Optical Ring Modulator (RING) - INTERCONNECT Element

modulates an optical signal depending on electrical signal

Keywords

electrical, optical, bidirectional

Ports

Properties

General Properties

Standard Properties

Waveguide Properties

Thermal Properties

Enhanced Properties

Numerical Properties

Numerical/Digital Filter Properties

Diagnostic Properties

Results

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Implementation Details

An optical ring modulator is usually used in wavelength division multiplexing (WDM) systems to isolate a wavelength out of the multiplexed signal. The INTERCONNECT ring modulator model can capture both quasi-static and time-varying effects through its transfer function and represents the relationship between the input and the through port.

This quasi-static behavior allows for using the ring modulator element not only as a modulator, but also as a filter or multiplexer device when cascading multiple ring modulators.

Theory

Basic schematic

The ring oscillator follows the schematic below. The two ports on the model represents only the input and the output (through) ports in the schematic.

Modulation

Modulation in the ring modulator is controlled by modulation tables specified under the “Standard” properties. You can specify the modulation either via absorption and phase directly or using the real and imaginary components of the effective refractive index. The formatting of the table is shown in the example section below.

The voltage-induced change in the effective index is scaled using the electrical fill factor under “Enhanced” properties.

Steady-state response

The ring modulator model is a 0-D model that relates field amplitudes at each port using the relations below under steady state excitation [1]:

$$
B = t_1 A + i k_1 D
$$

$$
D = t_2 C a e^{-i\phi}
$$

$$
C = i k_1 A + t_1 D
$$

$$
a = e^{-\alpha l}
$$

$$
\alpha = \text{loss} + \frac{2\pi}{\lambda} \Delta\kappa_{\text{eff}}
$$

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

where \(A\) is the input optical field at port 1 and \(B\) is the output optical field at port 2 as labeled in the schematic, \(t_{1,2}\), \(k_{1,2}\) represent respectively the amplitude transmission \(t\) and coupling \(k\) coefficients for each optical coupler, \(\phi\) represents the phase shift after each round trip around the ring, \(a\) is the attenuation for each round trip, and \(l \) is 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 is calculated by both the “loss” parameter under “Waveguide”, and the perturbation to imaginary component of the effective index, \(\Delta\kappa_{eff}\) as specified in the modulation table. In these equations, \(\lambda\) is the center wavelength.

Rearranging the equations above, we obtain the static transmission function as

$$
T = \frac{B}{A} = \frac{t_1 - t_2 a e^{-i\phi}}{1 - t_1 t_2 a e^{-i\phi}}
$$

Time-dependent response

This model has three modes to handle time-dependent responses. The mode of calculation is controlled via options in “Numerical/Digital Filter Properties”.

The ring modulator model uses time-dependent modeling when there is time-dependent electrical modulation connected to the modulation port.

Time variant digital filter: disabled

When the option “time variant digital filter” is set to “disabled”, the ring modulator model solves the 0-D time-dependent equations directly. This is the default option for this element.

The time-dependent equations are stated below [1], which assumes small contribution of index modulation to the final phase shift, and is accurate for typical materials. Further details of derivation are found in [1].

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

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

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

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

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

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

In these equations, \(\tau=\frac{ln_g}{c}\) is the roundtrip delay in the resonator, where \(n_g\) is the group index, and all other variables are as stated before, except with an explicit time dependence.

In this mode, INTERCONNECT solves these equations directly, updating the field amplitudes \(A(t)\), \(B(t)\), \(C(t)\), \(D(t)\), the attenuation \(a(t)\), and the phase \(\phi(t)\) at each time step. In this formulation, since the expression of \(D(t)\) depends on \(C(t)\) at a time that is \(\tau\) prior, the time step must be set such that it is much smaller than \(\tau\).

Time variant digital filter: interpolate and update

When the option “time variant digital filter” is set to “interpolate” or “update”, the ring modulator model is quasi static, and calculates the response using the steady-state transfer function for each time step using an approximation for the index given the input voltage. The transfer function is represented using a digital filter.

In this case, the steady-state transfer function from above is modified as

$$T(t)=\frac{B(t)}{A(t)}=\frac{t_1-t_2a(t)e^{-i\phi(t)}}{1-t_1t_2a(t)e^{-i\phi(t)}}$$

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

$$
a\left(t\right) = e^{-\alpha\left(t\right) l}
$$

$$
\alpha\left(t\right) = \text{loss} + \frac{2\pi}{\lambda} \Delta\kappa_{\text{eff}}(v t)
$$

where \(l\), \(\lambda\), \(v(t)\) is the instantaneous modulation signal at time \(t\), \(\Delta n_{\text{eff}}\) is the perturbation in the real part of the index change for the given voltage, and \(\Delta\kappa_{\text{eff}}\) is the perturbation in the imaginary part of the index change for the given voltage.

During the simulation, the instantaneous transfer function is used to calculate the output.

In the “interpolate” mode, the transfer function is pre-computed for multiple voltage points according to the modulation table, and the instantaneous transfer function is obtained by interpolation at each time step during the simulation.

$$
T\left(v_1\right) = \frac{t_1 - t_2 a(v_1) e^{-i\phi(v_1)}}{1 - t_1 t_2 a(v_1) e^{-i\phi(v_1)}},\quad
T\left(v_2\right) = \frac{t_1 - t_2 a(v_2) e^{-i\phi(v_2)}}{1 - t_1 t_2 a(v_2) e^{-i\phi(v_2)}},\quad
\ldots\quad
T\left(v_N\right) = \frac{t_1 - t_2 a(v_N) e^{-i\phi(v_N)}}{1 - t_1 t_2 a(v_N) e^{-i\phi(v_N)}}
$$

In the “update” mode, the transfer function is calculated for each time step.

Comparison of modes

To capture the widest range of dynamic effects, use the “disabled” mode to solve the time-dependent equations directly.

In the “update” or “interpolate” modes, the cavity dynamics are not taken into account. Therefore, if the quality factor (Q-factor) of the resonator is high, or if the modulation speed is fast, these modes cannot capture transient effects such as overshoot in the output signal. See the usage example section below for a comparison. Between “update” and “interpolate”, the “update” mode is generally more accurate, but is more computationally expensive due to the need to calculate the transfer function at each time step.

In the usage example section, we compare these three modes in terms of their time domain response.

Usage example

In this section, we characterize the time domain and frequency domain performances of the ring modulator. The example file Optical_Ring_Modulator.icp is used throughout.

Element settings

The following figure is the setting of the optical ring modulator.

The required input file measurement type could be selected from "effective index" and "absorption & phase". Please see the example measurement input file neff_vs_voltage_11.txt which goes under the "effective index" type. This is a 3-column table, with the first column being the amplitude, the second column being the perturbation to the real part of the effective index \(\Delta n_{\text{eff}}\), and the third column being the perturbation to the imaginary part of the effective index \(\Delta\kappa_{\text{eff}}\).

Frequency Domain Characterization

The frequency domain characterization of the optical ring modulator gives the following transmission response, blocking signals near a wavelength of 1309.8nm at an amplitude of 0.0285.

Time Domain Characterization

Results from the three numerical methods under time variant modulation are shown below.

The comparison of different digital filter modes is consistent with the descriptions of them earlier. The “disable” mode captures all dynamic effects, including overshoot and ringing. The “update” mode and “interpolate” mode results do not capture these effects and is different from each other in terms of the maximum amplitude. Both signals start later due to an inherent time delay from the digital filter. The “update” mode is accurate in capturing the correct amplitude compared to the dynamic model, whereas the “interpolate” mode shows inaccuracies even in the amplitude of the signal. A finer measurement table of the effective index versus amplitude will improve the accuracy of the results in the “interpolate” mode.

References

1. W.D.Sacher and J.K.S.Poon, “Dynamics of microring resonator modulators,” Opt. Exp., vol. 16, no.20, pp. 15741-15753, doi: 10.1364/OE.16.015741.