# Grating Waveguide Couplers

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This is a continuation from the previous tutorial - two-mode coupling.

Grating waveguide couplers have many useful applications and are one of the most important kinds of waveguide couplers. They consist of periodic fine structures that form gratings in waveguides.

The gratings in a waveguide can be either ** periodic index modulation** or

**.**

*periodic structural corrugation*Periodic index modulation can be permanently written in a waveguide by periodically modulating the doping concentration in the waveguide medium, for example, or it can be created by an electro-optic, acousto-optic, or nonlinear optical effect. In the latter case, the grating can be time dependent if the modulation is time varying. It can also be a moving grating if the modulation signal is a traveling wave.

In the case of periodic structural corrugation, the corrugation is a permanent structure of a waveguide. It is usually located at an interface between layers of different refractive indices, such as that between the guiding layer and the substrate or that between the guiding layer and the cover layer of a planar waveguide. It can also be placed away from the interfaces next to the guiding layer so long as the mode fields have sufficient penetration into the neighboring layers to see the corrugation.

Figure 1 below shows a few examples of grating structures in a planar waveguide.

In any event, the grating in a waveguide coupler can be considered as a periodic perturbation of \(\Delta\boldsymbol{\epsilon}\) that has a spatial periodicity characteristic of the grating.

In a ** coplanar coupler**, the grating can have a two-dimensional periodicity while the propagation vectors of the waves being coupled are in the same plane confined by the waveguide but not necessarily parallel to each other.

In a ** collinear coupler**, the waves being coupled are propagating either codirectionally or contradirectionally, and the grating is periodic only in the propagation direction of the guided waves.

We consider here only the case of collinear coupling in a waveguide along the \(z\) direction. Then, \(\Delta\boldsymbol{\epsilon}\) is periodic only in \(z\) with a period \(\Lambda\) of the grating, as shown in figure 1 above.

With this periodically \(z\)-dependent perturbation, the coupling coefficients as defined in (36) [refer to the coupled-mode theory tutorial] and used in (45) and (46) [refer to the two-mode coupling tutorial] are also periodic in \(z\). In addition, for coupling in a single waveguide, we have \(\kappa_{ab}=\kappa_{ba}^*\). They can be expressed in terms of the following Fourier series expansion:

\[\tag{1}\kappa_{ab}(z)=\omega\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\hat{\boldsymbol{\mathcal{E}}}_a^*\cdot\Delta\boldsymbol{\epsilon}(x,y,z)\cdot\hat{\boldsymbol{\mathcal{E}}}_b\text{d}x\text{d}y=\sum_q\kappa_{ab}(q)\exp(\text{i}qKz)\]\[\tag{2}\kappa_{ba}(z)=\kappa_{ab}^*(z)=\sum_q\kappa_{ab}^*(q)\exp(-\text{i}qKz)\]

where

\[\tag{3}K=\frac{2\pi}{\Lambda}\]

and

\[\tag{4}\kappa_{ab}(q)=\frac{1}{\Lambda}\displaystyle\int\limits_0^\Lambda\kappa_{ab}(z)\text{e}^{-\text{i}qKz}\text{d}z\]

Considering the fact learned from the two-mode coupling tutorial that efficient coupling exists only near phase matching, the coupled equations given in (45) and (46) [refer to the two-mode coupling tutorial] can be transformed into (50) and (51) [refer to the two-mode coupling tutorial] by the following approximations:

\[\tag{5}\pm\frac{\text{d}\tilde{A}}{\text{d}z}=\text{i}\kappa_{ab}(z)\tilde{B}\text{e}^{\text{i}\Delta\beta{z}}=\text{i}\tilde{B}\sum_q\kappa_{ab}(q)\text{e}^{\text{i}(\Delta\beta+qK)z}\approx\text{i}\kappa\tilde{B}\text{e}^{\text{i}(\Delta\beta+qK)z}\]\[\tag{6}\pm\frac{\text{d}\tilde{B}}{\text{d}z}=\text{i}\kappa_{ba}(z)\tilde{A}\text{e}^{-\text{i}\Delta\beta{z}}=\text{i}\tilde{A}\sum_q\kappa_{ab}^*(q)\text{e}^{-\text{i}(\Delta\beta+qK)z}\approx\text{i}\kappa^*\tilde{A}\text{e}^{-\text{i}(\Delta\beta+qK)z}\]

where we can identify the phase mismatch as that given in (56) [refer to the two-mode coupling tutorial]:

\[\tag{7}2\delta=\Delta\beta+qK=\beta_b-\beta_a+qK\]

Only one term in the Fourier series that yields a minimum value for \(|\delta|\) is kept in each of the two coupled-mode equations because only this term will effectively couple the two waves. To be consistent with the notation used in the discussions following (81) [refer to the two-mode coupling tutorial], we have also used

\[\tag{8}\kappa=\kappa_{ab}(q)\]

for the Fourier term that is kept in (5) and (6).

Note that although \(\kappa_{ba}(z)=\kappa_{ab}^*(z)\), as is indicated in (2), \(\kappa_{ba}(q)\) and \(\kappa_{ab}^*(q)\) are not necessarily the same unless both happen to be real quantities. Instead, we have \(\kappa_{ba}(q)=\kappa_{ab}^*(-q)\) among the Fourier components of \(\kappa_{ba}(z)\) and \(\kappa_{ab}^*(z)\). For this reason, the \(\kappa\) and \(\kappa^*\) defined above have the following relations: \(\kappa=\kappa_{ab}(q)=\kappa_{ba}^*(-q)\) and \(\kappa^*=\kappa_{ab}^*(q)=\kappa_{ba}(-q)\).

We see from the above discussion that (5) and (6) are identical to (50) and (51) [refer to the two-mode coupling tutorial], respectively, if we replace \(\kappa_{ab}\) and \(\kappa_{ba}\) in (50) and (51) respectively with \(\kappa\) and \(\kappa^*\). Therefore, the general results obtained in the two-mode coupling tutorial can be applied directly to the coupling of modes in a grating waveguide coupler with the coupling coefficients given by (8) and the phase mismatch given by (7).

**Example 1**

Find the periods of the first- and second-order gratings for phase-matched coupling between contrapropagating (a) TE_{0} and TE_{0} mode fields, (b) TE_{0} and TE_{1} mode fields, and (c) TE_{1} and TE_{1} mode fields for the waveguide described in the first example in the step-index planar waveguides tutorial.

For phase-matched coupling, it is required that \(qK=-\Delta\beta\) because \(\delta=0\) in (7). Therefore, the grating period is

\[\tag{9}\Lambda=-q\frac{2\pi}{\Delta\beta}\]

where \(q=1\) for the first-order grating and \(q=2\) for the second-order grating.

From the first example in the step-index planar waveguides tutorial, we find that \(\beta_0\) = 10.8432 μm^{-1} for the TE_{0} mode and \(\beta_1\) = 10.0036 μm^{-1} for the TE_{1} mode.

(a) For the coupling from a forward-propagating TE_{0} field to a backward-propagating TE_{0} field, \(\Delta\beta=-\beta_0-\beta_0=-21.6864\) μm^{-1}. We then find by using (9) that \(\Lambda_1=289.7\) nm for the first-order grating and \(\Lambda_2=579.4\) nm for the second-order grating.

(b) For the coupling from a forward-propagating TE_{0} field to a backward-propagating TE_{1} field, \(\Delta\beta=-\beta_1-\beta_0=-20.8468\) μm^{-1}. We find that \(\Lambda_1=301.4\) nm and \(\Lambda_2=602.8\) nm for the first- and second-order gratings, respectively.

(c) For the coupling from a forward-propagating TE_{1} field to a backward-propagating TE_{1} field, \(\Delta\beta=-\beta_1-\beta_1=-20.0072\) μm^{-1}. We find that \(\Lambda_1=314\) nm and \(\Lambda_2=628\) nm for the first- and second-order gratings, respectively.

**Coupling Coefficient**

As can be seen from the discussions in the preceding section and from (5), (6), and (7), the only important parameters for the coupling between two modes are the coupling coefficient \(\kappa\), or \(\kappa_{ab}(q)\) for the grating waveguide coupler discussed here, and the phase mismatch \(2\delta\).

The phase mismatch can be calculated using (7) once the propagation constants of the modes being coupled are known and the grating period is given.

The calculation of \(\kappa_{ab}(q)\) is less straightforward, however. It depends on exactly how the grating is created and where it is located in the waveguide. It also depends on the field distributions of the modes being coupled. In the following, we consider a few simple but important examples, including a grating produced by periodic index modulation, a sinusoidal corrugation grating, and a square corrugation grating, all in three-layer planar waveguides.

We assume that the unperturbed waveguides have the structure of the three-layer planar slab waveguide shown in the figure below.

Combining (1) and (4), we can write

\[\tag{10}\kappa=\kappa_{ab}(q)=\frac{\omega}{\Lambda}\displaystyle\int\limits_0^{\Lambda}\text{d}z\int\limits_{-\infty}^{\infty}\text{d}x\hat{\boldsymbol{\mathcal{E}}}_a^*(x)\cdot\Delta\boldsymbol{\epsilon}(x,z)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(x)\text{e}^{-\text{i}qKz}\]

for coupling in a planar waveguide. The guiding layer of index \(n_1\) has a thickness \(d\) located in the range of \(-d/2\lt x\lt d/2\). For the corrugation gratings, we consider the corrugation to be located at the interface between the guiding core and the cover layer. It is centered at the interface and has a depth of \(d_g\), extending a maximum distance of \(d_g/2\) into either side of the interface, as shown in figure 1(c) and 1(d) above.

1. **Index-modulation grating**. The geometric structure of the waveguide is not perturbed, but only the index of refraction is modulated. Because the index modulation is usually not localized but is distributed throughout the entire thickness of the guiding layer or a large portion of it, the integral in (10) has to be calculated with the complete field distributions \(\hat{\boldsymbol{\mathcal{E}}}_a^*(x)\) and \(\hat{\boldsymbol{\mathcal{E}}}_b(x)\) throughout the waveguide thickness.

2. **Sinusoidal corrugation grating**. For a sinusoidal corrugation grating in an isotropic planar waveguide as shown in figure 1(c) above, the perturbation susceptibility is

\[\tag{11}\Delta\epsilon(x,z)=\begin{cases}\epsilon_0(n_1^2-n_3^2), &\text{for }d/2\lt x\lt d/2+(d_g/2)\cos{Kz}, &\cos{Kz}\gt0\\-\epsilon_0(n_1^2-n_3^2), &\text{for }d/2+(d_g/2)\cos{Kz}\lt x\lt d/2, &\cos{Kz}\lt0\end{cases}\]

Substitution of (11) into (10) yields

\[\tag{12}\begin{align}\kappa_{ab}(q)=\frac{\omega}{\Lambda}\left[\displaystyle\int\limits_0^{\Lambda/4}\text{d}z\int\limits_{d/2}^{d/2+(d_g/2)\cos{Kz}}\text{d}x\epsilon_0(n_1^2-n_3^2)\hat{\boldsymbol{\mathcal{E}}}_a(x)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(x)\text{e}^{-\text{i}qKz}\right.\\-\displaystyle\int\limits_{\Lambda/4}^{3\Lambda/4}\text{d}z\int\limits_{d/2+(d_g/2)\cos{Kz}}^{d/2}\text{d}x\epsilon_0(n_1^2-n_3^2)\hat{\boldsymbol{\mathcal{E}}}_a(x)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(x)\text{e}^{-\text{i}qKz}\\\left.+\displaystyle\int\limits_{3\Lambda/4}^{\Lambda}\text{d}z\int\limits_{d/2}^{d/2+(d_g/2)\cos{Kz}}\text{d}x\epsilon_0(n_1^2-n_3^2)\hat{\boldsymbol{\mathcal{E}}}_a(x)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(x)\text{e}^{-\text{i}qKz}\right]\end{align}\]

In most practical devices, \(d_g\ll\lambda/n_1\). Then, we can approximate \(\hat{\boldsymbol{\mathcal{E}}}_a^*(x)\) and \(\hat{\boldsymbol{\mathcal{E}}}_b(x)\) in the range of \(d/2-d_g/2\lt x\lt d/2+d_g/2\) of the corrugation by \(\hat{\boldsymbol{\mathcal{E}}}_a^*(d/2)\) and \(\hat{\boldsymbol{\mathcal{E}}}_b(d/2)\), respectively, to obtain

\[\tag{13}\kappa_{ab}(q)\approx\omega\epsilon_0(n_1^2-n_3^2)\hat{\boldsymbol{\mathcal{E}}}_a^*(d/2)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(d/2)\frac{d_g}{4}(\delta_{q,1}+\delta_{q,-1})\]

where \(\delta_{q,1}\) and \(\delta_{q,-1}\) are Kronecker delta functions. Clearly, \(\kappa_{ab}(q)\ne0\) only for \(q=1\) or \(-1\).

Using the characteristics of the mode fields in planar slab waveguides discussed in the step-index planar waveguides tutorial, it can be shown that (13) yields

\[\tag{14}\kappa_{ab}(q)=\frac{h_ah_b}{\sqrt{\beta_a\beta_bd_a^Ed_b^E}}\frac{d_g}{4}(\delta_{q,1}+\delta_{q,-1})\]

for coupling between two TE modes, where \(h_a\) and \(h_b\) are the parameter \(h_1\) defined in (50) [refer to the step-index planar waveguides tutorial] and \(d_a^E\) and \(d_b^E\) are the effective waveguide thickness defined in (59) [refer to the step-index planar waveguides tutorial] for the TE_{a} and TE_{b} modes, respectively.

For coupling between two TM modes, we have

\[\tag{15}\begin{align}\kappa_{ab}(q)=&\frac{n_1^2-n_3^2}{2n_1^2}\frac{\beta_a\beta_b/n_1^4+\beta_a\beta_b/n_3^4+2\gamma_{3a}\gamma_{3b}/n_3^4}{\sqrt{h_a^2/n_1^4+\gamma_{3a}^2/n_3^4}\sqrt{h_b^2/n_1^4+\gamma_{3b}^2/n_3^4}}\frac{h_ah_b}{\sqrt{\beta_a\beta_bd_a^Md_b^M}}\\&\times\frac{d_g}{4}(\delta_{q,1}+\delta_{q,-1})\end{align}\]

where the parameters are relevant to the TM_{a} and TM_{b} modes.

Because TE and TM modes do not couple in an isotropic waveguide, it is necessary to introduce birefringence in order to couple them.

3. **Square corrugation grating**. The perturbation susceptibility of a square corrugation grating in an isotropic planar waveguide as shown in figure 1(d) above is

\[\tag{16}\Delta\epsilon(x,z)=\begin{cases}\epsilon_0(n_1^2-n_3^2), &\text{for }d/2\lt x\lt d/2+d_g/2, &0\lt z\lt\xi\Lambda,\\-\epsilon_0(n_1^2-n_3^2), &\text{for }d/2-d_g/2\lt x\lt d/2, &\xi\Lambda\lt z\lt\Lambda\end{cases}\]

where \(0\lt\xi\lt1\) is the duty factor of the square corrugation. Substitution of (16) into (10) yields

\[\tag{17}\begin{align}\kappa_{ab}(q)=\frac{\omega}{\Lambda}\left[\displaystyle\int\limits_0^{\xi\Lambda}\text{d}z\int\limits_{d/2}^{d/2+d_g/2}\text{d}x\epsilon_0(n_1^2-n_3^2)\hat{\boldsymbol{\mathcal{E}}}_a^*(x)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(x)\text{e}^{-\text{i}qKz}\right.\\\left.-\displaystyle\int\limits_{\xi\Lambda}^{\Lambda}\text{d}z\int\limits_{d/2-d_g/2}^{d/2}\text{d}x\epsilon_0(n_1^2-n_3^2)\hat{\boldsymbol{\mathcal{E}}}_a^*(x)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(x)\text{e}^{-\text{i}qKz}\right]\\\approx\omega\epsilon_0(n_1^2-n_3^2)\hat{\boldsymbol{\mathcal{E}}}_a^*(d/2)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(d/2)\frac{d_g}{2}\frac{1}{\Lambda}\left[\displaystyle\int\limits_0^{\xi\Lambda}\text{d}z\text{e}^{-\text{i}qKz}-\int\limits_{\xi\Lambda}^{\Lambda}\text{d}z\text{e}^{-\text{i}qKz}\right]\\=2\omega\epsilon_0(n_1^2-n_3^2)\hat{\boldsymbol{\mathcal{E}}}_a^*(d/2)\cdot\hat{\boldsymbol{\mathcal{E}}}_b(d/2)\frac{d_g}{2}\frac{\sin\xi q\pi}{q\pi}\text{e}^{-\text{i}\xi q\pi}\end{align}\]

For coupling between two TE modes, (17) yields

\[\tag{18}\kappa_{ab}(q)=\frac{h_ah_b}{\sqrt{\beta_a\beta_bd_a^Ed_b^E}}d_g\frac{\sin\xi q\pi}{q\pi}\text{e}^{-\text{i}\xi q\pi}\]

For coupling between two TM modes, we have

\[\tag{19}\begin{align}\kappa_{ab}(q)=\frac{n_1^2-n_3^2}{2n_1^2}\frac{\beta_a\beta_b/n_1^4+\beta_a\beta_b/n_3^4+2\gamma_{3a}\gamma_{3b}/n_3^4}{\sqrt{h_a^2/n_1^4+\gamma_{3a}^2/n_3^4}\sqrt{h_b^2/n_1^4+\gamma_{3b}^2/n_3^4}}\\\times\frac{h_ah_b}{\sqrt{\beta_a\beta_bd_a^Md_b^M}}d_g\frac{\sin\xi q\pi}{q\pi}\text{e}^{-\text{i}\xi q\pi}\end{align}\]

We have seen that in the case where the grating is a simple sinusoidal grating proportional to \(\cos Kz\) or, equivalently, \(\sin Kz\), the expansion in (1) has only two terms, with \(q=1\) and \(q=-1\). We also see from (18) and (19) that \(\kappa_{ab}(q)=0\) for a square grating if \(\xi q\) is an integer. In practical situations, the grating can be rectangular, triangular, or any other shape. Then \(q\) takes on any integer value for which \(\kappa_{ab}(q)\ne0\).

It can also be seen from (14) and (18) that coupling coefficient between two TE modes does not depend on any parameters of the cover layer but only on those of the guiding layer although the grating is located at the interface between the cover and the guiding layers. This means that we would get exactly the same coupling coefficient for the two given TE modes if we instead placed the grating with the same parameters, including its period \(\Lambda\), depth \(d_g\), and shape, at the interface between the substrate and the guiding layer.

This is also approximately, although not exactly, true for coupling between TM modes if the asymmetry of the waveguide is small so that \(\gamma_2\approx\gamma_3\) for each mode.

This is an important conclusion for practical applications. It indicates that the same grating can be placed at either interface next to the guiding layer for the same desired coupling coefficient. This conclusion does not apply, however, to coupling between TM modes in a highly asymmetric waveguide where \(\gamma_2\) and \(\gamma_3\) are significantly different for a given mode.

**Example 2**

A first-order square corrugation grating that has a depth of \(d_g=100\) nm and a duty factor of \(\xi=0.5\) is fabricated at either the upper or the lower core-cladding boundary of the waveguide described in the Example 1 above to make a grating waveguide coupler for \(\lambda=1\) μm wavelength. Find the following coupling coefficients: \(\kappa_{00}^{TE}\) between the forward-propagating \(TE_0\) mode field and the backward-propagating \(TE_0\) mode field, \(\kappa_{01}^{TE}\) between the forward-propagating \(TE_0\) mode field and the backward-propagating \(TE_1\) mode field, and \(\kappa_{00}^{TM}\) between the forward-propagating \(TM_0\) mode field and the backward-propagating \(TM_0\) mode field.

For the waveguide describe in Example 1 above, \(n_1=1.77\), \(n_2=1.45\), and \(n_3=1\) for \(\lambda=1\) µm. We first check the condition that \(d_g\ll\lambda/n_1\) for the approximation made in obtaining (18) and (19) to be valid. Because \(\lambda/n_1=565\) nm and \(d_g=100\) nm, this condition is satisfied. Therefore, we can use (18) to calculate the coupling coefficients for the TE modes and (19) for the TM modes with \(q=1\), \(\xi=0.5\), \(d_g=100\) nm, and the mode parameters given in the table in Example 1 above.

The coupling coefficient between two TE modes does not depend on whether the grating is placed in the upper or lower core-cladding boundary. From (18), we find that the coupling coefficient for \(TE_0-TE_0\) coupling is

\[\kappa_{00}^{TE}=-\text{i}\frac{h_0^2}{\beta_0d_0^E}\frac{d_g}{\pi}=-\text{i}0.014\text{ $\mu$m}^{-1}\]

and the coupling coefficient for \(TE_0-TE_1\) coupling is

\[\kappa_{01}^{TE}=-\text{i}\frac{h_0h_1}{\sqrt{\beta_0\beta_1d_0^Ed_1^E}}\frac{d_g}{\pi}=-\text{i}0.0277\text{ $\mu$m}^{-1}\]

where \(h_0\) and \(h_1\) are the \(h\) parameters for the \(TE_0\) and \(TE_1\) modes, respectively.

The coupling coefficient between two TM modes depends on the location of the grating. If the grating is placed at the upper boundary between the polymer core of index \(n_1=1.77\) and the air cover of index \(n_3=1\), the coupling coefficient for \(TM_0-TM_0\) is calculated using (19) as

\[\kappa_{00}^{TM}=-\text{i}\frac{n_1^2-n_3^2}{2n_1^2}\frac{\beta_0^2/n_1^4+\beta_0^2/n_3^4+2\gamma_{30}^2/n_3^4}{h_0^2/n_1^4+\gamma_{30}^2/n_3^4}\frac{h_0^2}{\beta_0d_0^M}\frac{d_g}{\pi}=-\text{i}0.0233\text{ $\mu$m}^{-1}\]

where all of the parameters belong to the \(TM_0\) mode.

If the grating is placed at the lower boundary between the polymer core of index \(n_1=1.77\) and the silica substrate of index \(n_2=1.45\), the coupling coefficient for \(TM_0-TM_0\) is calculated by replacing \(\gamma_3\) in (19) with \(\gamma_2\) as

\[\kappa_{00}^{TM}=-\text{i}\frac{n_1^2-n_2^2}{2n_1^2}\frac{\beta_0^2/n_1^4+\beta_0^2/n_2^4+2\gamma_{20}^2/n_2^4}{h_0^2/n_1^4+\gamma_{20}^2/n_2^4}\frac{h_0^2}{\beta_0d_0^M}\frac{d_g}{\pi}=-\text{i}0.0199\text{ $\mu$m}^{-1}\]

We see that there is about 17% difference between these two coefficients for TM mode coupling because of the difference between \(\gamma_2\) and \(\gamma_3\).

**Distributed Bragg Reflector**

We consider here a device of special interest that has very important applications. The function of this device is based on coupling between the forward- and backward-propagating fields of the same mode in a grating waveguide coupler. This is a special case of contradirectional coupling where \(\beta_b=-\beta_a\) and \(\Delta\beta=\beta_b-\beta_a\). In this case, we can define

\[\tag{20}\beta\equiv\beta_a=-\beta_b\]

Then, \(\Delta\beta=-2\beta\), and (7) becomes

\[\tag{21}2\delta=-2\beta+qK\]

Thus, the phase-matching condition (\(2\delta=0\)) can be stated as the following Bragg condition:

\[\tag{22}\beta_B=q\frac{K}{2}\]

where \(q\) is the integer that allows phase matching and is the ** order of coupling** between the two contrapropagating waves. The grating period required to satisfy this phase-matching condition is

\[\tag{23}\Lambda=q\frac{\pi}{\beta_B}=q\frac{\lambda_B}{2n_\beta}\]

where \(\lambda_B=2\pi n_\beta/\beta_B\) is the free-space ** Bragg wavelength** of the field and \(n_\beta\) is the effective refractive index of the mode field in the waveguide.

A grating with a period given by (23) for an integer \(q\) is called a ** qth-order grating** for the mode coupling under consideration. For example, it is a

*first-order grating*if \(\Lambda=\lambda_B/2n_\beta\) and is a

*second-order grating*if \(\Lambda=\lambda_B/n_\beta\). A simple sinusoidal grating can only be a first-order grating because, as mentioned above, \(q\) can only be 1 or -1 in this case and thus can only have the value 1 in (23).

To get an idea on the size of the grating period in a practical device structure, we consider as an example the grating in an InGaAsP waveguide for an optical wavelength of \(\lambda=\lambda_B=1.3\text{ $\mu$m}\) in free space. The index of refraction for InGaAsP with a bandgap energy corresponding to 1.3 µm optical wavelength is about 3.48. Taking \(n_\beta\approx3.48\), we find that \(\lambda_B/n_\beta\approx1.3\text{ $\mu$m}/3.48\approx374\text{ nm}\). We then have \(\Lambda=187\) nm for a first-order grating and \(\Lambda=374\) nm for a second-order grating. These are certainly very fine structures.

As discussed in the preceding section, the effect of this contradirectional coupling is an efficient transfer of power from the forward-propagating field to the backward-propagating field when \(\delta^2\lt|\kappa|^2\). From the input end of the grating waveguide coupler, it is seen that power is reflected back due to this coupling.

This type of reflector, which relies on the coupling of waves by a distributed periodic structure, is called the ** distributed Bragg reflector** (DBR). Its reflection coefficient \(r\) is that given in (78) [refer to the two-mode coupling tutorial] with an amplitude \(|r|=\eta^{1/2}\) and a phase \(\varphi_\text{DBR}\) given in (79) [refer to the two-mode coupling tutorial]. Thus, its reflectivity is simply the coupling efficiency \(\eta\) given by (91) [refer to the two-mode coupling tutorial]. The peak reflectivity at the Bragg wavelength is \(R_{DBR}=\eta_{PM}\) given by (87) under the phase-matched condition. The reflectance \(R_{DBR}\) and the transmittance \(T_{DBR}=1-R_{DBR}\) of such a reflector at its Bragg wavelength where the device has perfect phase matching are plotted in figure 2 below as a function of the effective coupler length \(|k|l\).

As can be seen from figure 6 [refer to the two-mode coupling tutorial], the reflection of power, or the transfer of power from the forward-propagating mode to the backward-propagating mode, is not localized but is distributed throughout the length of the grating coupler. The phase shift of Bragg reflection at the phase-matched ** Bragg frequency** \(\omega_B=2\pi\nu_B\), which corresponds to the Bragg wavelength \(\lambda_B\) through the relation \(\nu_B=c/\lambda_B\), is \(\varphi_{DBR}=\varphi_B\) for \(\delta=0\) in (79). When the optical frequency deviates from the Bragg frequency, the phase shift of Bragg reflection given in (79) [refer to the two-mode coupling tutorial] can be expressed in terms of the variation of the propagation constant \(\beta(\omega\)\) away from the phase-matched value of \(\beta(\omega_B)=\beta_B\) in the following form:

\[\tag{24}\varphi_{DBR}=\varphi_B+2[\beta(\omega)-\beta(\omega_B)]l_{DBR}^{eff}\]

where

\[\tag{25}l_{DBR}^{eff}=\frac{\tanh|\kappa|l}{2|\kappa|}=\frac{\eta_{PM}^{1/2}}{2|\kappa|}=\frac{R_{DBR}^{1/2}}{2|\kappa|}\]

The parameter \(l_{DBR}^{eff}\) is an effective length of the DBR for its reflection phase shift. These grating couplers can be used in a *distributed Bragg reflector laser* (DBR laser) or in a *distributed feedback laser* (DFB laser) to provide optical feedback without ordinary Fabry-Perot mirrors.

A DBR can be designed to function as a narrow-band *frequency filter*. Consequently, an important characteristic of a DBR or DFB laser is its frequency selectivity and stability, which results in stable single-frequency operation of the laser if the structure is properly designed.

This frequency selectivity of a DBR can be understood by considering the dispersion characteristics of a waveguide mode and the effect of phase matching, as shown in figure 3 below.

The dispersion relations \(\beta_a(\omega)\) and \(\beta_b(\omega)\) for the waveguide modes are determined by the waveguide parameters and the optical frequency \(\omega\). In the case under consideration, we have \(\beta_a(\omega)=\beta(\omega)\) and \(\beta_b(\omega)=-\beta(\omega)\). For phase matching, we need \(\beta_b+qK=\beta_a\), which can be found by shifting the dispersion curve of \(\omega\) versus \(\beta\) horizontally by an amount \(qK\) to find the intersection between the two curves representing \(\beta_b+qK\) and \(\beta_a\). This procedure is illustrated in figure 3 above.

The intersecting point of these two curves corresponds to a frequency \(\omega_B\) at which \(\delta(\omega_B)=0\) where phase matching is perfect. Away from this frequency, \(\delta\ne0\), and the coupling efficiency drops. The range of \(\delta\) within which the modes remain well coupled is \(-|\kappa|\lt\delta\lt|\kappa|\), which can be found by considering \(\alpha_c\) in (72) [refer to the two-mode coupling tutorial].

For \(|\delta|\lt|\kappa|\), \(\alpha_c\) remains real, and the coupling efficiency given by (87) or (91) [refer to the two-mode coupling tutorial] depends on hyperbolic functions that do not oscillate. Then, \(\eta\) increases monotonically as \(|\kappa|l\) increases, as can be see in figure 2 above.

For \(|\delta|\gt|\kappa|\), \(\alpha_c\) becomes pure imaginary. Then, \(\eta\) depends on sinusoidal functions and drops quickly as \(|\delta/\kappa|\) increases, as can be seen in figure 8 in the two-mode coupling tutorial. As a consequence, the forward- and backward-propagating modes gradually become decoupled.

The frequency bandwidth of a DBR can be found by considering the frequency dependence of \(\delta\). Using (21) above, we have

\[\tag{26}\delta(\omega)=-\beta(\omega)+\frac{qK}{2}=-\beta(\omega_B)-\frac{\text{d}\beta}{\text{d}\omega}(\omega-\omega_B)+...+\frac{qK}{2}\approx-\frac{\text{d}\beta}{\text{d}\omega}(\omega-\omega_B)\]

For a given value of \(|\kappa|l\), the maximum efficiency is \(\eta_{PM}\) given by (87) [refer to the two-mode coupling tutorial]. It can shown using (91) that at \(\eta=\eta_{PM}/2\), we have \(|\delta|>|\kappa|\) and

\[\tag{27}(2\coth^2|\kappa|l-1)\sin^2(|\kappa|l\sqrt{|\delta/\kappa|^2-1})=|\delta/\kappa|^2-1\]

The FWHM reflectivity bandwidth \(\Delta\omega\) for a DBR is given by

\[\tag{28}\Delta\omega=2\left|\delta_{1/2}\frac{\text{d}\omega}{\text{d}\beta}\right|\]

where \(|\delta_{1/2}|\) is the root of (27) for a given \(l\) and \(|\kappa|\). Its value can also be found by reading the value of \(|\delta/\kappa|\) for \(\eta=\eta_{PM}/2\) on the curve in figure 8 in the two-mode coupling tutorial for a given value of \(|\kappa|l\). For a given structure that has a fixed value of \(|\kappa|\), the bandwidth \(\Delta\omega\) decreases as the length \(l\) increases. However, for a fixed length \(l\), the bandwidth increases as the coupling becomes stronger, and the value of \(|\kappa|\) increases.

By taking \(\text{d}\beta/\text{d}\omega=N_\beta/c\), where \(N_\beta\) is the effective group refractive index of the waveguide mode at the Bragg wavelength, the bandwidth given in (28) is approximately bounded within the following range:

\[\tag{29}2\sqrt2\frac{|\kappa|c}{N_\beta}\coth|\kappa|l\ge\Delta\omega\gt2\frac{|\kappa|c}{N_\beta}\]

**Example 3**

A DBR is made with the grating waveguide coupler described in example 2 above that has a first-order grating for phase-matched coupling of the forward-propagating TE_{0} field to the backward-propagating TE_{0} field at \(\lambda=1\) μm.

_{1}field?

**(a)**

From the coupled-mode theory tutorial, we know that \(|\kappa|l\) = 0.88 for a phase-matched 3-dB contradirectional coupler of \(\eta\) = 50%. From example 2 above, we find that \(\kappa=\kappa_{00}^{TE}=-\text{i}0.014\,\mu m^{-1}\). Therefore, the required length of the DBR is

\[l=\frac{0.88}{|\kappa|}=63\,\mu m\]

From example 1 above, we know that the period of this first-order grating is \(\Lambda=289.7\) nm. The number of periods of the DBR is thus

\[N_{DBR}=\frac{l}{\Lambda}=217\]

**(b)**

For leakage coupling to the TE_{1} mode, \(\kappa_{01}^{TE}=-\text{i}0.0277\,\mu m^{-1}\) from example 2 above. This coupling is not phase matched. The phase mismatch can be found as

\[2\delta=\Delta\beta+qK=-\beta_1-\beta_0+K=0.8396\,\mu m^{-1}\]

for \(\beta_1=10.0036\,\mu m^{-1}\), \(\beta_0=10.8432\,\mu m^{-1}\), \(K=21.6864\,\mu m^{-1}\), and \(q=1\). We then find that \(|\delta|=15.16|\kappa_{01}^{TE}|\) and \(|\kappa_{01}^{TE}|l=1.745\) for \(l=63\,\mu m\). Plugging these numbers in (91) [refer to the two-mode coupling tutorial] for phase-mismatched contradirectional coupling, we find that the efficiency for leakage coupling is, for \(|\delta/\kappa_{01}^{TE}|\gt1\) in this case,

\[\eta_{01}=\frac{\sin^2\left(|\kappa_{01}^{TE}|\sqrt{|\delta/\kappa_{01}^{TE}|^2-1}\right)}{|\delta/\kappa_{01}^{TE}|^2-\cos^2\left(|\kappa_{01}^{TE}l|\sqrt{|\delta/\kappa_{01}^{TE}|^2-1}\right)}=0.004\]

Therefore, there is only about 0.4% of the TE_{0} mode power that is leaked into the TE_{1} mode because of the large phase mismatch in the TE_{0}-TE_{1} coupling.

**(c)**

To find the bandwidth without solving for the entire dispersion curve of the waveguide mode, we take \(N_\beta\approx n_\beta\) as an approximation. Then, for the TE_{0} mode under consideration, \(N_\beta\approx\beta_0/2\pi\lambda=10.8432/2\pi=1.7257\) because \(\lambda=1\,\mu m\). Using \(|\kappa|=0.014\,\mu m^{-1}\) and \(\Delta\omega=2\pi\Delta\nu\), we find by applying (29) that the bandwidth is bounded in the range

\[\sqrt{2}\frac{|\kappa|c}{\pi N_\beta}\coth|\kappa|l=1.5494\text{ THz}\ge\Delta\nu\gt\frac{|\kappa|c}{\pi N_\beta}=774.7\text{ GHz}\]

Solving (27) exactly for \(|\kappa|l=0.88\) yields \(|\delta_{1/2}|=1.998|\kappa|\). This results in an actual bandwidth of \(\Delta\nu=1.5478\text{ THz}\). Therefore, the actual bandwidth of this DBR is very close to its upper bound.

The next part continues with the Directional Couplers tutorial.