4.4: Arrayed waveguide grating

In the previous section, the add-drop multiplexer was discussed, which is a way to add and/or drop a single channel from a WDM system. It is certainly possible to cascade add-drops to create a WDM link with an ever-changing group of channels. It is also possible to do this in a single device called an arrayed waveguide grating. The arrayed waveguide grating (AWG) looks a bit like a very complex MZI, but it is easier to understand it as a type of diffraction grating.

Consider, as illustrated in Figure \(\PageIndex{1}\), light reflected from a grooved surface with period, \(a\). The light in direction \(\theta\) is the sum of fields from light scattered from each groove, and will be highest when those fields are in phase, which is given by

\[\begin{equation}m\lambda = a\left(\sin\phi + \sin\theta\right)\label{eqn:grating}\end{equation}\]

where \(\phi\) is the angle of incidence of the light, and \(m\) is an integer. What equation \(\eqref{eqn:grating}\) tells us is that light will be separated by wavelength, as each diffracts at a different angle. But, underlying that is that each groove contributes a different "path", and all paths have to result in constructive interference. In Figure \(\PageIndex{1}\), this is illustrated on the right hand side. Two incident light rays are shown, but one of them has to travel an additional distance \(d_i\) before it hits the grating surface. Two diffracted rays are also shown, and one of them must travel an additional distance \(d_r\). These two distances must add to an integer number of wavelengths for constructive interference to occur, which results in equation \(\eqref{eqn:grating}\). With a "real" diffraction grating in free space, the paths continuously vary between some minimum value to some maximum value. But, we don't necessarily need all those paths.

Figure \(\PageIndex{1}\): diffraction grating. White light (multicolored arrow) is incident from the left at angle \(\phi\) and a specific color leaves the grating at angle \(\theta\) (left hand side). The grooves each act as a source of light (scattering the incoming light) with a phase delay due to their separation (right handside). The result is that each source constructively interferes with all the others at an angle that depends on the wavelength of light. This separates the light into its separate colors.

An AWG, as illustrated in Figure \(\PageIndex{2}\), first spreads the light, then captures the light in a number of different waveguides, which are all different length. The light exits these waveguides into a common space where it can recombine. For each channel, constructive interference will occur in a unique direction, so, by placing a receiving waveguide at those angles, the AWG demultiplexes all channels in a single step. Since the AWG is a linear device, the reverse is also true: all channels can be multiplexed into a single waveguide in one step.

Figure \(\PageIndex{2}\): Arrayed waveguide grating. Light enters from the left and expands evenly in a slab waveguide (light yellow), each waveguide captures a small amount of the light and carries it to a second slab waveguide where the light mixes. Due to the different paths, the interference between the different light fields ensures taht each output waveguide (only one is shown) will only receive a small range of wavelengths. Since an AWG is symmetrical, inputs and outputs can be swapped, thus an AWG can demultiplex from a single input, or multiplex from multiple inputs.

The transmission function of an AWG is given by

\[\begin{equation}T = \frac{\sin^2\left(\frac{\pi L \Delta d}{\lambda}\right)}{\sin^2\left(\frac{\pi\Delta d}{\lambda}\right)}\label{eqn:awg}\end{equation}\]

where \(L\) is the number of waveguides and \(\Delta d\) is the incremental increase in waveguide length. Figures \(\PageIndex{3}\)a - \(\PageIndex{3}\)d show the transfer function of four AWGs with a varying number of waveguides and a varying incremental increase in length. Note that the transmission appears to increase with wavelength. This is because the grating is not centered. To center the grating, we need to satisfy \(\lambda_c = \frac{n_{eff}\Delta d}{m}\), where for simplicity, \(n_{eff} = 1\). Not included in

The code that produced Figures \(\(\PageIndex{3}\)a - \(\PageIndex{3}\)d is available in Exercise \(\PageIndex{1}\), where you can design your own AWG.

Figure \(\PageIndex{3}\): AWG transfer function for four different combinations of \(L\) and \(\Delta d\)

Exercise \(\PageIndex{1}\)

The code below calculates the transfer function of an AWG. Set \(\Delta d\) to 1550 nm. Is the transfer function of the AWG centered on 1550 nm as expected (you may need to adjust \(L\) to see whole pattern). In a real AWG, the number of waveguides is limited to about 100. Set the effective refractive index to 2.0 (corresponding to a silicon nitride waveguide) and limit the number of waveguides to 100. Adjust \(\Delta d\) to obtain a multiplexer that can multiplex at least 10 channels from the ITU grid. What is the channel spacing do you obtain?

import numpy as np
from matplotlib import pylab as plt
from scipy.constants import c, h

ld = np.linspace(1500e-9, 1600e-9, 2000) #wavelength range
n_eff = 1
L = 2000 #number of waveguides in the array
dd = 250e-9 #incremental increase in distance
ul = (np.sin(np.pi*L*n_eff*dd/ld))**2
ll = (np.sin(np.pi*n_eff*dd/ld))**2
T = ul/ll
T = T/T.max()
plt.figure()
plt.plot(ld*1e9, T, color="xkcd:fuchsia")
plt.xlabel("Wavelength (nm)")
plt.ylabel("Transmitted power (au)")

Text(0, 0.5, 'Transmitted power (au)')

Answer

Yes, setting \(\Delta d\) to 1550 nm centers the pattern on 1550 nm. To get 10 channels within the ITU channel range, a \(\Delta d\) of about 250 nm is required, then by careful movement of \(\Delta d\), you can come close to a good spacing and centering.

A basic understanding of AWGs is given by Equation \(\eqref{eqn:awg}\), however, there is another degree of freedom that provides a bit more freedom. The input and output waveguides are placed on the circumference of a circle (radius \(R\)) with a spacing of \(s\), which leads to a wavelength resolution of \(\delta\lambda= \frac{n_cs^2}{mR^2}\), where \(n_c\) is the refractive index of the material that makes up the circle. Note that for a small \(\delta\lambda\), a large circle is required, and the number of waveguides between the input and output also needs to be large, making for an unfeasibly large and complicated design. Thus, often more sophisticated designs with cascaded AWGs are often used.