Integrated nonlinear interferometer with wavelength multicasting functionality

Integrated nonlinear interferometer with wavelength multicasting functionality

Optics Express

1. Introduction

2. Operation principle

3. Experiment and results

4. Simulation for potential application based on the proposed scheme

5. Conclusions

Funding

References and links

Abstract

1. Introduction

2. Operation principle

3. Experiment and results

4. Simulation for potential application based on the proposed scheme

5. Conclusions

Funding

References and links

Cited By

Figures (9)

Tables (4)

Equations (3)

Weili Yang; Yu Yu; Xinliang Zhang

doi:10.1364/OE.24.018217

In recent years, the researches on nonlinear interferometer (NI) have attracted considerable attention, owing to its phase sensitivity. This unique feature leads to a vast array of capabilities, spanning from signal de-multiplexing [1], all-optical logic exclusive-OR (XOR) gate [2,3], eye diagram sampling [4], low-noise amplification [5], all-optical regeneration [6–9] and many others. Depending on different applications, the NI can be constructed with various architectures, e.g. a Mach-Zehnder interferometer [3], a Sagnac loop mirror [8] and an in-line structure through wavelength conversion based on four wave mixing (FWM) [5–7]. Among these schemes, the FWM-based one is considered as an attractive candidate owing to its transparency to data rate and capability of simultaneous wavelength multicasting or conversion [10]. However, most of the reported FWM-based NI schemes were achieved utilizing cascaded FWM processes. Recently, a NI using single-stage FWM with a dual-pump and dual-signal input was proposed and experimentally demonstrated in [11], where the interferential outputs between the signals were obtained at two different wavelengths exploiting the highly nonlinear fiber (HNLF). Compared with cascaded FWMs, single-stage FWM offers less complexity and lower cost, despite the HNLF is usually large-scale, not integratable and may need to suppress the stimulated Brillouin scattering (SBS) effect.

The photonic integration has remarkable advantages, including small footprint, low power consumption, low cost and many others [12]. Silicon is widely considered as a very promising platform for optoelectronic integration, thanks to the fabrication processes compatible with complementary metal oxide semiconductor (CMOS) technologies [13]. Due to the ultrahigh nonlinear coefficient and immunity of the SBS effect, silicon waveguide has emerged as an alternative competitor to HNLF for all-optical signal processing. The reported applications include wavelength conversion [14], all-optical logic gate [15], all-optical regeneration [16], optical parametric amplification [17], etc. On the other hand, wavelength multicasting, which replicates one wavelength channel onto many different wavelength channels thus data can be parallelly transmitted to different destinations in the wavelength-division multiplexing systems, is a key functionality to effectively increase the network efficiency and simplify the transmitter and receiver [18–22]. Thus, it is urgently desirable to construct a silicon-based NI with wavelength multicasting functionality utilizing single-stage FWM. The combination of the NI and wavelength multicasting can improve the network efficiency and add the diversity of network functionalities.

In this paper, we thus propose and experimentally demonstrate a NI based on single-stage FWM in a silicon nonlinear waveguide. With elaborate frequency selection for three pumps and dual signals, four interferential outputs at different wavelengths are obtained. Two independent frequency combs are utilized to generate the pumps and signals. Limited by the performance of the utilized optical processor, extra filters are used to suppress the unneeded sidebands. This would increase the complexity of the scheme compared with [11]. The multiple outputs can potentially realize both the interference and wavelength multicasting functionality. For a proof-of-concept application, we further theoretically demonstrate the multicasting logic XOR gate based on the proposed scheme. The logic operation is feasible for both return-to-zero on-off keying (RZ-OOK) and binary phase-shift keying (RZ-BPSK) signals. This is quite superior compared with those reported XOR schemes, which were rarely available for both intensity and phase modulated formats. The proposed scheme enables the on-chip construction of NI and potential applications.

The schematic of the proposed NI with wavelength multicasting functionality is shown in Fig. 1. With three pumps (P₁-P₃) and dual signals (S₁, S₂) launched into a silicon waveguide, both phase insensitive and phase sensitive (PS) idlers are generated through the FWM processes. Among them, only the PS idler generated from the interference of two frequency-degenerate idlers can be considered as the interferential output between the dual signals. Furthermore, the two frequency-degenerate idlers should be separately generated by two different FWM processes (each involving one of the signals) with identical conversion efficiencies (CEs). It is important to have the CEs of the two FWM processes approximatively identical. Otherwise, the ratios from the dual signals converted to corresponding idlers are different. As a result, the interferential output of the frequency-degenerate idlers cannot correctly reflect the interference between the dual input signals. As shown in Fig. 1, the frequency spacings between P₁/P₂ and P₂/P₃ are selected to be $2 Δ ω$ and $Δ ω$ , while the value for S₁/S₂ is $4 Δ ω$ . Based on the specified frequency distribution, four interferential outputs, i.e. I₁-I₄, are generated. The I₁-I₃ are located between the S₁ and S₂ with an identical spacing of $Δ ω$ , while the I₄ is located at the frequency $2 ω_{P_{1}} - 2 ω_{S_{1}}$ (or $2 ω_{P_{2}} - 2 ω_{S_{2}}$ ).

Fig. 1 The operation principle of the proposed NI with wavelength multicasting functionality. P: pump; S: signal; I: idler.

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The detailed FWM processes for the generation of the I₁-I₄ are listed in Table 1. The generation of the I₁ is described as an example to clarify the phase sensitivity. It is correlated to two FWM processes (i.e. P₃ + S₁→P₂ + I₁ and P₁ + S₂→P₃ + I₁), as marked by the orange curves in Fig. 1. In order to ensure identical CEs of the two FWM processes, one can optimize the powers of the involved pumps. The complex amplitudes of the two frequency-degenerate idlers generated from each FWM process can be represented as $2 A_{P_{1}}^{*} A_{P_{3}} A_{S}_{_{1}}$ and $2 A_{P_{1}}^{*} A_{P_{3}} A_{S}_{_{2}}$ , while the phases of the two idlers are $φ_{S_{1}} + φ_{P_{3}} - φ_{P_{2}}$ and $φ_{S_{1}} + φ_{P_{3}} - φ_{P_{2}}$ , respectively. As a result, their relative phase difference $θ_{r e l}^{}$ is $φ_{S_{1}} - φ_{S_{2}} +(2 φ_{P_{3}} - φ_{P_{1}} - φ_{P_{2}})$ , which consists of the signal phase item $φ_{S_{1}} - φ_{S_{2}}$ (denoted as $θ_{r e l}^{s}$ ) and the pump phase item $2 φ_{S_{1}} + φ_{P_{3}} - φ_{P_{2}}$ (denoted as $θ_{r e l}^{p}$ ). The latter one is constant, provided the pumps are coherent. Similarly, when the signals are coherent, the former one is also constant unless additional phase shift is introduced. Given the two idlers are coherent, they will interfere with the relative phase difference $θ_{r e l}^{}$ and the PS output I₁ is generated. The processes of other PS idlers are similar to the I₁. It is worth mentioning that the coherence between pump and signal is unnecessary for the construction of the NI.

Table 1. Details for the Generation of the PS Idlers

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The scheme is then experimentally examined, with the setup depicted in Fig. 2(a). The three pumps and dual signals are separately obtained from two frequency combs with 20 and 40 GHz line spacings, assisted by appropriate filtering. The details are presented as follows: a continuous wave (CW: Laser1) at 1548.576 nm is phase modulated with a 20 GHz RF signal, generating the frequency comb (FC1) with 20 GHz line spacing. The FC1 is filtered by a tunable bandpass filter (TBPF1) to suppress those unneeded sidebands, obtaining the 0, ± 1, ± 2 order ones. Another CW (Laser2) at 1551.570 nm is modulated with a Mach-Zehnder modulator (MZM) driven by a frequency-doubled RF signal at 40 GHz, generating the frequency comb (FC2) with 40 GHz line spacing. The FC2 is then filtered with a bandstop filter (BSF) to suppress the carrier.

Fig. 2 (a) Experimental setup. Red line: electrical circuit; Black: optical circuit. (b) The cross-section diagram of the silicon ridge waveguide.

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Through optical amplification and out-of-band noise filtering for the reserved sidebands of the FC1 and FC2, they are combined and then input into an optical processor. The optical processor is used to select the ± 1 and + 2 order sidebands from the FC1 as the pumps, and the ± 1 order sidebands from the FC2 as the signals. It is also used to adjust the relative power and phase between the waves, as well as further reject the unneeded sidebands. The output of the processor is amplified by a high-power erbium doped fiber amplifier (EDFA) and launched into a 4 cm silicon ridge waveguide through a grating coupler.

The silicon device is fabricated on a 220 nm silicon-on-insulator (SOI) wafer with 2 μm buried oxide (SiO₂) at the Institute of Microelectronics in Singapore. The waveguide has a width and slab height of 550 and 150 nm, with the cross-section diagram shown in Fig. 2(b). The measured propagation and coupling losses are 2.5 dB/cm and 5 dB per grating coupler for the fundamental transverse electric (TE₀) mode. The dispersion and nonlinear coefficient are calculated to be −1674 ps/nm/km and 110 /W/m at 1550 nm with the commercial software COMSOL. The output of the waveguide is split with a 50:50 coupler to monitor the spectrum and power, using an optical signal analyzer (OSA) and a power meter.

To characterize the proposed scheme, single signal and the three pumps are launched into the silicon waveguide to examine the FWM CE. The S₁ and three pumps are first launched, and the S₂ is switched off by the optical processor. The input powers of the pumps and S₁ are 20.3, 19.8, 19.1 and 13.3 dBm, as listed in Table 2. The measured FWM spectrum is shown in Fig. 3(a), where the numbers mark the PS idlers I₁-I₄. The output powers (measured by the OSA) of the I₁-I₄ are −35.6, −34.8, −36 and −40.5 dBm, corresponding to the CEs of −20.8, −20, −21.2 and −25.7 dB, respectively. The CE is defined by

C E = 10 \log \frac{P_{i, o u t}}{P_{s, o u t}}

where

P_{i, o u t}

and

P_{s, o u t}

denote the output powers of idler and signal, respectively.

Table 2. Input and Output Powers for the FWM Processes with Single-signal Input

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Fig. 3 The measured FWM output spectra with (a) the three pumps and S₁, (b) the three pumps and S₂ input.

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Subsequently, the S₂ is switched on and the S₁ is switched off. The input power of the S₂ is 8.2 dBm. The corresponding FWM output spectrum is shown in Fig. 3(b). The output powers of the I₁-I₄ are listed in Table 2. The CEs are measured to be −20.2, −19.1, −20.9 and −25.3 dB, respectively. For each idler, the CE difference between the two FWM processes is less than 1 dB. We suppose that the slight difference may be caused by the waveguide dispersion and fiber-to-waveguide coupling. Hence, it is concluded that the two FWM processes for the generation of each PS idler have approximatively identical CEs.

The S₁ and S₂ are then both switched on to examine the interferometer output. The relative phase difference between the S₁ and S₂, i.e. $θ_{r e l}^{s}$ , is varied by the optical processor. The output powers of the I₁-I₄ are measured as functions of the $θ_{r e l}^{s}$ , as shown in Fig. 4. The measured maximum and minimum output powers correspond to the constructive and destructive interferences between the S₁ and S₂. One can find that when the I₂ and I₄ are constructive interference, the I₁ and I₃ are destructive interference, and vice versa. The difference is caused by the $θ_{r e l}^{p}$ correlated to the generation of the I₁-I₄. As shown in Table 1, besides the $θ_{r e l}^{s}$ , the $θ_{r e l}^{p}$ also affects the interferential output. The pumps are obtained from a frequency comb, resulting in different $θ_{r e l}^{p}$ for the I₁/I₃ and I₂/I₄. Thus, the I₁/I₃ is out-of-phase compared to the I₂/I₄.

Fig. 4 The measured output powers of the I₁-I₄ as functions of the relative phase difference between the S₁ and S₂, $θ_{r e l}^{s}$ .

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On the other hand, a modest extinction ratio (ER, which is defined as the difference between the constructive and destructive interferences) of ~12 dB can be observed in Fig. 4. The ER of the interferometer can be improved, provided that high CEs of the involved FWM processes, small CE difference between the two FWM processes (which generate the frequency-degenerate idlers) and small power difference between the frequency-degenerate idlers are ensured. In order to avoid serious nonlinear absorption induced by the two photon absorption (TPA) and free-carrier effects, a moderate pump power is utilized in the experiment. The FWM efficiency can be increased by launching a larger pump power into a waveguide with p-i-n junction which can effectively remove the free carriers [16]. A waveguide with flatter dispersion can be utilized to decrease the wavelength dependence of the FWM, as well as the CE difference between the two FWM processes. Theoretically, the ER of the interferometer would be much larger when the frequency-degenerate idlers are equal in power, benefitting from the thorough destructive interference. For this case, the output power of the destructive interference is very low. In the experiment, the low power is easily affected by the noise originated from the EDFAs. Hence, the measured result would deviate from the actual value. In order to avoid the influence of the noise and obtain the actual interferential results, a power difference of 5 dB for the dual signals is intentionally utilized. Using experimental setup with low noise, the power difference between the dual signals, and thus the power difference between the frequency-degenerate idlers, can be decreased.

Figure 5 shows the FWM spectra for the $θ_{r e l}^{s}$ of 0.125π (blue dash line) and 1.125π (red solid line), one can observe that the I₁-I₄ all exhibit obvious phase sensitivity.

Fig. 5 The measured FWM spectra for the relative phase differences (between the S₁ and S₂) of 0.125π and 1.125π.

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The measured constructive and destructive output powers of the I₁-I₄ as well as the theoretically calculated values are shown in Table 3. The theoretical values are the constructive and destructive outputs calculated from the powers (shown in Table 2) of the frequency-degenerate idlers. The maximum diversity between the theoretical and measured values is 1.1 dB, indicating that the experimental results are well consistent with the theoretical prediction.

Table 3. Comparison Between the Theoretical and Measured Interferential Outputs.

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In the measurements, the total input power before the grating coupler is ~26 dBm. In order to evaluate the nonlinear losses induced by the TPA and free-carrier effects for the utilized power level, the insertion loss variation of the waveguide is measured by launching a CW with different powers into the waveguide. The results show that the insertion loss for the input power (before the grating coupler) of 26 dBm is ~3.4 dB larger than that for a low input power of 11.7 dBm. The additional loss of 3.4 dB is caused by the nonlinear losses. A larger input power can be used to increase the CE of FWM when the nonlinear absorption is weak. However, when the launched power is overlarge, the nonlinear losses become obvious, and thus the pump depletion as well as the FWM CE saturation get serious. As a result, the ER of the interferometer is limited. The maximum CE of the utilized waveguide, which is measured by launching two CWs at 1550 and 1552 nm into the waveguide serving as pump and signal, is about −12 dB. The input pump power before the grating coupler is 29.5 dBm. This value of CE should be sufficiently high for the applications of the proposed scheme, when the signals are data-modulated. Unfortunately, further experimental demonstration with modulated signals cannot be performed, due to lack of phase-locked loop, which is very necessary to compensate for the slow relative phase drifts between two separately modulated signals.

There are several potential applications based on the proposed scheme, e.g. a basic building block of photonic correlator and optical logic XOR gate. Logic XOR gate is an essential network element, which is widely used for data encoding, decision making, complex computing and so on [23]. Combining with the multicasting functionality, the data processed by the XOR gate can be parallelly transmitted to different destinations where different wavelengths are utilized. Utilizing the destructive interference of the proposed NI scheme, the multicasting logic XOR operation can be realized for both intensity and phase modulated signals. The truth tables of the logic XOR function for OOK and BPSK formats are shown in Table 4. It can be observed that the output powers of the I₁-I₄ directly carry the XOR results between the two signals.

Table 4. Truth Tables of the Logic XOR Gate for (a) OOK and (b) BPSK Formats

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The multicasting logic XOR gate based on the proposed scheme is then investigated by calculating the nonlinear Schrödinger equation (NLSE) as follows [24–27]:

\frac{\partial E}{\partial z} = - \frac{α}{2} E - i \frac{β_{2}}{2} \frac{\partial^{2} E}{\partial t^{2}} + i γ {| E |}^{2} E - \frac{β_{T P A}}{2 A_{e f f}} {| E |}^{2} E - N_{c} (\frac{σ}{2} + i k_{c} k_{0}) E

In the equation, the Kerr effects, TPA effect, free-carrier absorption (FCA) effect and free-carrier dispersion (FCD) effect are considered. E is the electric field envelope,

α

= 2.5 dB/cm is the propagation loss,

β_{2}

= 2.13 ps²/m is the second-order dispersion parameter at 1550 nm,

γ

= 110 /W/m is the nonlinear coefficient,

A_{e f f}

= 0.222 μm² is the effective mode area,

k_{0}

is the wave vector corresponding to 1550 nm. The parameter values are consistent with those of the fabricated waveguide, which are achieved from the waveguide transmission test and character calculation by the COMSOL. In addition,

β_{T P A}

= 5 × 10⁻¹² m/W is the TPA coefficient,

σ

= 1.45 × 10⁻²¹ m² is the FCA coefficient,

k_{c}

= 1.35 × 10⁻²⁷ m³ is the FCD coefficient. The free carriers N_c generated by the TPA is given by [26, 27]

\frac{\partial N_{c} (t)}{\partial z} = \frac{β_{T P A}}{2 h ν_{0} A_{e f f}^{2}} {| E |}^{4} - \frac{N_{c}}{τ}

where h is the Plank’s constant,

ν_{0}

is the frequency corresponding to 1550 nm,

τ

= 1 ns is the carrier lifetime.

The equation is numerically solved by the split step Fourier method. The dual signals are modulated with distinct information at 5 Gbit/s (determined by $Δ ω$ ) in the RZ-OOK format with a duty cycle of 33% (PRBS 2⁹-1). An additional phase shift of π is added to one of the signals in order to realize the nonlinear destructive interference. The power of each signal is 10 dBm. Three CWs with identical power of 15 dBm and phase of 0 are employed as the pumps. $Δ ω$ is set to be 40 GHz in order to avoid the channel crosstalk. The signal wavelengths are 1553.36 and 1554.64 nm, and the pump wavelengths are 1549.68, 1550.32 and 1550.64 nm.

Figure 6 shows the partial waveforms of the input signals and output PS idlers I₁-I₄. The results show that the XOR operation between the two OOK signals is obtained at four different wavelengths. Figure 7 shows the eye diagrams of the I₁-I₄, and slight eyelid fluctuation can be observed. The fluctuation might be caused by the slightly different CEs of the two FWM processes correlated to the generation of the PS idler. The calculated ERs for the I₁-I₄ are 40.6, 24.9, 19 and 22 dB. The much higher ER of the I₁ is caused by the smaller CE difference between the two FWM processes, whose efficiencies are wavelength correlated. The simulated ERs are much larger than the experimental results, mainly benefitting from the identical signal power employed for the simulation. Figures 8 and 9 show the calculated results for the BPSK case, indicating successful demonstration of the multicasting logic XOR gate. The calculated ERs for the I₁-I₄ are 41.1, 28, 21.9 and 25 dB.

Fig. 6 The simulated multicasting logic XOR gate for the OOK signals.

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Fig. 7 The simulated eye diagrams of the (a) I₁, (b) I₂, (c) I₃, and (d) I₄.

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Fig. 8 The simulated multicasting logic XOR gate for the BPSK signals.

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Fig. 9 The simulated eye diagrams of the (a) I₁, (b) I₂, (c) I₃, and (d) I₄.

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In summary, we propose and experimentally demonstrate a nonlinear interferometer with wavelength multicasting functionality based on a silicon waveguide. With a three-pump and dual-signal four wave mixing configuration, the interferential outputs between the signals are obtained at four different wavelengths. In addition, the multicasting logic exclusive-OR gate based on the proposed scheme for both intensity and phase modulated signals is theoretically demonstrated, as a potential application. The proposed scheme is meaningful and promising for the on-chip construction of nonlinear interferometer and corresponding applications.

This work was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 61475050 and 61275072), the New Century Excellent Talent Project in Ministry of Education of China (NCET-13-0240), the Natural Science Foundation of Hubei Province (2014CFA004) and the Fundamental Research Funds for the Central Universities (HUST2015TS079).

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PS idler	FWM processes	Idler complex amplitude	Idler phase	Relative phase difference
I₁	P₃ + S₁→P₂ + I₁	$2 A_{P_{2}}^{*} A_{P_{3}} A_{S}_{_{1}}$	$φ_{I_{1}} = φ_{S_{1}} + φ_{P_{3}} - φ_{P_{2}}$	$θ_{r e l} = φ_{S_{1}} - φ_{S_{2}} +(2 φ_{P_{3}} - φ_{P_{1}} - φ_{P_{2}})$
I₁	P₁ + S₂→P₃ + I₁	$2 A_{P}_{1} A_{P_{3}}^{*} A_{S_{2}}$	$φ_{I_{1}} = φ_{S_{2}} + φ_{P_{1}} - φ_{P_{3}}$
I₂	P₂ + S₁→P₁ + I₂	$2 A_{P_{1}}^{*} A_{P}_{_{2}} A_{S_{1}}$	$φ_{I_{2}} = φ_{S_{1}} + φ_{P_{2}} - φ_{P_{1}}$	$θ_{r e l} = φ_{S_{1}} - φ_{S_{2}} +(2 φ_{P_{2}} - 2 φ_{P_{1}})$
I₂	P₁ + S₂→P₂ + I₂	$2 A_{P_{1}} A_{P_{2}}^{*} A_{S_{2}}$	$φ_{I_{2}} = φ_{S_{2}} + φ_{P_{1}} - φ_{P_{2}}$
I₃	P₃ + S₁→P₁ + I₃	$2 A_{P_{1}}^{*} A_{P_{3}} A_{S}_{_{1}}$	$φ_{I_{3}} = φ_{S_{1}} + φ_{P_{3}} - φ_{P_{1}}$	$θ_{r e l} = φ_{S_{1}} - φ_{S_{2}} +(2 φ_{P_{3}} - φ_{P_{1}} - φ_{P_{2}})$
I₃	P₂ + S₂→P₃ + I₃	$2 A_{P_{2}} A_{P_{3}}^{*} A_{S_{2}}$	$φ_{I_{3}} = φ_{S_{2}} + φ_{P_{2}} - φ_{P_{3}}$
I₄	2P₁→S₁ + I₄	$A_{P_{1}}^{2} A_{S_{1}}^{*}$	$φ_{I_{4}} = 2 φ_{P_{1}} - φ_{S_{1}}$	$θ_{r e l} = φ_{S_{2}} - φ_{S_{1}} +(2 φ_{P_{1}} - 2 φ_{P_{2}})$
I₄	2P₂→S₂ + I₄	$A_{P_{2}}^{2} A_{S_{2}}^{*}$	$φ_{I_{4}} = 2 φ_{P_{2}} - φ_{S_{2}}$

		I₁	I₂	I₃	I₄
Max. (dBm)	Theory	−31.7	−30.8	−32.2	−36.7
Max. (dBm)	Experiment	−32.3	−31.2	−32.7	−37
Min. (dBm)	Theory	−42.8	−42.4	−42.8	−47.4
Min. (dBm)	Experiment	−43	−41.4	−41.7	−46.7

PS idler	FWM processes	Idler complex amplitude	Idler phase	Relative phase difference
I₁	P₃ + S₁→P₂ + I₁	$2 A_{P_{2}}^{*} A_{P_{3}} A_{S}_{_{1}}$	$φ_{I_{1}} = φ_{S_{1}} + φ_{P_{3}} - φ_{P_{2}}$	$θ_{r e l} = φ_{S_{1}} - φ_{S_{2}} +(2 φ_{P_{3}} - φ_{P_{1}} - φ_{P_{2}})$
I₁	P₁ + S₂→P₃ + I₁	$2 A_{P}_{1} A_{P_{3}}^{*} A_{S_{2}}$	$φ_{I_{1}} = φ_{S_{2}} + φ_{P_{1}} - φ_{P_{3}}$
I₂	P₂ + S₁→P₁ + I₂	$2 A_{P_{1}}^{*} A_{P}_{_{2}} A_{S_{1}}$	$φ_{I_{2}} = φ_{S_{1}} + φ_{P_{2}} - φ_{P_{1}}$	$θ_{r e l} = φ_{S_{1}} - φ_{S_{2}} +(2 φ_{P_{2}} - 2 φ_{P_{1}})$
I₂	P₁ + S₂→P₂ + I₂	$2 A_{P_{1}} A_{P_{2}}^{*} A_{S_{2}}$	$φ_{I_{2}} = φ_{S_{2}} + φ_{P_{1}} - φ_{P_{2}}$
I₃	P₃ + S₁→P₁ + I₃	$2 A_{P_{1}}^{*} A_{P_{3}} A_{S}_{_{1}}$	$φ_{I_{3}} = φ_{S_{1}} + φ_{P_{3}} - φ_{P_{1}}$	$θ_{r e l} = φ_{S_{1}} - φ_{S_{2}} +(2 φ_{P_{3}} - φ_{P_{1}} - φ_{P_{2}})$
I₃	P₂ + S₂→P₃ + I₃	$2 A_{P_{2}} A_{P_{3}}^{*} A_{S_{2}}$	$φ_{I_{3}} = φ_{S_{2}} + φ_{P_{2}} - φ_{P_{3}}$
I₄	2P₁→S₁ + I₄	$A_{P_{1}}^{2} A_{S_{1}}^{*}$	$φ_{I_{4}} = 2 φ_{P_{1}} - φ_{S_{1}}$	$θ_{r e l} = φ_{S_{2}} - φ_{S_{1}} +(2 φ_{P_{1}} - 2 φ_{P_{2}})$
I₄	2P₂→S₂ + I₄	$A_{P_{2}}^{2} A_{S_{2}}^{*}$	$φ_{I_{4}} = 2 φ_{P_{2}} - φ_{S_{2}}$

		I₁	I₂	I₃	I₄
Max. (dBm)	Theory	−31.7	−30.8	−32.2	−36.7
Max. (dBm)	Experiment	−32.3	−31.2	−32.7	−37
Min. (dBm)	Theory	−42.8	−42.4	−42.8	−47.4
Min. (dBm)	Experiment	−43	−41.4	−41.7	−46.7

PS idlers (I₁-I₄)

PS idlers (I₁-I₄)

P₁	P₂	P₃	S₁	S₂	I₁	I₂	I₃	I₄	I₁	I₂	I₃	I₄
P_in (dBm)					P_out (dBm)				CE (dB)
20.3	19.8	19.1	13.3	–	−35.6	−34.8	−36.0	−40.5	−20.8	−20	−21.2	−25.7
20.3	19.8	19.1	–	8.2	−40.6	−39.5	−41.3	−45.7	−20.2	−19.1	−20.9	−25.3