Analysis of photonic noise generated due to Kerr nonlinearity in silicon ring resonators

Tomer Yeminy; Dan Sadot; Zeev Zalevsky

doi:10.1364/OE.26.000284

1. Introduction

Silicon photonic ring resonators (RRs) are extensively used in optical communications and sensing applications. Their high quality (Q) factors and small dimensions enable them to attain nonlinear effects using modest optical input power [1–4]. As a result, the Kerr effect in RRs has been applied to implement many silicon photonic components such as switches, isolators, logical gates, and modulation format converters [4–11]. High Q factor RRs are also used for label-free sensing in order to reduce the intensity noise [12–18]. However, their performance is limited by the Kerr effect which gives rise to a determinate deviation in the output signal's intensity [18].

While the signal in silicon Kerr RRs has been thoroughly analyzed [1–4], their noise power and distribution have not been discussed. As a result, the performances of RR-based components have been evaluated by Monte-Carlo simulations and experiments, thus lacking rigorous mathematical modeling [5–14]. Having the noise distribution in hand will allow performance analysis and design of silicon Kerr RR-based devices. Moreover, this noise distribution can be further used to calculate its impact on the lower bound of channel capacity [19].

Here, we analytically derive the noise distribution in silicon Kerr RRs for notch configuration. The noise of the input signal is assumed to be circularly Gaussian [20,21], such as amplified spontaneous emission (ASE) and relative intensity noise (RIN). We show that for input signals with sufficiently high input signal to noise ratio (SNR) the output noise distribution is still Gaussian, but not circular. Furthermore, the output noise depends on the input signal power, and is therefore not wide sense stationary (WSS). In addition, the noise power, the signal power, and the optical signal to noise ratio (OSNR) at the RR's output increase and then saturate with the enhancement of the Kerr effect in the RR.

We use the derived noise distribution to analyze the noise at the output of silicon Kerr RRs for intensity detection sensing [18] and optical communications applications. We show that the dominant noise in intensity detection sensing at the presence of the Kerr effect is the determinate deviation in the output signal's intensity rather than the output stochastic noise. On the other hand, in optical communications applications the output signal's determinate deviation is a part of the signal, and the stochastic noise distribution variation due to the Kerr effect plays a significant role in the performance analysis and design of silicon Kerr RR based components.

This paper is organized as follows. In section 2 we derive the analytical model for the noise. Then, in section 3 we numerically simulate and discuss various noise related parameters, and section 4 concludes the paper.

2. Analytical noise model

The silicon RR to be discussed is illustrated in Fig. 1. Although the following analysis is performed for notch RRs, it can be easily adapted to add-drop RRs.

Fig. 1 Silicon notch ring resonator.

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The RR's directional coupler obeys the following field equations [1]:

E_{3} (t) = r E_{1} (t) + i s E_{2} (t)

E_{4} (t) = i s E_{1} (t) + r E_{2} (t)

where

r^{2} + s^{2} = 1

. In addition:

E_{2} (t) = a (t) e^{i ϕ (t)} E_{4} (t - Δ t)

where

a (t)

,

ϕ (t)

, and

Δ t

are the attenuation, phase shift, and time delay of the light due to a single round trip in the RR. The single round trip attenuation and phase shift can be obtained from the following propagation equation [2,22–24]:

\frac{\partial E}{\partial z} = i k_{0} \frac{n_{2}}{n_{0}} (1 + i K) {| E |}^{2} E - α_{l} \frac{E}{2}

with:

k_{0} = 2 π n_{0} / λ

,

K = β_{T P A} n_{0} / (2 k_{0} n_{2})

, where

λ, β_{T P A}, n_{0}, n_{2}

, and

α_{l}

are the light's wavelength, two photon absorption (TPA) coefficient, refractive index, Kerr index, and linear loss coefficient respectively. We assume that the free carrier absorption (FCA) and carrier plasma effect can be neglected due to a reverse biased diode structure implementation which sweeps the free carriers out of the waveguide [25,26]. In addition, thermal noise resulting from the thermo-optic effect is assumed to be resolved by using thermal heaters to control the RR's temperature or by employing athermal waveguide design [18,27]. Furthermore, the laser linewidth effect [28] of the input signal can be neglected since typical laser coherence times are significantly longer than the photon’s lifetime in the RR (micro seconds versus nano seconds, respectively). Solving this equation and including the linear phase shift we get:

a (t) = \frac{\exp (- α_{l} L / 2)}{\sqrt{1 + b I_{4} (t - Δ t)}}

ϕ (t) = k_{0} L + \frac{1}{2 K} \ln [1 + b I_{4} (t - Δ t)]

where

b = β_{T P A} L_{e f f}

,

I_{4} = {| E_{4} |}^{2} / A_{e f f}

, and

L_{e f f} = (1 - \exp (- α_{l} L)) / α_{l}

. The parameters L and

A_{e f f}

are the RR's perimeter and effective mode cross-section respectively. Hence,

a (t)

results from the linear and TPA losses, and similarly

ϕ (t)

stems from the linear as well as self-phase modulation (SPM) phase shifts. Using Eq. (2) and substituting

a (t)

and

ϕ (t)

in Eq. (3) yields the following recursive equation:

\begin{array}{l} E_{4} (t) = i s E_{1} (t) + r a (t) e^{i ϕ (t)} E_{4} (t - Δ t) \\ = i s E_{1} (t) + v \frac{\exp {\frac{j}{2 K} \ln [1 + b I_{4} (t - Δ t)]}}{\sqrt{1 + b I_{4} (t - Δ t)}} E_{4} (t - Δ t) \end{array}

The constant

v = r \exp [- α L / 2 + i k_{0} L]

includes the linear loss terms i.e. the coupler's field reflection and linear propagation loss, as well as the linear phase shift term. We now define the following nonlinear propagation function:

g (x) ≜ \frac{\exp {\frac{i}{2 K} \ln [1 + x]}}{\sqrt{1 + x}}

and expand

g (x)

to a Taylor series in order to simplify the noise analysis:

g (x) = \sum_{m = 0}^{M} a_{m} x^{m}

.

Thus, Eq. (7) turns to:

E_{4} (t) = i s E_{1} (t) + v \sum_{m = 0}^{M} {\tilde{a}}_{m} {| E_{4} (t - Δ t) |}^{2 m} E_{4} (t - Δ t)

where:

{\tilde{a}}_{m} = a_{m} {(b / A_{e f f})}^{m}

. Let us write the input field as:

E_{1} (t) = {\bar{E}}_{1} (t) + n_{1} (t)

where

{\bar{E}}_{k} (t)

and

n_{k} (t)

are the ensemble average and noise of the k^th RR's field respectively. We assume that

n_{1} (t)

is zero-mean and stationary with circularly Gaussian distribution. Our goal is to find

{\bar{E}}_{4} (t)

and the distribution of

n_{4} (t)

as a function of the input signal and noise. We first use binomial expansion for the term

{| E_{4} (t - Δ t) |}^{2 m} E_{4} (t - Δ t)

in Eq. (9), omitting the time notation for convenience:

\begin{array}{l} {| E_{4} |}^{2 m} E_{4} = {({\bar{E}}_{4} + n_{4})}^{m + 1} {({\bar{E}}_{4}^{*} + n_{4}^{*})}^{m} \\ = [\sum_{l = 0}^{m + 1} (\begin{matrix} m + 1 \\ l \end{matrix}) {\bar{E}}_{4}^{m + 1 - l} n_{4}^{l}] [\sum_{p = 0}^{m} (\begin{matrix} m \\ p \end{matrix}) {({\bar{E}}_{4}^{*})}^{m - p} {(n_{4}^{*})}^{p}] \end{array}

Assuming sufficiently high SNR, we neglect terms with power of

n_{4} (t)

higher than 1. Thus:

{| E_{4} |}^{2 m} E_{4} \approx {| {\bar{E}}_{4} |}^{2 m} {\bar{E}}_{4} + m {| {\bar{E}}_{4} |}^{2 (m - 1)} {\bar{E}}_{4}^{2} n_{4}^{*} + (m + 1) {| {\bar{E}}_{4} |}^{2 m} n_{4}

Using Eq. (11) in Eq. (9) yields:

\begin{array}{l} E_{4} (t) = i s E_{1} (t) + v \sum_{m = 0}^{M} {\tilde{a}}_{m} [{| {\bar{E}}_{4} (t - Δ t) |}^{2 m} {\bar{E}}_{4} (t - Δ t) + \\ + m {| {\bar{E}}_{4} (t - Δ t) |}^{2 (m - 1)} {\bar{E}}_{4}^{2} (t - Δ t) n_{4}^{*} (t - Δ t) \\ + (m + 1) {| {\bar{E}}_{4} (t - Δ t) |}^{2 m} n_{4} (t - Δ t)] \end{array}

Taking the ensemble average of Eq. (12) we have:

{\bar{E}}_{4} (t) = i s {\bar{E}}_{1} (t) + v \sum_{m = 0}^{M} {\tilde{a}}_{m} {| {\bar{E}}_{4} (t - Δ t) |}^{2 m} {\bar{E}}_{4} (t - Δ t)

Equation (13) is identical to Eq. (9) with the difference of using the ensemble average fields in Eq. (13) instead of the fields themselves. Thus, assuming sufficiently high SNR, the ensemble average fields can be computed based on

{\bar{E}}_{1} (t)

, which is the first moment of the input field, in spite of the nonlinearity of the Kerr RR.

The noise term of Eq. (12) is:

\begin{array}{l} n_{4} (t) = i s n_{1} (t) + v \sum_{m = 0}^{M} {\tilde{a}}_{m} [m {| {\bar{E}}_{4} (t - Δ t) |}^{2 (m - 1)} {\bar{E}}_{4}^{2} (t - Δ t) n_{4}^{*} (t - Δ t) + \\ + (m + 1) {| {\bar{E}}_{4} (t - Δ t) |}^{2 m} n_{4} (t - Δ t)] \end{array}

We now define the following notations:

\begin{array}{l} d_{1} = i s \\ d_{2, l} (t) = v \sum_{m = 0}^{M} {\tilde{a}}_{m} m {| {\bar{E}}_{4} (t - l Δ t) |}^{2 (m - 1)} {\bar{E}}_{4}^{2} (t - l Δ t) \\ d_{3, l} (t) = v \sum_{m = 0}^{M} {\tilde{a}}_{m} (m + 1) {| {\bar{E}}_{4} (t - l Δ t) |}^{2 m} \end{array}

Thus, for low input power:

\begin{array}{l} \lim_{{| E_{1} (t - l Δ t) |}^{2} \to 0} d_{2, l} (t) = 0 \\ \lim_{{| E_{1} (t - l Δ t) |}^{2} \to 0} d_{3, l} (t) = v \end{array}

Using Eq. (15) in Eq. (14) yields:

n_{_{4}} (t) = d_{1} n_{_{1}} (t) + d_{2, 1} (t) n_{4}^{*} (t - Δ t) + d_{3, 1} (t) n_{4} (t - Δ t)

For zero initial power inside the RR, it can be shown from Eqs. (15) and (17) by induction that:

n_{4} (t) = \sum_{p = 0}^{N} α_{p} (t) n_{1} (t - p Δ t) + β_{p} (t) n_{1}^{*} (t - p Δ t)

where:

\begin{array}{l} α_{p} (t) = α_{p - 1} (t) d_{3, p} (t) - β_{p - 1} (t) d_{2, p}^{*} (t), α_{0} (t) = d_{1} \\ β_{p} (t) = β_{p - 1} (t) d_{_{3, p}}^{*} (t) - α_{p - 1} (t) d_{2, p} (t), β_{0} (t) = 0 \end{array}

and N is the number of a photon's round trips in the RR:

N = ⌈ \frac{τ_{p} c}{L n_{0}} ⌉

with

τ_{p}

being the photon lifetime in the RR.

Equations (16) and (19) imply that:

\begin{array}{l} \lim_{{| E_{1} |}^{2} \to 0} α_{p} (t) = i s \cdot v^{p} \\ \lim_{{| E_{1} |}^{2} \to 0} β_{p} (t) = 0 \end{array}

Hence, for low input power where the Kerr effect is negligible:

n_{4, l i n} (t) = \lim_{{| E_{1} |}^{2} \to 0} n_{4} (t) = i s \sum_{p = 0}^{N} v^{p} n_{1} (t - p Δ t)

Since v is the attenuation and phase shift factor due to a photon's single round trip in the RR and the term is corresponds to the coupler's transmission,

n_{4, l i n} (t)

is indeed the noise in the RR right after the coupler when only linear effects take place.

Having $n_{4} (t)$ as a function of the input noise by Eq. (18), we can find the noise distributions of the RR's fields. The distribution of $n_{4} (t)$ is zero-mean Gaussian since $n_{1} (t)$ has zero-mean complex normal distribution. The autocorrelation function of $n_{4} (t)$ is [30]:

\begin{array}{l} R_{4} (t_{1}, t_{2}) = E [n_{4} (t_{1}) n_{_{4}}^{*} (t_{2})] \\ = \sum_{p, k = 0}^{N} α_{p} (t_{1}) α_{_{k}}^{*} (t_{2}) R_{1} ((k - p) Δ t + τ) + β_{p} (t_{1}) β_{_{k}}^{*} (t_{2}) R_{_{1}}^{*} ((k - p) Δ t + τ) \end{array}

where we used the circularity of

n_{1} (t)

and

τ = t_{1} - t_{2}

. Since

R_{4} (t_{1}, t_{2})

is a function of

t_{1}

and

t_{2}

,

n_{4} (t)

is not wide sense stationary (WSS). The dependence on

t_{1}

and

t_{2}

stems from the terms:

α_{p} (t_{1}), α_{_{k}}^{*} (t_{2}), β_{p} (t_{1}),

and

β_{_{k}}^{*} (t_{2})

, which are time dependent due to the variations of the input power as implied by Eqs. (15) and (19). Thus, the non-stationarity of

n_{4} (t)

results from the changes in the RR's nonlinearity due to the input power variations.

The variance of $n_{4} (t)$ is:

\begin{array}{l} σ_{4}^{2} (t) = E [{| n_{4} (t) |}^{2}] \\ = \sum_{p, k = 0}^{N} α_{p} (t) α_{_{k}}^{*} (t) R_{1} ((k - p) Δ t) + β_{p} (t) β_{_{k}}^{*} (t) R_{_{1}}^{*} ((k - p) Δ t) \end{array}

Similarly, the autocovariance function of

n_{4} (t)

is:

\begin{array}{l} C_{4} (t_{1}, t_{2}) = E [n_{4} (t_{1}) n_{4} (t_{2})] \\ = \sum_{p, k = 0}^{N} α_{p} (t_{1}) β_{k} (t_{2}) R_{1} ((k - p) Δ t + τ) + α_{k} (t_{2}) β_{p} (t_{1}) R_{1}^{*} ((k - p) Δ t + τ) \end{array}

Since

C_{4} (t_{1}, t_{2})

is nonzero,

n_{4} (t)

is not circular [20,21]. The cross-correlation and cross-covariance of

n_{1} (t)

and

n_{4} (t)

are [30]:

R_{1, 4} (t_{1}, t_{2}) = E [n_{1} (t_{1}) n_{4}^{*} (t_{2})] = \sum_{p = 0}^{N} α_{p}^{*} (t_{2}) R_{1} (τ + p Δ t)

C_{1, 4} (t_{1}, t_{2}) = E [n_{1} (t_{1}) n_{4} (t_{2})] = \sum_{p = 0}^{N} β_{p} (t_{2}) R_{1} (τ + p Δ t)

respectively.

Knowing the distribution of $n_{4} (t)$ , we can derive the distribution of $n_{2} (t)$ using Eq. (2). Since $n_{2} (t) = (n_{4} (t) - i s n_{1} (t)) / r$ , $n_{2} (t)$ is zero-mean Gaussian with the following second order statistics:

R_{2} (t_{1}, t_{2}) = \frac{1}{r^{2}} [R_{4} (t_{1}, t_{2}) + i s R_{_{1, 4}}^{*} (t_{2}, t_{1}) - i s R_{1, 4} (t_{1}, t_{2}) + s^{2} R_{1} (τ)]

σ_{_{2}}^{2} (t) = \frac{1}{r^{2}} [σ_{_{4}}^{2} (t) + 2 s Im {R_{1, 4} (t, t)} + s^{2} σ_{_{1}}^{2}]

C_{2} (t_{1}, t_{2}) = \frac{1}{r^{2}} [C_{4} (t_{1}, t_{2}) - i s C_{1, 4} (t_{2}, t_{1}) - i s C_{1, 4} (t_{1}, t_{2})]

Using Eq. (1) and the statistics of $n_{1} (t)$ and $n_{2} (t)$ we obtain that the RR's output noise $n_{3} (t)$ is zero-mean Gaussian with:

R_{3} (t_{1}, t_{2}) = \frac{1}{r^{2}} [R_{1} (τ) - i s R_{1, 4} (t_{1}, t_{2}) + i s R_{1, 4}^{*} (t_{2}, t_{1}) + s^{2} R_{4} (t_{1}, t_{2})]

σ_{3}^{2} (t) = \frac{1}{r^{2}} [σ_{1}^{2} + 2 s Im {R_{1, 4} (t, t)} + s^{2} σ_{4}^{2} (t)]

When only linear effects take place, the noises of the RR's fields can be found by using Eq. (21) in Eqs. (23)-(33). Since the noises in this case have zero autocovariance functions, their distributions are zero-mean circularly Gaussian. Furthermore, these noises are stationary due to their lack of dependence on the input power. Equations (31)-(33) form the second order statistics of the RR's output noise. Since the output noise is zero-mean Gaussian, we therefore have the probability density function (PDF) of the output noise in hand. The output noise is not circular, nor WSS.

3. Simulation results and discussion

We used Monte-Carlo Matlab simulations to verify the above calculated signals and noise distributions in a notch RR. The examined applications were optical communications [4–11] and sensing [12–18].

3.1 Optical communications applications

In optical signal processing applications the input power usually ranges from −20dBm to 20dBm [29]. The common signal bandwidth is 10-30GHz, and the noise is usually filtered at the input of the silicon chip, hence having the same bandwidth as the signal. The dominant optical noise is the ASE of optical amplifiers, which is a circularly Gaussian white noise with an OSNR of 15-30dB/0.1nm. The simulation parameters are given in Table 1. The coupling to the RR was 110% of the critical coupling [3] to account for fabrication tolerances. In addition, the input pulses were Gaussian and the RR's full width at half maximum (FWHM) was set to the input signal's bandwidth. The RR's FWHM and the input signal's OSNR were 10GHz and 20dB/0.1nm, unless noted otherwise.

Table 1. Simulated ring resonator parameters

View Table

Figure 2(a) shows the power and phase of the ensemble average output field at the presence of the Kerr effect for an input signal with peak power, bandwidth, and OSNR of 10dBm, 10GHz, and 20dB/0.1nm respectively. The output pulse power is distorted and nonlinear phase shift is observed. The theory and simulation results agree, indicating that the mean fields in a Kerr RR can indeed be computed by Eqs. (1)-(6), using the ensemble average input signal ${\bar{E}}_{1} (t)$ as concluded after Eq. (13). Figure 2(b) presents the power and phase of the ensemble average output field without the Kerr effect for the same input signal and RR parameters. It can be seen that signal is significantly attenuated by the RR, as expected.

Fig. 2 (a). Power and phase of the RR's ensemble average output field at the presence of the Kerr effect. (b) without the Kerr effect.

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We now show the signal and noise distribution at the output pulse's peak versus the input power for a RR with 10GHz bandwidth and 20dB/0.1nm input OSNR. Figure 3(a) illustrates the output pulse power enhancement ratio and phase shift relative to the output signal of a linear RR. Similarly, Fig. 3(b) presents the noise power enhancement ratio with respect to the output noise power of a RR without the Kerr effect. The theory and Monte Carlo results show a good agreement.

Fig. 3 (a) Output pulse's power enhancement ratio and phase shift with respect to a RR without Kerr effect. (b). Output noise power enhancement ratio with respect to a RR without Kerr effect.

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We define the noise figure (NF) of the RR as the ratio between the input and output OSNRs. Figure 4 illustrates the RR's NF versus the input power with and without the Kerr effect. When only linear effects take place the NF is not affected by the input power, as expected. However the NF at the presence of the Kerr effect raises for 0-7dBm input power as the noise is enhanced more than the signal. For input power above 7dBm the NF decreases until the output OSNR equals the input OSNR.

Fig. 4 RR's noise figure with Kerr effect and when only linear effects take place.

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The results in Figs. 3(a) and 3(b), as well as in Fig. 4 can be explained both in time and frequency domains. In time domain, the output pulse power increases with the input power due to the Kerr effect's enhancement, which changes the RR's refractive index. The output noise power also rises owing to the signal-noise multiplication resulting from the Kerr effect. As the input pulse power further increases, the output pulse and noise power enhancements saturate due to the saturation of the power inside the RR, since more power goes into the output port owing to the dominant Kerr effect.

For input power up to 7dBm, the output noise increases more significantly than the output pulse power since the Kerr effect is mild enough to let only little power leak into the RR's output port. Thus, the noise is enhanced due to the strong signal-noise multiplication inside the RR. As the input power raises above 7dBm, the output pulse increases more significantly than the noise as shown in Figs. 3(a) and 3(b), since without the Kerr effect the input pulse's wavelength obeys the RR's resonance condition and is thus strongly attenuated by the RR, whereas the input noise power is equally distributed in the RR's wavelengths range and is thus less attenuated. Hence, the OSNR at the RR's output improves until it recovers to the input OSNR for strong Kerr effect.

In the frequency domain, the Kerr effect red shifts the spectrum of the RR's output signal. Thus, the signal and noise output powers increase versus the input power. Due to the spectral profiles of the signal, noise, and RR, the noise is enhanced more than the signal for input power up to 7dBm. Above this input power, the signal will be enhanced more than the noise and the output OSNR will increasingly improve. This improvement will continue until the notch filter is completely shifted from the signal's bandwidth and hence passes the signal and the noise, which results in recovery to the input OSNR.

Figure 5 illustrates the output noise distribution parameters versus the input power. The inset shows the noise distribution for a 10dBm input signal using Monte Carlo simulation. It can be observed that the distribution is zero-mean and elliptic. This indicates that the output noise is indeed zero-mean Gaussian, but not circular since its autocorrelation function is not zero due to the Kerr effect as derived at Eq. (25). Thus, the equal-probability density contours of the output noise's PDF are elliptic rather than circular [32].

Fig. 5 The axes ratio and angle rotation of the ellipse defined by the equal-probability density contours of the output noise's probability density function. Inset: Output noise distribution obtained from the Monte Carlo simulation for input power of 10dBm.

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The axes ratio and rotation angle of the major axis of the ellipse are also presented in Fig. 5 and define the Gaussian PDF of the output noise. At low input power the ellipse's axes ratio is 0dB, as the noise is circularly Gaussian for a negligible Kerr effect. The ellipse's axes ratio increases with the input power and peaks at about 12.5dBm due to the Kerr effect enhancements. Then, it declines as the input power further rises. It can be seen that there is a difference in the rotation angle between the theory and the Monte Carlo results at input powers lower than −6.5dBm. This difference results from the theoretical axes ratio at these input powers which is exactly 0dB, implying that the ellipse is actually a perfect circle. In this case, the theoretical rotation angle has no meaning. The Monte Carlo simulation also yields very low axes ratio for these low input powers, indicating that the ellipse turns into a circle.

In order to explain the decrease of the ellipse's axes ratio for input power above 12.5dBm we define the coupling ratio as the power coupled from the RR to the output port divided by the power coupled from the input port to the output port:

C o u p l i n g r a t i o = \frac{{| s E_{2} |}^{2}}{{| r E_{1} |}^{2}}

Figure 6 presents the coupling ratio versus the input power. It shows that starting from 12.5dBm input power the coupling ratio declines due to the saturation of the power inside the RR. Thus, the increase of the input power makes the RR's input signal more dominant than the signal inside the RR in determining the PDF of the output noise. Hence, the output noise becomes more circularly Gaussian, which results in the elliptic equal-probability density contours of the PDF becoming circular. Consequently, the ellipse's axes ratio goes to unity as the input power increases above 12.5dBm. The complementary explanation in frequency domain is that the increasing red shift of the notch filter above 12.5dBm input power gradually turns the RR to an all-pass filter that does not affect the input signal and noise. Hence, the output noise becomes more circular when the input power is further increased.

Fig. 6 (a). Coupling ratio of the RR vs. the input power. (b) Deviation of the phase between $E_{2}$ and $E_{1}$ from its value at the absence of the Kerr effect vs. the input power.

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The enhancement of the output signal and noise as well as the increasing phase shift of the output signal shown in Figs. 3(a) and 3(b) can also be explained in light of the drop of the coupling ratio at high input power. At low coupling ratio $E_{1}$ is more dominant than $E_{2}$ in determining $E_{3}$ . Hence, the input and output signal and noise become more similar as the coupling ratio decreases, until the output signal and noise reach the values of the input signal and noise. Thus, for high input power where the coupling ratio is low, the output signal and noise enhancement is high and is saturated as the coupling ratio further drops and the output signal and noise reach the input signal and noise values.

The decline of the RR's NF in Fig. 4 can be explained similarly. The OSNR of $E_{2}$ is lower than the OSNR of $E_{1}$ due to the signal-noise multiplication resulting from the Kerr effect. At low coupling ratio $E_{1}$ dominates, thus the output OSNR reaches the input OSNR value. However, for moderate coupling ratio $E_{2}$ has considerable influence on the output signal. Hence, both the coupling ratio and the phase between $E_{2}$ and $E_{1}$ determine the signal and noise enhancement as well the NF. Figure 6(b) shows the deviation of the phase between $E_{2}$ and $E_{1}$ from the $π / 2$ value obtained at the absence of the Kerr effect. Since this phase deviation monotonically increases with the input power due to the Kerr effect as shown in Fig. 6(b), the NF and axis ratio do not peak at the same input power.

The influence of the RR's bandwidth on the results shown in Figs. 2-6 was also examined for input power and OSNR of 10dBm and 20dB/0.1nm respectively. As the RR's bandwidth increases, the output signal's power, phase, noise power and noise distribution converged to those attained without the Kerr effect, due to the lower power inside the RR. Thus, the RR's NF also converged to the NF obtained at the absence of the Kerr effect. The NF for Kerr RR is a little higher than for linear RR starting from FWHM of about 14GHz since the Kerr effect is mild enough such that the power leakage into the RR's output port is small, as detailed at the explanation to Fig. 4. The NF and ellipse parameters vs. the RR's FWHM are presented in Figs. 7(a) and 7(b) respectively.

Fig. 7 (a). RR's noise figure with Kerr effect and when only linear effects take place, vs. the RR's FWHM. (b). The parameters of the ellipse defined by the output noise's PDF, vs. the RR's FWHM

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In addition to the effect of the input power and RR's bandwidth, we also checked the impact of the input signal's OSNR on the compatibility of the theory and the Monte Carlo results for input power of 10dBm and RR bandwidth of 10GHz. Since we assumed in Eq. (11) that the input SNR should be high enough to neglect noise terms with power higher than 1, this sensitivity analysis is desirable in order to know the limitations of the theory. The NF and ellipse parameters vs. the input OSNR are shown in Figs. 8(a) and 8(b) respectively.

Fig. 8 (a). RR's noise figure with Kerr effect and when only linear effects take place, vs. the input OSNR. (b). The parameters of the ellipse defined by the output noise's PDF, vs. the input OSNR. Inset: Output noise distribution obtained by the Monte Carlo simulation for input OSNR of 5dB/0.1nm.

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The results in Figs. 8(a) and 8(b) imply that the theory is not very sensitive to the input OSNR. A very good agreement between the theory and Monte Carlo simulation is achieved starting from OSNR of 15dB/0.1nm. The smaller NF for low input OSNR is due to the terms with high powers of noise, which increase the ensemble average of the noise above zero. Thus the power in the RR increases, and consequently the NF decreases similarly to Fig. 4. The inset in Fig. 8(b) shows the noise distribution for 5dB/0.1nm input OSNR using the Monte Carlo simulation. The ellipse is distorted, which indicates that the distribution is not Gaussian. In addition, the noise mean is nonzero due to the terms with high powers of the noise in Eq. (11), which cannot be neglected at low input OSNRs .

3.2 Sensing applications

The typical input power in sensing applications is between −30dBm up to 10dBm and the input signal is a continuous wave (CW) [33]. The dominant optical noise is the RIN of laser, which is a circularly Gaussian white noise with typical values in the rage −110 to −160 dB/Hz, corresponding to OSNR of 10-60 dB/0.1nm [18]. High Q factor RRs are preferred to mitigate the intensity noise, but have been shown to suffer from deterministic measurement errors due to the Kerr effect [18]. Please note that in sensing applications, unlike optical communications applications, the power deviation is a part of the noise and not of the signal. Hence, in addition to the stochastics noise we have a deterministic noise defined as the difference between the RR’s output in the linear and nonlinear cases.

The simulation parameters are the same as Table 1 apart from the RR's radius which was 250µm in order to increase the time steps of the simulation and alleviate the computation power required for the Monte Carlo simulations due to the high Q factors involved. The coupling to the RR was 110% of the critical coupling [3] to account for fabrication tolerances. The FWHM of the RR was 1GHz with Q factor of 1.2x10⁶. In addition, the frequency of the input signal was higher than the RR's central frequency in 20% of the RR's bandwidth. The input power and OSNR were 0dBm and 40dB/0.1nm respectively. Since the input noise is white, we measured the output noise around the frequency of the input CW signal within the bandwidth of the RR in order to examine the influence of the Kerr effect.

Figures 9(a)-9(c) present the output noise power enhancement ratio defined as the output noise power with the Kerr effect divided by the output noise power without the Kerr effect. The Monte Carlo and theoretical graphs include both the deterministic and the stochastic noise, while the third graph excludes the stochastic noise. Figure 9(a) indicates that the stochastic noise is negligible to the deterministic noise above −25dBm input power where the noise enhancement ratio > 0dB, which is when the Kerr effect takes part. Similarly, Figs. 9(b) and 9(c) show that the stochastic noise is negligible to the deterministic noise at the presence of the Kerr effect for wide ranges of RR's FWHM and input OSNR. Thus, when the Kerr effect applies, the deterministic noise is the relevant design parameter rather that the laser's RIN.

Fig. 9 (a). Output noise power enhancement ratio vs. the input power. (b). Output noise power enhancement ratio vs. the RR's FWHM. (c). Output noise power enhancement ratio vs. the input OSNR.

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Figure 10 presents the power spectral density of the stochastic noise for 10dBm input power and 40dB/0.1nm input OSNR. The red shift due to the increase of the RR's refractive index resulting from the Kerr effect is clearly seen. In addition, the extinction ratio of the stochastic noise due to the notch is smaller at the presence of Kerr effect. This stems from the decrease of the coupling ratio defined in Eq. (34) and the change in the ring's output phase, which weaken the destructive interference at the output port between the input noise and the noise from the RR.

Fig. 10 Power spectral density of the output noise for the linear and nonlinear cases.

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4. Conclusion

We analytically and numerically analyzed the noise properties in silicon photonic RRs at the presence of the Kerr effect. The analysis shows that for circularly Gaussian input noise such as ASE or RIN, input OSNR ≥ 15dB/0.1nm, and typical RR design parameters, the output noise is Gaussian but neither circular nor WSS. We derived the probability density function of the output noise and showed that the output power of the noise, output power of the signal, and output OSNR increase and saturate due to the Kerr effect. Using the developed noise model we evaluated the RR's NF and output noise distribution for optical communications and sensing applications. This evaluation indicates that in optical communications applications the stochastic noise's distribution is significantly changed due to Kerr effect. Hence, this distribution variation is an important design parameter. On the other hand, ring resonator sensors are mainly affected by the deterministic noise caused by the Kerr effect, which is thus the significant noise in the design of sensing applications.

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Parameter	Description	Value	Units	Reference
λ	Wavelength	1550	nm
n₀	Refractive index	2.43		[10]
n₂	Kerr index	7.9x10⁻¹⁸	m²/W	[31]
_{$β$ TPA}	TPA coefficient	0.5	cm/GW	[21]
R	RR radius	10	µm	[5]
h	Waveguide height	220	nm	[5]
w	Waveguide width	450	nm	[5]

Analysis of photonic noise generated due to Kerr nonlinearity in silicon ring resonators

Abstract

1. Introduction

2. Analytical noise model

3. Simulation results and discussion

3.1 Optical communications applications

3.2 Sensing applications

4. Conclusion

References and links

Cited By

Figures (10)

Tables (1)

Equations (34)

Optics Express