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Multifunctional switching unit for add/drop, wavelength conversion, format conversion, and WDM multicast based on bidirectional LCoS and SOA-loop architecture

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Abstract

We propose a multifunctional optical switching unit based on the bidirectional liquid crystal on silicon (LCoS) and semiconductor optical amplifier (SOA) architecture. Add/drop, wavelength conversion, format conversion, and WDM multicast are experimentally demonstrated. Due to the bidirectional characteristic, the LCoS device cannot only multiplex the input signals, but also de-multiplex the converted signals. Dual-channel wavelength conversion and format conversion from 2 × 25Gbps differential quadrature phase-shift-keying (DQPSK) to 2 × 12.5Gbps differential phase-shift-keying (DPSK) based on four-wave mixing (FWM) in SOA is obtained with only one pump. One-to-six WDM multicast of 25Gbps DQPSK signals with two pumps is also achieved. All of the multicast channels are with a power penalty less than 1.1 dB at FEC threshold of 3.8 × 10−3.

© 2014 Optical Society of America

1. Introduction

With the exponential growth of network capacity and traffic rates, data traffic grooming becomes increasingly important to improve the efficiency and flexibility of networks. At network switching nodes, a grooming switch will likely be required to simultaneously perform multiple optical signal processing functions [1, 2]. In flexible optical network, it might be advantageous for a reconfigurable network node to achieve: (a) wavelength add/drop [3]; (b) all-optical wavelength conversion [4]; (c) format conversion [5]; (d) all-optical WDM multicast [6]. However, even though all these functions have been demonstrated separately, there is little reported research on aggregating all these functions together into a multifunctional unit at a network switching node.

Previous research work performed the separate functions of wavelength add/drop, wavelength conversion, format conversion, and WDM multicast. For example, a reconfigurable optical add/drop multiplexer was demonstrated in [3]. Wavelength converters based on four-wave mixing (FWM) were demonstrated in highly nonlinear fiber (HNLF) [7], periodically poled lithium niobate (PPLN) [8], or silicon waveguide [9]. The single-channel format conversions from DQPSK to DPSK through FWM-based optical phase erasure in HNLF were proposed in [1013]. However, to the best of our knowledge, the simultaneous multi-channel format conversion from DQPSK to DPSK in other optical materials and devices has not been demonstrated. The WDM multicast was performed in HNLF [1416], PPLN [17, 18], or silicon waveguide [6, 19]. In addition to the mentioned techniques, a semiconductor optical amplifier (SOA) is another choice for optical signal processing for its evident advantages, e.g. ease of integration, low power consumption, and high conversion efficiency (CE). Recently, 16 QAM and 64 QAM wavelength conversions have been demonstrated in SOA [20, 21]. However, the SOA-based multicast is still limited to OOK or DPSK [2224].

Liquid crystal on silicon (LCoS) opens new possibilities and applications for the liquid crystal in different fields where faster response is required, especially in optics area [25]. A mode converter was employed on a LCoS-based spatial light modulator (SLM) in [26]; the capability of a LCoS phase modulator for wave-aberration manipulation was investigated and the response speed could reach up to 60 GHz [27]; an integrated wavelength cross-connect was presented by utilizing LCoS-based wavelength blocker array in [28]; flexible and grid-less wavelength selective switch (WSS) based on the intrinsic grid-free capabilities of LCoS was presented in [29].

In this paper, with the help of bidirectional LCoS technique, we propose and experimentally demonstrate a reconfigurable network grooming switch unit that can simultaneously implements add/drop, wavelength conversion, format conversion, and WDM multicast based on the designed SOA-loop architecture. We believe this is the first demonstration of dual-channel format conversion from 2 × 25Gbps DQPSK signals to 2 × 12.5Gbps DPSK based on FWM in SOA using only one pump. One-to-six WDM multicast of 25Gbps QPSK signals is also achieved by applying two pumps. All the newly generated channels exhibit high optical signal-to-noise ratio (OSNR), reach a maximum power penalty of 1.1 dB and a relatively high multicast efficiency.

2. Operation principle and concept

As one of the key optical signal processing technologies, optical format conversion has been extensively studied, but mainly focused on conversion between conventional on-off keying (OOK) signals like the conversion between NRZ to RZ [3032]. Recently, the single-channel format conversion from DQPSK to DPSK based on FWM was comprehensively investigated in HNLF for various application scenarios [1013]. However, compared with HNLF, a SOA is more suitable for network switching nodes because of its compact size and photonics integration potential. In some situations, format conversion of multi-channel signals is important for WDM system. Here, we achieve simultaneous wavelength conversion and format conversion for dual-channel signals through FWM in SOA with only one pump.

As shown in Fig. 1(a), pump at ω2, DQPSK signal1 (S1) at ω1, and signal2 (S2) at ω3 are combined together, then fed into a SOA. After the FWM in SOA, the converted signals at ω223, ω112, ω221, ω332 are obtained with the electrical field, E223, E112, E221, E332, given by:

E223=k223A22A3exp[j(2ω2ω3)t+(2θ2θ3)]
E112=k112A12A2exp[j(2ω1ω2)t+(2θ1θ2)]
E221=k221A22A1exp[j(2ω2ω1)t+(2θ2θ1)]
E332=k332A32A2exp[j(2ω3ω2)t+(2θ3θ2)]
where ωi, Ai, and θi (i[1,2,3]) are the angular frequency and the corresponding input field amplitude and phase of amplified pump and two input DQPSK signals, respectively, and k is a proportional constant related to FWM efficiency. As shown in Fig. 1(a), with the DQPSK encoding of input signals at ω1and ω3, the phase information could be transparently transferred to the generated idlers at ω221 and ω223, acting as wavelength conversions of dual-channel DQPSK signals. However, because the phase modulation depths for the phase information carried at ω1 and ω3are doubled in the resultant phase pattern at ω112and ω332, i.e., θ112=2θ1θ2 and θ332=2θ3θ2, two converted DPSK signals are obtained.

 figure: Fig. 1

Fig. 1 Operation principle: (a) Simultaneous wavelength and format conversion for dual-channel DQPSK signals; (b) input DQPSK using serially-cascaded modulator; (c) input DQPSK using parallel IQ modulator; (d) one-to-six WDM multicast.

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The schemes of DQPSK-to-DPSK format conversion can be roughly divided into two categories according to the generating methods of DQPSK: (a) serially-cascaded modulator [12]; (b) parallel in-phase/quadrature (IQ) modulator [33]. Different modulators have different generation methods in which the different data is modulated to the different optical phase. Therefore, different transmitter configurations or structures results in different logical phase mapping. The DQPSK signal generated by a serially-cascaded transmitter is shown in Fig. 1(b). The binary data carried in the converted DPSK is logically equal to the Q-component of the input DQPSK and the I-component is erased, while the converted DQPSK can carry the original data to another wavelength. The logic and phase mapping between input and output signals are summarized in Table 1. All of the previous works were implemented in this case [1013]. In this paper we adopt another method, parallel IQ modulator, to achieve DQPSK to DPSK based on FWM for the first time shown in Fig. 1(c). Compared with the serially-cascaded modulator, the DQPSK generated by the parallel IQ modulator conforms to Gray code, which contributes to lower bit-error rate (BER) and, in addition, the IQ modulator is more suitable for coherent system to achieve further ultra-high speed and long-haul transmission. Using the generated DQPSK from the IQ modulator as input signal, logically, the converted binary DPSK signal is the XOR (or XNOR) logical operation result between I-and Q-components of the input DQPSK, as listed in Table 1. The two format conversion schemes above-mentioned can be used in different applications. For example, the serially-cascaded method could be applied to erase the redundant overhead tributary of the signal implementing Trellis coded modulation, as suggested in [34]; moreover, it could be utilized to “update” the information carried in the incoming DQPSK in an optical manner since the converted DPSK signal could be modulated by a new OOK signals through cross-phase modulation, and then it could be converted to another DQPSK [35]; the scheme could also be exploited to “drop” one tributary and optically “add” another by phase modulation [10]. While the parallel IQ modulator method has the potential to be applied in the optical label switching (OLS) networks for optical label processing or recognition [36]; furthermore, it could be devoted to all-optical logic gates such as XOR [37]; and with the cooperation of a pre-coder, it could also be applied in optical signal encryption [38, 39].

Tables Icon

Table 1. Logic and phase mapping between input and output signals

Based on FWM in SOA, the WDM multicast can also be achieved with two pumps, as shown in Fig. 1(d). The frequencies of the two pumps (P1 and P2) and the original DQPSK signal (S) are represented by ω1,ω2, and ω3, respectively. After a FWM process, five new frequencies preserving the phase information of the original signal will be generated [6]. As the above analysis, to preserve the phase information of the original DQPSK signal, it requires that only one wave carries the original DQPSK data (ω3) among the three wave (ω1, ω2, ω3) participating in the FWM process. The three non-degenerate FWM converted idlers (ω123, ω312, ω321) and two degenerate FWM idlers (ω113,ω223) satisfy the phase-preserving requirement and thus can copy all the information of the original DQPSK signal. Combined with the original input signal, one-to-six optical WDM multicast is achieved.

In order to simultaneously perform the aforementioned functions at a network switching node, the bidirectional LCoS technology is introduced to compose a multifunctional switch unit, as shown in Fig. 2.The key part of the setup is a two-dimensional (2D) array of LCoS containing a large number of individual pixels [25]. In the horizontal axis, the WDM signals are sent from an input/output fiber array (port A) to the imaging optics and a diffraction grating, which can angularly disperses each wavelength channel to a different portion of the LCoS along the horizontal “wavelength” axis. Vertically, the light diverges to overlap a large number of pixels. Independent attenuation control of optical power and spatial switch of individual wavelength change to the desired fiber array ports (from port B to port H) are achieved by controlling the phase front of the 2D array of LCoS pixels along the vertical “displacement” axis. Therefore, the add/drop function can be achieved by the LCoS-based devices [29].

 figure: Fig. 2

Fig. 2 Multi-function optical switch unit based on bidirectional LCoS and SOA-loop architecture

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As shown in Fig. 2, combined with the bidirectional LCoS technology, the multifunctional unit is constituted by the designed SOA-loop architecture at port A. The input S1 and S2 are sent into LCoS through port C and port D. In the case of wavelength conversion and format conversion, only one pump is required and selected by port B. Then the three input wavelengths are coupled together via the reflection from LCoS and switched to the same fiber at port A. The output from port A is sent into the SOA. After FWM in SOA, the desired converted signals are amplified by an erbium-doped fiber amplifier (EDFA) and sent back into port A through a circulator. Due to the bidirectional transmission characteristic, the generated signals could be delivered back to the LCoS. After the spectral dispersion and wavelength switch by LCoS, the signals at different wavelengths can be de-multiplexed into the desired fiber port. The programmable LCoS can be flexibly controlled by the software. We can see clearly the process of how the six signals including the original S1, S2, the converted DQPSK1, DPSK1, DPSK2, and DQPSK2 are routed from port C to port H in Fig. 2.

In the case of WDM multicast, two pumps coupled with the original DQPSK signal are required for correct multicast operation. The following operations are similar to the format conversion scheme: three combined signals are sent into SOA; after FWM, the WDM multicast signals are delivered back to LCoS; then all the multicast signals are de-multiplexed and sent into the corresponding fiber ports. Therefore, considering the proposed setup in Fig. 2 as an “LCoS& SOA-based switching unit”, a reconfigurable multifunctional grooming switch (add/drop, wavelength conversion, format conversion and WDM multicast) is enabled by the bidirectional programmable LCoS and FWM in a SOA-loop architecture.

3. Experiment and results

The experimental setup for the LCoS & SOA-based switching unit is shown in Fig. 3. We adopt a bandwidth variable wavelength selective switch (BV-WSS Finisar DWP9F) as the LCoS device, which has 10 ports (9 switching ports and 1 common port), a maximum insertion loss of 6.5 dBm, and the variable channel bandwidth of 12.5 to 500 GHz with 12.5 GHz resolution according to ITU-T G.694.1. Firstly, we demonstrate the wavelength conversion and format conversion schemes in which only single pump1 is required. The light sources of pump1, S1, and S2 are from three tunable external cavity lasers (ECL-EXFO-FLS2800) with linewidths of less than 100 kHz. The coherent optical IQ modulator (Tektronix OM5110) capable of generating NRZ-DQPSK signals is driven by an arbitrary waveform generator (Tektronix AWG7001A) to provide a repeated PRBS with the length of 215-1 at 25 Gbps. In this proof-of-concept experiment, two CW lasers (ECL1, ECL2) are coupled into a single IQ modulator to generate the dual-channel DQPSK signals which are separated out to different fibers using a 3 dB coupler and two optical band-pass filters (BPF), respectively. In order to emulate two different DQPSK signals, the two filtered signals are separately passed through an optical delay line (ODL) and a variable optical attenuator (VOA). After amplification, the two DQPSK signals are combined with pump1 using three of the LCoS switching ports. These three coupled input wavelengths output from the common port are injected into the SOA (CIP-NL-OEC-1550) with a small signal gain of 34 dB and a saturation output power of 6 dBm. In order to achieve the higher FWM efficiency, we adjust the signals and pumps into the co-polarized state [24]. The polarization sensitivity is mainly due to the fact that FWM efficiency is dependent on the relative polarization between the injected signals. Then the desired wavelength conversion and format conversion are simultaneously achieved via FWM in the SOA. With the assist of a circulator, the converted signals are delivered back into the LCoS. Then the signals are de-multiplexed and filtered to the corresponding output ports. The selected signal is detected by a coherent receiver (Tektronix OM4106D). The receiver sensitivity is measured before EDFA at the coherent receiver. Here, the EDFA as a pre-amplifier is set to the constant power mode to keep the input power of coherent receiver constant. Finally, constellation diagram, eye-diagram and bit-error rate (BER) measurements are obtained by a 33-GHz bandwidth oscilloscope (Tektronix MOS73304DX) and the optical spectra is observed by an optical spectra analyzer (YOKOGAWA AQ6370B).

 figure: Fig. 3

Fig. 3 Experiment setup: PC: polarization controller; OC: optical coupler; AWG: arbitrary waveform generator; EDFA: erbium-doped fiber amplifier; VOA: variable optical attenuator; BPF: band-pass filter; ODL: optical delay line; A/D: analog to digital converter; LO: local oscillation. Insets: the constellations and eye-diagrams of the original input signals before SOA for wavelength/format conversion and WDM multicast.

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In order to avoid the crosstalk, we analyze the wavelengths of all possible new idlers generated through FWM and arrange the wavelength and interval for pump and signals properly. The wavelengths and powers of input signals (S1, S2) before LCoS are set to 1548.52 nm (193.6 THz) with 3.5 dBm and1551.72 nm (193.2 THz) with 4.2 dBm, while the pump is set to 1549.32 nm (193.5 THz) with 6.5 dBm. The insets of Fig. 3 display the constellations and eye-diagrams of S1 and S2 before the LCoS. After the LCoS, these two signals and one pump are combined together and injected into the SOA which has a bias current of 300 mA. The spectra measured at the input of SOA is shown in Fig. 4(a). After the FWM in SOA, two wavelength conversion parts (DQPSK1 and DQPSK2) and two format conversions (DPSK1 and DPSK2) are simultaneously carried out as shown in Fig. 4(b). S1 and S2 separately interact with the pump to implement the degenerate FWM. In SOA, the FWM performs the complicated energy transfer process between the electric and optical energy, which results in the power ratio change between the pump and the signals at the input and output of SOA.

 figure: Fig. 4

Fig. 4 Optical spectra of wavelength and format conversion at the (a) input; (b) output of SOA (resolution: 0.02 nm).

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Owing to the tunable bandwidth of the LCoS, the generated idler signals can be de-multiplexed and filtered out by the LCoS, as shown in Fig. 5. Compared with arrayed waveguide grating (AWG), the BV-WSS is more flexible and suitable for the optical switching nodes. However, the filtered signals have to suffer from larger power loss. Therefore, the pre-amplifier before the coherent receiver is necessary. The BER performance of both original signals and converted signals as a function of the received power is shown in Fig. 6. All of the signals exhibit better BER performance than FEC threshold of 3.8 × 10−3.

 figure: Fig. 5

Fig. 5 Optical spectra of converted signals de-multiplexed by LCoS: (a) converted DQPSK2 at 1546.92nm; (b) converted DPSK1 at 1547.72 nm; (c) converted DQPSK1 at 1550.12 nm; (d) converted DPSK2 at 1554.12 nm (resolution: 0.02 nm)

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 figure: Fig. 6

Fig. 6 Measured BER performance and corresponding constellations and eye-diagrams (measured at −35 dBm) of input signals (S1, S2) and converted signals (DPSK1, DPSK2, DQPSK1, DQPSK2).

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The characteristics of the converted signals including wavelength, frequency, conversion efficiency (CE), optical signal-to-noise ratio (OSNR), and power penalty at FEC threshold are summarized in Table2. The OSNR is measured at the point after SOA, but in front of LCoS. Compared with the original signal S1, around 0.5 dB sensitivity improvement at FEC threshold is observed for converted DPSK1, which is mainly attributed to the reduced bit rate and enlarged symbol distance after the format conversion. However, in order to avoid crosstalk, the wavelength range between S2 and the pump is stretched to 2.4 nm, which results in the lower conversion efficiency. Therefore, about a 0.4 dB power penalty is paid for the converted DPSK2 in comparison with S2. The performance could be enhanced by optimizing the launched power and increasing the bias current to further improve the conversion efficiency. The insets of Fig. 6 display the concentrated constellations and clear eye-diagrams for both I and Q channels of the DQPSK and the DPSK, which verifies the feasibility of the scheme. Here, one point need to be emphasized: according to our analysis, the converted DPSK should have the phase of (π/2, 3π/2); however, the phase of DPSK in Fig. 6 is (0, π), and this is due to the relative phase of the signal as compared to the local oscillator. Therefore, the final displayed constellation of DPSK is just rotated π/2.

Tables Icon

Table2. Summary of wavelength, frequency, conversion efficiency, OSNR, and power penalty

Next, the pump2 is activated to implement the WDM multicast function. We use S2 as the original DQPSK to be multicast. The pumps, P1 and P2, are set to 1548.52 nm (193.6 THz) and 1549.32 nm (193.5 THz) with optical powers of 2.5 dBm and 4.5 dBm, respectively. The original signal, S2, is set to 1551.72 nm (193.2 THz) with a power of −2.5 dBm. The insets of Fig. 3 show the constellation and eye-diagrams of input signal for WDM multicast before the LCoS. The signal and two pumps are coupled together and S1 is blocked by the LCoS. The combined signals are injected into the SOA, which has a bias current of 280 mA, and through FWM generates the six multicast signals ranging from channel1 (Ch1) to channel6 (Ch6). The optical spectra measured at the input and output of the SOA are presented in Fig. 7. Ch5 is the original signal, while the others are the converted idlers. The six multicast signals are sent back into the LCoS and filtered out to the corresponding ports as shown in Fig. 8. The BER performance of the six multicast signals as function of the received power is investigated in Fig. 9. All of the six multicast signals achieve better performance than FEC threshold.

 figure: Fig. 7

Fig. 7 Optical spectra of WDM multicast at the (a) input; (b) output of SOA (resolution: 0.02 nm).

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 figure: Fig. 8

Fig. 8 Optical spectra of WDM multicast de-multiplexed by LCoS: (a) Ch1 at 1545.32 nm; (b) Ch2 at 1546.12 nm; (c) Ch3 at 1546.92 nm; (d) Ch4 at 1549.92 nm; (e) Ch5 at 1551.72 nm (f) Ch6 at 1552.52 nm (resolution: 0.02 nm).

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 figure: Fig. 9

Fig. 9 Measured BER performance and corresponding constellations and eye-diagrams (measured at −35 dBm) of six WDM multicast signals

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The characteristics of the multicast signals are summarized in Table3. From the obvious comparison among the six signals, it is seen that the power penalty at FEC threshold is inversely proportional to the OSNR which is dependent on the CE and signal power. The maximum power penalty is 1.1 dB for Ch1 due to the lowest CE of −27.5 dB. The performance of multicast signals can be improved by adjusting the bias current and pump power, and properly controlling the polarizations between pumps and original signal. The insets of Fig. 9 present the constellation and I/Q eye-diagrams of six multicast signals and the concentrated constellations and clear opening eye-diagrams demonstrate that the scheme is feasible in the proposed architecture.

Tables Icon

Table3. Summary of wavelength, frequency, conversion efficiency, OSNR, and power penalty of six multicast signals.

In the future, the multi-channel WDM multicast and format conversion are worth investigating. We think it is possible to implement multi-channel WDM multicast and format conversion based on our architecture. However, there are some details need to be considered: (a) With the increase of channel number, the wavelength arrangement will become more complicated. Accurate wavelength analysis of newly generated idlers is important. (b) Multi-channel conversion may greatly enlarge the wavelength conversion range which may result in lower conversion efficiency. In order to achieve higher conversion efficiency, bigger bias current, larger pump power, and precise control of polarization state are necessary. (c) Most SOAs have upper limit of the input optical power. However, multi-channel scheme with more input signals needs higher input power. Therefore, maybe it could also be performed in quantum-dot (QD) SOA, HNLF, PPLN, or silicon waveguide which can tolerate higher power. In addition, our format conversion scheme is capable of extracting one component (either I or Q). Recently, several studies have demonstrated to simultaneously extract both I- and Q-components in different ways [4042]. We think our simple and multifunctional grooming switching unit has the potential to be applied in the other schemes and it is valuable to expand the research in the future.

4. Conclusion

We propose and demonstrate a reconfigurable LCoS & SOA-based switching unit that performs multiple functions, including wavelength add/drop, wavelength conversion, format conversion, and WDM multicast. Due to the bidirectional characteristic, the LCoS device cannot only multiplex the input signals, but also de-multiplexed the converted signals. In the case of dual-channel format conversion from 2 × 25Gpbs DQPSK to 2 × 12.5Gbps DPSK, about 0.5 dB sensitivity improvement at FEC threshold for DPSK1 is achieved and 0.4 dB power penalty is paid for DPSK2; in the case of WDM multicast from one 25Gbps DQPSK signal to 6 × 25Gbps signals, less than 1.1 dB power penalty is observed for all the six multicast signals.

Acknowledgment

This study is supported by NSFC Project No.61372119, 863 Program No. 2012AA011302, Doctoral Scientific Fund Project of the Ministry of Education of China (No. 20120005110010).

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Figures (9)

Fig. 1
Fig. 1 Operation principle: (a) Simultaneous wavelength and format conversion for dual-channel DQPSK signals; (b) input DQPSK using serially-cascaded modulator; (c) input DQPSK using parallel IQ modulator; (d) one-to-six WDM multicast.
Fig. 2
Fig. 2 Multi-function optical switch unit based on bidirectional LCoS and SOA-loop architecture
Fig. 3
Fig. 3 Experiment setup: PC: polarization controller; OC: optical coupler; AWG: arbitrary waveform generator; EDFA: erbium-doped fiber amplifier; VOA: variable optical attenuator; BPF: band-pass filter; ODL: optical delay line; A/D: analog to digital converter; LO: local oscillation. Insets: the constellations and eye-diagrams of the original input signals before SOA for wavelength/format conversion and WDM multicast.
Fig. 4
Fig. 4 Optical spectra of wavelength and format conversion at the (a) input; (b) output of SOA (resolution: 0.02 nm).
Fig. 5
Fig. 5 Optical spectra of converted signals de-multiplexed by LCoS: (a) converted DQPSK2 at 1546.92nm; (b) converted DPSK1 at 1547.72 nm; (c) converted DQPSK1 at 1550.12 nm; (d) converted DPSK2 at 1554.12 nm (resolution: 0.02 nm)
Fig. 6
Fig. 6 Measured BER performance and corresponding constellations and eye-diagrams (measured at −35 dBm) of input signals (S1, S2) and converted signals (DPSK1, DPSK2, DQPSK1, DQPSK2).
Fig. 7
Fig. 7 Optical spectra of WDM multicast at the (a) input; (b) output of SOA (resolution: 0.02 nm).
Fig. 8
Fig. 8 Optical spectra of WDM multicast de-multiplexed by LCoS: (a) Ch1 at 1545.32 nm; (b) Ch2 at 1546.12 nm; (c) Ch3 at 1546.92 nm; (d) Ch4 at 1549.92 nm; (e) Ch5 at 1551.72 nm (f) Ch6 at 1552.52 nm (resolution: 0.02 nm).
Fig. 9
Fig. 9 Measured BER performance and corresponding constellations and eye-diagrams (measured at −35 dBm) of six WDM multicast signals

Tables (3)

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Table 1 Logic and phase mapping between input and output signals

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Table2 Summary of wavelength, frequency, conversion efficiency, OSNR, and power penalty

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Table3 Summary of wavelength, frequency, conversion efficiency, OSNR, and power penalty of six multicast signals.

Equations (4)

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E 223 = k 223 A 2 2 A 3 exp[j(2 ω 2 ω 3 )t+(2 θ 2 θ 3 )]
E 112 = k 112 A 1 2 A 2 exp[j(2 ω 1 ω 2 )t+(2 θ 1 θ 2 )]
E 221 = k 221 A 2 2 A 1 exp[j(2 ω 2 ω 1 )t+(2 θ 2 θ 1 )]
E 332 = k 332 A 3 2 A 2 exp[j(2 ω 3 ω 2 )t+(2 θ 3 θ 2 )]
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