Optical Nonlinear Ring Modulator (NL_RING) - INTERCONNECT Element

modulates an optical signal depending on electrical signal incorporating nonlinear effects

Keywords

electrical, optical, unidirectional

Ports

Properties

General Properties

Standard Properties

Standard/Table Properties

Waveguide Properties

Waveguide/Nonlinearities Properties

Enhanced Properties

Numerical Properties

Numerical/Digital Filter Properties

Diagnostic Properties

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Implementation Details

The nonlinear ring modulator builds upon the linear optical ring modulator element in INTERCONNECT, extending its functionality with additional equations that model nonlinear effects such as self-heating and free carrier dynamics.

Linear Model

When self-heating and free carrier effects are disabled, this model behaves similarly to the linear optical ring modulator element. However, it includes an additional “drop” port and extra input parameters, which extends its functionality beyond the basic linear model.  

Basic equations

The basic schematic and 0-D relations between fields within the ring is shown below.

$$B(t)=t_1A(t)+ik_1D(t)$$

$$D(t)=t_2C(t-\tau)a(t)e^{-i\phi(t)}$$

$$C(t)=ik_1A(t)+t_1D(t)$$

$$E(t)=ik_2a(t)^{\frac{1}{2}}e^{-\frac{i\phi}{2}}C\left(t-\frac{\tau}{2}\right)$$

$$a(t)=e^{-\alpha_{lin}(t)l}$$

$$\alpha_{lin}(t)=\mathrm{loss}+\frac{2\pi}{\lambda}\Delta\kappa_{eff}(t)$$

$$\phi=\frac{2\pi l}{\lambda}\left(n_{eff}+\Delta n_{eff}(t)\right)$$

where \(A(t)\), \(B(t)\), \(C(t)\), \(D(t)\), and \(E(t)\) are electric field amplitudes labeled in the schematic, and \(a(t)\) is the attenuation for each round trip.

In the equations for \(A(t)\), \(B(t)\), \(C(t)\), and \(D(t)\), the variables \(t_{1,2}\) and \(k_{1,2}\) are respectively the amplitude transmission (\(t\)) and coupling (\(k\)) coefficients for each optical coupler, \(\phi(t)\) is the phase shift after each round trip around the ring, \(\tau=\frac{ln_g}{c}\) is the roundtrip delay around the ring, with \(n_g\) being the group index, and \(l\) being the roundtrip length around the ring.

The phase shift is calculated using the "effective index" parameter under "Waveguide", and the perturbation to the real component of the effective index, \(\Delta n_{eff}\), as specified in the modulation table.

The attenuation for each round trip, \(a(t)\), is calculated based on the amplitude loss per unit length of the linear model, the loss per unit length, \(\alpha_{lin}(t)\), which is calculated by converting and summing the power loss parameter "non-absorption loss", the power loss parameter "linear absorption loss", and \(\Delta\kappa_{eff}\), which is the perturbation to the imaginary component of the effective index as specified in the modulation table. The wavelength \(\lambda\) is the center wavelength of the modulator. For power-dependent loss calculations, this 0-D model assumes uniform power around the ring structure.

Additions to linear ring modulator

When nonlinear effects are disabled, the nonlinear ring modulator behaves similarly to the linear ring modulator. However, it includes the following enhancements:

1. A drop port is added as an additional output optical port.
2. A second modulation port is introduced as an input electrical port. The two modulation ports are interchangeable; one can be used for modulation while the other can model thermal tuning effects. 

Because of the addition of the “drop” port, the sample rate of the simulation must be set to be much lower than half the roundtrip delay time to ensure that the results are accurate in the time domain.

With the addition of the second modulation port, a second modulation file can also be inputted under the “Standard/Table” properties, with “measurement filename 1” used for modulation port 1, and “modulation filename 2” used for modulation port 2. For each of the ports, the voltage-induced change in the effective index is scaled using their respective electrical fill factors under “Enhanced” properties.

Nonlinear Model

The self-heating (thermal) and free carrier effects in the nonlinear ring modulator model are modeled through parameters under “Waveguide/Nonlinearities Properties” shown above. These effects are also inherently 0-D, with an assumption of uniform temperature inside the ring, and uniform power for the purposes of power-dependent loss calculations.

The contribution of nonlinear effects appears through the loss per unit length and the phase change of the wave as it makes a roundtrip in the ring. These effects are based on the equations in [1] and we describe below how both quantities are calculated.

Note: When the nonlinear effects are enabled, simulation with an ONA element is not supported and will result in an error.

Carrier Density and Temperature Change

$$\frac{d\Delta N}{dt}=-\frac{\Delta N}{\tau_{FC}}+G\left|C\left(t-\tau\right)\right|^2$$

$$\frac{d\Delta T}{dt}=-\frac{\Delta T}{\tau_{th}}+\frac{1}{H_c}P_{abs}$$

where \(\Delta N\) is the change in carrier density from equilibrium, \(\Delta T\) is the change in temperature from the operating temperature, \(\tau_{FC}\) (free carrier time constant), \(\tau_{th}\) (thermal time constant), and \(H_c\) (heat capacity) are input parameters to the model. \(P_{abs}\) is the total absorbed power in the ring automatically calculated by INTERCONNECT, assuming constant loss per unit length \(\alpha\) throughout the ring, and includes linear, free carrier, and two-photon absorption. \(C\left(t-\tau\right)\) is the optical field as marked in the schematic above, at one round trip delay earlier in the simulation.

The coefficient \(G\) is the free carrier generation rate calculated as seen below.

$$G=\Gamma_{FC}\frac{\beta_{TPA}}{h\cdot f_{res}\cdot A_{eff-FC}^2}\cdot\left|C\left(t-\tau\right)\right|^2$$

where \(h\) is Planck's constant, \(\Gamma_{FC}\) (free carrier confinement factor), \(\beta_{TPA}\) (two-photon absorption coefficient), and \(A_{eff-FC}\) (free carrier effective area) are input parameters to the model, and \(f_{res}\) is the resonance frequency that is closest to the selected center frequency of the modulator via the "Frequency" field. The value of \(f_{res}\) is automatically calculated by INTERCONNECT.

Loss Parameter

When free carrier effects are enabled, two-photon absorption and free carrier absorption contribute to additional loss in the ring.

The model calculates the total loss per unit length using the following formula:

$$\alpha(t)=\alpha_{lin}(t)+\alpha_{TPA}(t)+\alpha_{FCA}(t)$$

where \(\alpha_{lin}(t)\) is the amplitude loss per unit length of the linear model, \(\alpha_{TPA}(t)\) is the amplitude loss from two-photon absorption effects, and \(\alpha_{FCA}(t)\) is the amplitude loss from free carrier absorption effects.

The two-photon absorption loss is calculated as follows:

$$\alpha_{TPA}(t)=\Gamma_{TPA}\frac{\beta_{TPA}}{A_{eff-TPA}}\cdot\left|C\left(t-\tau\right)\right|^2$$

where \(\Gamma_{TPA}\) (two-photon absorption confinement factor), \(\beta_{TPA}\) (two-photon absorption coefficient), and \(A_{eff-TPA}\) (two-photon absorption effective area) are input parameters to the model, and \(C\left(t-\tau\right)\) is the optical field as marked in the schematic above, at one round trip delay earlier in the simulation.

The free carrier absorption loss is calculated as follows:

$$\alpha_{FCA}(t)=\sigma_{FCA}\Delta N$$

where \(\sigma_{FCA}\) (free carrier absorption coefficient) is an input parameter, and \(\Delta N\) is the change in carrier density as calculated above.

Phase Change

The total phase change from a round-trip is calculated as

$$
\phi = \phi_{lin}\left(t\right) + \frac{2\pi l}{\lambda} \left( \frac{dn_{eff}}{dT} \Delta T + \frac{dn_{eff}}{dN} \Delta N \right)
$$

where \(\phi_{lin}(t)\) is the phase change from the linear model, accounting for those from the modulation signal and the natural roundtrip delay, and \(\frac{dn_{eff}}{dT}\) (effective index temperature sensitivity) and \(\frac{dn_{eff}}{dN}\) (effective index free carrier sensitivity) are input parameters to the model.

Usage Examples

Sample rate in nonlinear ring modulator

In this example, the non-linear effects are disabled. Since the nonlinear ring modulator has an additional output for the "drop" port, the sample rate must be adjusted accordingly, such that the time step is smaller than half the round-trip delay, with the full round-trip delay calculated by \(\tau=\frac{l}{v_g}\).

In the example below, we use the optical network analyzer (ONA) element to illustrate this effect. The simulation is set up with two ONAs, one in the scattering analysis mode (frequency-domain), and one in the impulse response mode (time-domain). The modulation ports are connected to DC electrical sources, such that the only element that controls the sample rate of the ONA is the “frequency range” setting in the ONA. 

In the plot below, we compare the frequency-domain results and the time-domain results for the drop port.

As seen from the figure above, as the frequency range value in the ONA is increased, corresponding to a larger sample rate, the results converge to the scattering analysis spectrum. The simulation file for this example is attached to the article as nonlinearRing_ONA_samplerate_example.icp. The plot is generated manually by sending the results to the same visualizer.

Note:  When running an impulse response analysis where an ONA and additional signal source(s) are present, the simulation “sample rate” needs to match the ONA’s “frequency range”. One way to ensure these values are identical is to set the “frequency range” of the ONA component to the sample rate of the simulation using the %sample rate% expression, as seen below.

Temperature and carrier density profiles

Under the diagnostic settings, you can also record the time evolution of the temperature and carrier density profiles of the nonlinear ring modulator by changing the options for “record temperature profile” and “record carrier density profile”, respectively, to “true”. These results then appear as a part of the results for the element. The recorded temperature profile and carrier density profiles are the values of \(\Delta T\) and \(\Delta N\) described in the equations above.

The “downsample factor” determines how frequently data is recorded during the time domain simulation. The total number of samples recorded is the total number of time steps divided by the downsample factor.

The figures below show the time evolution of the temperature and carrier density profiles. For typical ring modulators, the temperature time constant is large, as such, when interpreting your results, it is important to first ensure that the simulation time window is long enough such that both temperature and carrier density have reached steady state.

The simulation file for this example is attached to the article as nonlinearRing_diagnostic_example.icp. The plot is generated manually by sending the results to the same visualizer.

References

1. T.J.Johnson et al., “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator”, Opt. Exp., vol. 14, no. 2, pp. 817-831, doi: 10.1364/OPEX.14.000817.