Abstract

We present an analytical and numerical description of optical Nyquist pulse propagation in optical fibers in the presence of dispersion and nonlinearity. An optical Nyquist pulse has a profile given by the sinc-like impulse response of a Nyquist filter, which has periodic zero-crossing points at every symbol interval. This property makes it possible to interleave bits to an ultrahigh symbol rate with no intersymbol interference in spite of the strong overlap between adjacent pulses. We analyze how this periodic zero-crossing property is maintained or affected by the fiber dispersion and nonlinearity, and show that it is better maintained against nonlinearity in the presence of normal dispersion.

© 2012 OSA

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References

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  1. H. G. Weber and M. Nakazawa, eds., Ultrahigh-Speed Optical Transmission Technology (Springer, 2007).
  2. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett.23(3), 142–144 (1973).
    [CrossRef]
  3. M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, “Marked increase in the power margin through the use of a dispersion-allocated soliton,” IEEE Photon. Technol. Lett.8(8), 1088–1090 (1996).
    [CrossRef]
  4. D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser – Part I: Theory,” IEEE J. Quantum Electron.6(11), 694–708 (1970).
    [CrossRef]
  5. H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys.46(7), 3049–3058 (1975).
    [CrossRef]
  6. M. Nakazawa, T. Hirooka, P. Ruan, and P. Guan, “Ultrahigh-speed “orthogonal” TDM transmission with an optical Nyquist pulse train,” Opt. Express20(2), 1129–1140 (2012).
    [CrossRef] [PubMed]
  7. H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans.47, 617–644 (1928).
  8. K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express5(1), 6–10 (2008).
    [CrossRef]
  9. G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett.22(15), 1129–1131 (2010).
    [CrossRef]
  10. X. Zhou, L. E. Nelson, P. Magill, B. Zhu, and D. W. Peckham, “8x450-Gb/s, 50-GHz spaced, PDM-32QAM transmission over 400 km and one 50 GHz-grid ROADM,” in Optical Fiber Communication Conference (OFC 2011), paper PDPB3.
  11. R. Schmogrow, M. Meyer, S. Wolf, B. Nebendahl, D. Hillerkuss, B. Baeuerle, M. Dreschmann, J. Meyer, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “150 Gbit/s real-time Nyquist pulse transmission over 150 km SSMF enhanced by DSP with dynamic precision,” in Optical Fiber Communication Conference (OFC 2012), paper OM2A.6.
  12. J. G. Proakis, Digital Transmission, 4th ed. (McGraw Hill, 2000).
  13. T. Hirooka, P. Ruan, P. Guan, and M. Nakazawa, “Highly dispersion-tolerant 160 Gbaud optical Nyquist pulse TDM transmission over 525 km,” Opt. Express20(14), 15001–15007 (2012).
    [CrossRef] [PubMed]
  14. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).

2012

2010

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett.22(15), 1129–1131 (2010).
[CrossRef]

2008

K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express5(1), 6–10 (2008).
[CrossRef]

1996

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, “Marked increase in the power margin through the use of a dispersion-allocated soliton,” IEEE Photon. Technol. Lett.8(8), 1088–1090 (1996).
[CrossRef]

1975

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys.46(7), 3049–3058 (1975).
[CrossRef]

1973

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett.23(3), 142–144 (1973).
[CrossRef]

1970

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser – Part I: Theory,” IEEE J. Quantum Electron.6(11), 694–708 (1970).
[CrossRef]

1928

H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans.47, 617–644 (1928).

Bosco, G.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett.22(15), 1129–1131 (2010).
[CrossRef]

Carena, A.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett.22(15), 1129–1131 (2010).
[CrossRef]

Curri, V.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett.22(15), 1129–1131 (2010).
[CrossRef]

Forghieri, F.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett.22(15), 1129–1131 (2010).
[CrossRef]

Goto, H.

K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express5(1), 6–10 (2008).
[CrossRef]

Guan, P.

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett.23(3), 142–144 (1973).
[CrossRef]

Haus, H. A.

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys.46(7), 3049–3058 (1975).
[CrossRef]

Hirooka, T.

Hongo, J.

K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express5(1), 6–10 (2008).
[CrossRef]

Kasai, K.

K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express5(1), 6–10 (2008).
[CrossRef]

Kubota, H.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, “Marked increase in the power margin through the use of a dispersion-allocated soliton,” IEEE Photon. Technol. Lett.8(8), 1088–1090 (1996).
[CrossRef]

Kuizenga, D. J.

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser – Part I: Theory,” IEEE J. Quantum Electron.6(11), 694–708 (1970).
[CrossRef]

Nakazawa, M.

T. Hirooka, P. Ruan, P. Guan, and M. Nakazawa, “Highly dispersion-tolerant 160 Gbaud optical Nyquist pulse TDM transmission over 525 km,” Opt. Express20(14), 15001–15007 (2012).
[CrossRef] [PubMed]

M. Nakazawa, T. Hirooka, P. Ruan, and P. Guan, “Ultrahigh-speed “orthogonal” TDM transmission with an optical Nyquist pulse train,” Opt. Express20(2), 1129–1140 (2012).
[CrossRef] [PubMed]

K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express5(1), 6–10 (2008).
[CrossRef]

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, “Marked increase in the power margin through the use of a dispersion-allocated soliton,” IEEE Photon. Technol. Lett.8(8), 1088–1090 (1996).
[CrossRef]

Nyquist, H.

H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans.47, 617–644 (1928).

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett.22(15), 1129–1131 (2010).
[CrossRef]

Ruan, P.

Sahara, A.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, “Marked increase in the power margin through the use of a dispersion-allocated soliton,” IEEE Photon. Technol. Lett.8(8), 1088–1090 (1996).
[CrossRef]

Siegman, A. E.

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser – Part I: Theory,” IEEE J. Quantum Electron.6(11), 694–708 (1970).
[CrossRef]

Tamura, K.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, “Marked increase in the power margin through the use of a dispersion-allocated soliton,” IEEE Photon. Technol. Lett.8(8), 1088–1090 (1996).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett.23(3), 142–144 (1973).
[CrossRef]

Yoshida, M.

K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express5(1), 6–10 (2008).
[CrossRef]

AIEE Trans.

H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans.47, 617–644 (1928).

Appl. Phys. Lett.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett.23(3), 142–144 (1973).
[CrossRef]

IEEE J. Quantum Electron.

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser – Part I: Theory,” IEEE J. Quantum Electron.6(11), 694–708 (1970).
[CrossRef]

IEEE Photon. Technol. Lett.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, “Marked increase in the power margin through the use of a dispersion-allocated soliton,” IEEE Photon. Technol. Lett.8(8), 1088–1090 (1996).
[CrossRef]

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett.22(15), 1129–1131 (2010).
[CrossRef]

IEICE Electron. Express

K. Kasai, J. Hongo, H. Goto, M. Yoshida, and M. Nakazawa, “The use of a Nyquist filter for reducing an optical signal bandwidth in a coherent QAM optical transmission,” IEICE Electron. Express5(1), 6–10 (2008).
[CrossRef]

J. Appl. Phys.

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys.46(7), 3049–3058 (1975).
[CrossRef]

Opt. Express

Other

H. G. Weber and M. Nakazawa, eds., Ultrahigh-Speed Optical Transmission Technology (Springer, 2007).

X. Zhou, L. E. Nelson, P. Magill, B. Zhu, and D. W. Peckham, “8x450-Gb/s, 50-GHz spaced, PDM-32QAM transmission over 400 km and one 50 GHz-grid ROADM,” in Optical Fiber Communication Conference (OFC 2011), paper PDPB3.

R. Schmogrow, M. Meyer, S. Wolf, B. Nebendahl, D. Hillerkuss, B. Baeuerle, M. Dreschmann, J. Meyer, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “150 Gbit/s real-time Nyquist pulse transmission over 150 km SSMF enhanced by DSP with dynamic precision,” in Optical Fiber Communication Conference (OFC 2012), paper OM2A.6.

J. G. Proakis, Digital Transmission, 4th ed. (McGraw Hill, 2000).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).

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Figures (10)

Fig. 1
Fig. 1

Propagation of an optical Nyquist pulse in the presence of GVD with T = 6.25 ps and α = 0 (a), 0.5 (b), and 1 (c). The left figures show the pulse evolution as a function of the cumulative dispersion |β2L|, and the right figures are the waveforms for |β2L| = 0 (black) and 10 ps2 (red). The dots in the right figure of (a) show the analytical result given by Eq. (10).

Fig. 2
Fig. 2

Propagation of an optical Nyquist pulse (T = 6.25 ps and α = 0.5) in a periodically dispersion-managed fiber. The left figure shows the pulse evolution within a 75 km span, and the right figure shows the pulse waveform at each period.

Fig. 3
Fig. 3

Extinction ratio of the pulse intensity between the center and the adjacent symbol point, η, as a function of the cumulative dispersion |β2L|. Red, blue, and green curves are the results obtained with optical Nyquist, Gaussian, and sech pulses, respectively. (b) An expanded view of the dotted area in (a).

Fig. 4
Fig. 4

Comparison of η in the numerical and analytical results (Eq. (17)) for an optical Nyquist pulse with α = 0.

Fig. 5
Fig. 5

Eye diagram of a 160 Gbaud OOK optical Nyquist pulse train with α = 0.5 (left) and the demultiplexed 40 Gbaud signal (right) in the presence of GVD. The power is normalized to the peak intensity of a single Nyquist pulse without dispersion as indicated by the blue lines.

Fig. 6
Fig. 6

Eye diagram of a 160 Gbaud OOK Gaussian pulse train (left) and the demultiplexed 40 Gbaud signal (right) in the presence of GVD.

Fig. 7
Fig. 7

Evolution of an optical Nyquist pulse spectrum with T = 6.25 ps and α = 0.5 in the presence of SPM. The left figures show the spectrum evolution as a function of distance, and the right figures are the input spectrum (black) and the spectrum after 525 km propagation (red).

Fig. 8
Fig. 8

Evolution of an optical Nyquist pulse waveform (left) and spectrum (right) with T = 6.25 ps and α = 0.5 in the presence of SPM and anomalous GVD (0.1 ps/nm/km).

Fig. 9
Fig. 9

Evolution of an optical Nyquist pulse waveform (left) and spectrum (right) with T = 6.25 ps and α = 0.5 in the presence of SPM and a normal GVD (−0.1 ps/nm/km).

Fig. 10
Fig. 10

(a) The change in the zero-crossing point around t = T (6.25 ps) and its deviation for various transmission power values. The open and closed circles correspond to normal and anomalous dispersion (−0.1 and + 0.1 ps/nm/km), respectively. (b) Extinction ratio η corresponding to (a).

Equations (19)

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r(t)= sin(πt/T) πt/T cos(απt/T) 1 (2αt/T) 2 ,R(f)={ T,0|f| 1α 2T T 2 { 1sin[ π 2α (2T|f|1) ] }, 1α 2T |f| 1+α 2T 0,|f| 1+α 2T ,
u(nT)={ 1, n=0 0, n0 ,
m= U(f+m/T)=T .
i u z β 2 2 2 u t 2 =0,
u(L,t)= 1 2π U(0,ω)exp( i β 2 L 2 ω 2 iωt )dω = T 4π π T (1+α) π T (1α) { 1+sin( Tω+π 2α ) }exp ( i β 2 L 2 ω 2 iωt )dω + T 2π π T (1α) π T (1α) exp ( i β 2 L 2 ω 2 iωt )dω+ T 4π π T (1α) π T (1+α) { 1sin( Tωπ 2α ) }exp ( i β 2 L 2 ω 2 iωt )dω,
u(L,t)= T 4π π T (1+α) π T (1+α) exp ( i β 2 L 2 ω 2 iωt )dω+ T 4π π T (1α) π T (1α) exp ( i β 2 L 2 ω 2 iωt )dω T 2π π T (1α) π T (1+α) sin( Tωπ 2α )cosωtexp ( i β 2 L 2 ω 2 )dω.
u(L,t)= T 8 1 iπ β 2 L/2 { F( t; π T (1+α) )F( t; π T (1+α) )+F( t; π T (1α) )F( t; π T (1α) ) } + T 16 i π β 2 L/2 [ exp( iπ 2α ){ F( t+ α 2 ; π T (1+α) )F( t+ α 2 ; π T (1α) ) } +exp( iπ 2α ){ F( ( t+ α 2 ); π T (1+α) )F( ( t+ α 2 ); π T (1α) ) } +exp( iπ 2α ){ F( t α 2 ; π T (1+α) )F( t α 2 ; π T (1α) ) } +exp( iπ 2α ){ F( ( t α 2 ); π T (1+α) )F( ( t α 2 ); π T (1α) ) } ].
F(τ;β)=exp( i τ 2 2 β 2 L )erfi[ i β 2 L 2 ( τ β 2 L +β ) ],
erfi(z)= erf(iz) i = 2 i π 0 iz exp( t 2 )dt
u(L,t)= T 4 1 iπ β 2 L/2 { F( t; π T )F( t; π T ) }.
d dz erfi(z)= 2exp( z 2 ) π .
t | u(L,t) | 2 = i T 2 2 β 2 Lπ π { erfi[ i β 2 L 2 ( t β 2 L + π T ) ]erfi[ i β 2 L 2 ( t β 2 L π T ) ] } ×exp[ i t 2 2 β 2 L i β 2 L 2 ( π T ) 2 ]sin πt T .
erfi(z)=i+ exp( z 2 ) π ( 1 z + 1 2 z 3 + )forz>> 1.
u(t)~ Texp( i β 2 L 2 ( π T ) 2 ) 4π i β 2 L/2 { exp( i π T t ) i 2 β 2 L t+ i β 2 L 2 π T + exp( i π T t ) i 2 β 2 L t+ i β 2 L 2 π T }.
|u(nT) | 2 = ( β 2 L) 2 { (nT) 2 ( β 2 L π T ) 2 } 2 .
η= | u(T) | 2 | u(0) | 2 ,
η= ( β 2 L) 2 { T 2 ( β 2 L π T ) 2 } 2 .
i u z +γ | u | 2 u=i Γ 2 u.
i u z β 2 2 2 u t 2 +γ | u | 2 u=i Γ 2 u.

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