Abstract

This paper focuses on the studies on the polarization- modulator-based single-side-band modulator (PSSBM) as well as its implementation in generation of frequency-locked multicarrier. The principle and properties of PSSBM have been analyzed in detail with theoretical and simulation results. Then, the PSSBM-based frequency-locked multicarrier generator (PSMCG) with recirculating frequency shifting loop has also been demonstrated via simulation. The results have a good agreement with the theoretical analysis. According to the results, multiple frequency-locked carriers with high quality can be achieved based on the proposed PSMCG. The generated carriers have the potential applications in different scenarios such as optical transmission, microwave photonics and so on.

© 2014 Optical Society of America

1. Introduction

Optical frequency comb is broadly applied to optical arbitrary waveform generation [1], precise optical metrology [2], high accuracy optical sensor [3], ultrafast optical signal processing [4], photonic microwave signal processing [5], high data rate optical fiber transmission [69], and so on. In those applications except optical transmission, optical combs must have the equal frequency spacing and fixed phase-relation between the comb lines. Hence, optical comb generators in optical transmission can be referred as frequency-locked multicarrier generator (MCG) [10]. To achieve the high-speed large capacity optical transmission for the increasing demand of data traffic, there are some technologies such as coherent wavelength-division-multiplexing (CoWDM) [11], and especially coherent optical orthogonal-frequency-division- multiplexing (CO-OFDM) or direct-detect (DD) OFDM superchannels with different modulation formats [1216] investigated widely in recent years.

In those superchannels, MCG is a key component. There are many methods to implement the MCG, such as typical overdrive single and cascaded Mach-Zenhder modulators (MZMs) and/or phase modulators (abbr., PM-MZM-based) [17,18], mode-locked lasers (MFL) [19], supercontinuum [20], cascaded polarization modulators (CPolM) [21], CW-seeded Shock-Wave Mixer (CSWM) [22] and the recirculating frequency shifter (RFS)-based MCG [68,12,13, 2332]. Among those methods, the PM-MZM-based method can generate high tone-to-noise- ratio (TNR) carriers with the required high-power electrical amplifiers which decide the number of generated carriers. As for the MFL, it can generate multiple carriers but the stable operation should be guaranteed and the frequency spacing is difficult to tune in a wide range. The weak-point of the supercontinuum is the poor wavelength tunability. Although the CPolM can achieve high-flatness output carriers, the TNR is low and carrier number should be improved if more carriers are required. Although there are amazing 1000 combs demonstrated based on the CSWM, the precise dispersion control and in-line regeneration should be applied. The advantages of the RFS-based MCG, having been studied extensively, are of easing control of carrier number and frequency spacing, low radio frequency (RF) drive power, and high stability. There are two basic configurations of RFS-based MCG, one is single-side-band (SSB) modulator-based MCG (SMCG) [6,7, 12,13,2328], the other is multi-side-band (MSB) modulator-based (MMCG) [8, 29,30]. Although the MMCG can achieve hundreds of optical carriers, the output stability is poor compared with the SMCG configuration. However, the quality of output carriers, such as the number and TNR of carriers, should be further improved by applying the SMCG configuration. One can use the configuration of multiseed-SMCG [13,27] and dual- [25,26] or multi-loop [28] SMCG to achieve the generation of hundreds of output carriers. Meanwhile, one can also use the improved configuration of SMCG [32] based on the suppressed-third-crosstalk scheme [33] to obtain higher TNR. However, the SMCG implemented by the conventional SSB modulator (CSSBM) based on the integrated LiNbO3 dual-parallel MZM or IQ modulator, has the requirement of three DC bias controls to make the modulator operate at the right point. Thus, the auto bias control is necessary [34] due to the property of the modulator, sensitive to the changes of environmental conditions [35].

In this paper, aim to achieve SSB frequency shifting so as to implement high quality multicarrier generation with recirculating frequency shifting loop, we propose a polarization-modulator-based SSB modulator (PSSBM) with only one DC bias control required. The detailed theoretical analysis and simulation results show that the proposed configurations can be used to achieve the desired frequency shift and generation of multiple frequency-locked multicarriers.

2. Theoretical analysis

The schematic of PSSBM is shown in Fig. 1.It consists of a power splitter/combiner, two polarization modulators (PolMs), a phase modulator (PM), two polarization controllers (PC), a polarizer (Pol), an RF signal and a DC bias. The input RF signal f(t) with its frequency of fm is split into two equal parts with π/2 phase shift and then drives respective PolMs. As illustrated in Fig. 1, when the frequency of input seed laser Ein(t) is f0, the novel PSSBM can achieve a desired frequency-spacing SSB-shift with frequency up-shift (f0 + fm) or down-shift (f0fm) of seed laser by applying the DC bias control and tuning the two PCs. In general, the input linearly polarized seed laser from a tunable laser (TLS) can be represented as Ein(t) = E0exp(0t). The light-wave with its state of polarization (SOP) is aligned at an angle of α1 tuned by PC1 relative to one principal axis of the PolM and then projected to the two orthogonal directions. At the output of each PolM, the two modulated light waves are combined with a power combiner and then sent to the polarizer via PC2. The principal axis of the polarizer is aligned at α2 relative to the x axis of the PolMs by PC2. The two angles α1 and α2 can be tuned by the PCs before the PolMs and Pol. We also represent the input RF signals used to drive PolM1 and PolM2 as f1(t) = Vppcos(ωmt) and f2(t) = Vppsin(ωmt) respectively. VPP is the amplitude of the RF drive signals. Based on the modulation property of polarization modulator shown in [36], the outputs of the two PolMs can be expressed as follows

Eo1(t)=c1[Einx(t)exp(jπf1(t)Vπ)+Einy(t)exp(jπf1(t)Vπ)]Eo2(t)=c2[Einx(t)exp(jπf2(t)Vπ)+Einy(t)exp(jπf2(t)Vπ)]
where coefficients c1 and c2 represent the portion of input seed laser projected to the two orthogonal directions of PolMs. To obtain the transfer function of PSSBM, we adopt the transfer- matrix-based method, i.e., using Hin, HPolM, HPM, HPC and HDPol to represent the SOP of input linearly polarized signal and functions of PolMs, PM, PC and polarizer, as shown in Eq. (2),
Hin=[cosα1sinα1],HPol=[cosα2sinα2],HPC=[cosθsinθsinθcosθ][exp(jϕ2)00exp(jϕ2)];HPolM=[H1x+HPMH2x00H1y+HPMH2y];HPM=exp[jπVDC(t)/Vπ];H1x=H1y*=exp[jπf1(t)/Vπ]=exp[jδmcos(ωmt)];H2x=H2y*=exp[jπf2(t)/Vπ]=exp[jδmsin(ωmt)]
where Vπ is the half-wave voltage of PolM and δm = (πVpp)/(Vπ) denotes for the phase modulation depth. The parameters θ and ϕ are the rotation angle and static phase difference induced by the PC2 between the x- and y-polarization components respectively and can be tuned by the PC2 before the Pol. Note that the obvious difference is the PC model used here is a fully-tunable wave-plate as used in [37] when compared with our previous work [26]. Then, the normalized transfer function of PSSBM can be expressed mathematically as follows
TPSSBM=HPolHPCHPolMHin={C1exp[jδmcos(ωmt)]C2exp[jδmcos(ωmt)]}+exp(jϕDC){C1exp[jδmsin(ωmt)]C2exp[jδmsin(ωmt)]}C1=[cosα2cosα1cosθ+sinα2cosα1sinθ]exp(jϕ2);C2=[cosα2sinα1sinθsinα2sinα1cosθ]exp(jϕ2)
where C1/2 is the amplitude variation coefficient induced by the PCs and Pol. Obviously, to achieve the desired SSB frequency shift, the required condition deduced from Eq. (3) should be satisfied as follows
 C1=C2=C[cosα2cosα1cosθ+sinα2cosα1sinθ]exp(jϕ2)=[cosα2sinα1sinθsinα2sinα1cosθ]exp(jϕ2)
To satisfy the above condition, the two simplest cases can be deduced and shown as: case1, θ = kπ, ϕ = 2kπ (k is an integer), then cosα1cosα2 = −sinα1sinα2; or case2, θ = (kπ + π/2), ϕ = 2kπ (k is an integer), then cosα1sinα2 = sinα1cosα2. Namely, α1α2 should be equal to ± π/2 or 0. In other words, the SOPs of the output and input signals should be orthogonal or identical corresponding to the two cases by carefully adjusting the two PCs manually or electrically depending on the types of them. By the way, the stated condition could be also satisfied even if the input seed has more than one wavelength. We then denote the amplitude variation coefficient in this case as an brief function of α1 and α2 as |Cavc(α)| = |sin(2α)|/2. By applying this guiding result and setting VDC(t) = ± Vπ/2 (‘ + ’ and ‘−’ represent the desired SSB frequency up- and down-shift respectively), the final transfer function of PSSBM can be expressed as follows
 TPSSBM=|Cavc(α)|[sin(δmcos(ωmt))±jsin(δmsin(ωmt))]
Obviously, the above transfer function is coincident with that of the conventional IQ modulator which is usually used to realize the SSB modulation [24]. Thus, the desired SSB frequency shift can be achieved by the proposed PSSBM. Besides, the unwanted carrier will be also fully-suppressed by tuning the PCs under the ideal case. After using the Jacobi-Anger expansion, Eq. (5) can be expressed as
 TPSSBM=|Cavc(α)|[sin(δmcos(ωmt))±jsin(δmsin(ωmt))]=|Cavc(α)|{k=1j2k2J2k1(δm)cos[(2k1)ωmt]±jk=1J2k1(δm)sin[(2k1)ωmt]}=|C(α)avc|{[J1(δm)exp(±jωmt)J3(δm)exp(j3ωmt)])+[J5(δm)exp(j±5ωmt)]}
where J2k-1(δm) are the odd-order Bessel functions of the first kind. Ignoring all the high order harmonics beyond the third order, then the normalized transfer function of PSSBM obtained from Eq. (6) has the same expression as in [24]
 TPSSBM=[exp(±jωmt)+bexp(j3ωmt)]
where b = −J3/J1, stands for the third-order crosstalk coefficient depended on the modulator drive voltage Vpp, which can affect the output property of PSSBM. In order to decrease the crosstalk, we must apply proper Vpp to make |b|<<1. In addition, we should also take the optimal value of |Cavc(α,Vpp)| to maximize the conversion efficiency of desired frequency SSB shifted signal.

 

Fig. 1 Schematic of PSSBM

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Figure 2 shows the amplitude variation coefficient corresponding to the converted power of desired SSB signal as a function of angle α and drive voltage Vpp. There are three features that can be found from these figures: (1) Given a fixed Vpp, the amplitude variation coefficient of desired signal shows a periodic change with the variation of angle α. The extremum will appear at α = kπ/4 corresponding to the two cases. (2) Given a fixed angle α, the amplitude variation coefficient of desired signal shows a symmetric change with the variation of Vpp where the extremum appears nearby Vpp = 0.6Vπ. (3) The amplitude variation coefficient shows a contrary variation of the two cases. Hence, the maximum conversion efficiency of PSSBM can be obtained by tuning α = π/4 and Vpp = 0.6Vπ.

 

Fig. 2 The |Cavc| vs. α and Vpp. for case1 (a), (b) and case2 (c), (d).

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3. Modeling results

3.1 Modeling results of PSSBM

To validate our proposed PSSBM scheme, we do the simulation through VPI software [38]. Figure 3 shows the output spectra of the PSSBM with frequency up- and down-shifted respectively with different SOPs under case1 (θ = 0, ϕ = 0) and case2 (θ = π/2, ϕ = 0). The global parameters we used here are, fm = 12.5GHz and Vpp = 0.27 Vπ of RF signal, λ0 = 1552.52 nm and Pin = 0 dBm for the input seed laser, and VDC = −Vπ/2. In these figures, Fn = f0 + nfm (n is an integer) represents the generated components of PSSBM (under the condition, the frequency down-shifted signal is the desired SSB signal F−1). The α1 and α2 equal 0 and −π/2 in Fig. 3(a), and 0 and −π/3 in Fig. 3(b) respectively. Obviously, the desired SSB frequency shift can be achieved as shown in Fig. 3(a) when the condition shown in Eq. (4) is satisfied. However, if the condition does not meet the requirement of the SSB frequency shift, the output will contain other unwanted crosstalk components as shown in Fig. 3(b), such as 0th-order F0 and 2nd-order F ± 2. Figures 3(i) and 3(ii) display the SOPs of the input (red point) and output (blue point) signals. We can see that the simulation results are in agreement with the above theoretical analysis where the SOPs of the input and output signals should be kept orthogonal. For case2 (θ = π/2, ϕ = 0), the same feature has been shown in Figs. 3(c) and 3(d). To achieve desired SSB frequency shift, the SOPs of the input and output signals should be kept identical as shown in Figs. 3(iii) and 3(iv). If not, the other unwanted components F0, F ± 2 and F ± 4 are existed.

 

Fig. 3 Output spectra of the PSSBM under case1 (θ = 0, ϕ = 0) and case2 (θ = π/2, ϕ = 0).

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3.2 Modeling results of PSMCG

As analyzed above, the PSSBM can be achieved. In this section, we mainly give the configuration and simulation results of the proposed PSSBM-based frequency-locked multicarrier generator (PSMCG) with recirculating frequency loop as shown in Fig. 4. The principle of PSMCG is the same as the typical SMCG with recirculating frequency loop [24]. The seed light from the TLS is fed into the PSSBM via a polarization-maintained optical coupler (PMOC). After the desired SSB frequency shifted by the PSSBM, the generated signals are amplified by the optical amplifier (OA) and then filtered by the optical bandpass filter (OBF) and finally coupled into the PSSBM again to form a recirculating frequency loop so as to generate the desired multiple frequency-locked carriers. PC1 and PC2 are used to adjust the SOPs of seed light and signals in the loop to maximize the output power. The generated carriers can be observed by the optical spectrum analyzer (OSA). If the high-flatness of output signals is required, the waveshaping devices, such as wavelength-selective-switch (WSS) shown in the dash-box in Fig. 4, can be used to further improve the output quality. By changing the bias control voltage of PSSBM, the frequency up- or down- converted carriers can be obtained.

 

Fig. 4 The configuration of PSMCG.

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The output spectra of PSMCG have been also simulated through the VPI software. Figure 5 shows the output spectra of PSMCG with 50 and 80 carriers corresponding to the bandwidth of 5nm and 8nm respectively. Here, the drive voltage Vpp = 0.27 Vπ, and the optical bandpass filter used in simulation is an ideal rectangle model with proper bandwidth which covers the desired 50- and 80-carrier frequency ranges. The assumed stopband attenuation is 40dB which is a common parameter and can be realized by the commercial products. The optical amplifier we used is an EDFA with black-box model which is operated at saturation mode with 23 dBm output power and 4 dB noise figure (NF). By sweeping the PC in the loop, the final 50- and 80-carrier outputs with an acceptable flatness have been obtained by the PSMCG as shown in Figs. 5(a) and 5(b) respectively. The insets show the SOPs of the output signals. Although the generated power of TNR is just 20 dB for 80-carrier output, there is no doubt that the proposed PSSBM can be used to achieve the generation of high-quality frequency-locked multicariers under further improved configuration. However, to implement the proposed scheme in practice, the polarization extinction ratio, which is the most critical property of the polarization modulator, is desired to be as high as possible. Meanwhile, some practical factors should be considered to improve the TNR of the generated carriers, including the SOPs of the input and output signals, the crosstalk of the undesired sideband components, the DC bias control, the gain and noise figure of the optical amplifier and the qualities of the optical bandpass filter etc.

 

Fig. 5 The output spectra of PSMCG with (a) 50 carriers and (b) 80 carriers.

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4. Conclusions

In conclusion, we have theoretically studied the proposed polarization-modulator-based SSB modulator as well as the frequency-locked multicarrier generator by utilizing it with recirculating frequency shifting loop, including the theoretical analysis and simulation of basic principle and output properties. Compared with the conventional integrated LiNbO3-based SSBM, the proposed PSSBM will only require two phase modulators and one DC bias control and also can be fabricated with integrated structure. Meanwhile, the typical operation frequency range of PolM is larger than 40 GHz which means the output carriers with large tunable frequency spacing can be achieved. According to the theoretical analysis and simulation results, the proposed PSSBM would find some potential applications in various scenarios.

Acknowledgments

This work is supported by the National Basic Research Programme of China (973) Project (No. 2012CB315603), the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), National Natural Science Foundation of China (NSFC) under Grant No. 61307092, the Planned Science and Technology Project of Guangzhou (Grant No. 2012J5100028), the Fundamental Research Funds for the Central Universities and the Program for New Century Excellent Talents in University (NCET-12-0679) in China.

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38. VPIsystemsTM, “VPltransmissionMakerTM”.

References

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  1. Z. Jiang, C. B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
    [Crossref]
  2. I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
    [Crossref]
  3. G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
    [Crossref] [PubMed]
  4. P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
    [Crossref]
  5. E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
    [Crossref]
  6. Y. Ma, Q. Yang, Y. Tang, S. Chen, W. Shieh, “1-Tb/s per channel coherent optical OFDM transmission with subwavelength bandwidth access,” in OFC (2009), PDPC1.
  7. S. Chandrasekhar, X. Liu, B. Zhu, D. W. Peckham, “Transmission of a 1.2-Tbps 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” in ECOC (2009), PD2.6.
  8. R. Dischler, F. Buchali, “Transmission of 1.2 Tb/s Continuous Waveband PDM-OFDM-FDM signal with Spectral Efficiency of 3.3 bit/s/Hz over 400 km of SSMF,” in OFC (2009), PDPC2.
  9. J. Yu, Z. Dong, X. Xiao, Y. Xin, S. Shi, C. Ge, W. Zhou, N. Chi, Y. Shao, “Generation, transmission and coherent detection of 11.2Tb/s (112 100 Gb/s) single source optical OFDM superchannel,” in OFC (2011), PDPA6.
  10. S. Chandrasekhar and X. Liu, “OFDM based superchannel transmission technology,” J. Lightwave Technol. 30(24), 3816–3823 (2012).
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  11. A. Ellis, F. Gunning, B. Cuenot, T. Healy, E. Pincemin, “Towards 1TbE using coherent WDM,” in OECC (2008), WeA-1.
  12. F. Tian, X. Zhang, L. Xi, A. J. Stark, S. E. Ralph, and G.-K. Chang, “Experiment of 2.56-Tb/s, polarization division multiplexing return-to-zero 16-ary quadrature amplitude modulation, 25 GHz grid coherent optical wavelength division multiplexing, 800 km transmission based on optical comb in standard single-mode fiber,” Opt. Eng. 52(11), 116103 (2013).
  13. Q. Yang, Z. He, W. Liu, Z. Yang, S. Yu; W. Shieh and I. B. Djordjevic, “1-Tb/s large girth LDPC-coded coherent optical OFDM transmission over 1040-km standard single-mode fiber,” in OFC (2011), ThA35.
  14. R. Hu, Q. Yang, X. Xiao, T. Gui, Z. Li, M. Luo, S. Yu, and S. You, “Direct-detection optical OFDM superchannel for long-reach PON using pilot regeneration,” Opt. Express 21(22), 26513–26519 (2013).
    [Crossref] [PubMed]
  15. Z. Li, X. Xiao, T. Gui, Q. Yang, R. Hu, Z. He, M. Luo, C. Li, X. Zhang, D. Xue, S. You, and S. Yu, “432-Gb/s direct-detection optical OFDM superchannel transmission over 3040-km SSMF,” IEEE Photon. Technol. 25(15), 1524–1526 (2013).
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  16. W. Peng, I. Morita, H. Takahashi, and T. Tsuritani, “Transmission of high-speed (> 100 Gb/s) direct-detection optical OFDM superchannel,” J. Lightwave Technol. 30(12), 2025–2034 (2012).
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  17. T. Healy, F. C. Garcia Gunning, A. D. Ellis, and J. D. Bull, “Multi-wavelength source using low drive-voltage amplitude modulators for optical communications,” Opt. Express 15(6), 2981–2986 (2007).
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  18. R. Wu, V. R. Supradeepa, C. M. Long, D. E. Leaird, and A. M. Weiner, “Generation of very flat optical frequency combs from continuous-wave lasers using cascaded intensity and phase modulators driven by tailored radio frequency waveforms,” Opt. Lett. 35(19), 3234–3236 (2010).
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  20. T. Yamamoto, T. Komukai, K. Suzuki, and A. Takada, “Multicarrier light source with flattened spectrum using phase modulators and dispersion medium,” J. Lightwave Technol. 27(19), 4297–4305 (2009).
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  21. C. He, S. Pan, R. Guo, Y. Zhao, and M. Pan, “Ultraflat optical frequency comb generated based on cascaded polarization modulators,” Opt. Lett. 37(18), 3834–3836 (2012).
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  22. V. Ataie, B. P.-P. Kuo, E. Myslivets and S. Radic, “Generation of 1500-tone, 120nm-wide ultraflat frequency comb by single CW source,” in OFC (2013), PDP5C.1.
  23. F. Tian, X. Zhang, J. Li, and L. Xi, “Generation of 50 stable frequency-locked optical carriers for Tb/s multicarrier optical transmission using a recirculating frequency shifter,” J. Lightwave Technol. 29(8), 1085–1091 (2011).
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  24. 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).
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  25. J. Li and Z. Li, “Frequency-locked multicarrier generator based on a complementary frequency shifter with double recirculating frequency-shifting loops,” Opt. Lett. 38(3), 359–361 (2013).
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  26. J. Li, C. Yu, and Z. Li, “Complementary frequency shifter based on polarization modulator used for generation of a high-quality frequency-locked multicarrier,” Opt. Lett. 39(6), 1513–1516 (2014).
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  27. J. Zhang, J. Yu, N. Chi, Y. Shao, L. Tao, J. Zhu, and Y. Wang, “Stable optical frequency-locked multi-carriers generation by double recirculating frequency shifter loops for Tb/s communication,” J. Lightwave Technol. 30(24), 3938–3945 (2012).
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  28. X. Li, J. Yu, Z. Dong, J. Zhang, Y. Shao, and N. Chi, “Multi-channel multi-carrier generation using multi-wavelength frequency shifting recirculating loop,” Opt. Express 20(20), 21833–21839 (2012).
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  29. J. Zhang, J. Yu, N. Chi, Y. Shao, L. Tao, Y. Wang, and X. Li, “Improved multicarriers generation by using multifrequency shifting recirculating loop,” IEEE Photon. Technol. 24(16), 1405–1408 (2012).
    [Crossref]
  30. J. Yu, Z. Dong, J. Zhang, X. Xiao, H. Chien, and N. Chi, “Generation of coherent and frequency-locked multi-carriers using cascaded phase modulators for 10 Tb/s optical transmission system,” J. Lightwave Technol. 30(4), 458–465 (2012).
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  31. J. Zhang, J. Yu, Z. Dong, Y. Shao, and N. Chi, “Generation of full C-band coherent and frequency-lock multi-carriers by using recirculating frequency shifter loops based on phase modulator with external injection,” Opt. Express 19(27), 26370–26381 (2011).
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  33. T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
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  34. T. Gui, C. Li, Q. Yang, X. Xiao, L. Meng, C. Li, X. Yi, C. Jin, and Z. Li, “Auto bias control technique for optical OFDM transmitter with bias dithering,” Opt. Express 21(5), 5833–5841 (2013).
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  35. J. P. Salvestrini, L. Guilbert, M. Fontana, M. Abarkan, and S. Gille, “Analysis and control of the DC drift in LiNbO3-based Mach-Zehnder modulators,” J. Lightwave Technol. 29(10), 1522–1534 (2011).
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  36. J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
    [Crossref]
  37. W. Liu, M. Wang, and J. Yao, “Tunable microwave and sub-terahertz generation based on frequency quadrupling using a single polarization modulator,” J. Lightwave Technol. 31(10), 1636–1644 (2013).
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  38. VPIsystemsTM, “VPltransmissionMakerTM”.

2014 (1)

2013 (6)

F. Tian, X. Zhang, L. Xi, A. J. Stark, S. E. Ralph, and G.-K. Chang, “Experiment of 2.56-Tb/s, polarization division multiplexing return-to-zero 16-ary quadrature amplitude modulation, 25 GHz grid coherent optical wavelength division multiplexing, 800 km transmission based on optical comb in standard single-mode fiber,” Opt. Eng. 52(11), 116103 (2013).

R. Hu, Q. Yang, X. Xiao, T. Gui, Z. Li, M. Luo, S. Yu, and S. You, “Direct-detection optical OFDM superchannel for long-reach PON using pilot regeneration,” Opt. Express 21(22), 26513–26519 (2013).
[Crossref] [PubMed]

Z. Li, X. Xiao, T. Gui, Q. Yang, R. Hu, Z. He, M. Luo, C. Li, X. Zhang, D. Xue, S. You, and S. Yu, “432-Gb/s direct-detection optical OFDM superchannel transmission over 3040-km SSMF,” IEEE Photon. Technol. 25(15), 1524–1526 (2013).
[Crossref]

T. Gui, C. Li, Q. Yang, X. Xiao, L. Meng, C. Li, X. Yi, C. Jin, and Z. Li, “Auto bias control technique for optical OFDM transmitter with bias dithering,” Opt. Express 21(5), 5833–5841 (2013).
[Crossref] [PubMed]

J. Li and Z. Li, “Frequency-locked multicarrier generator based on a complementary frequency shifter with double recirculating frequency-shifting loops,” Opt. Lett. 38(3), 359–361 (2013).
[Crossref] [PubMed]

W. Liu, M. Wang, and J. Yao, “Tunable microwave and sub-terahertz generation based on frequency quadrupling using a single polarization modulator,” J. Lightwave Technol. 31(10), 1636–1644 (2013).
[Crossref]

2012 (9)

C. He, S. Pan, R. Guo, Y. Zhao, and M. Pan, “Ultraflat optical frequency comb generated based on cascaded polarization modulators,” Opt. Lett. 37(18), 3834–3836 (2012).
[Crossref] [PubMed]

J. Li, X. Zhang, and L. Xi, “Generation of stable and high-quality frequency-locked carriers based on improved re-circulating frequency shifter,” Opt. Commun. 285(20), 4072–4075 (2012).
[Crossref]

W. Peng, I. Morita, H. Takahashi, and T. Tsuritani, “Transmission of high-speed (> 100 Gb/s) direct-detection optical OFDM superchannel,” J. Lightwave Technol. 30(12), 2025–2034 (2012).
[Crossref]

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

J. Zhang, J. Yu, N. Chi, Y. Shao, L. Tao, J. Zhu, and Y. Wang, “Stable optical frequency-locked multi-carriers generation by double recirculating frequency shifter loops for Tb/s communication,” J. Lightwave Technol. 30(24), 3938–3945 (2012).
[Crossref]

X. Li, J. Yu, Z. Dong, J. Zhang, Y. Shao, and N. Chi, “Multi-channel multi-carrier generation using multi-wavelength frequency shifting recirculating loop,” Opt. Express 20(20), 21833–21839 (2012).
[Crossref] [PubMed]

J. Zhang, J. Yu, N. Chi, Y. Shao, L. Tao, Y. Wang, and X. Li, “Improved multicarriers generation by using multifrequency shifting recirculating loop,” IEEE Photon. Technol. 24(16), 1405–1408 (2012).
[Crossref]

J. Yu, Z. Dong, J. Zhang, X. Xiao, H. Chien, and N. Chi, “Generation of coherent and frequency-locked multi-carriers using cascaded phase modulators for 10 Tb/s optical transmission system,” J. Lightwave Technol. 30(4), 458–465 (2012).
[Crossref]

S. Chandrasekhar and X. Liu, “OFDM based superchannel transmission technology,” J. Lightwave Technol. 30(24), 3816–3823 (2012).
[Crossref]

2011 (3)

2010 (4)

2009 (2)

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

T. Yamamoto, T. Komukai, K. Suzuki, and A. Takada, “Multicarrier light source with flattened spectrum using phase modulators and dispersion medium,” J. Lightwave Technol. 27(19), 4297–4305 (2009).
[Crossref]

2007 (2)

Z. Jiang, C. B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

T. Healy, F. C. Garcia Gunning, A. D. Ellis, and J. D. Bull, “Multi-wavelength source using low drive-voltage amplitude modulators for optical communications,” Opt. Express 15(6), 2981–2986 (2007).
[Crossref] [PubMed]

2004 (2)

T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
[Crossref]

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Abarkan, M.

Akbulut, M.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Avino, S.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

Bhooplapur, S.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Bull, J. D.

T. Healy, F. C. Garcia Gunning, A. D. Ellis, and J. D. Bull, “Multi-wavelength source using low drive-voltage amplitude modulators for optical communications,” Opt. Express 15(6), 2981–2986 (2007).
[Crossref] [PubMed]

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Chandrasekhar, S.

Chang, G.-K.

F. Tian, X. Zhang, L. Xi, A. J. Stark, S. E. Ralph, and G.-K. Chang, “Experiment of 2.56-Tb/s, polarization division multiplexing return-to-zero 16-ary quadrature amplitude modulation, 25 GHz grid coherent optical wavelength division multiplexing, 800 km transmission based on optical comb in standard single-mode fiber,” Opt. Eng. 52(11), 116103 (2013).

Chi, N.

Chien, H.

Coddington, I.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Davila-Rodriguez, J.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

De Natale, P.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

Delfyett, P. J.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Dong, Z.

Ellis, A. D.

Fairburn, M.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Ferraro, P.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

Fontana, M.

Gagliardi, G.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

Garcia Gunning, F. C.

Ghanipour, P.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Gille, S.

Gui, T.

Guilbert, L.

Guo, R.

Hamidi, E.

E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
[Crossref]

He, C.

He, Z.

Z. Li, X. Xiao, T. Gui, Q. Yang, R. Hu, Z. He, M. Luo, C. Li, X. Zhang, D. Xue, S. You, and S. Yu, “432-Gb/s direct-detection optical OFDM superchannel transmission over 3040-km SSMF,” IEEE Photon. Technol. 25(15), 1524–1526 (2013).
[Crossref]

Healy, T.

Hoghooghi, N.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Hu, R.

R. Hu, Q. Yang, X. Xiao, T. Gui, Z. Li, M. Luo, S. Yu, and S. You, “Direct-detection optical OFDM superchannel for long-reach PON using pilot regeneration,” Opt. Express 21(22), 26513–26519 (2013).
[Crossref] [PubMed]

Z. Li, X. Xiao, T. Gui, Q. Yang, R. Hu, Z. He, M. Luo, C. Li, X. Zhang, D. Xue, S. You, and S. Yu, “432-Gb/s direct-detection optical OFDM superchannel transmission over 3040-km SSMF,” IEEE Photon. Technol. 25(15), 1524–1526 (2013).
[Crossref]

Huang, C. B.

Z. Jiang, C. B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Izutsu, M.

T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
[Crossref]

Jaeger, N. A. F.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Jiang, Z.

Z. Jiang, C. B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Jin, C.

Kato, H.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Kawanishi, T.

T. Kawanishi and M. Izutsu, “Linear single-sideband modulation for high-SNR wavelength conversion,” IEEE Photon. Technol. Lett. 16(6), 1534–1536 (2004).
[Crossref]

Komukai, T.

Leaird, D. E.

R. Wu, V. R. Supradeepa, C. M. Long, D. E. Leaird, and A. M. Weiner, “Generation of very flat optical frequency combs from continuous-wave lasers using cascaded intensity and phase modulators driven by tailored radio frequency waveforms,” Opt. Lett. 35(19), 3234–3236 (2010).
[Crossref] [PubMed]

E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
[Crossref]

Z. Jiang, C. B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Li, C.

Li, J.

Li, X.

Li, Z.

Liu, W.

Liu, X.

Long, C. M.

Luo, M.

Z. Li, X. Xiao, T. Gui, Q. Yang, R. Hu, Z. He, M. Luo, C. Li, X. Zhang, D. Xue, S. You, and S. Yu, “432-Gb/s direct-detection optical OFDM superchannel transmission over 3040-km SSMF,” IEEE Photon. Technol. 25(15), 1524–1526 (2013).
[Crossref]

R. Hu, Q. Yang, X. Xiao, T. Gui, Z. Li, M. Luo, S. Yu, and S. You, “Direct-detection optical OFDM superchannel for long-reach PON using pilot regeneration,” Opt. Express 21(22), 26513–26519 (2013).
[Crossref] [PubMed]

Meng, L.

Morita, I.

Nenadovic, L.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Newbury, N. R.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Ozdur, I.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Pan, M.

Pan, S.

Peng, W.

Ralph, S. E.

F. Tian, X. Zhang, L. Xi, A. J. Stark, S. E. Ralph, and G.-K. Chang, “Experiment of 2.56-Tb/s, polarization division multiplexing return-to-zero 16-ary quadrature amplitude modulation, 25 GHz grid coherent optical wavelength division multiplexing, 800 km transmission based on optical comb in standard single-mode fiber,” Opt. Eng. 52(11), 116103 (2013).

Reid, A.

J. D. Bull, N. A. F. Jaeger, H. Kato, M. Fairburn, A. Reid, and P. Ghanipour, “40 GHz electro-optic polarization modulator for fiber optic communications systems,” Proc. SPIE 5577, 133–143 (2004).
[Crossref]

Salvestrini, J. P.

Salza, M.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

Shao, Y.

Stark, A. J.

F. Tian, X. Zhang, L. Xi, A. J. Stark, S. E. Ralph, and G.-K. Chang, “Experiment of 2.56-Tb/s, polarization division multiplexing return-to-zero 16-ary quadrature amplitude modulation, 25 GHz grid coherent optical wavelength division multiplexing, 800 km transmission based on optical comb in standard single-mode fiber,” Opt. Eng. 52(11), 116103 (2013).

Supradeepa, V. R.

Suzuki, K.

Swann, W. C.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Takada, A.

Takahashi, H.

Tao, L.

J. Zhang, J. Yu, N. Chi, Y. Shao, L. Tao, Y. Wang, and X. Li, “Improved multicarriers generation by using multifrequency shifting recirculating loop,” IEEE Photon. Technol. 24(16), 1405–1408 (2012).
[Crossref]

J. Zhang, J. Yu, N. Chi, Y. Shao, L. Tao, J. Zhu, and Y. Wang, “Stable optical frequency-locked multi-carriers generation by double recirculating frequency shifter loops for Tb/s communication,” J. Lightwave Technol. 30(24), 3938–3945 (2012).
[Crossref]

Tian, F.

F. Tian, X. Zhang, L. Xi, A. J. Stark, S. E. Ralph, and G.-K. Chang, “Experiment of 2.56-Tb/s, polarization division multiplexing return-to-zero 16-ary quadrature amplitude modulation, 25 GHz grid coherent optical wavelength division multiplexing, 800 km transmission based on optical comb in standard single-mode fiber,” Opt. Eng. 52(11), 116103 (2013).

F. Tian, X. Zhang, J. Li, and L. Xi, “Generation of 50 stable frequency-locked optical carriers for Tb/s multicarrier optical transmission using a recirculating frequency shifter,” J. Lightwave Technol. 29(8), 1085–1091 (2011).
[Crossref]

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]

Tsuritani, T.

Wang, M.

Wang, Y.

J. Zhang, J. Yu, N. Chi, Y. Shao, L. Tao, Y. Wang, and X. Li, “Improved multicarriers generation by using multifrequency shifting recirculating loop,” IEEE Photon. Technol. 24(16), 1405–1408 (2012).
[Crossref]

J. Zhang, J. Yu, N. Chi, Y. Shao, L. Tao, J. Zhu, and Y. Wang, “Stable optical frequency-locked multi-carriers generation by double recirculating frequency shifter loops for Tb/s communication,” J. Lightwave Technol. 30(24), 3938–3945 (2012).
[Crossref]

Weiner, A. M.

R. Wu, V. R. Supradeepa, C. M. Long, D. E. Leaird, and A. M. Weiner, “Generation of very flat optical frequency combs from continuous-wave lasers using cascaded intensity and phase modulators driven by tailored radio frequency waveforms,” Opt. Lett. 35(19), 3234–3236 (2010).
[Crossref] [PubMed]

E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable programmable microwave photonic filters based on an optical frequency comb,” IEEE Trans. Microw. Theory Tech. 58(11), 3269–3278 (2010).
[Crossref]

Z. Jiang, C. B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Wu, R.

Xi, L.

F. Tian, X. Zhang, L. Xi, A. J. Stark, S. E. Ralph, and G.-K. Chang, “Experiment of 2.56-Tb/s, polarization division multiplexing return-to-zero 16-ary quadrature amplitude modulation, 25 GHz grid coherent optical wavelength division multiplexing, 800 km transmission based on optical comb in standard single-mode fiber,” Opt. Eng. 52(11), 116103 (2013).

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[Crossref]

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).
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[Crossref]

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[Crossref]

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).
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Figures (5)

Fig. 1
Fig. 1 Schematic of PSSBM
Fig. 2
Fig. 2 The |Cavc| vs. α and Vpp. for case1 (a), (b) and case2 (c), (d).
Fig. 3
Fig. 3 Output spectra of the PSSBM under case1 (θ = 0, ϕ = 0) and case2 (θ = π/2, ϕ = 0).
Fig. 4
Fig. 4 The configuration of PSMCG.
Fig. 5
Fig. 5 The output spectra of PSMCG with (a) 50 carriers and (b) 80 carriers.

Equations (7)

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E o 1 ( t ) = c 1 [ E i n x ( t ) exp ( j π f 1 ( t ) V π ) + E i n y ( t ) exp ( j π f 1 ( t ) V π ) ] E o 2 ( t ) = c 2 [ E i n x ( t ) exp ( j π f 2 ( t ) V π ) + E i n y ( t ) exp ( j π f 2 ( t ) V π ) ]
H i n = [ cos α 1 sin α 1 ] , H P o l = [ cos α 2 sin α 2 ] , H P C = [ cos θ sin θ sin θ cos θ ] [ exp ( j ϕ 2 ) 0 0 exp ( j ϕ 2 ) ] ; H P o l M = [ H 1 x + H P M H 2 x 0 0 H 1 y + H P M H 2 y ] ; H P M = exp [ j π V D C ( t ) / V π ] ; H 1 x = H 1 y * = exp [ j π f 1 ( t ) / V π ] = exp [ j δ m cos ( ω m t ) ] ; H 2 x = H 2 y * = exp [ j π f 2 ( t ) / V π ] = exp [ j δ m sin ( ω m t ) ]
T P S S B M = H P o l H P C H P o l M H i n = { C 1 exp [ j δ m cos ( ω m t ) ] C 2 exp [ j δ m cos ( ω m t ) ] } + exp ( j ϕ D C ) { C 1 exp [ j δ m sin ( ω m t ) ] C 2 exp [ j δ m sin ( ω m t ) ] } C 1 = [ cos α 2 cos α 1 cos θ + sin α 2 cos α 1 sin θ ] exp ( j ϕ 2 ) ; C 2 = [ cos α 2 sin α 1 sin θ sin α 2 sin α 1 cos θ ] exp ( j ϕ 2 )
  C 1 = C 2 = C [ cos α 2 cos α 1 cos θ + sin α 2 cos α 1 sin θ ] exp ( j ϕ 2 ) = [ cos α 2 sin α 1 sin θ sin α 2 sin α 1 cos θ ] exp ( j ϕ 2 )
  T P S S B M = | C a v c ( α ) | [ sin ( δ m cos ( ω m t ) ) ± j sin ( δ m sin ( ω m t ) ) ]
  T P S S B M = | C a v c ( α ) | [ sin ( δ m cos ( ω m t ) ) ± j sin ( δ m sin ( ω m t ) ) ] = | C a v c ( α ) | { k = 1 j 2 k 2 J 2 k 1 ( δ m ) co s [ ( 2 k 1 ) ω m t ] ± j k = 1 J 2 k 1 ( δ m ) sin [ ( 2 k 1 ) ω m t ] } = | C ( α ) a v c | { [ J 1 ( δ m ) exp ( ± j ω m t ) J 3 ( δ m ) exp ( j 3 ω m t ) ] ) + [ J 5 ( δ m ) exp ( j ± 5 ω m t ) ] }
  T P S S B M = [ exp ( ± j ω m t ) + b exp ( j 3 ω m t ) ]

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