The Traveling Wave Laser Model supports edge-emitting lasers with gratings, such as Distributed Bragg Reflector (DBR) lasers and Distributed Feedback (DFB) lasers. These parameters can be adjusted with options under “Waveguide/Grating” and “Waveguide/Mode 1” in the TWLM element. This article will describe the theory of the grating model in the TWLM, and how the grating coupling coefficient can be calculated.
Grating Parameters
The effect of the grating is determined by both settings under the “Waveguide/Grating” parameters and “Waveguide/Mode 1” parameters. The “Waveguide/Grating” parameters determine the physical structure of the grating such as the grating period and phase slip. Under “Waveguide/Mode 1”, the parameters that control the effects of the grating on the laser are the real and imaginary part of the grating coupling coefficient, “grating coupling real coefficient 1” and “grating coupling imag coefficient 1”, respectively.
The section below explains the grating model in TWLM. Next, the method of calculating the grating coupling coefficient is described, with examples for two common grating profiles.
Grating Model
Wave Propagation in the Cavity
In the laser cavity, the total field can be expressed as a sum of the forward propagating and backward propagating fields. Here, we assume that the field propagation is in the \(z\) direction and the total electric field can be expressed with the following ansatz:
$$
\mathbf{E}\left(x, y, z, t\right) = \mathbf{U}\left(x, y\right)\left[E_f e^{j\left(\omega t - \beta z\right)} + E_b e^{j\left(\omega t + \beta z\right)}\right]
$$
$$
\beta = \frac{2\pi n}{\lambda}
$$
where \(\mathbf{U}(x, y)\) is the transverse mode profile, \(\beta\) is the propagation constant in the medium, \(n\) is the refractive index, and \(\lambda\) is the optical wavelength.
The overall propagation of the field in the cavity is governed by the wave equation:
$$
\nabla^2 \mathbf{E} + \epsilon\left(x, y, z\right)k_0^2 \mathbf{E} = 0
$$
$$
k_0 = \frac{2\pi}{\lambda}
$$
where \(k_0\) is the free space propagation constant, and \(\epsilon\) is the permittivity of the propagating medium. The permittivity is related to the refractive index \(n\) via \(n^2 = \epsilon\).
Effects of Grating on the Wave Propagation
When a grating is present it introduces a perturbation in the permittivity
$$
\varepsilon\left(x, y, z\right) \rightarrow \varepsilon\left(x, y, z\right) + \Delta\varepsilon\left(x, y, z\right)
$$
$$
\mathbf{U} \rightarrow \mathbf{U} + \Delta\mathbf{U}
$$
$$
\beta \rightarrow \beta + \Delta\beta.
$$
Specifically, the perturbation to the permittivity can be expanded as a Fourier series:
$$
\Delta\varepsilon\left(x, y, z\right) = \sum_{l \neq 0}{{\Delta\varepsilon}_l\left(x, y\right)e^{-jl\frac{2\pi}{\Lambda}z}},
$$
where \(\Lambda\) is the grating period and \(l\) is the grating order. Substituting the ansatz for the total electric field into the wave equation, and neglecting the second-order perturbation terms and the second-order field derivatives in the \(z\) direction, the following coupled wave equation is obtained for the forward and backward propagating fields:
$$
\frac{dE_f\left(z\right)}{dz} = -j\kappa_l E_b\left(z\right)e^{2j\delta z}
$$
$$
\frac{dE_b\left(z\right)}{dz} = j\kappa_{-l} E_f\left(z\right)e^{-2j\delta z}
$$
where the grating coupling coefficient, \(\kappa\), and the detuning parameter, \(\delta\), are defined as:
$$
\kappa_{\pm l} \equiv \frac{k_0^2}{2\beta} \frac{\int{{\Delta\varepsilon}_{\pm l}\left(x, y\right)|\mathbf{U}|^2 dA}}{\int{\left|\mathbf{U}\right|^2 dA}} = \frac{k_0^2}{2\beta} \cdot \Delta{\bar{\varepsilon}}_{\pm l}
$$
$$
\delta \equiv \beta - \beta_0
$$
$$
\beta_0 = \frac{\pi l}{\Lambda}
$$
where \(\beta_0\) is the propagation constant at the Bragg condition for coupling \(E_f\) and \(E_b\), \(l\) is the grating order as before, and all other quantities are as previously defined. Here, the bars on top of quantities represent an effective modal quantity.
In TWLM, the propagation of fields along the longitudinal direction is calculated by propagating the mode envelope through segments of the laser, as discussed in the introductory article. In the case of segments where grating is present, the coupling coefficient \(\kappa\) and the detuning parameter \(\delta\) are used to capture the propagation by relating them to the transmission and reflection through the grating segment.
Transmission and Reflection
The reflection and transmission coefficients through the grating segment can be found by analytically solving the coupled wave equations above:
$$
\tilde{r} = -j\frac{\tilde{\kappa}_{-l} \text{tanh}{\left(\tilde{\sigma}L\right)}}{\tilde{\sigma} + j\cdot\tilde{\delta} \text{tanh}{\left(\tilde{\sigma}L\right)}}
$$
$$
\tilde{t} = \frac{\tilde{\sigma} \text{sech}{\left(\tilde{\sigma}L\right)}}{\tilde{\sigma} + j\tilde{\delta} \text{tanh}{\left(\tilde{\sigma}L\right)}} e^{-j\beta_0L}
$$
$$
\sigma = \kappa_l \kappa_{-l} - \delta^2 = \kappa^2 - \delta^2
$$
where \(L\) is the grating length and the tilde over the symbols designates complex quantities.
In principle, both \(\delta\) and \(\kappa\) can be complex, and represent cases where the grating introduces either gain or attenuation to the field. When there is gain or loss in the grating waveguide, the propagation constant \(\beta\) is complex, and \(\delta\) can be expressed as:
$$
\tilde{\delta} = \tilde{\beta} - \beta_0 = \frac{2\pi\overline{n}}{\lambda} + j\frac{\left\langle g\right\rangle_{xy} - \alpha}{2} - \frac{\pi l}{\Lambda}
$$
where \(\left\langle g\right\rangle_{xy}\) is the modal gain, calculated from the gain spectrum of the material multiplied by the mode confinement factor, and \(\alpha\) is the modal loss.
The case of complex \(\kappa\) will be explained below in the gain modulation section.
Summary and Limitations
The approximations made in the derivation above lead to a larger error in the reflection and transmission coefficients for larger coupling coefficients, which occurs for larger grating index perturbations or larger reflectivities between different grating periods. However, these approximations are generally valid for DFB and DBR edge-emitting lasers, and results are generally indistinguishable from the more exact (and slower) solutions. For a more in-depth discussion of the accuracy of these assumptions, see [1, Ch.6] and [1, Fig. 6].
Overall, the quantities \(\tilde{\kappa}\) and \(\tilde{\delta}\) capture the effects of a grating structure on the fields in a cavity. The quantity \(\tilde{\delta}\) only depends on the grating material and period, which are inputs to the TWLM. Therefore, this quantity is automatically calculated in the model from the inputs. However, the value \(\tilde{\kappa}\) depends on the exact shape of the grating index perturbation and on the grating order and therefore needs to be precalculated. The following section details the calculation of \(\kappa\).
Calculating Grating Coupling Coefficient \(\kappa\)
Calculation Procedure
Continuing from the previous derivations, \(\kappa\) is expressed as:
$$
\kappa_{\pm l} = \frac{k_0^2}{2\beta} \cdot \Delta{\overline{\varepsilon}}_{\pm l}.
$$
Using the definition \(\epsilon = n^2\), the following expression can be obtained for the perturbed permittivity \(\Delta\overline{\epsilon}\) in terms of the perturbed index \(\Delta\overline{n}\). The grating order \(l\) will be omitted here for brevity:
$$
\overline{\epsilon} + \Delta\overline{\epsilon} = \overline{n}^2 + 2\overline{n}\Delta\overline{n} + \Delta{\overline{n}}^2.
$$
Ignoring the non-linear terms in \(\Delta\overline{n}\), \(\beta = \frac{2\pi\overline{n}}{\lambda}\), and \(k_0 = \frac{2\pi}{\lambda}\), the following result is obtained:
$$
\Delta\overline{\epsilon} \cong 2\overline{n}\Delta\overline{n}
$$
$$
\kappa_{\pm l} = \frac{\left(\frac{2\pi}{\lambda}\right)^2}{2\left(\frac{2\pi\overline{n}}{\lambda}\right)} \cdot 2\overline{n}\Delta\overline{n}
$$
and can be simplified into:
$$
\kappa_{\pm l} = \frac{\pi}{\lambda} 2\Delta{\overline{n}}_{\pm l}
$$
where \(\Delta{\overline{n}}_{\pm l}\) is the refractive index perturbation for the given grating order \(l\), which in general depends on the spatial index profile in the grating.
TWLM supports only one grating order, so the user should precalculate and supply the coupling coefficient for the required grating perturbation shape and grating order. To calculate the coupling coefficient, we can expand the spatial index profile of the grating structure as a Fourier series:
$$
\delta\overline{n}\left(z\right) = \sum_{l \neq 0}{\Delta{\overline{n}}_l e^{-jl\frac{2\pi}{\Lambda}z}}.
$$
For example, for the first order, we take the first Fourier component by multiplying both sides by \(e^{j\frac{2\pi}{\Lambda}z}\) and integrating over the period of the grating, \(\Lambda\):
$$
\Delta{\overline{n}}_1 = \frac{1}{\Lambda} \int_{0}^{\Lambda}{\delta\overline{n}\left(z\right)e^{\frac{j2\pi}{\Lambda}z}dz}.
$$
where \(\delta\overline{n}\left(z\right)\) is the expression for the change in refractive index in the grating, and \(\Delta{\overline{n}}_1\) can be directly substituted into the earlier equation for \(\kappa\) to calculate \(\kappa_{\pm1}\), which is then to be entered into the “grating coupling real coefficient 1” field.
Example Cases – Square and Cosine Grating
We have calculated the case for a square and cosine grating using the procedure outlined above. The gratings are described with the diagrams below.
Actual values of \(\delta\overline{n}\) for the desired mode can be obtained with Lumerical MODE, or the FEEM solver in Lumerical Multiphysics. For an approximative procedure that does not require simulating the effective modal index, please see [1, Ch. 6.3.2.1].
Cosine Grating
To calculate the coupling coefficient for the cosine grating, the expression for \(\delta\overline{n}\left(z\right)\) can be written as:
$$
\delta\overline{n}\left(z\right) = \delta n \cdot \cos{\left(\frac{2\pi z}{\Lambda}\right)}.
$$
Applying the formula for \(\Delta{\overline{n}}_1\) above, and Euler’s formula, the following is obtained:
$$
\Delta{\overline{n}}_1 = \frac{1}{\Lambda} \int_{0}^{\Lambda}{\delta n \left(\frac{1}{2}e^{\frac{j2\pi}{\Lambda}z} + \frac{1}{2}e^{-\frac{j2\pi}{\Lambda}z}\right)e^{\frac{j2\pi}{\Lambda}z}dz}.
$$
Evaluating the integral, the result for \(\Delta{\overline{n}}_1\) is:
$$
\Delta{\overline{n}}_1 = \frac{\delta\overline{n}}{2}
$$
and can be substituted into the formula for \(\kappa_{\pm1}\) above (repeated here for convenience), obtaining the following final results for the cosine grating:
$$
\kappa_{\pm1} = \frac{\pi}{\lambda} 2\Delta{\overline{n}}_{\pm l}
$$
$$
\kappa_{\pm1} = \frac{\pi\delta\overline{n}}{\lambda}.
$$
Square Grating
To calculate the grating coupling coefficient for the square grating, the expression for \(\delta\overline{n}\left(z\right)\) can be written as:
$$
\delta\overline{n}\left(z\right) =
\begin{cases}
\delta n & 0 \leq z < \frac{\Lambda}{4} \\
-\delta n & \frac{\Lambda}{4} \leq z < \frac{3\Lambda}{4} \\
\delta n & \frac{3\Lambda}{4} \leq z \leq \Lambda
\end{cases}
$$
and substituted into the formula for \(\Delta{\overline{n}}_1\) above, obtaining:
$$
\Delta{\overline{n}}_1 = \frac{1}{\Lambda} \delta n \left( \int_{0}^{\frac{1}{4}\Lambda}{e^{\frac{j2\pi}{\Lambda}z}dz} - \int_{\frac{1}{4}\Lambda}^{\frac{3}{4}\Lambda}{e^{\frac{j2\pi}{\Lambda}z}dz} + \int_{\frac{3}{4}\Lambda}^{\Lambda}{e^{\frac{j2\pi}{\Lambda}z}dz} \right).
$$
Evaluating the integral, the expression for \(\Delta{\overline{n}}_1\) is:
$$
\Delta{\overline{n}}_1 = \frac{2\delta n}{\pi}
$$
and the final result for the square grating, using the formula for \(\kappa_{\pm1}\) above, is:
$$
\kappa_{\pm1} = \frac{4\delta\overline{n}}{\lambda}.
$$
For non-square and non-cosine shapes, the same procedure can be used; however, numerical integration may be necessary.
Gain Modulation
In some gratings, there could be a modulation in both the refractive index as well as the optical gain. In this case, the refractive index perturbation is expressed as a complex quantity. The complex refractive index can be expressed as
$$
\tilde{n} = \text{Re}\{\tilde{n}\} + j\text{Im}\{\tilde{n}\}
$$
with a modal gain \(\overline{g}\) defined as:
$$
\overline{g} = -\frac{2\omega}{c} \text{Im}\{\tilde{n}\} \rightarrow \text{Im}\{\tilde{n}\} = -\frac{c}{2\omega}g = -\frac{\text{Re}\{\tilde{n}\}}{2k}g = -\frac{g}{2k_0}
$$
where \(\omega\) is the optical angular frequency.
In this case, the change in index \(\delta\overline{n}(z)\) can also be broken down into real and imaginary components. The same procedure can be followed as above, and the final result depends on both \(\Delta \text{Re}\{\tilde{n}\}\), expressed as \(\Delta{\overline{n}}_{\pm l}\) in the example section above, as well as \(\Delta\overline{g}\), which can be calculated using, for example, CHARGE and MQW solvers.
For example, if both the real part and gain (i.e., imaginary part) have the same cosine perturbation, the final expression is analogous to the real cosine perturbation derived above:
$$
\kappa_{\pm1} = \frac{\pi}{2\lambda}\left(2\delta\overline{n} - j\frac{\delta\overline{g}}{k_0}\right),
$$
where \(\delta\overline{g}\) is the amplitude of the gain perturbation.
References
- L.Coldren, S.Corzine, and M.Mashanovitch, Diode Lasers and Photonic Integrated Circuits, 1st Edition, Wiley, 1995.
See Also
INTERCONNECT as a Laser Design Platform, Laser TW – INTERCONNECT Element