# Guided-wave magneto-optic devices

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

It is possible to implement various kinds of guided-wave magneto-optic devices for optical modulation, switching, and many other functions. Nevertheless, there has been very little interest in developing such devices because equal or better performance of the functions provided by such devices can be accomplished by their electro-optic or acousto-optic counterparts.

Among devices of equal performance, the magneto-optic ones have certain disadvantages. Magneto-optic waveguides are not compatible with the dielectric and semiconductor waveguides used in most photonic devices because they have to be fabricated with magnetic materials, most commonly garnets, on special substrates that can support such waveguides.

Furthermore, magneto-optic modulators and switches compare less favorably to the voltage-controlled electro-optic modulators and switches of the same function, particularly those in waveguide structures, because they have to be controlled by currents.

However, magneto-optic devices utilizing the linear magneto-optic effect have the unique advantage of nonreciprocity, which is not possible for device utilizing electro-optic or acousto-optic effects.

Therefore, the most important guided-wave magneto-optic devices are nonreciprocal devices, including guided-wave optical isolators and circulators. In such devices, the core of the waveguide consists of a material that has a spontaneous magnetization. No controlling current is needed. Most of them utilize YIG or Bi-substituted YIG waveguides on GGG substrates.

Nonreciprocal TE-TM mode converters

There are some fundamental differences between guided-wave devices and bulk devices. Polarization rotation in a waveguide is accomplished through coupling between orthogonally polarized modes, such as the TE and TM modes in a planar waveguide and the TE-like and TM-like modes in a three-dimensional waveguide.

Figure 7-21 below shows a YIG waveguide on a GGG substrate with a magnetization $$\pmb{M}_0=M_{0z}\hat{z}$$ and a permittivity tensor described by (7-16) [refer to the magneto-optic effects tutorial].

From the expression in (7-7) [refer to the magneto-optic effects tutorial], this permittivity tensor can be divided into two parts by writing $$\boldsymbol{\epsilon}(\pmb{M}_0)=\boldsymbol{\epsilon}(0)+\Delta\boldsymbol{\epsilon}(\pmb{M}_0)$$. The waveguide modes are defined by the permittivity tensor:

$\tag{7-59}\boldsymbol{\epsilon}(0)=\epsilon_0\begin{bmatrix}n_\perp^2&0&0\\0&n_\perp^2&0\\0&0&n_\parallel^2\end{bmatrix}$

which does not include the circular birefringence caused by the magnetization.

These modes are coupled through the linear magneto-optic effect caused by the perturbing permittivity tensor:

$\tag{7-60}\Delta\boldsymbol{\epsilon}(\pmb{M}_0)=\epsilon_0\begin{bmatrix}0&\text{i}\xi&0\\-\text{i}\xi&0&0\\0&0&0\end{bmatrix}$

which is responsible for circular birefringence.

For a waveguide that supports only fundamental TE-like and TM-like modes, the coupling coefficients can be found using (36) [refer to the coupled-mode theory tutorial].

With $$\Delta\boldsymbol{\epsilon}$$ given in (7-60) for $$\pmb{M}_0=M_{0z}\hat{z}$$, the self-coupling coefficients, $$\kappa_\text{EE}$$ and $$\kappa_\text{MM}$$ for TE-like and TM-like modes, respectively, are both found to be zero. Consequently, the propagation constants of both modes are not influenced by the perturbation caused by $$\Delta\boldsymbol{\epsilon}$$, and the phase mismatch between them is simply given by

$\tag{7-61}2\delta=\Delta\beta=\beta_\text{TM}-\beta_\text{TE}$

where $$\beta_\text{TE}$$ and $$\beta_\text{TM}$$ are determined by taking the dielectric tensor of the waveguide core to be only $$\boldsymbol{\epsilon}(0)$$ given in (7-59).

However, there is a nonvanishing coupling coefficient between these two modes given by

\tag{7-62}\begin{align}\kappa&=\kappa_\text{EM}=\kappa^*_\text{ME}\\&=\omega\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}y\hat{\boldsymbol{\mathcal{E}}}_\text{TE}^*(x,y)\cdot\Delta\boldsymbol{\epsilon}(x,y)\cdot\hat{\boldsymbol{\mathcal{E}}}_\text{TM}(x,y)\\&\approx-\text{i}\rho_\text{F}\Gamma_\text{EM}\end{align}

where $$\rho_\text{F}$$ is the specific Faraday rotation, given by (7-28) [refer to the Faraday effect tutorial], of the waveguide material and

\tag{7-63}\begin{align}\Gamma_\text{EM}&=\frac{2\beta_\text{TE}^{1/2}\beta_\text{TM}^{1/2}}{\omega\mu_0}\frac{1}{M_{0z}}\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\text{d}x\text{d}yM_{0z}(x,y)\hat{\mathcal{E}}_{\text{TE},y}^*(x,y)\hat{\mathcal{E}}_{\text{TM},x}(x,y)\\&\approx\frac{1}{M_{0z}}\frac{\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\text{d}x\text{d}yM_{0z}(x,y)\hat{\mathcal{E}}_{\text{TE},y}^*(x,y)\hat{\mathcal{E}}_{\text{TM},x}(x,y)}{\left[\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}y|\hat{\mathcal{E}}_{\text{TE},y}|^2\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}y|\hat{\mathcal{E}}_{\text{TM},x}|^2\right]^{1/2}}\end{align}

is the overlap factor for the magneto-optic coupling of TE-like and TM-like modes through the linear magneto-optic effect.

The distribution of the magnetization in the entire structure is described by $$M_{0z}(x,y)$$, which has a value of $$M_{0z}(x,y)=M_{0z}$$ in the magnetic core region and a value of $$M_{0z}(x,y)=0$$ in the nonmagnetic regions.

The coupling efficiency from one mode to another in a nonreciprocal magneto-optic TE-TM mode converter follows that of the codirectional coupler discussed in the two-mode coupling tutorial.

However, unlike the electro-optic TE-TM mode converter discussed in the guide-wave electro-optic modulators tutorial, a magneto-optic TE-TM mode converter is a nonreciprocal device. If an optical wave is allowed to travel multiple passes back and forth in a nonreciprocal magneto-optic TE-TM mode converter, the net coupling efficiency from one mode to the other depends only on the total distance, but not on the direction, over which the wave has traveled within the waveguide.

When the TE and TM modes are perfectly phase matched, the conversion efficiency is simply

$\tag{7-64}\eta=\sin^2|\kappa|l$

where $$l$$ is the cumulative distance traveled in both directions.

An optical wave that makes a round trip in a nonreciprocal magneto-optic TE-TM mode converter due to reflection at the far end of the waveguide does not return to the input end in its original polarization state unless the value of $$|\kappa|l$$, with $$l$$ being twice the length of the waveguide for a round trip, happens to be an integral multiple of $$\pi$$.

In contrast, in a reciprocal TE-TM mode converter, irrespective of how many round-trip passes an optical wave has traveled, the length $$l$$ used in (7-64) for calculation of the coupling efficiency is the physical distance measured from the input end to the point where the conversion is being evaluated, not the cumulative distance traveled by the optical wave. Consequently, an optical wave that makes a round trip in a reciprocal TE-TM mode converter always returns to the input end in its original polarization state.

According to the discussions in the magneto-optic effects tutorial, the materials used in a magneto-optic device must have no other birefringence that dominates the magneto-optic effect used for the operation of the device. The materials, such as YIG and GGG, used in magneto-optic waveguides are generally isotropic materials.

The phase mismatch between TE-like and TM-like modes in such waveguides is caused by the structural birefringence due to the difference in the boundary conditions imposed by the waveguide structure on the TE-like and TM-like modes.

This structural birefringence is not very large in an ordinary waveguide, but in a magneto-optic waveguide it can easily dominate the circular birefringence caused by the linear magneto-optic effect because the value of $$\rho_\text{F}$$ is generally very small. Therefore, the phase mismatch caused by the structural birefringence, which always results in $$\Delta\beta=\beta_\text{TM}-\beta_\text{TE}\lt0$$ according to (69) [refer to the step-index planar waveguides tutorial], is generally too large for a magneto-optic TE-TM mode converter.

The general concept for reducing the phase mismatch and attaining phase matching in a nonreciprocal magneto-optic TE-TM mode converter is to introduce other birefringence of opposite sign in the waveguide to counterbalance the structural birefringence.

There are several approaches to implementing this concept. The simplest is to incorporate in the magnetic waveguide core a proper amount of stress-induced birefringence, which is caused by stress in the waveguide due to a slight lattice mismatch between the waveguide core and the substrate, or growth-induced birefringence, which is not caused by lattice mismatch but by the dopants in the waveguide core such as the bismuth atoms in a Bi-substituted YIG layer.

Another approach is to have a layer of anisotropic crystal, such as LiIO3, grown on top of the magnetic waveguide core in such a way that the fields of the orthogonally polarized TE-like and TM-like modes penetrating into this layer see a proper amount of difference in the refractive index to counterbalance the difference in their boundary conditions.

The desired counterbalancing birefringence for the elimination of phase mismatch can also be introduced by artificial structures, such as periodic grooves, on the waveguide, as the discussions in the grating waveguide couplers tutorial demonstrate.

Nonreciprocal phase shifters

A unique nonreciprocal waveguide device that has no counterpart among bulk devices is the nonreciprocal phase shifter.

The function of this device depends on nonreciprocal coupling between the transverse and longitudinal electric field components of a waveguide mode through a magnetization that has a component perpendicular to both of them.

For simplicity, we consider a planar magneto-optic waveguide whose magnetic core layer has a magnetization, $$\pmb{M}_0=M_{0y}\hat{y}$$, in a direction perpendicular to both longitudinal and transverse field components, $$\hat{\mathcal{E}}_{\text{TM},z}\hat{z}$$ and $$\hat{\mathcal{E}}_{\text{TM},x}\hat{x}$$, respectively, of the TM waveguide mode, as shown in figure 7-22 below.

The permittivity tensor of this magnetic layer can be written as $$\boldsymbol{\epsilon}(\pmb{M}_0)=\boldsymbol{\epsilon}(0)+\Delta\boldsymbol{\epsilon}(\pmb{M}_0)$$ with

$\tag{7-65}\boldsymbol{\epsilon}(0)=\epsilon_0\begin{bmatrix}n_\perp^2&0&0\\0&n_\parallel^2&0\\0&0&n_\perp^2\end{bmatrix}$

and

$\tag{7-66}\Delta\boldsymbol{\epsilon}(\pmb{M}_0)=\epsilon_0\begin{bmatrix}0&0&-\text{i}\xi\\0&0&0\\\text{i}\xi&0&0\end{bmatrix}$

where $$\xi$$ is linearly proportional to $$M_{0y}$$ and $$\xi(-M_{0y})=-\xi(M_{0y})$$.

In this case, the propagation constants of the waveguide modes without perturbation of the linear magneto-optic effect due to $$\xi(M_{0y})$$ are determined by taking $$\boldsymbol{\epsilon}(0)$$ given in (7-65) alone as the permittivity tensor of the waveguide core.

The effect of $$\Delta\boldsymbol{\epsilon}$$ on the propagation characteristics of the waveguide modes is again evaluated using coupled-mode theory [refer to the coupled-mode theory tutorial] in a procedure similar to that used above in the treatment of the TE-TM mode converter.

For the planar waveguide shown in figure 7-22 above, it is found that $$\kappa_\text{EM}=\kappa_\text{ME}=\kappa_\text{EE}=0$$ and

\tag{7-67}\begin{align}\kappa_\text{MM}&=\omega\displaystyle\int\limits_{-\infty}^\infty\text{d}x\hat{\boldsymbol{\mathcal{E}}}_\text{TM}^*(x)\cdot\Delta\boldsymbol{\epsilon}(x)\cdot\hat{\boldsymbol{\mathcal{E}}}_\text{TM}(x)\\&=2\omega\epsilon_0\frac{\xi}{M_{0y}}\text{Im}\left[\displaystyle\int\limits_{-\infty}^\infty\text{d}xM_{0y}(x)\hat{\mathcal{E}}^*_{\text{TM},x}(x)\hat{\mathcal{E}}_{\text{TM},z}(x)\right]\end{align}

where $$M_{0y}(x)=M_{0y}$$ in the magnetic core layer and $$M_{0y}(x)=0$$ outside the magnetic core.

There is no TE-TM mode coupling in this waveguide, and the propagation constant of a TE mode is not perturbed by the linear magneto-optic effect.

However, because $$\hat{\mathcal{E}}_x$$ and $$\hat{\mathcal{E}}_z$$ of a TM mode are $$90^\circ$$ out-of-phase, as can be seen by comparing (35) with (36) [refer to to the wave equations for optical waveguides tutorial], $$\kappa_\text{MM}$$ exists if the integral in (7-67) yields a nonzero value.

It can be shown that this integral is zero if the waveguide is symmetric but is nonzero if the two boundaries of the magnetic core layer are different. Consequently, the linear magneto-optic effect can induce a change in the propagation constant of a TM mode in an asymmetric waveguide.

If the designation of the $$x$$, $$y$$, and $$z$$ coordinate axes is fixed, the sign of $$\xi$$ is fixed in a given waveguide with a fixed magnetization.

However, the product $$\hat{\mathcal{E}}^*_{\text{TM},x}\hat{\mathcal{E}}_{\text{TM},z}$$ changes sign together with the propagation constant when the direction of propagation of a TM mode is reversed, as can be seen from figure 7-22.

According to (7-67), this sign change leads to a corresponding sign change in $$\kappa_\text{MM}$$ with the reversal of the propagation direction:

$\tag{7-68}\kappa_\text{MM}^\text{b}=-\kappa_\text{MM}^\text{f}$

Taking into account these changes caused by the linear magneto-optic effect, the propagation constants of a TM mode in forward and backward directions of propagation are

$\tag{7-69}\beta_\text{TM}^\text{f}=\beta_\text{TM}+\kappa_\text{MM}^\text{f}$

and

$\tag{7-70}\beta_\text{TM}^\text{b}=-\beta_\text{TM}-\kappa_\text{MM}^\text{b}=-\beta_\text{TM}+\kappa_\text{MM}^\text{f}$

respectively, where $$\beta_\text{TM}$$ represents the absolute value of the propagation constant of a TM mode in the absence of the linear magneto-optic effect.

In (7-70), the value of the backward-propagation constant is chosen to be negative by following the convention defined in the coupled-mode theory of the coupled-mode theory tutorial and the two-mode coupling tutorial.

Because $$\beta_\text{TM}^\text{b}\ne-\beta_\text{TM}^\text{f}$$ in an asymmetric waveguide where $$\kappa_\text{MM}\ne0$$, the phase shift experienced by a TM mode propagating in such a waveguide is nonreciprocal.

The nonreciprocal phase shift results from coupling between the longitudinal and transverse electric field components of a TM mode through the linear magneto-optic effect. Such a phenomenon does not exist for TE mode fields, nor does it exist for fields in bulk homogeneous media. It does not exist for TM modes of a symmetric waveguide, either.

Optical Isolators

Guided-wave optical isolator can be implemented by using either a nonreciprocal TE-TM mode converter or a nonreciprocal phase shifter.

A waveguide isolator using a nonreciprocal TE-TM mode converter follows the same basic concept of a polarization-dependent optical isolator in bulk form. The key component is a $$45^\circ$$ Faraday rotator, which has the function of turning the direction of polarization of a linearly polarized wave by $$45^\circ$$ in a single pass and by $$90^\circ$$ in double passes.

One major difference between a waveguide Faraday rotator in the form of a nonreciprocal TE-TM mode converter and a bulk Faraday rotator has to be recognized, however.

An optical wave that is linearly polarized at the input remains linearly polarized along its path through a bulk Faraday rotator. Only its direction of polarization is rotated. A $$45^\circ$$ Faraday rotator made in a bulk material is easily obtained by simply choosing the length $$l_\text{F}$$ of the Faraday rotator properly so that $$\theta_\text{F}=\rho_\text{F}l_\text{F}=\pi/4$$.

In comparison, implementation of a $$45^\circ$$ Faraday rotator in a waveguide structure using a nonreciprocal magneto-optic TE-TM mode converter is not so straightforward if phase mismatch between the TE-like and TM-like modes is not completely eliminated.

If perfect phase matching is accomplished so that $$\Delta\beta=0$$, it can be shown from solution of the coupled-mode equations that the TE-like and TM-like modes excited by any linearly polarized input wave and coupled by the coefficient given in (7-62) remain in phase throughout the waveguide.

In this situation, a $$45^\circ$$ Faraday rotator in a waveguide structure is accomplished by choosing the length of the waveguide to be

$\tag{7-71}l_\text{F}=\frac{\pi}{4|\rho_\text{F}|\Gamma_\text{EM}}$

so that the coupling efficiency given in (7-64) has a value of $$\eta(l=l_\text{F})=1/2$$ for a single pass and a value of $$\eta(l=2l_\text{F})=1$$ for a round-trip pass.

A TE-polarized input wave will be completely converted to the orthogonal TM polarization after a round trip through such a waveguide, and vice versa, as shown in figure 7-23(a) below.

If any phase mismatch exists so that $$\Delta\beta\ne0$$, the phase between the TE-like and TM-like components of a guided field varies along the waveguide for any input polarization.

Although it is possible, in the case that $$|\Delta\beta|\le2|\rho_\text{F}|\Gamma_\text{EM}$$, to choose a length of the waveguide such that $$\eta=1/2$$ for a single pass, it still does not make the waveguide a $$45^\circ$$ Faraday rotator due to the fact that its output in a single pass is elliptically polarized.

Therefore, it is not possible to have a $$45^\circ$$ Faraday rotator with a purely TE-polarized or purely TM-polarized input wave if phase mismatch exists.

To accomplish a high reverse isolation for an isolator, however, any light propagating in the reverse direction must be blocked by a polarizer at the input end of the device. Because an elliptically polarized wave cannot be completely blocked by a polarizer, it is necessary that a $$45^\circ$$ Faraday rotator be used, at least for the reverse propagation direction.

A $$45^\circ$$ Faraday rotator using a waveguide that has a phase mismatch of $$\Delta\beta\ne0$$ is possible, however, if the length of the waveguide is chosen to be

$\tag{7-72}l_\text{F}=\frac{1}{[\rho_\text{F}^2\Gamma_\text{EM}^2+(\Delta\beta)^2/4]^{1/2}}\tan^{-1}\left[1+\frac{(\Delta\beta)^2}{4\rho_\text{F}^2\Gamma_\text{EM}^2}\right]^{1/2}$

while the input polarization is chosen to be $$22.5^\circ$$ off the TE- or TM-polarization direction on the proper side determined by the sign of $$\rho_\text{F}$$, as shown in figure 7-23(b).

To make an isolator, the orientation of the output polarizer is chosen at $$22.5^\circ$$ on the proper side so that any light that is back coupled from the output end returns to the input end linearly polarized. It can then be blocked by the input polarizer whose axis is chosen to be orthogonal to this polarization and at $$45^\circ$$ with respect to that of the output polarizer. Therefore, perfect isolation can always be accomplished for any values of $$\Delta\beta$$ and $$\rho_\text{F}$$.

However, because of the simultaneous presence of a nonreciprocal effect caused by $$\rho_\text{F}$$ and a reciprocal effect caused by $$\Delta\beta$$, the waveguide does not function as a $$45^\circ$$ Faraday rotator for a wave propagating in the forward direction with its input polarization defined by the input polarizer.

As a result, the wave becomes elliptically polarized at the output, resulting in an insertion loss given by

$\tag{7-73}\text{Insertion loss}=L_0-10\log\frac{8\rho_\text{F}^2\Gamma_\text{EM}^2}{8\rho_\text{F}^2\Gamma_\text{EM}^2+(\Delta\beta)^2}$

where $$L_0$$ accounts for all background insertion loss including coupling losses to the waveguide and absorption losses in the waveguide and polarizers.

Example 7-8

A magneto-optic waveguide has the simple structure of a magnetic Bi:YIG film of 4 μm thickness on a nonmagnetic GGG substrate. The top of the Bi:YIG film is exposed to the air. The waveguide is used as a nonreciprocal TE-TM converter in a guided-wave optical isolator for 1.15 μm wavelength. At this wavelength, $$n_1=2.178$$ for the Bi:YIG film, $$n_2=1.945$$ for the GGG substrate, and $$\rho_\text{F}=280^\circ\text{ cm}^{-1}$$, equivalent to $$0.49\text{ rad mm}^{-1}$$. The device is operated in the fundamental $$\text{TE}_0$$ and $$\text{TM}_0$$ modes of the waveguide. It has a background insertion loss of $$L_0=3\text{ dB}$$. Find the required length of the waveguide and the total insertion loss of the isolator.

First, we solve for the $$\text{TE}_0$$ and $$\text{TM}_0$$ mode parameters of the asymmetric slab waveguide with $$n_1=2.178$$, $$n_2=1.945$$, $$n_3=1$$, and $$d=4$$ μm to find that $$\beta_\text{TE}=11.87717\text{ μm}^{-1}$$, $$\beta_\text{TM}=11.87593\text{ μm}^{-1}$$, and $$\Gamma_\text{EM}=0.999$$. Therefore, $$\Delta\beta=\beta_\text{TM}-\beta_\text{TE}=-1.24\text{ mm}^{-1}$$. Using these parameters, we find from (7-72) that the required length for the waveguide is

$l_\text{F}=\frac{1}{(0.49^2\times0.999^2+1.24^2/4)^{1/2}}\tan^{-1}\left(1+\frac{1.24^2}{4\times0.49^2\times0.999^2}\right)^{1/2}\text{ mm}=1.29\text{ mm}$

Using (7-73), we find that

$\text{Insertion loss}=3\text{ dB}-10\times\log\frac{8\times0.49^2\times0.999^2}{8\times0.49^2\times0.999^2+1.24^2}\text{ dB}=5.6\text{ dB}$

The phase mismatch in the waveguide contributes an additional 2.6 dB to the insertion loss.

A practical problem arises in the use of a $$45^\circ$$ Faraday rotator in a waveguide. This rotation angle poses no problem for a bulk device, but it results in a mixture of orthogonally polarized modes at the input or the output, or both, of the waveguide that carries out this essential function for an isolator.

This situation is not consistent with that in the applications of most guided-wave devices, which normally have a well-defined single TE-like or TM-like mode at both input and output ends. Some guided-wave devices are designed to function only properly for a particular mode of polarization.

Therefore, a practical guided-wave optical isolator that can be integrated with other guided-wave devices must operate with single-polarization mode fields at both its input and output ends, meaning that the total amount of polarization rotation between its input and output ends has to be $$0^\circ$$ or an integral multiple of $$90^\circ$$.

Because nonreciprocity is required of an isolator, such a guided-wave optical isolator must be a unidirectional TE-TM mode converter, which converts a TE-like mode into a TM-like mode, and vice versa, in one direction of propagation but has no net polarization conversion in the opposite direction of propagation.

It is not possible to construct such a unidirectional TE-TM mode converter using the linear magneto-optic effect alone, nor is it possible without using the linear magneto-optic effect.

A similar function in bulk devices, shown in figure 7-7(b) [refer to the optical isolators and circulators tutorial], is accomplished by combination of a $$45^\circ$$ Faraday rotator and a quarter-wave plate. In guided-wave devices, such a function can be accomplished by combination of a nonreciprocal magneto-optic TE-TM mode converter functioning as a $$45^\circ$$ Faraday rotator and a reciprocal TE-TM mode converter functioning as a $$45^\circ$$ linear polarization rotator analogous to a quarter-wave plate. These two mode converters can be either placed in tandem or distributedly mixed.

A unidirectional TE-TM mode converter can be realized by using the reciprocal linear magnetic birefringence of the Cotton-Mouton effect for the reciprocal TE-TM mode converter. In this approach, shown in figure 7-24(a) below, both nonreciprocal and reciprocal TE-TM mode converters are realized using the same magnetic waveguide materials.

In the nonreciprocal section the magnetization is parallel to the longitudinal direction of the waveguide, whereas in the reciprocal section it lies in the transverse plane and is tilted at an angle, $$\theta_\text{m}$$, with respect to the transverse TM electric field polarization.

By choosing a proper value of the angle $$\theta_\text{m}$$ for the magnetization direction and a proper length for the reciprocal section of the waveguide, the desired reciprocal $$45^\circ$$ linear polarization rotation can be realized even when $$\Delta\beta\ne0$$.

A second approach uses an anisotropic crystal, such as LiNbO3 or LiIO3, for the reciprocal function. This approach can be carried out by using such an anisotropic crystal for a layer on top of the magnetic waveguide, as shown in figure 7-24(b).

In this arrangement, the nonreciprocal and reciprocal TE-TM mode converters are distributedly mixed. Coupling between TE-like and TM-like modes for reciprocal conversion is caused by penetration of the mode fields into the nonmagnetic top layer.

The optical axis of the anisotropic crystal has to be tilted at a proper angle away from the transverse TE and TM electric field polarization directions in order for the TE-like and TM-like modes to have a nonzero coupling coefficient of the amount needed for the reciprocal $$45^\circ$$ linear polarization rotation.

Other components needed in completing a guided-wave optical isolator using a unidirectional TE-TM mode converter are the input and output mode selectors analogous to the polarizers used in a bulk isolator.

These components can be easily implemented with metal-loaded strips at both input and output ends of the device, as also shown in figure 7-24.

With proper design, the metallic films on top of the waveguide can have strong attenuation for the TM-like mode but very little attenuation for the TE-like mode, based on the fact that the transverse electric field components of the TM-like mode is perpendicular to the metallic film surface but that of the TE-like mode is parallel to it. Thus the metallic film sections function as mode filters for transmitting only TE-like modes.

When LiNbO3 on YIG is used to implement the concept shown in figure 7-24(b), it is also possible to design the waveguide structure such that the TE-like mode is a guided mode while the TM-like mode is a leaky mode. There is no need for additional mode selectors in an isolator using such a semileaky waveguide, which already has a built-in mode-filtering function.

A very different type of guided-wave optical isolator that has no bulk counterpart uses a nonreciprocal phase shifter in an asymmetric Mach-Zehnder waveguide interferometer, as shown in figure 7-25 below.

Because nonreciprocal phase shifter is possible only for the TM-like mode, this optical isolator functions only in the TM-like mode.

This device consists of two asymmetric waveguides. One waveguide has a properly oriented transverse magnetization and functions as a nonreciprocal phase shifter. The other is not magnetized and functions as a reciprocal phase shifter.

In the absence of the perturbation from the linear magneto-optic effect, there is a reciprocal difference of $$\Delta\beta_\text{TM}$$ between the upper and lower waveguides in the propagation constants of the TM-like mode due to asymmetry between the two waveguides.

The TM-like mode field propagating through the magnetized arm has a net reciprocal phase advance of $$\Delta\varphi_\text{rec}=\Delta\beta_\text{TM}l=\pi/2$$ over that propagating through the nonmagnetized arm, where $$l$$ is the length of the phase-shifter section of the interferometer.

The magnetized waveguide is further designed to have in the forward-propagation direction a net nonreciprocal phase shift of $$\Delta\varphi_\text{nonrec}^\text{f}=\kappa_\text{MM}^\text{f}l=-\pi/2$$ due to the linear magneto-optic effect and a corresponding nonreciprocal phase shift of $$\Delta\varphi_\text{nonrec}^\text{b}=\kappa_\text{MM}^\text{b}l=\pi/2$$ in the backward direction.

The two waveguide arms are connected at both input and output ends with 3-dB Y-junctions. Consequently, in the forward direction, the combined reciprocal and nonreciprocal phase difference between the two arms is $$\Delta\varphi^\text{f}=\Delta\varphi_\text{rec}+\Delta\varphi_\text{nonrec}^\text{f}=0$$, resulting in total transmission of the TM-like mode launched into the device. In the backward direction, a combined phase difference of $$\Delta\varphi^\text{b}=\Delta\varphi_\text{rec}+\Delta\varphi_\text{nonrec}^\text{b}=\pi$$ causes destructive interference to completely block the transmission of the TM-like mode.

Optical Circulators

A guided-wave optical circulator can be realized with a nonreciprocal balanced-bridge interferometer by simply replacing the Y-junctions at the input and output ends of the nonreciprocal Mach-Zehnder interferometer shown in figure 7-25 with ordinary 3-dB directional couplers. The resulting device and its function are shown in figure 7-26 below.

From the discussions regarding the operation of a balanced-bridge interferometer in the guided-wave electro-optic modulators tutorial, it can be easily seen by applying (6-79) and (6-80) that this nonreciprocal interferometer is in the cross state for forward propagation and is in the parallel state for backward propagation when it is designed to have $$\Delta\varphi^\text{f}=0$$ and $$\Delta\varphi^\text{b}=\pi$$. Consequently, it functions as a four-port optical circulator with a $$1\rightarrow4\rightarrow2\rightarrow3\rightarrow1$$ looping sequence.

It is also possible to implement an optical circulator using a nonreciprocal directional coupler switch that consists of a nonreciprocal phase-shifter in one of its two coupling arms and a nonmagnetized reciprocal waveguide in another, as shown in figure 7-27 below.

Because the cross state of a simple directional coupler switch is accessible only with a total phase difference of $$\Delta\varphi=0$$ between its two arms, the length of the device has to be chosen according to (6-83) [refer to the guided-wave electro-optic modulators tutorial], and the phase difference for reaching the parallel state is determined by (6-84).

Consequently, the reciprocal and nonreciprocal phase shifts needed for proper operation of this device are different from those needed for the nonreciprocal Mach-Zehnder and balanced-bridge interferometers.

The shortest length that can be chosen for the phase-shifter section is $$l=l_\text{c}^\text{PM}$$ with a corresponding phase difference of $$\Delta\varphi=\sqrt3\pi$$ between the two coupling arms for the parallel state.

With these parameters, this device should be designed to have $$\Delta\varphi_\text{rec}=\sqrt3\pi/2$$ and $$\Delta\varphi_\text{nonrec}^\text{f}=-\Delta\varphi_\text{nonrec}^\text{b}=-\sqrt3\pi/2$$ so that $$\Delta\varphi^\text{f}=0$$ and $$\Delta\varphi^\text{b}=\sqrt3\pi$$ for it to operate in the cross state in the forward direction and in the parallel state in the backward direction, as also shown in figure 7-27.

Both devices shown in figure 7-26 and 7-27 function only with the TM-like mode because of the use of the nonreciprocal phase shifter.

The next part continues with the elastic waves tutorial.