Chirped gratings are gratings that have spatially varying pitches. The Traveling Wave Laser Model (TWLM) has chirped grating settings that can be used to model these structures. These values are defined in the Waveguide/Chirp Properties section of the element. This article will outline how to set up chirped gratings with example files.
Model Parameters
The main parameter in chirped gratings is chip, which is defined as the difference between the original grating propagation constant and the perturbed grating propagation constant due to grating pitch changes.
In the TWLM, there are two approaches to set a user defined chirped grating following the chirp definition above. Both of them would give identical results. The first approach is the chirp coefficient, which is defined as:
$$
-\mathrm{chirp} \cdot \frac{\lambda_{\mathrm{bragg}}^2}{4\pi n_{\mathrm{eff}}z}.
$$
where \(\lambda_{\mathrm{bragg}}\) is the Bragg wavelength of the grating, \(n_{\mathrm{eff}}\) is the effective index, and \(z\) is the discretized segment position in TWLM and is defined by \(z = [0, dz, 2dz, \dots, L]\), and \(\mathrm{chirp}\) is the difference of grating propagation constants.
The second approach is the chirp parameter defined as:
$$
\mathrm{chirp} \cdot \frac{L^2}{z},
$$
where \(L\) is the length of TWLM, \(z\) is the discretized segment position in TWLM and is defined by \(z = [0, dz, 2dz, \dots, L]\), and \(\mathrm{chirp}\) is the difference of grating propagation constants.
Example
In the attached demo scripts, the TWLM is "passive", that is, it is set with zero gain and zero spontaneous emission, so that only the passive chirped grating structure is considered in the simulation. This simplification is chosen to clearly illustrate and validate the effect of chirp on the Bragg wavelength.
The first scriptset_chirp_from_grating_period_variation.lsfillustrates how to set the grating chirp for the specified spatial variation of the grating period, which in this case is just a simple grating period shift by a constant value. The script calculates the chirp and by using the equation above, the chirp property is then set in our TWLM model.
After the script is finished, it would generate one plot which shows the parameters along the TWLM positions. The left-hand side figure is for "chirp parameter", while the right hand side figure is for "chirp coefficient".
You can then run the INTERCONNECT simulation and check the result of ONA1.
From the figure above we can see that the new shifted Bragg wavelength, \(\lambda_{\mathrm{bragg-new}}\), due to the grating chirp is 1305.9 nm, which agrees with the calculation based on the analytical formula at the end of the script using the equation:
$$
\lambda_{\mathrm{bragg-new}} = 2 \cdot n_{\mathrm{eff}}\left(\lambda_{\mathrm{bragg-new}}\right) \cdot \Lambda_{\mathrm{perturbed}}.
$$
where the new \(n_{\mathrm{eff}}\) is the effective index at the new Bragg wavelength and \(\Lambda_{\mathrm{perturbed}}\) is the perturbed grating period.
This is a nonlinear equation since \(n_{\mathrm{eff}}\) depends on the Bragg wavelength. Therefore, instead of solving it directly, we show that the new Bragg wavelength from simulation satisfies this equation. The grating spectrum shows a double dip, because we introduced a quarter-wave phase slip in the middle of the grating.
The second script set_chirp_for_constant_bragg_wavelength_shift.lsf illustrates, on a simple example, how to set the grating chirp to achieve the target Bragg wavelength shift. In this script, the dispersion relation of \(n_{\mathrm{eff}}\) is included.
See Also
INTERCONNECT as a Laser Design Platform, Laser TW – INTERCONNECT Element