Noise Cancellation Effects in Integrated Photonics with Wilkinson Power Dividers

Wilkinson power dividers (WPDs) are a popular element in RF and microwave technologies known for providing isolation capabilities. However, the benefits that WPDs could offer to integrated photonic systems are far less studied. Here, we investigate the thermal emission from and the noise performance of silicon-on-insulator (SOI) WPDs. We find that WPDs exhibit a noiseless port, with important implications for receiving systems and absorption-based quantum state transformations. At the same time, the thermal signals exiting noisy ports exhibit nontrivial correlations, opening the possibility for noise cancellation. We analyze passive and active networks containing WPDs showing how such nontrivial correlations can prevent the amplification of the thermal noise introduced by WPDs while benefiting from their isolation capabilities. Using this insight, we propose a modified ring-resonator amplifier that improves by N times the SNR in comparison with conventional traveling wave and ring-resonator amplifiers, with N being the number of inputs/outputs of the WPD. We believe that our results represent an important step forward in the implementation of SOI-WPDs and their integration in complex photonic networks, particularly for mid-IR and quantum photonics applications.


Section 2: Thermal radiation in back-to-back WPDs
In this section we analyze the noise performance of a back-to-back conguration of two 1 × N -port WPDs as shown in Figure 2. To this end, we will evaluate the SNR in the c-port.
First, applying the input-output relations for a WPD we have that the signals after the rst WPD are given by Figure S1 Scattering matrix of the ideal WPD (a) and designed SOI-WPD (b).
Figure S2 Sketch of the integrated back-to-back conguration. and the signal after the second WPD can be dened as: where n T i is the thermal noise in the i port.
Next, the signal to noise ratio is given by where , as the average of thermal noise is dened as zero. In order to obtain the previous equation, we have calculated the thermal power emission by the noise correlation matrix: Thus, we obtain that where N 0a is the noise of the input signal. We highlight that the contribution from the thermal emission to the total power cancels out the rst term corresponds to the contribution of the diagonal elements, and the second term corresponds to the contribution of the o-diagonal elements.
Finally, the signal to noise ratio reduces to It can be concluded from this result that the SNR at the output is the same than at the input, due to the thermal noise cancellation because of the combination of the diagonal elements and all the nontrivial correlations (o-diagonal elements).

Section 3: Travelling wave amplier
Here we show the noise performance analysis of the TWA (see Figure S 3). The scattering matrix of this device is as follows: Figure S3 Scheme of TWA implemented in a silicon-on-insulator (SOI) platform.
When α > 0, the TWA behaves as amplier, and when α < 0, it behaves as a lossy device. Notice that α = 0 refers to a neutral element with no gain and loss.
The output can be expressed as: being n a the noise generated by the amplier. Thus, the SNR can be calculated as: ⟨b * b⟩ = e 2αL ⟨a * a⟩ + ⟨n * a n a ⟩ where: Finally, the SNR at the output of the amplier can be expressed as: We note that more realistic ampliers present additional noise signals in addition to the minimum thermal noise. In order to model this case, the amplier's noise can be expressed as: where n aT corresponds with the one studied previously and n A models additional noise signals, with the following properties: ⟨n * a n a ⟩ = e 2αL − 1 N T I + r A I Where r A I corresponds with the correlation factor of n A and the two parts of the amplier's noise are uncorrelated between them. Then, Eq. (14) is generalized by substituting the minimal noise (G − 1)N T by e 2αL − 1 N T + r A .
Section 4: Wilkinson power divider (WPD) within a ring resonator network Here, we show the theoretical analysis of the device depicted in Figure 5 on the main text for the case N = 2. In this conguration, the TWAs are located within the ring resonators.
Therefore, the input-output equations that describe this device are the following: where a s and n i are the amplitude and noise distribution of the ith input signal.
Consequently, the output signal is given by 1 − e αL t * a 2 + 1 2 |k| 2 e αL 1 − e αL t * (n 1 − n 2 ) + 1 √ 2 k 1 − e αL t * (n a1 + n a2 ) + 1 2 kt * e αL 1 − e αL t * (n T 1 + n T 2 ) + kn T 2 (24) By using Eqs.(19-24), we can calculate the SNR, being the intensity and the cross products the following: |k| 4 e 2αL |1 − e αL t * | 2 ⟨n * 1 n 1 ⟩ + 1 4 |k| 4 e 2αL |1 − e αL t * | 2 ⟨n * 2 n 2 ⟩ + 1 2 Assuming the following statements: ⟨n * T 1 n T 1 ⟩+⟨n * T 2 n T 2 ⟩+⟨n * T 1 n T 2 ⟩+⟨n * T 2 n T 1 ⟩ = 0,⟨a * a⟩ = |⟨a 2 ⟩| 2 + ⟨a * 2 a 2 ⟩, ⟨n * i n j ⟩ = 0, ⟨n * 2 n 2 ⟩ = ⟨n * 1 n 1 ⟩ = ⟨a * 2 n 2 ⟩ = ⟨n * 2 a 2 ⟩ = N 0 ; the gain and the SNR of the amplifying state can be written as That is the same result that the device analyzed in the main text with only one amplier but located between the two WPDs. Now, we can approach the SNR for high gain, considering that the gain tends to innite when e αL = 1 t * . On the one hand, according with the term related to the thermal noise when G → ∞ we have: On the other hand, the term related to the signal noise can be approached as: Thus, the SNR can be written for high gain as Nevertheless, as we demonstrate in Figure 5 (main text), with a gain greater that 2 (in lineal terms), the improvement in the noise is signicant.