Group Index

Group Index of Silicon Waveguide

This function returns group index of the fundamental mode for given waveguide width and wavelength.

Group Index is calculated using the following equation where \(n_\mathrm{eff}\) is the effective index, \(n_g\) is the group index, \(w\) is the waveguide width, and \(\lambda\) is the wavelength.

\[n_g(w, \lambda) = n_\mathrm{eff}(w, \lambda) - \lambda \cdot \frac{\partial n_\mathrm{eff}(w, \lambda)}{\partial \lambda}\]

This is what is implemented in the built-in function ng.

Note: ng function implementation is based on finite difference method. Therefore, the result is not exact. However, the result is accurate enough for most of the cases. If you need exact result, please use grad_neff function to calculate the derivative of neff function.

import sipkit
import matplotlib.pyplot as plt
from jax import numpy as jnp
from jax import numpy as jnp

%matplotlib inline

wavelengths = jnp.linspace(1.5, 1.6, 1000)
group_index_400 = jnp.array([sipkit.ng(0.4, wl) for wl in wavelengths])
group_index_450 = jnp.array([sipkit.ng(0.45, wl) for wl in wavelengths])
group_index_500 = jnp.array([sipkit.ng(0.5, wl) for wl in wavelengths])
group_index_550 = jnp.array([sipkit.ng(0.55, wl) for wl in wavelengths])
group_index_600 = jnp.array([sipkit.ng(0.6, wl) for wl in wavelengths])

WARNING:jax._src.lib.xla_bridge:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

plt.plot(wavelengths, group_index_400, label='400 nm')
plt.plot(wavelengths, group_index_450, label='450 nm')
plt.plot(wavelengths, group_index_500, label='500 nm')
plt.plot(wavelengths, group_index_550, label='550 nm')
plt.plot(wavelengths, group_index_600, label='600 nm')
plt.legend()
plt.xlabel('Wavelength (μm)')
plt.ylabel('Group index')
plt.title('Group index of a SOI strip waveguide (SiO2/Si)')
plt.ylim(4.0, 4.4)
plt.xlim(1.5, 1.6)
plt.grid(axis='y', color='0.9', linestyle='-', linewidth=1)
plt.show()

Free Spectral Range (FSR)

As an example, the group index can be used to calculate the free spectral range (FSR) of a resonator, the spectral spacing between two consecutive resonant wavelengths. The FSR can be expressed as the following equation where \(L\) is the round-trip cavity length, \(\lambda\) is the wavelength, and \(n_g\) is the group index.

\[FSR = \frac{\lambda^2}{n_{g}L}\]

def fsr(width, wavelength, length):
    """
    Calculates the free spectral range of a waveguide. This is the distance
    between two consecutive resonances.
    :param width: Waveguide width in micrometers.
    :param wavelength: Wavelength in micrometers.
    :param length: Propagation length in micrometers.
    :return: Free spectral range in micrometers.
    """
    return wavelength ** 2 / (sipkit.ng(width, wavelength) * length)


my_fsr = fsr(width=0.5, wavelength=1.55, length=30)
my_fsr

DeviceArray(0.0190803, dtype=float64)