Abstract

We demonstrate the generation of full C-band coherent and frequency-lock multi-carriers by using two recirculating frequency shifter (RFS) EDFA loops based on phase modulator with external injection-seeding. In our proposed novel scheme, only one phase modulator is used in each loop and two loops are used to generate long wavelength and short wavelength subcarriers, respectively. Full C-band with totally 178 subcarriers and 26 GHz subcarrier frequency spacing covering nearly 36.66 nm from 1527.54 to 1564.20 nm are obtained. The performance of 178 subcarriers with superior flatness less than 3dB and tone-to-noise ratio (TNR) larger than 20 dB after a wavelength selective switch (WSS) shows that this novel scheme is a potential technique for the future 10Tb/s optical communication.

© 2011 OSA

1. Introduction

In order to achieve Tb/s per-channel optical communications to support the rapid and huge increasing demand of the capacity in data communication, multi-carriers or multi-tone generation has attracted increasing research attentions these days [112]. Generally, there are several methods to generate multi-carriers optical source including the optical frequency comb or supercontinuum technique [1,2], the cascaded phase modulator and intensity modulator [6,7,13], single-sideband (SSB) modulator with recirculation frequency shifter (RFS) [35,14] and multi-wavelength erbium doped fiber laser (EDFL) [1518]. Recently, 5.4 Tb/s OFDM PDM-QPSK optical signal has been generated by comb generation or supercontinuum technique [19]. However, due to the limited optical signal noise ratio (OSNR) of the OFDM optical signal generation by supercontinuum technique, the transmission distance is quite limited. Also, 1.96 Tb/s PDM- QPSK OOFDM [6] optical signal is reported by using multi-carriers generated by cascaded modulators. However, only 21 subcarriers (25 GHz spacing) covering about 525GHz bandwidth with flat spectrum amplitude [6] is generated due to the limited amplitude of the RF signals on the phase modulator. Recirculating frequency shifter (RFS) based on the frequency shifting in a SSB modulator has been reported to generate 36 and 50 subcarrier, which shows the potential capability to generate more subcarriers [35,14]. However, the total number of subcarriers is strictly depended on the recirculating times because only one subcarrier generated each time for SSB modulation. Also, the SSB modulator based RFS has the significant stability problem due to the nonlinearity, the imbalance of DC bias, and the imbalance of I/Q RF signals of optical I/Q modulator (IQM) [14]. Due to the mode competition, the amount of multi-carriers is also limited less than 26 in the schemes in multi-wavelength EDFL in [1518]. On the other hand, phase modulation in amplified re-circulating loop has shown the possibility to generate optical multi-carriers with theoretical prediction in [2023]. However, a sawtooth RF drive signal is needed for SSB modulation in [20] which is complex and high-cost. This scheme is first proposed and predicted by [21]. Also, experiment results of [22,23] show 1.8THz optical comb generation by this scheme with absolute frequency accuracy and only one active mode-locked loop is used. Recently we have reported a novel scheme to generate about 112 coherent optical subcarriers with the subcarriers spacing of 25GHz using two cascade phase modulators and RFS based on one EDFA loop [24,25]. The feasibility and the good performance of this scheme have been demonstrated by the generation and transmission of 11.2 Tb/s OOFDM signal per channel [24]. However, due to the mode competition and gain limitation in EDFA, the number of the generated carriers is limited. Also, the phase deviation of RF drive signals on two phase modulators should be carefully controlled to generate more subcarriers [25].

In this paper, we improve the previous work and demonstrate the generation of full C-band coherent and frequency-lock multi-carriers by using two recirculating frequency shifter (RFS) EDFA loops based on phase modulator with external injection seeding. The two EDFA loops are actively mode-locked by phase modulators and also frequency-locked by external injection locking. In our proposed novel scheme, only one phase modulator is used in each loop reducing the loss and complexity and two loops are used to generate long wavelength and short wavelength subcarriers respectively. Finally, full C-band with 178 subcarriers and 26GHz subcarrier frequency spacing covering 36.66 nm from 1527.54 to 1564.20 nm are obtained. The performance of 178 subcarriers with superior flatness less than 3dB and tone-to-noise ratio (TNR) larger than 20dB after a wavelength selective switch (WSS) shows that this novel scheme is a potential technique for the future over 10Tb/s optical communication [24].

2. The principle of full C-band multi-carriers generation

Figure 1 shows the principle of the full C-band multi-carriers generation by using two RFS EDFA loops based on phase modulator. The configuration of the proposed scheme consists of two closed loops, and each loop comprises with two 50:50 polarization maintaining optical couplers, one phase modulator to generate multi-carriers, a polarization maintaining EDFA amplifier to compensate the loop loss and a polarization maintaining optical tunable delay line which is used to adjust the recirculating loop length for synchronization [21]. Different from the previous work in [24,25], in this novel scheme, we use two RFS EDFA loops (Loop1 and Loop 2 shown in Fig. 1) to generate long wavelength and short wavelength subcarriers, respectively. Also, only one phase modulator is used in each loop which reduces the cost and complexity and most importantly there is no RF signals phase deviation problem. The CW lightwave generated from one narrow linewidth ECL is split by one optical coupler as the seed source for each RFS EDFA loop. The upper loop is free run to generate the long wavelength subcarriers. In order to lock the generated multi-carriers of Loop2 in the short wavelengths band, one tunable optical band-pass filter in short wavelength with a polarization controller for polarization maintaining is used in the lower loop (Loop2) as shown in Fig. 1. By combining the two outputs of Loop1 and Loop2 to pass through a wavelength selective switch (WSS) or waveform shaper, we can shape the amplitude of each subcarrier and cut off the overlapped spectrum from one loop to obtain wanted subcarriers. Finally, we can generate flattened subcarriers covering the whole C-band.

 

Fig. 1 Schematic configuration of the full C-band multi-carriers generation by using two RFS EDFA loops based on phase modulator (ECL: external cavity laser; OC: optical coupler (polarization maintaining); PM: phase modulator; TD: tunable delay line (polarization maintaining); PC: polarization controller; TOF: tunable optical filter; WSS: wavelength selective switch)

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Considering the EDFA loops with the RFS configuration in Fig. 1 as Loop 1 and Loop 2, in each round the output of phase modulator is split into two branches by optical coupler, one coupled out and the other recirculating back into a EDFA to compensate the losses in the closed loop. This scheme induces power transferring to subcarriers in the loop round by round as analyzed in previous [21,25]. The principle for multi-carriers generation by RFS EDFA loop is shown in Fig. 2 , in which the output limited by EDFA gain spectrum or band pass filter is considered.

 

Fig. 2 The principle for multi-carriers generation in RFS EDFA loop without and with EDFA gain spectrum or band pass filter limit

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The CW lightwave generated from one narrow linewidth laser as seed source can be represented asEc=Eoexp(j2πfct). The two outputs from the optical coupler are coupled into each loop and modulated by the phase modulators. Each phase modulator is driven by a RF clock signal with a fixed frequency offs. To obtain more subcarriers, one booster electrical amplifier (EA) with high output power is used in each loop. The phase modulators used here are identical and the RF drive signal represented asfs(t)=RVπsin(2πfst). As analyzed in [23], the output optical signal can be expanded by the well-known Jacobi-Anger expansion to give the harmonics format with the multi-carriers expression as

Eout=Eoexp(j2πfct)exp(jπRsin2πfst)=Eon=+Jn(πR)exp[j2π(fc+nfs)]Eon=N1/2N1/2Jn(πR)exp[j2π(fc+nfs)]
here Jn(πR) is the first kind Bessel function of ordern, R is the modulation index representing the rate of RF signal amplitude to the half-wave voltage Vπand in this Eq. (1) we have Jn(πR)=(1)nJn(πR). The amplitude of nth harmonics optical subcarrier Jn(πR)exp[j2π(fc+nfs)t] is determined by the first kind Bessel functionJn(πR). In General, increasing the modulation indexRincreases the power transferred to the higher order subcarriers, and increases the number of higher order subcarriers. In this way, the maximum number of subcarriers is depended on the modulation indexR. According to the Bessel expansion of the output of Phase modulators, the power of high orders subcarriers drop sharply, we assume N1 subcarriers are generated in each round by the phase modulator. After a round trip, theN1 subcarriers amplified recirculate back into the cascaded phase modulators again to generate more subcarriers by frequency shifting along with the CW seed light coupled by OC1, the final generated subcarriers are coupled out by OC2.

The subcarriers generated by single phase modulator are frequency-locked and coherent. However, in order to obtain frequency-locked and coherent subcarriers by these EDFA loops, the recirculating loop length should be adjusted and keep synchronization for harmonic actively mode-locked. Assuming the running round-trip delay is T, the loop delay should satisfy the following two conditions as [21]

2πfsT=2pπ
2πfcT=2qπ
here p is an small integer and q is a large integer, which means the loop resonance frequency must be an integral subharmonic of both the RF driving frequency and the input optical frequency. These conditions can be satisfied by carefully adjusting the loop to the proper length and keep synchronization. In this way, the loop length of the ring is adjusted so that the round trip time of the loop length matches the modulator frequency to achieve stable active mode-locking.

Under above phase conditions, the phase delay for each subcarrier after every round should be integral multiple of 2π. Therefore, the transfer function considering EDFA and cascaded phase modulators for each round trip can be express as

F(t)=grexp(ar)n=N1/2N1/2Jn(πR)exp(j2πnfst)
where gris the EDFA amplify gain factor and the exp(αr)represents the total insertion losses in the loop for each round trip. As shown in Eq. (4), the gain of EDFA should be large enough to compensate all the insertion losses and modulation loss for N1/2 order subcarriers to run following round trips to generate more subcarriers. For saturation and stable case, we havegrexp(ar)=1. All the generated subcarriers still satisfy the synchronization loop length conditions as shown in Eqs. (2) and (3). Therefore, the newly generated nth subcarrier is the coherent addition of all the subcarriers at frequency of fc+nfs generated from different frequency subcarriers with frequency shifting based on the phase modulation. Assuming the losses in the closed loop are compensated and making some simplifications, the normalized outputs after each round trip can be expressed as

Eout_1=Eogrexp(ar)exp(j2πfct)n=N1/2N1/2Jn(πRc)exp(j2πnfst)Eout_2=Eoexp(j2πfct)n=N1/2N1/2Jn(πRc)exp(j2πnfst)+Eout_1F=Eout_1(1+F)Eout_3=Eout_1+Eout_2F=Eout_1(1+F+F2)......

Assuming after K round through the recirculating loop, the amount of generated subcarriers reach the largest and the bandwidth covering by the total subcarriers reach the largest band which is limited by EDFA gain spectrum or band-pass filter. The output can be expressed in a simple format as

Eout_K=Eout_1(1+F+F2+......+FK1)=Eoexp(j2πfct)F(1FK)1F

As shown in Eq. (6), when the loop input is derived from a laser locked to an absolute frequency, each of the output subcarriers frequencies is locked and has an absolute accuracy approaching that of the input. Here, K is the number of round times. For free run loops, K can approach infinity if the gain is constant for infinite bandwidth and can compensate all the insertion losses as well as the modulation loss theoretically. However, in practice, the effective K which determines the amount of output subcarrier is limited by the amplifier gain spectrum and also can be limited by the induced band pass filter. As analyzed in [23], due to the power limit of EDFA and the mode competition, the final output bandwidth is limited by the EDFA gain spectrum.

Considering the phase modulation, single subcarriers is frequency shifted and power transferred from adjacent subcarriers which is the coherent addition. However, the resonance process which is similar to the conventional ring laser should also be considered as analyzed above in Eq. (2) and Eq. (3). Assuming grexp(ar)=1, the optical filed of nth (n≠0) subcarrier after the M (M can be smaller or larger than K) round can be expressed as

EMn=exp(jϕr)i=N1/2N1/2EM1niJi(πR)+EM1ni
where ϕr=2πnfsT is the round-trip phase delay. Equation (7) shows contribution of both frequency shifting and power transferring by phase modulation and also the resonance process with phase delay. When Eq. (2) and Eq. (3) are satisfied, we have exp(jϕr)=1 and the whole process is the coherent addition.

Above analysis gives the operating principle and conditions for coherent and frequency-locked multi-carriers generation by the RFS EDFA loop based on the phase modulator. We give the theoretical analysis based on the simplified mathematical derivation with equations. The impact factors that affect the stability of the system can be analyzed by above Eqs. (2)~7, including the round-trip delay for loop synchronization, the gain to compensate the loop loss and the bandwidth limited by gain spectrum or filter.

Note that the noise figure performance and mode competition are not considered in above analysis. Compared to the conventional active mode-locked fiber ring laser reported in [2628], we apply external injection locking with CW seed light to the two harmonic actively mode-locked EDFA loops in our scheme. Typical setup in active mode-locked fiber ring laser including gain medium with pumped EDFA and modulators driven by sinusoidal RF signal are also used for the loops setup in our scheme. The loop length of our scheme should be adjusted for synchronization which is the same to the principle of harmonic active mode-locked. As reported in [2123,29] with injection locking, in order to generate frequency-locked multi-carriers with narrow linewidth, we use a CW source from an ECL as external injection seeding with phase modulators driven by large amplitude RF source for wide bandwidth locked to an absolute frequency. In this way, we can generate more than 100 subcarriers of which the frequencies are locked and have an absolute accuracy approaching that of the input based on one loop. Experimental results in [25] show the generation of long wavelength of subcarriers when the loop runs with the only limitation of EDFA gain spectrum. However, a band pass filter can be used to limit the subcarriers in appointed gain bandwidth with special selected CW source wavelength and filter bandwidth as shown in Fig. 2. We can choose a CW source in the middle of C-band and use a band pass filter to lock the frequency of generated subcarriers in short wavelength. In this way, it gives a possible to generate the whole C-band with two RFS EDFA loops which generate the long wavelength and short wavelength respectively.

3. Experimental results

The full C-band multi-carriers generation experiment is carried out as shown in Fig. 3 . The seed CW light source is from a narrow linewidth C-band ECL. The wavelength of ECL we chose here is 1545.55 nm which is around the middle of C-band. With this wavelength CW seed light, we can generate both long and short wavelength subcarriers. The output power of the ECL is 14dBm and the linewidth is less than 100 kHz. The RF clock frequency used here is 26 GH and the RF peak to peak voltage after the booster electrical amplifier is 17V. The half-wave voltage of the phase modulator is 4V. In this way, the amplitude of the drive RF signal is 8.5V which is about 2.1 to the half-wave voltage of the phase modulator and we can generate about 13 subcarriers at one time. The CW seed light coupled into two RFS EDFA loops and one optical band pass filter is used in lower loop.

 

Fig. 3 Experiment setup for full C-band multi-carriers generation with two EDFA recirculating loops (OSA: optical spectrum analyzer; EDFA Power: 14dBm; RF driven signal frequency: 26GHz).

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In our experiment, all the optical couplers, tunable optical delay lines and EDFAs used are polarization maintaining for stable performance. Also, a polarization controller is used in the lower loop before the non-polarization maintaining band pass filter. There is no any DC bias controller in the phase modulator and only one phase modulator is used in each loop, which is more stable. The phase modulators used have the identical performance. The insertion loss of phase modulator is 4 dB. The polarization maintaining optical couplers (PM-OC) are also identical with the insertion loss of 3.1dB and coupling ratio of 50:50. The optical time delay has an insertion loss of 1.8 dB, which is used for synchronization in the recirculation loop. The polarization controller used in the lower loop for short wavelength generation has an insertion loss of 0.5dB. For the lower loop, a band pass filter of 19 nm band width with tunable wavelength is used with an insertion loss of 6 dB. The lower loop output A and upper loop output B are combined and pass through a WSS to shape the subcarriers. The impact of EDFA output power is analyzed in [25]. Therefore, the EDFA output power is larger enough for compensation the loop loss. In our experiment, we carefully adjust the loop length with the optical delay line to obtain stable output for active mode-locking loop synchronization condition. For practice use with long-term stability, these operation factors should be controlled carefully. The small drifting of RF frequency or small perturbation such as a mechanical vibration or thermal expansion applied to the loop length in a long time can cause unstable output and phase mismatch for mode-locking synchronization condition. However, this problem can be overcome by structures such as regenerative mode-locking feedback circuit in [26,28]. Further research of the long term stability is under way.

After the phase modulator driven by RF sinuous signal, about 13 subcarriers are generated with subcarrier frequency spacing of 26GHz as shown in Fig. 4 . The power of CW source seed light transfers to the generated side band subcarriers in the phase modulation. The optical spectrum of the stable outputs of each loop (output A of the lower loop and output B of the upper loop) are shown in Fig. 5(a) and (b) , respectively. The generated short wavelength subcarriers by the lower EDFA loop with band pass filter are shown in Fig. 5(a). About 95 subcarriers from 1527.54 to 1547.21 nm are generated with TNR larger than 21dB. As analyzed in section 2, the final generated subcarriers in the lower loop are determined by the EDFA gain spectrum and band pass filter together. As shown in Fig. 5(b), the long wavelength subcarriers are generated from about 1540.18 to 1564.20 nm with 26GHz frequency spacing. In this way, 115 subcarriers generated by upper loop with TNR larger than 31dB are obtained. Here, the noise bandwidth is 0.02nm, and bandwidth for the tone is also 0.02nm. For the upper loop, the final stable generated subcarriers bandwidth is determined by the EDFA gain spectrum. Due to the mode competition, the number of generated subcarriers is limited. Due to the large ASE noise in the short wavelength of EDFA, the TNR of output A is about 9dB worse than the long wavelength subcarriers from output B. Also, the total number of generated subcarriers of lower loop with band pass filter is less than the upper loop which is free run. The optical power of output A and output B is 0.56 and 5.50dBm, respectively. The time domain waveform of entire output from a single ring of free run loop B on a high speed oscilloscope within 1ns is shown in Fig. 6 . We can see that small power fluctuations exist in the output which is caused by the small drifting of RF frequency or small perturbation such as a mechanical vibration or thermal expansion applied to the loop length. The regenerative feedback loop [26,28] is required in further research.

 

Fig. 4 The optical spectrum after phase modulator with 13 subcarriers generated.

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Fig. 5 The optical spectrum of (a) output A and (b) output B

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Fig. 6 the observation of the entire output from a single ring on a high speed oscilloscope within 1ns

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Combining output A and output B to pass through the WSS we can obtain the full C-band subcarriers. The WSS is fully programmed. To avoid the overlap subcarriers from A and B, we cut off the overlap part of A which is the output of shot wavelength subcarriers and only preserve subcarriers from 1527.54 to 1541.52 nm. Figure 7(a) is the optical spectrum of combined subcarriers from A and B passing through WSS as one optical band pass filter. From Fig. 7(a), we can see the total subcarriers occupy the full C-band covering both short and long wavelength. However, due to the non-flatness of EDFA gain spectrum, subcarriers generated are not so flat. One C-band programed WSS is used to shape the amplitude of each subcarrier. The amplitude and wavelength bandwidth of each channel in the WSS are all fully tunable. The minimal bandwidth of the WSS for each channel is 0.15 nm. The optical spectrum of amplitude equalized subcarriers after WSS is shown in Fig. 7(b). Finally, after the WSS, the amplitude difference between different subcarriers is smaller than 3dB which can be even smaller with carefully adjusting. The residual power difference is probably the result of long term slow fluctuations which can be improved by loop length control. The shortest and longest wavelength is 1527.54 nm and 1564.20 nm, respectively. There are totally 178 subcarriers obtained and the subcarrier frequency spacing is 26GHz. The TNR of the total subcarriers after WSS is 20dB with 1 dB penalty due to the WSS amplitude shaping. We can see that multi-carriers obtained after WSS cover the full C-band nearly 36.66 nm. Noting that the optical power for each subcarrier is small and about −40 to −37dBm, we need optical amplifier to obtain large optical power for each subcarrier before the data signal modulation and transmission. The resolution of all optical spectra in Fig. 6 is 0.02 nm.

 

Fig. 7 The optical spectrum of the combining of output A and output B (a) passing through a optical band pass filter; (b) passing through a WSS.

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The detailed optical spectrum of subcarriers of combined output A and output B passing WSS as a pass-band filter without flattening from 1528 to1532 nm, 1533 to1537 nm, 1538 to1542 nm, 1543 to1547 nm, 1548 to1552 nm and 1558 to1561 nm are illustrated in Fig. 8(a), (b), (c), (d), (e) and (f) , respectively. The cutoff wavelength for output A to avoid overlap is 1541.52 nm, which is marked out in Fig. 8(c). We can see a clear noise falling at this point. The resolution of all optical spectra in Fig. 8 is 0.02 nm. From Fig. 8, we can see that the subcarriers from 1543 to 1552 nm in (d) and (e) have the best TNR which is larger than 30dB. Subcarriers from 1528 to 1532 nm have the worst TNR due to the lower EDFA gain and ASE. For subcarriers from 1558 to 1561 nm, EDFA loop resonance ASE noise is clear but still with good TNR. The model number of this OSA is Yokogawa AQ6370B which cannot show the correct spectrum, especially when the optical power is small, such as at −60dBm where the spectral width of each subcarrier seems wider. The performance of generated 178 subcarriers with the subcarrier frequency spacing of 26GHz covering full C-band nearly 36.66 nm shows that this scheme is a potential technique for the future 10Tb/s optical communication.

 

Fig. 8 The detailed optical spectrum of subcarriers passing through WSS without flattening (a) from 1528 to1532 nm; (b) from 1533 to1537 nm; (c) from 1538 to1542 nm; (d) from 1543 to1547 nm; (e) from 1548 to1552 nm; (f) from 1558 to1561 nm.

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4. Conclusion

We demonstrate the generation of full C-band coherent and frequency-lock multi-carriers by using two recirculating frequency shifter (RFS) EDFA loops based on phase modulator. In our proposed novel scheme, only one phase modulator is used in each loop reducing the loss and complexity and the two loops are used to generate long wavelength and short wavelength subcarriers respectively. Finally, full C-band totally 178 subcarriers with 26GHz subcarrier frequency spacing covering nearly 36.66 nm from 1527.54 to 1564.20 nm are obtained. The performance of 178 subcarriers with superior flatness less than 3dB and tone-to-noise ratio larger than 20dB after a WSS shows that this novel scheme is a potential technique for the future 10Tb/s optical communication.

Acknowledgment

This work is partially supported by the National High Technology Research and Development Program (973) of China (Grant No. 2010CB328300), National Natural Science Foundation of China (No. 61107064, No. 61177071, No. 600837004), Chinese Postdoctoral Science Foundation funded project (No. 20090460593), Doctoral Fund of Ministry of Education, Pujiang Fund, Shuguang fund and the Creative Talent Project Foundation for Key Disciplines of Fudan University.

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23. H. Y. Ryu, H. S. Moon, and H. S. Suh, “Optical frequency comb generator based on actively mode-locked fiber ring laser using an acousto-optic modulator with injection-seeding,” Opt. Express 15(18), 11396–11401 (2007). [CrossRef]   [PubMed]  

24. J. Yu, Z. Dong, X. Xiao, Y. Xia, S. Shi, C. Ge, W. Zhou, N. Chi and Y. Shao, “Generation of 112 coherent multi-carriers and transmission of 10 Tb/s (112x100Gb/s) single optical OFDM superchannel over 640 km SMF,” OFC2011, PDPA6 (2011).

25. J. Zhang, N. Chi, J. Yu, Y. Shao, J. Zhu, B. Huang, and L. Tao, “Generation of coherent and frequency-lock multi-carriers using cascaded phase modulators and recirculating frequency shifter for Tb/s optical communication,” Opt. Express 19(14), 12891–12902 (2011). [CrossRef]   [PubMed]  

26. M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarisation-maintaining erbium fibre ring laser,” Electron. Lett. 30(19), 1603–1605 (1994). [CrossRef]  

27. R. Kiyan, O. Deparis, O. Pottiez, P. Mégret, and M. Blondel, “Stabilization of actively mode-locked Er-doped fiber lasers in the rational-harmonic frequency-doubling mode-locking regime,” Opt. Lett. 24(15), 1029–1031 (1999). [CrossRef]   [PubMed]  

28. B. Bakhshi and P. A. Andrekson, “40GHz actively modelocked polarisationmaintaining erbium fibre ring laser,” Electron. Lett. 36(5), 411–413 (2000). [CrossRef]  

29. A. Takada and W. Imajuku, “Linewidth narrowing and optical phase control of mode-locked semiconductor ring laser employing optical injection locking,” IEEE Photon. Technol. Lett. 9(10), 1328–1330 (1997). [CrossRef]  

References

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    [CrossRef]
  7. J. Yu, “1.2 Tbit/s orthogonal PDM-RZ-QPSK DWDM signal transmission over 1040 km SMF-28,” Electron. Lett. 46(11), 775–777 (2010).
    [CrossRef]
  8. G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
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  9. B. Zhu, X. Liu, S. Chandrasekhar, D. Peckham, and R. Lingle, “Ultra-long-haul transmission of 1.2-Tb/s multicarrier no-guard-interval CO-OFDM superchannel using ultra-large-area fiber,” IEEE Photon. Technol. Lett. 22(11), 826–828 (2010).
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  12. H. Masuda, E. Yamazaki, A. Sano, T. Yoshimatsu, T. Kobayashi, E. Yoshida, Y. Miyamoto, S. Matsuoka, Y. Takatori, M. Mizoguchi, K. Okada, K. Hagimoto, T. Yamada, S. Kamei, “13.5-Tb/s (135 × 111-Gb/s/ch) no-guard-interval coherent OFDM transmission over 6,248 km using SNR maximized second-order DRA in the extended L-band,” OFC. PDPB5 (2009).
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  17. J. Yao, J. Yao, Z. Deng, and J. Liu, “Multiwavelength erbium-doped Fiber ring laser incorporating an SOA-based phase modulator,” IEEE Photon. Technol. Lett. 17(4), 756–758 (2005).
    [CrossRef]
  18. M. A. Mirza and G. Stewart, “Multiwavelength operation of erbium-doped fiber lasers by period filtering and phase modulation,” J. Lightwave Technol. 27(8), 1034–1044 (2009).
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    [CrossRef]
  21. K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
    [CrossRef]
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    [CrossRef]
  23. H. Y. Ryu, H. S. Moon, and H. S. Suh, “Optical frequency comb generator based on actively mode-locked fiber ring laser using an acousto-optic modulator with injection-seeding,” Opt. Express 15(18), 11396–11401 (2007).
    [CrossRef] [PubMed]
  24. J. Yu, Z. Dong, X. Xiao, Y. Xia, S. Shi, C. Ge, W. Zhou, N. Chi and Y. Shao, “Generation of 112 coherent multi-carriers and transmission of 10 Tb/s (112x100Gb/s) single optical OFDM superchannel over 640 km SMF,” OFC2011, PDPA6 (2011).
  25. J. Zhang, N. Chi, J. Yu, Y. Shao, J. Zhu, B. Huang, and L. Tao, “Generation of coherent and frequency-lock multi-carriers using cascaded phase modulators and recirculating frequency shifter for Tb/s optical communication,” Opt. Express 19(14), 12891–12902 (2011).
    [CrossRef] [PubMed]
  26. M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarisation-maintaining erbium fibre ring laser,” Electron. Lett. 30(19), 1603–1605 (1994).
    [CrossRef]
  27. R. Kiyan, O. Deparis, O. Pottiez, P. Mégret, and M. Blondel, “Stabilization of actively mode-locked Er-doped fiber lasers in the rational-harmonic frequency-doubling mode-locking regime,” Opt. Lett. 24(15), 1029–1031 (1999).
    [CrossRef] [PubMed]
  28. B. Bakhshi and P. A. Andrekson, “40GHz actively modelocked polarisationmaintaining erbium fibre ring laser,” Electron. Lett. 36(5), 411–413 (2000).
    [CrossRef]
  29. A. Takada and W. Imajuku, “Linewidth narrowing and optical phase control of mode-locked semiconductor ring laser employing optical injection locking,” IEEE Photon. Technol. Lett. 9(10), 1328–1330 (1997).
    [CrossRef]

2011

2010

J. Li, X. Li, X. Zhang, F. Tian, and L. Xi, “Analysis of the stability and optimizing operation of the single-side-band modulator based on re-circulating frequency shifter used for the T-bit/s optical communication transmission,” Opt. Express 18(17), 17597–17609 (2010).
[CrossRef] [PubMed]

J. Yu, “1.2 Tbit/s orthogonal PDM-RZ-QPSK DWDM signal transmission over 1040 km SMF-28,” Electron. Lett. 46(11), 775–777 (2010).
[CrossRef]

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
[CrossRef]

B. Zhu, X. Liu, S. Chandrasekhar, D. Peckham, and R. Lingle, “Ultra-long-haul transmission of 1.2-Tb/s multicarrier no-guard-interval CO-OFDM superchannel using ultra-large-area fiber,” IEEE Photon. Technol. Lett. 22(11), 826–828 (2010).
[CrossRef]

2009

2007

2005

A. Lowery, “Performance predictions and topology improvements for optical serrodyne comb generators,” J. Lightwave Technol. 23(8), 2371–2379 (2005).
[CrossRef]

J. Yao, J. Yao, Z. Deng, and J. Liu, “Multiwavelength erbium-doped Fiber ring laser incorporating an SOA-based phase modulator,” IEEE Photon. Technol. Lett. 17(4), 756–758 (2005).
[CrossRef]

2003

2000

1999

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[CrossRef]

R. Kiyan, O. Deparis, O. Pottiez, P. Mégret, and M. Blondel, “Stabilization of actively mode-locked Er-doped fiber lasers in the rational-harmonic frequency-doubling mode-locking regime,” Opt. Lett. 24(15), 1029–1031 (1999).
[CrossRef] [PubMed]

1997

A. Takada and W. Imajuku, “Linewidth narrowing and optical phase control of mode-locked semiconductor ring laser employing optical injection locking,” IEEE Photon. Technol. Lett. 9(10), 1328–1330 (1997).
[CrossRef]

1994

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarisation-maintaining erbium fibre ring laser,” Electron. Lett. 30(19), 1603–1605 (1994).
[CrossRef]

1993

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
[CrossRef]

Andrekson, P. A.

B. Bakhshi and P. A. Andrekson, “40GHz actively modelocked polarisationmaintaining erbium fibre ring laser,” Electron. Lett. 36(5), 411–413 (2000).
[CrossRef]

Bakhshi, B.

B. Bakhshi and P. A. Andrekson, “40GHz actively modelocked polarisationmaintaining erbium fibre ring laser,” Electron. Lett. 36(5), 411–413 (2000).
[CrossRef]

Bellemare, A.

Bennett, S.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[CrossRef]

Blondel, M.

Bosco, G.

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
[CrossRef]

Bull, J. D.

Burr, E.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[CrossRef]

Cai, B.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[CrossRef]

Carena, A.

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
[CrossRef]

Chandrasekhar, S.

B. Zhu, X. Liu, S. Chandrasekhar, D. Peckham, and R. Lingle, “Ultra-long-haul transmission of 1.2-Tb/s multicarrier no-guard-interval CO-OFDM superchannel using ultra-large-area fiber,” IEEE Photon. Technol. Lett. 22(11), 826–828 (2010).
[CrossRef]

Chen, S.

Chi, N.

J. Zhang, N. Chi, J. Yu, Y. Shao, J. Zhu, B. Huang, and L. Tao, “Generation of coherent and frequency-lock multi-carriers using cascaded phase modulators and recirculating frequency shifter for Tb/s optical communication,” Opt. Express 19(14), 12891–12902 (2011).
[CrossRef] [PubMed]

J. Yu, Z. Dong, and N. Chi, “1.96-Tb/s (21x100 Gb/s) optical OFDM superchannel generation and transmission over 3200-km SMF-28 with EDFA-only,” IEEE Photon. Technol. Lett. 23, 1061–1063 (2011).
[CrossRef]

Deng, Z.

J. Yao, J. Yao, Z. Deng, and J. Liu, “Multiwavelength erbium-doped Fiber ring laser incorporating an SOA-based phase modulator,” IEEE Photon. Technol. Lett. 17(4), 756–758 (2005).
[CrossRef]

Deparis, O.

Dong, F.

Dong, Z.

J. Yu, Z. Dong, and N. Chi, “1.96-Tb/s (21x100 Gb/s) optical OFDM superchannel generation and transmission over 3200-km SMF-28 with EDFA-only,” IEEE Photon. Technol. Lett. 23, 1061–1063 (2011).
[CrossRef]

Ellis, A. D.

Forghieri, F.

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
[CrossRef]

Garcia Gunning, F. C.

Gavioli, G.

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
[CrossRef]

Gough, O.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[CrossRef]

Healy, T.

Ho, K. P.

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
[CrossRef]

Huang, B.

Imajuku, W.

A. Takada and W. Imajuku, “Linewidth narrowing and optical phase control of mode-locked semiconductor ring laser employing optical injection locking,” IEEE Photon. Technol. Lett. 9(10), 1328–1330 (1997).
[CrossRef]

Kahn, J. M.

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
[CrossRef]

Karasek, M.

Kimura, Y.

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarisation-maintaining erbium fibre ring laser,” Electron. Lett. 30(19), 1603–1605 (1994).
[CrossRef]

Kiyan, R.

Kurokawa, K.

T. Sakamoto, T. Yamamoto, K. Kurokawa, and S. Tomita, “DWDM transmission in O-band over 24 km PCF using optical frequency comb based multicarrier source,” Electron. Lett. 45(16), 850–851 (2009).
[CrossRef]

Li, J.

Li, X.

Lingle, R.

B. Zhu, X. Liu, S. Chandrasekhar, D. Peckham, and R. Lingle, “Ultra-long-haul transmission of 1.2-Tb/s multicarrier no-guard-interval CO-OFDM superchannel using ultra-large-area fiber,” IEEE Photon. Technol. Lett. 22(11), 826–828 (2010).
[CrossRef]

Liu, J.

J. Yao, J. Yao, Z. Deng, and J. Liu, “Multiwavelength erbium-doped Fiber ring laser incorporating an SOA-based phase modulator,” IEEE Photon. Technol. Lett. 17(4), 756–758 (2005).
[CrossRef]

Liu, X.

B. Zhu, X. Liu, S. Chandrasekhar, D. Peckham, and R. Lingle, “Ultra-long-haul transmission of 1.2-Tb/s multicarrier no-guard-interval CO-OFDM superchannel using ultra-large-area fiber,” IEEE Photon. Technol. Lett. 22(11), 826–828 (2010).
[CrossRef]

Lowery, A.

LRochelle, S.

Ma, Y.

Mégret, P.

Mirza, M. A.

Moon, H. S.

Nakazawa, M.

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarisation-maintaining erbium fibre ring laser,” Electron. Lett. 30(19), 1603–1605 (1994).
[CrossRef]

Ngo, N. Q.

Peckham, D.

B. Zhu, X. Liu, S. Chandrasekhar, D. Peckham, and R. Lingle, “Ultra-long-haul transmission of 1.2-Tb/s multicarrier no-guard-interval CO-OFDM superchannel using ultra-large-area fiber,” IEEE Photon. Technol. Lett. 22(11), 826–828 (2010).
[CrossRef]

Poggiolin, P.

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
[CrossRef]

Pottiez, O.

Rochette, M.

Ryu, H. Y.

Sakamoto, T.

T. Sakamoto, T. Yamamoto, K. Kurokawa, and S. Tomita, “DWDM transmission in O-band over 24 km PCF using optical frequency comb based multicarrier source,” Electron. Lett. 45(16), 850–851 (2009).
[CrossRef]

Savory, S.

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
[CrossRef]

Seeds, A. J.

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[CrossRef]

Shao, Y.

Shieh, W.

Stewart, G.

Suh, H. S.

Takada, A.

A. Takada and W. Imajuku, “Linewidth narrowing and optical phase control of mode-locked semiconductor ring laser employing optical injection locking,” IEEE Photon. Technol. Lett. 9(10), 1328–1330 (1997).
[CrossRef]

Tang, Y.

Tao, L.

Tetu, M.

Tian, F.

Tomita, S.

T. Sakamoto, T. Yamamoto, K. Kurokawa, and S. Tomita, “DWDM transmission in O-band over 24 km PCF using optical frequency comb based multicarrier source,” Electron. Lett. 45(16), 850–851 (2009).
[CrossRef]

Torrengo, E.

G. Gavioli, E. Torrengo, G. Bosco, A. Carena, S. Savory, F. Forghieri, and P. Poggiolin, “Ultra-narrow-spacing 10-channel 1.12 Tb/s D-WDM long-haul transmission over uncompensated SMF and NZDSF,” IEEE Photon. Technol. Lett. 22(19), 1419–1421 (2010).
[CrossRef]

Xi, L.

Yamamoto, T.

T. Sakamoto, T. Yamamoto, K. Kurokawa, and S. Tomita, “DWDM transmission in O-band over 24 km PCF using optical frequency comb based multicarrier source,” Electron. Lett. 45(16), 850–851 (2009).
[CrossRef]

Yang, Q.

Yao, J.

J. Yao, J. Yao, Z. Deng, and J. Liu, “Multiwavelength erbium-doped Fiber ring laser incorporating an SOA-based phase modulator,” IEEE Photon. Technol. Lett. 17(4), 756–758 (2005).
[CrossRef]

J. Yao, J. Yao, Z. Deng, and J. Liu, “Multiwavelength erbium-doped Fiber ring laser incorporating an SOA-based phase modulator,” IEEE Photon. Technol. Lett. 17(4), 756–758 (2005).
[CrossRef]

Yoshida, E.

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarisation-maintaining erbium fibre ring laser,” Electron. Lett. 30(19), 1603–1605 (1994).
[CrossRef]

Yu, J.

J. Zhang, N. Chi, J. Yu, Y. Shao, J. Zhu, B. Huang, and L. Tao, “Generation of coherent and frequency-lock multi-carriers using cascaded phase modulators and recirculating frequency shifter for Tb/s optical communication,” Opt. Express 19(14), 12891–12902 (2011).
[CrossRef] [PubMed]

J. Yu, Z. Dong, and N. Chi, “1.96-Tb/s (21x100 Gb/s) optical OFDM superchannel generation and transmission over 3200-km SMF-28 with EDFA-only,” IEEE Photon. Technol. Lett. 23, 1061–1063 (2011).
[CrossRef]

J. Yu, “1.2 Tbit/s orthogonal PDM-RZ-QPSK DWDM signal transmission over 1040 km SMF-28,” Electron. Lett. 46(11), 775–777 (2010).
[CrossRef]

Zhang, J.

Zhang, X.

Zhou, D.

Zhou, K.

Zhu, B.

B. Zhu, X. Liu, S. Chandrasekhar, D. Peckham, and R. Lingle, “Ultra-long-haul transmission of 1.2-Tb/s multicarrier no-guard-interval CO-OFDM superchannel using ultra-large-area fiber,” IEEE Photon. Technol. Lett. 22(11), 826–828 (2010).
[CrossRef]

Zhu, J.

Electron. Lett.

J. Yu, “1.2 Tbit/s orthogonal PDM-RZ-QPSK DWDM signal transmission over 1040 km SMF-28,” Electron. Lett. 46(11), 775–777 (2010).
[CrossRef]

T. Sakamoto, T. Yamamoto, K. Kurokawa, and S. Tomita, “DWDM transmission in O-band over 24 km PCF using optical frequency comb based multicarrier source,” Electron. Lett. 45(16), 850–851 (2009).
[CrossRef]

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarisation-maintaining erbium fibre ring laser,” Electron. Lett. 30(19), 1603–1605 (1994).
[CrossRef]

B. Bakhshi and P. A. Andrekson, “40GHz actively modelocked polarisationmaintaining erbium fibre ring laser,” Electron. Lett. 36(5), 411–413 (2000).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Takada and W. Imajuku, “Linewidth narrowing and optical phase control of mode-locked semiconductor ring laser employing optical injection locking,” IEEE Photon. Technol. Lett. 9(10), 1328–1330 (1997).
[CrossRef]

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop,” IEEE Photon. Technol. Lett. 5(6), 721–725 (1993).
[CrossRef]

S. Bennett, B. Cai, E. Burr, O. Gough, and A. J. Seeds, “1.8-THz bandwidth, zero-frequency error, tunable optical comb generator for DWDM applications,” IEEE Photon. Technol. Lett. 11(5), 551–553 (1999).
[CrossRef]

J. Yao, J. Yao, Z. Deng, and J. Liu, “Multiwavelength erbium-doped Fiber ring laser incorporating an SOA-based phase modulator,” IEEE Photon. Technol. Lett. 17(4), 756–758 (2005).
[CrossRef]

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B. Zhu, X. Liu, S. Chandrasekhar, D. Peckham, and R. Lingle, “Ultra-long-haul transmission of 1.2-Tb/s multicarrier no-guard-interval CO-OFDM superchannel using ultra-large-area fiber,” IEEE Photon. Technol. Lett. 22(11), 826–828 (2010).
[CrossRef]

J. Yu, Z. Dong, and N. Chi, “1.96-Tb/s (21x100 Gb/s) optical OFDM superchannel generation and transmission over 3200-km SMF-28 with EDFA-only,” IEEE Photon. Technol. Lett. 23, 1061–1063 (2011).
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S. Chandrasekhar, X. Liu, B. Zhu, D. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area Fiber,” ECOC. PD2.6 (2009).

H. Masuda, E. Yamazaki, A. Sano, T. Yoshimatsu, T. Kobayashi, E. Yoshida, Y. Miyamoto, S. Matsuoka, Y. Takatori, M. Mizoguchi, K. Okada, K. Hagimoto, T. Yamada, S. Kamei, “13.5-Tb/s (135 × 111-Gb/s/ch) no-guard-interval coherent OFDM transmission over 6,248 km using SNR maximized second-order DRA in the extended L-band,” OFC. PDPB5 (2009).

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Figures (8)

Fig. 1
Fig. 1

Schematic configuration of the full C-band multi-carriers generation by using two RFS EDFA loops based on phase modulator (ECL: external cavity laser; OC: optical coupler (polarization maintaining); PM: phase modulator; TD: tunable delay line (polarization maintaining); PC: polarization controller; TOF: tunable optical filter; WSS: wavelength selective switch)

Fig. 2
Fig. 2

The principle for multi-carriers generation in RFS EDFA loop without and with EDFA gain spectrum or band pass filter limit

Fig. 3
Fig. 3

Experiment setup for full C-band multi-carriers generation with two EDFA recirculating loops (OSA: optical spectrum analyzer; EDFA Power: 14dBm; RF driven signal frequency: 26GHz).

Fig. 4
Fig. 4

The optical spectrum after phase modulator with 13 subcarriers generated.

Fig. 5
Fig. 5

The optical spectrum of (a) output A and (b) output B

Fig. 6
Fig. 6

the observation of the entire output from a single ring on a high speed oscilloscope within 1ns

Fig. 7
Fig. 7

The optical spectrum of the combining of output A and output B (a) passing through a optical band pass filter; (b) passing through a WSS.

Fig. 8
Fig. 8

The detailed optical spectrum of subcarriers passing through WSS without flattening (a) from 1528 to1532 nm; (b) from 1533 to1537 nm; (c) from 1538 to1542 nm; (d) from 1543 to1547 nm; (e) from 1548 to1552 nm; (f) from 1558 to1561 nm.

Equations (7)

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E out = E o exp(j2π f c t)exp(jπRsin2π f s t) = E o n= + J n (πR) exp[j2π( f c +n f s )] E o n= N 1 /2 N 1 /2 J n (πR)exp[j2π( f c +n f s )]
2π f s T=2pπ
2π f c T=2qπ
F(t)= g r exp( a r ) n= N 1 /2 N 1 /2 J n (πR)exp(j2πn f s t)
E out_1 = E o g r exp( a r )exp(j2π f c t) n= N 1 /2 N 1 /2 J n (π R c ) exp(j2πn f s t) E out_2 = E o exp(j2π f c t) n= N 1 /2 N 1 /2 J n (π R c ) exp(j2πn f s t)+ E out_1 F= E out_1 (1+F) E out_3 = E out_1 + E out_2 F= E out_1 (1+F+ F 2 ) ......
E out_K = E out_1 (1+F+ F 2 +......+ F K1 ) = E o exp(j2π f c t) F(1 F K ) 1F
E M n =exp(j ϕ r ) i= N 1 /2 N 1 /2 E M1 ni J i (πR)+ E M1 ni

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