difference between the two possibilities that causes the time-reversal symmetry to break down. In a nonreciprocal medium, there is no symmetry in the interchange of the source and the detector of an optical signal. In contrast, the symmetry in such an interchange exists in a reciprocal medium that has a loss or gain. Therefore, only the magneto-optic system, being nonreciprocal, can provide the function of ** optical isolation** discussed in later tutorials. A lossy dielectric system, despite its lack of time-reversal symmetry, is not capable of such a function.

4. **Magnetic symmetry**. The symmetry properties of dielectric materials discussed in earlier tutorials are based on the considerations of spatial transformations only. They are in fact the ** electric symmetry properties** of materials. The

**of materials have to be determined by considering spatial transformations in combination with time-reversal transformation because magnetic structures do not have time-reversal symmetry. The result is**

*magnetic symmetry properties***, called**

*magnetic symmetry groups***, which are much more complicated than the ordinary symmetry groups based solely on the electric structures of crystals. Therefore, general symmetry considerations for the magneto-optic effects in magnetically ordered crystals, particularly in anisotropic ones, are quite complicated.**

*Shubnikov's groups*

We first consider the magneto-optic effects in a material that is not magnetically ordered, i.e., a paramagnet or a diamagnet. In such a material, operation of the time-reversal transformation yields

\[\tag{7-9}\epsilon_{ij}(\omega,\pmb{H}_0)=\epsilon_{ji}(\omega,-\pmb{H}_0)\]

when the material is subject to an external magnetic field \(\pmb{H}_0\). This relation characterizes the magneto-optic effects in a magnetically non-ordered material. It is generally true regardless of the symmetry property of the material. If the material is lossless, then its dielectric permittivity tensor is Hermitian:

\[\tag{7-10}\epsilon_{ij}(\omega,\pmb{H}_0)=\epsilon_{ji}^*(\omega,\pmb{H}_0)\]

If we express the real and imaginary parts of \(\boldsymbol{\epsilon}\) explicitly by writing \(\epsilon_{ij}=\epsilon_{ij}'+\text{i}\epsilon_{ij}''\), we find from combination of these two relations that

\[\tag{7-11}\epsilon_{ij}'(\omega,\pmb{H}_0)=\epsilon_{ij}'(\omega,-\pmb{H}_0)=\epsilon_{ji}'(\omega,\pmb{H}_0)=\epsilon_{ji}'(\omega,-\pmb{H}_0)\]

\[\tag{7-12}\epsilon_{ij}''(\omega,\pmb{H}_0)=-\epsilon_{ij}''(\omega,-\pmb{H}_0)=-\epsilon_{ji}''(\omega,\pmb{H}_0)=\epsilon_{ji}''(\omega,-\pmb{H}_0)\]

As a result, the magneto-optic effects in a magnetically non-ordered, lossless material can be generally described as

\[\tag{7-13}\epsilon_{ij}(\pmb{H}_0)=\epsilon_{ij}+\Delta\epsilon_{ij}(\pmb{H}_0)=\epsilon_{ij}+\text{i}\epsilon_0\sum_kf_{ijk}H_{0k}+\epsilon_0\sum_{k,l}c_{ijkl}H_{0k}H_{0l}+\ldots,\]

where \(f_{ijk}\) and \(c_{ijkl}\) are real quantities that satisfy the following relations:

\[\tag{7-14}f_{ijk}=-f_{jik},\qquad{c_{ijkl}=c_{jikl}=c_{ijlk}=c_{jilk}}\]

The linear dependence of \(\epsilon_{ij}(\pmb{H}_0)\) on the magnetic field appears only as antisymmetric, imaginary components in the off-diagonal elements of the permittivity tensor. This first-order magneto-optic effect results in ** circular birefringence**; it manifests itself as, notably, the

**and the**

*Faraday effect***discussed in the following tutorials.**

*magneto-optic Kerr effect*** The first-order magneto-optic effect and the phenomena resulting from it are nonreciprocal**. In any material, even a centrosymmetric one, that has a spontaneous magnetization or is subject to an external magnetic field, the first-order magneto-optic effect always exists, resulting in the nonreciprocity of such a material.

In contrast, the quadratic dependence on the magnetic field appears as symmetric, real components in the permittivity tensor elements. ** This second-order magneto-optic effect is reciprocal** and is called the

**. It causes a**

*Cotton-Mouton effect***in the material and is analogous to, but much weaker than, the electro-optic Kerr effect.**

*linear birefringence*Note that in expressing the magneto-optic effects in terms of (7-13) we have expanded the elements of the permittivity tensor \(\boldsymbol{\epsilon}\) instead of expanding those of the relative impermeability tensor \(\boldsymbol{\eta}\), as done in (6-14) [refer to the electro-optic effects tutorial] for electro-optic effects.

The reason is that it is convenient to use the index ellipsoid in dealing with electro-optic effects but not in treating magneto-optic effects. Instead, as we shall see in later tutorials, it is convenient to use the permittivity tensor directly to treat magneto-optic effects.

As demonstrated in the electro-optic effects tutorial, the choice of using \(\boldsymbol{\epsilon}\) or \(\boldsymbol{\eta}\) does not make a difference in the final result and is only a matter of convenience.

The magneto-optic effects are relatively weak in comparison to, and tend to be obscured by, any natural or structural birefringence that might exist in a material.

Fortunately, both first- and second-order magneto-optic effects exist in isotropic materials, including non-crystals and cubic crystals. For these reasons, materials of particular interest and practical importance for magneto-optic effects and their applications are those in which any birefringence originated from other effects, such as material anisotropy or inhomogeneity, does not exist or, if it exists, does not dominate the particular magneto-optic effect of interest.

Such materials include isotropic materials and, in some cases, uniaxial crystals subject to a magnetic field that is parallel to the optical axis. For magneto-optic effects in these materials, we can take the direction of \(\pmb{H}_0\) to be the \(z\) direction without loss of generality. Then, \(\pmb{H}_0=H_{0z}\hat{z}\), and (7-13) yields

\[\tag{7-15}\boldsymbol{\epsilon}(\pmb{H}_0)=\epsilon_0\begin{bmatrix}n_o^2+c_{\perp}H_{0z}^2&\text{i}fH_{0z}&0\\-\text{i}fH_{0z}&n_o^2+c_{\perp}H_{0z}^2&0\\0&0&n_e^2+c_{\parallel}H_{0z}^2\end{bmatrix}\]

where \(f=f_{123},c_{\perp}=c_{1133}=c_{2233},\) and \(c_{\parallel}=c_{3333}\).

Clearly, the Cotton-Mouton effect results in a linear birefringence, which is insignificant unless \((c_{\parallel}-c_{\perp})H_{0z}^2\ge{n_e^2-n_o^2}\). This condition usually means that the material has to be isotropic or nearly isotropic in order for the Cotton-Mouton effect to be observable. Note also that in order to observe the Cotton-Mouton effect, the propagation direction of an optical wave has to be perpendicular to the magnetic field direction so that an optical field component parallel to \(\pmb{H}_0\) exists.

The magneto-optic effects in magnetically ordered crystals can be rather complicated due to the magnetic symmetry properties of such crystals. However, for the same reasons as describe above, magnetically ordered materials of practical importance for device applications are also isotropic materials and, in some cases, uniaxial crystals with the magnetic field or the magnetization parallel to the optical axis.

For a magnetically ordered, lossless material that falls into one of these categories, (7-15) applies if it is antiferromagnetic.

In a ferromagnetic or ferrimagnetic material, the magneto-optic effects are determined by its magnetization \(\pmb{M}_0\), rather than by any external applied magnetic field \(\pmb{H}_0\), despite the fact that the value and direction of \(\pmb{M}_0\) can be varied by \(\pmb{H}_0\). Then, for \(\pmb{M}_0=M_{0z}\hat{z}\), \(\boldsymbol{\epsilon}(\pmb{M}_0)\) can be expressed as

\[\tag{7-16}\boldsymbol{\epsilon}(\pmb{M}_0)=\epsilon_0\begin{bmatrix}n_{\perp}^2&\text{i}\xi&0\\-\text{i}\xi&n_{\perp}^2&0\\0&0&n_\parallel^2\end{bmatrix}\]

where \(\xi\), representing the first-order effect, is linearly dependent on \(M_{0z}\) with the symmetry of \(\xi(M_{0z})=-\xi(-M_{0z})\), and \(n_\perp^2\) and \(n_\parallel^2\), accounting for the second-order effect, are functions of \(M_{0z}^2\).

In the case of an isotropic material, \(n_\parallel^2-n_\perp^2\) is proportional to \(M_{0z}^2\), creating a magnetically induced linear birefringence. Because of this ** magnetic linear birefringence**, a ferromagnet or ferrimagnet that has an isotropic structure does not really have isotropic optical properties. For the same reason, the so-called cubic ferromagnetic or cubic ferrimagnetic crystals, such as YIG and other magnetic garnets, are never really cubic.

However, the magnetic linear birefringence is generally very small, with \(n_\parallel-n_\perp\) on the order of a few \(\times10^{-5}\) at room temperature for most magnetic garnets. In the case of a uniaxial crystal, the magnetic linear birefringence is dominated by, and is difficult to separate from, the nonmagnetic natural birefringence of the crystal.

In the rest of the following tutorials, we shall restrict our discussions to magneto-optic effects in isotropic materials or uniaxial crystals with \(\pmb{H}_0\), or \(\pmb{M}_0\), parallel to the optical axis.

The next part continues with the Faraday effect tutorial.