Tutorial Model: Directional Coupler
Introduction
Directional couplers are used for coupling a light wave from one waveguide to another waveguide. By controlling the refractive index in the two waveguides (for instance, by heating or current injection), it is possible to control the amount of coupling between the waveguides.
Figure 13: Schematic drawing of the waveguide structure. The structure consists of the two waveguide cores and the surrounding cladding. Port 1 and 2 are used for exciting the waveguides and Port 3 and 4 absorb the waves. Notice that the waveguide structure is not drawn to scale. The length of the waveguides is around 2 mm, whereas the waveguide cross section is square with a side length of 3 μm. The waveguide separation is 3 μm.
Light that propagates through a dielectric waveguide has most of the power concentrated within the central core of the waveguide. Outside the waveguide core, in the cladding, the electric field decays exponentially with the distance from the core. However, if you put another waveguide core close to the first waveguide (see Figure 14), that second waveguide will perturb the mode of the first waveguide (and vice versa). Thus, instead of having two modes with the same effective index, one localized in the first waveguide and the second mode in the second waveguide, the modes and their respective effective indices split and you get a symmetric supermode (see Figure 14 and Figure 16 below), with an effective index that is slightly larger than the effective index of the unperturbed waveguide mode, and an antisymmetric supermode (see Figure 15 and Figure 17), with an effective index that is slightly lower than the effective index of the unperturbed waveguide mode.
Since the supermodes are the solution to the wave equation, if you excite one of them, it will propagate unperturbed through the waveguide. However, if you excite both the symmetric and the antisymmetric mode, that have different propagation constants, there will be a beating between these two waves. Thus, you will see that the power fluctuates back and forth between the two waveguides, as the waves propagate through the waveguide structure. You can adjust the length of the waveguide structure to get coupling from one waveguide to the other waveguide. By adjusting the phase difference between the fields of the two supermodes, you can decide which waveguide that initially will be excited.
Model Definition
The directional coupler, as shown in Figure 13, consists of two waveguide cores embedded in a cladding material. The cladding material is GaAs, with ion-implanted GaAs for the waveguide cores. The structure is modeled after Ref. 1.
The core cross section is square, with a side length of 3 μm. The two waveguides are separated 3 μm. The length of the waveguide structure is 2 mm. Thus, given the narrow cross section compared to the length, it is advantageous to use a view that does not preserve the aspect ratio for the geometry.
For this kind of problem, where the propagation length is much longer than the wavelength, the Electromagnetic Waves, Beam Envelopes interface is particularly suitable, as the mesh does not need to resolve the wave on a wavelength scale, but rather the beating between the two waves.
The model is set up to factor out the fast phase variation that occurs in synchronism with the first mode. Mathematically, we write the total electric field as the sum of the electric fields of the two modes,
The expression within the square parentheses is what will be solved for. It will have a beat length L defined by
or
.
In the simulation, this beat length must be well resolved. Since the waveguide length is half of the beat length and the waveguide length is discretized into 20 subdivisions, the beat length will be very well resolved in the model.
The model uses two numeric ports per input and exit boundary (see Figure 13). The two ports define the lowest symmetric and antisymmetric modes of the waveguide structure.
In the second part of the modeling procedure, the bidirectional formulation is used. In this case, the two wave vectors are codirectional — they point in the same direction. However, the magnitude of the wave vectors are given by the propagation constants of the two beating modes. Thus, you expect the two waves to have almost constant amplitudes, so the mesh can be very coarse in the propagation direction.
A problem with the first two procedures is that the numerical procedure returns mode fields with an arbitrary phase. Thus, when you superpose the two input port modes, the result can be different on different computers. In the last part of the modeling procedure, it is shown how you can form a summation of the mode fields, with expansion coefficients that are calculated to minimize the difference between the summed mode fields and a target field. Thereby, independently of the mode field phases, the resulting superposition will be stable.
The summation of the two input port mode fields should approximate the target field, as expressed in
(1),
where ET,target is the tangential target electric field and ci and ET0,i are the expansion coefficients and the tangential non-normalized electric mode fields for mode i, respectively. Taking the cross product with the complex conjugate of the tangential magnetic mode field for mode j, multiplying with the port normal and integrating over the port boundary, we get
(2),
where Pj is the mode power for mode j. Thus, the expansion coefficients are given by the overlap integral
(3).
COMSOL defines the normalized mode field as
(4),
where Pin,i is the specified input power for mode i and θin,i is the corresponding specified mode phase.
Comparing Equation 1, Equation 3, and Equation 4, we can deduce that
(5)
or
(6)
and
(7).
Equation 6 and Equation 7 can now be used for specifying the input power and mode phase for the two exciting ports.
Results and Discussion
Figure 14 to Figure 17 show the results of the initial boundary mode analysis. The first two modes (those with the largest effective mode index) are both symmetric. Figure 14 shows the first mode. This mode has the transverse polarization component along the z direction. The second mode, shown in Figure 16, has transverse polarization along the y direction.
Figure 14: The symmetric mode for z-polarization. Notice that the returned solution can also show the electric field as positive values in the peaks at the cores.
Figure 15: The antisymmetric mode for z-polarization.
Figure 15 and Figure 17 show the antisymmetric modes. Those have effective indices that are slightly smaller than those of the symmetric modes. Figure 15 shows the mode for z-polarization and Figure 17 shows the mode for y-polarization.
Figure 16: The symmetric mode for y-polarization. Notice that the returned solution can also show the electric field as positive values in the peaks at the cores.
Figure 17: The antisymmetric mode for y-polarization.
Figure 18 shows how the electric field increases in the receiving waveguide and decreases in the exciting waveguide. If the waveguide had been longer, the waves would switch back and forth between the waveguides.
Figure 18: Excitation of the symmetric and the antisymmetric modes. The wave couples from the input waveguide to the output waveguide. Notice that your result may show that the wave is excited in the other waveguide core, if your mode fields have different signs than what is displayed in Figure 14 to Figure 17.
Figure 19 shows the result, when there is a π phase difference between the fields of the exciting ports. In this case, the superposition of the two modes results in excitation of the other waveguides (as compared to the case in Figure 18).
Figure 19: The same excitation conditions as in Figure 18, except that there is a phase difference between the two ports of π radians. Notice that your result may show that the wave is excited in the other waveguide core, if your mode fields have different signs than what is displayed in Figure 14 to Figure 17.
Figure 20 and Figure 21 display the amplitudes of the first and second wave, respectively, when the bidirectional formulation is used. As expected, the amplitudes are almost constant.
Figure 20: The amplitude of the first wave, when the bidirectional formulation is used.
Figure 21: The amplitude of the second wave, when the bidirectional formulation is used.
Figure 22 shows that the input field approximates the square target field, centered on the left waveguide, when the input power and the mode phase for the two exciting ports are calculated using an overlap integral between the target field and the mode field (c.f. Equation 6 and Equation 7).
Figure 22: The red square with amplitude 1 V/m represents the target function. This target function has an amplitude of1 V/m in the left waveguide core and 0 V/m everywhere else. The total input field is shown to be localized in the left waveguide, as expected, with an amplitude that approaches the target amplitude.
Reference
1. S. Somekh, E. Garmire, A. Yariv, H.L. Garvin, and R.G. Hunsperger, “Channel Optical Waveguides and Directional Couplers in GaAs-Imbedded and Ridged,” Applied Optics, vol. 13, no. 2, pp. 327–30, 1974.