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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 23 — Nov. 17, 2003
  • pp: 3122–3127
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Simultaneous residual chromatic dispersion monitoring and frequency conversion with gain using a parametric amplifier

T. T. Ng, J. L. Blows, J. T. Mok, P. Hu, J. A. Bolger, P. Hambley, and B. J. Eggleton  »View Author Affiliations


Optics Express, Vol. 11, Issue 23, pp. 3122-3127 (2003)
http://dx.doi.org/10.1364/OE.11.003122


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Abstract

We demonstrate that chromatic dispersion induced pulse-width broadening can be effectively monitored by a simple average power measurement of the filtered output from a parametric amplifier when additional four-wave mixing interactions are introduced. This all-optical technique also provides all-optical frequency conversion of the signal being monitored and signal gain.

© 2003 Optical Society of America

1. Introduction

Future optical networks will operate at bit rates of around 100 Gbits/s thus requiring return-to-zero modulation and pulse widths of a few picoseconds. Constant monitoring and subsequent correction of residual chromatic dispersion will become essential because the pulse degradation resulting from chromatic dispersion is much worse for picosecond pulses compared to the relatively longer pulses now used [1

1. B.J. Eggleton, et al, “Integrated tunable fiber gratings for dispersion management in high-bit rate systems” J. Lightwave Technol. 18, 1418–1432 (2000). [CrossRef]

,2

2. P.S. Westbrook, B.J. Eggleton, G. Raybon, S. Hunsche, and T.H. Her, “Measurement of Residual Chromatic dispersion of a 40-Gb/s RZ Signal via Spectral Broadening,” IEEE Photon. Tech. Lett. 14, 346–348 (2002) [CrossRef]

]. While there are various electrical methods to measure the residual chromatic dispersion including bit error rate measurements [3

3. H. Ohita, S. Nogiwa, Y. Kawaguchi, and Y. Endo, “Measurement of 200 Gbit/s optical eye diagram by optical sampling with gain-switched optical pulses,” Electron. Lett 36, 737 (2000) [CrossRef]

] and RF spectrum monitoring [4

4. Z. Panet al., “Chromatic dispersion monitoring and automated compensation for NRZ and RZ data using clock regeneration and fading without adding signaling,” in Optical Fiber Communication Conference, (Optical Society of America, Washington, DC, 2001), WH5

] it is preferable to perform monitoring and signal processing functions all optically because electrical signal processing will not be economical at these high bit rates.

In this paper we demonstrate an all-optical technique that monitors pulse broadening as a result of residual chromatic dispersion. The proposed scheme relies on four-wave mixing in a nonlinear optical fiber and involves a simple average power measurement of the filtered output from a parametric amplifier. While all-optical dispersion monitoring using nonlinear optics has been previously demonstrated [2

2. P.S. Westbrook, B.J. Eggleton, G. Raybon, S. Hunsche, and T.H. Her, “Measurement of Residual Chromatic dispersion of a 40-Gb/s RZ Signal via Spectral Broadening,” IEEE Photon. Tech. Lett. 14, 346–348 (2002) [CrossRef]

,5

5. S. Wielandyet al., “Real-time measurement of accumulated chromatic dispersion for automatic dispersion compensation,” Electon. Lett. 38, 1198–1199 (2002) [CrossRef]

], here we present a dispersion monitoring device with the additional functionalities of gain with low noise figure [6

6. J. Blows and S. French, “Low noise figure optical parametric amplifier with a continuous-wave frequency-modulated pump,” Opt. Lett. 27, 491–493 (2002) [CrossRef]

] and the provision of a copy of the signal at a new frequency [7

7. R. McKerracher, J. Blows, and M.C. deSterke, “Wavelength conversion bandwidth in fibre based optical parametric amplifiers” Opt. Express 11, 1002–1007 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1002. [CrossRef] [PubMed]

].

Recently, a technique for multiplying the repetition rate of a periodic train of pulses has been proposed and demonstrated [15

15. J. Azana and M.A. Muriel, “Technique for multiplying the repetition rate of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. 24, 1672–1674 (1999) [CrossRef]

17

17. J. Azana and M.A. Muriel, “Temportal self-imaging effects: Theory and Application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001) [CrossRef]

] using chirped fiber gratings. We demonstrate three to seven times frequency multiplication of the input 10GHz pulse train using a single tunable dispersion compensator, which is an adjustable chirped fiber grating.

2. Device concept

The device utilizes multiple and simultaneous degenerate wave-mixing processes in optical fiber [8

8. E. Ciaramella, F. Curti, and S. Trillo, “All-optical reshaping by means of Four-Wave Mixing in Optical Fibers,” IEEE Phton. Technol. Lett. 13, 142–144 (2001) [CrossRef]

10

10. J. Blows, “Crosstalk in a fibre parametric wavelength converter,” Trends in Optics and Photonics. 86, Optical Fiber Communication Conference, (Optical Society of America, Washington, DC, 2003), pp 565

]. In degenerate wave mixing two ‘pump’ photons are annihilated and one signal and one idler photons are created simultaneously. Because energy is conserved the signal and idler frequencies are located symmetrically on either side of the ‘pump’. We divide the wave mixing processes that occur in the device into degenerate parametric amplification and degenerate four-wave-mixing. Parametric amplification occurs when the pump is intense and the linear phase mismatch between the pump, signal and idler waves are small. In this case, the interaction is strong and significant power is transferred to the signal and idler waves from the pump wave. Signal gains of hundreds or thousands of times are common [6

6. J. Blows and S. French, “Low noise figure optical parametric amplifier with a continuous-wave frequency-modulated pump,” Opt. Lett. 27, 491–493 (2002) [CrossRef]

,11

11. J.H. Hansryd and P.A. Andrekson, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Selec. Top. Quat. Electon. 8, 506–520 (2002) [CrossRef]

12

12. K.K.Y. Wong, M.E. Mahic, K. Uesaka, and L.G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Tech. Lett. 7, 911–913 (2002) [CrossRef]

] because the gain is at best exponentially dependant on the pump power. Degenerate four-wave-mixing however provides very little signal gain because the ‘pump’ is weak and the linear phase matching is poor. The growth of the signal is at best only proportional to the pump power squared [13

13. F. Forghieri, R.W. Tkach, and A.R. Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications IIIA, (1997) Academic Press (San Diego)

].

The degenerate four-wave-mixing creates new frequencies with instantaneous powers that are nonlinearly dependant on the instantaneous ‘pump’ power. Consequently, an average power measurement of the new frequency will decrease with decreasing peak power of the ‘pump’, even though the average power of the ‘pump’ is unchanged. We can use this relationship to measure chromatic dispersion. Because the effect of chromatic dispersion is to reduce the peak power of the pulsed pum a simple average power measurement of the new frequency would monitor the residual net dispersion experienced by the pulse train. Because the technique is more sensitive to peak power than pulse shape it works equally as well for Gaussian or sech2 pulses. However, the power in the new frequency peak is too small to be practically measured and thus parametric amplification is used to conveniently boost these seeded peaks to practical levels [10

10. J. Blows, “Crosstalk in a fibre parametric wavelength converter,” Trends in Optics and Photonics. 86, Optical Fiber Communication Conference, (Optical Society of America, Washington, DC, 2003), pp 565

]. The parametric amplification also provides strong gain for the signal and simultaneously provides an idler which contains the same data as the signal but at a new frequency.

A schematic of the device concept is shown in Fig. 1. The signal passes through the tunable dispersion compensator, followed by a parametric amplifier. Degenerate four wave mixing combined with parametric gain in the amplifier creates a new frequency with an average power which is dependant on the residual dispersion. This new frequency is separated and measured using a slow detector. The signal from this slow detector is then used to determine a new operational point for the tunable dispersion compensator that minimizes the residual chromatic dispersion. The dispersion compensated and now amplified signal is allowed to continue its journey together with a copy of the signal at a new frequency. We demonstrate the monitoring concept is ready for inclusion into a feedback loop such as shown in Fig. 1.

Fig. 1. (Top) The signal undergoes dispersion compensation and then amplification and simultaneous frequency conversion. A simple average power measurement of a new monitor frequency is used to minimize the residual dispersion. (Bottom) The optical spectrum of points (I) through (IV). The peaks are signal (s), pump (p), idler (i) and monitor (m)

3. Experimental setup

A schematic diagram of the experiment is shown in Fig. 2. The parametric amplifier is pumped with the amplified output of a 1550.2 nm DFB laser diode. The laser is driven with a current that oscillates between 25 mA and 75 mA at a frequency of 300 kHz, the modulation being necessary to suppress the detrimental effects of stimulated Brillouin scattering in the fiber used as the nonlinear gain medium [6

6. J. Blows and S. French, “Low noise figure optical parametric amplifier with a continuous-wave frequency-modulated pump,” Opt. Lett. 27, 491–493 (2002) [CrossRef]

]. The output from the DFB laser passes through a polarization controller and then is amplified with a high power erbium doped fiber amplifier. To ensure low noise operation of the parametric amplifier the output of the EDFA is filtered with a Bragg fiber grating-circulator pair [6

6. J. Blows and S. French, “Low noise figure optical parametric amplifier with a continuous-wave frequency-modulated pump,” Opt. Lett. 27, 491–493 (2002) [CrossRef]

].

At point ‘A’ in Fig. 2 the pulses have a near-Gaussian optical power spectrum envelope with deep modulations at the repetition frequency. The FWHM of the optical spectrum was measured to be 0.57 nm with 10% of the power in the spectrum lost due to clipping by the tunable dispersion compensator. The pulse temporal full-width-half-maximum was measured to be 7.86 ps and thus the time-bandwidth product is 0.56.

Fig. 2. Schematic diagram of the experimental setup. PC-Polarization controller, FBG-Fiber Bragg grating and circulator based filter, MFL-Modelocked fiber laser, TDC-Tuneable dispersion compensator, VA-Variable attenuator, Attn-Attenuator, TBF-Tuneable bandpass filter, PM-Power meter, OSA-Optical Spectrum Analyzer.

The parametric amplifier pump and probe signal are coupled with a 90/10 coupler and then launched into 1.5km of Corning dispersion shifted fiber with a zero dispersion wavelength of 1549±2 nm, nonlinearity of approximately 2W-1Km-1 and a dispersion slope of 0.0816 ps/nm2/km. Strong four-wave mixing and parametric gain take place within this fiber. The output from the fiber is then attenuated and measured with an optical spectrum analyzer (0.06nm resolution) and simultaneously the average power of the desired peak measured after spectral filtering.

4. Results

Fig. 3 (left) shows the measured optical spectrum analyzer trace with the parametric amplifier pump, corresponding to peak (c), both on and off. Peak (d) corresponds to the launched signal. The signal experienced a net gain of 12.8dB when the pump is on. Peak (b) is the idler generated by the parametric gain, where two pump photons (c) are annihilated and one signal (d) and one idler (b) photons are created. The idler (b) contains the same data as carried by (d), and is thus the frequency converted signal. Parametric gain is experienced over a wide range of wavelengths as indicated by the measured small signal gain spectrum plotted in Fig. 3 (right). This data was obtained by replacing all the components upstream from point ‘A’ with a tunable external cavity laser with an output of 2 mW. The gain was determined by comparing the signal power with and without the pump for one wavelength at a time. Peak (e) is seeded when two photons from the signal (d) are annihilated and one photon each at (c) and (e) are created. This is a degenerate four-wave mixing process because the phase matching is poor and the ‘pump’ peak (d) is weak. This seeded peak rapidly grows in power because it experiences parametric gain of 25 dB. Peak (a) is generated by the parametric gain process where two photons from (c) are annihilated and a photon is created at (a) and (e).

The instantaneous power in peak (e) is nonlinearly dependant on the instantaneous power in the signal peak (d). We measured the power transfer function between peaks (d) and (e) by launching a 1550.0 nm signal with a 24% duty cycle 200 kHz square wave modulation into point ‘A’. The modulation enabled us to realize peak powers similar to that of the signal to be monitored. Fig. 4 (Left) is a plot of the transfer function data with the peak power on both axes. For small signal (d) the peak (e) power has a power dependence of index 2. Due to pump saturation this becomes more linear and then sub-linear at higher power levels.

The power in peak (e) against dispersion from the tuneable dispersion compensator was easily measured using a laboratory average power meter and the data are plotted in Fig. 4 (right). Two different sets of data are shown for average signal powers of 7.42 mW and 3.71 mW measured at point ‘A’. The signal pulse width was measured using an auto-correlator at point ‘A’ and is also plotted against dispersion in Fig. 4 (Right). A curve has been fitted to the pulse width data to guide the eye. The power in the peak (e) is strongly dependant on the dispersion induced pulse broadening, with a well defined peak centered around zero dispersion. The peak is well defined between -70 to +70 ps/nm. Thus, the dispersion monitor may be used to measure residual dispersion in this range.

Fig. 3. (Left) The output from the device measured on the optical spectrum analyzer. (Right) The measured gain spectrum.

A numerical model was also devised that predicts the measured power in peak (e) as a function of dispersion. The model uses a fast-Fourier transform routine to solve the basic wave propagation equation [14

14. G.P. Agrawal “Wave propagation in Optical Fibers,” in Nonlinear Fiber Optics, (Academic Press, San Diego, 1995)

]. The envelope of the optical power spectrum measured at ‘A’ is used as an initial condition. We incorporate the measured characteristics of the tunable dispersion compensator, including the varying reflectivity and group delay ripple. A cubic fit to the measured transfer function in Fig. 4 (left) is then applied to the pulse train for point ‘A’ to give a calculation of the measured power. The predicted power as a function of dispersion from this model is also plotted in Fig. 4 (right) and agrees well with the experimental data. We have taken into account the 8% power clipping by the band pass filter placed before the power meter. Because of practical limitations it was necessary to extrapolate the transfer function to higher powers which resulted in a worse prediction than expected around the zero dispersion wavelength.

Fig. 4. (Left) The measured power transfer function between points A and B in Fig. 2 and (Right) the measured power and predicted power in peak (e) as a function of dispersion.

5. Discussion

To better understand the curves in Fig. 4 (right) we present an animation in Fig. 5. Each frame shows simulated and experimental results for a particular dispersion value setting on the tunable dispersion compensator. In each frame, the top left panel is the optical power spectrum measured by the optical spectrum analyzer in dBm. The top right panel is the spectrum of peak (e) measured on the optical spectrum analyzer and plotted using µW power units. The bottom right panel is the power of peak (e) measured using the average power meter. The bottom left panel is the simulated signal pulse train of (d) at point ‘A’.

The static Fig. 5 is the frame for a zero dispersion setting. In this case, the spectrum of peak (e) is at a maximum, the measured power is at a maximum and there is a single well defined pulse in the centre of the simulation panel. The animation clearly demonstrates that the peak signal power decreases for increasing dispersion magnitude, even though the signal power does not change.

We also note the generation of pulse trains with repetition rates that are integer multiples of the input 10 GHz pulse train, which can be seen in the animated Fig. 5. We clearly see multiplication of between three and seven times at -413 ps/nm, -310 ps/nm, -248 ps/nm, -207 ps/nm and -177 ps/nm. This effect is well understood and is because the dispersed pulses interfere with each other [15

15. J. Azana and M.A. Muriel, “Technique for multiplying the repetition rate of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. 24, 1672–1674 (1999) [CrossRef]

17

17. J. Azana and M.A. Muriel, “Temportal self-imaging effects: Theory and Application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001) [CrossRef]

]. The measured power of peak (e) reach a local maximum at dispersion values where frequency multiplication occurs. The effect of the group delay ripple is also evident in distortion of the pulse shape. From our simulations we have noted that the measured power of peak (e) would be ~10% higher if the tunable dispersion compensator has no group delay ripple.

Fig. 5. (1.1 MB) Animation showing the evolution of the pulse train at ‘A’ and the corresponding spectral and power measurements.

6. Conclusion

We have demonstrated that pulse-width broadening resulting from chromatic dispersion can be monitored by a simple average power measurement of the filtered output from an optical parametric amplifier. The device also provides gain for the signal and creates a copy of the signal at a new frequency. Both the dispersion-monitoring and frequency conversion functions are all-optical and are appropriate for application to systems operating near and beyond 100 GBits/s. Because the underlying degenerate four-wave-mixing mechanism is dependant on the peak pulse power we expect that this technique could also be used to measure other pulse broadening mechanisms.

Acknowledgments

This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence, Linkage and Large Grant Programmes. Justin Blows thanks Bishop Innovation for support.

References and Links

1.

B.J. Eggleton, et al, “Integrated tunable fiber gratings for dispersion management in high-bit rate systems” J. Lightwave Technol. 18, 1418–1432 (2000). [CrossRef]

2.

P.S. Westbrook, B.J. Eggleton, G. Raybon, S. Hunsche, and T.H. Her, “Measurement of Residual Chromatic dispersion of a 40-Gb/s RZ Signal via Spectral Broadening,” IEEE Photon. Tech. Lett. 14, 346–348 (2002) [CrossRef]

3.

H. Ohita, S. Nogiwa, Y. Kawaguchi, and Y. Endo, “Measurement of 200 Gbit/s optical eye diagram by optical sampling with gain-switched optical pulses,” Electron. Lett 36, 737 (2000) [CrossRef]

4.

Z. Panet al., “Chromatic dispersion monitoring and automated compensation for NRZ and RZ data using clock regeneration and fading without adding signaling,” in Optical Fiber Communication Conference, (Optical Society of America, Washington, DC, 2001), WH5

5.

S. Wielandyet al., “Real-time measurement of accumulated chromatic dispersion for automatic dispersion compensation,” Electon. Lett. 38, 1198–1199 (2002) [CrossRef]

6.

J. Blows and S. French, “Low noise figure optical parametric amplifier with a continuous-wave frequency-modulated pump,” Opt. Lett. 27, 491–493 (2002) [CrossRef]

7.

R. McKerracher, J. Blows, and M.C. deSterke, “Wavelength conversion bandwidth in fibre based optical parametric amplifiers” Opt. Express 11, 1002–1007 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1002. [CrossRef] [PubMed]

8.

E. Ciaramella, F. Curti, and S. Trillo, “All-optical reshaping by means of Four-Wave Mixing in Optical Fibers,” IEEE Phton. Technol. Lett. 13, 142–144 (2001) [CrossRef]

9.

K. Inoue, “Suppression of level fluctuation without extinction ratio degradation based on output saturation in higher order optical parametric interaction in fiber,” IEEE Photon. Technol. Lett. 13, 338–340 (2001) [CrossRef]

10.

J. Blows, “Crosstalk in a fibre parametric wavelength converter,” Trends in Optics and Photonics. 86, Optical Fiber Communication Conference, (Optical Society of America, Washington, DC, 2003), pp 565

11.

J.H. Hansryd and P.A. Andrekson, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Selec. Top. Quat. Electon. 8, 506–520 (2002) [CrossRef]

12.

K.K.Y. Wong, M.E. Mahic, K. Uesaka, and L.G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Tech. Lett. 7, 911–913 (2002) [CrossRef]

13.

F. Forghieri, R.W. Tkach, and A.R. Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications IIIA, (1997) Academic Press (San Diego)

14.

G.P. Agrawal “Wave propagation in Optical Fibers,” in Nonlinear Fiber Optics, (Academic Press, San Diego, 1995)

15.

J. Azana and M.A. Muriel, “Technique for multiplying the repetition rate of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. 24, 1672–1674 (1999) [CrossRef]

16.

S. Longhiet al., “50-GHz pulse-train generation at 1.5 mm with a chirped fiber grating as a frequency multiplier,” Opt. Lett. 25, 1481–1483 (2000) [CrossRef]

17.

J. Azana and M.A. Muriel, “Temportal self-imaging effects: Theory and Application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001) [CrossRef]

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(060.4510) Fiber optics and optical communications : Optical communications
(190.4370) Nonlinear optics : Nonlinear optics, fibers

ToC Category:
Research Papers

History
Original Manuscript: October 9, 2003
Revised Manuscript: November 6, 2003
Published: November 17, 2003

Citation
T. Ng, J. Blows, J. Mok, P. Hu, J. Bolger, P. Hambley, and Benjamin Eggleton, "Simultaneous residual chromatic dispersion monitoring and frequency conversion with gain using a parametric amplifier," Opt. Express 11, 3122-3127 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-23-3122


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References

  1. B.J. Eggleton, et al, �??Integrated tunable fiber gratings for dispersion management in high-bit rate systems�?? J. Lightwave Technol. 18, 1418-1432 (2000). [CrossRef]
  2. P.S. Westbrook, B.J. Eggleton, G. Raybon, S. Hunsche and T.H. Her, �??Measurement of Residual Chromatic dispersion of a 40-Gb/s RZ Signal via Spectral Broadening,�?? IEEE Photon. Tech. Lett. 14, 346-348 (2002) [CrossRef]
  3. H. Ohita, S. Nogiwa, Y. Kawaguchi and Y. Endo, �??Measurement of 200 Gbit/s optical eye diagram by optical sampling with gain-switched optical pulses,�?? Electron. Lett 36, 737 (2000) [CrossRef]
  4. Z. Pan et al., �??Chromatic dispersion monitoring and automated compensation for NRZ and RZ data using clock regeneration and fading without adding signaling,�?? in Optical Fiber Communication Conference, (Optical Society of America, Washington, DC, 2001), WH5.
  5. S. Wielandy et al., �??Real-time measurement of accumulated chromatic dispersion for automatic dispersion compensation,�?? Electon. Lett. 38, 1198-1199 (2002) [CrossRef]
  6. J. Blows, S. French, �??Low noise figure optical parametric amplifier with a continuous-wave frequency-modulated pump,�?? Opt. Lett. 27, 491-493 (2002) [CrossRef]
  7. R. McKerracher, J. Blows and M.C. deSterke, �??Wavelength conversion bandwidth in fibre based optical parametric amplifiers�?? Opt. Express 11, 1002-1007 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1002.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1002</a>. [CrossRef] [PubMed]
  8. E. Ciaramella, F. Curti, and S. Trillo, �??All-optical reshaping by means of Four-Wave Mixing in Optical Fibers,�?? IEEE Phton. Technol. Lett. 13, 142-144 (2001) [CrossRef]
  9. K. Inoue, �??Suppression of level fluctuation without extinction ratio degradation based on output saturation in higher order optical parametric interaction in fiber,�?? IEEE Photon. Technol. Lett. 13, 338-340 (2001) [CrossRef]
  10. J. Blows, �??Crosstalk in a fibre parametric wavelength converter,�?? Trends in Optics and Photonics. 86, Optical Fiber Communication Conference, (Optical Society of America, Washington, DC, 2003), pp 565
  11. J.H. Hansryd, P.A. Andrekson, �??Fiber-based optical parametric amplifiers and their applications,�?? IEEE J. Selec. Top. Quat. Electon. 8, 506-520 (2002) [CrossRef]
  12. K.K.Y. Wong, M.E. Marhic, K. Uesaka, L.G. Kazovsky, �??Polarization-independent two-pump fiber optical parametric amplifier,�?? IEEE Photon. Tech. Lett. 7, 911-913 (2002) [CrossRef]
  13. F. Forghieri, R.W. Tkach and A.R. Chraplyvy, �??Fiber nonlinearities and their impact on transmission systems,�?? in Optical Fiber Telecommunications IIIA, (1997) Academic Press (San Diego)
  14. G.P. Agrawal �??Wave propagation in Optical Fibers,�?? in Nonlinear Fiber Optics, (Academic Press, San Diego, 1995)
  15. J. Azana, M.A. Muriel, �??Technique for multiplying the repetition rate of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings, Opt. Lett. 24, 1672-1674 (1999) [CrossRef]
  16. S. Longhi et al., �??50-GHz pulse-train generation at 1.5 mm with a chirped fiber grating as a frequency multiplier,�?? Opt. Lett. 25, 1481-1483 (2000) [CrossRef]
  17. J. Azana, M.A. Muriel, Temportal self-imaging effects: Theory and Application for multiplying pulse repetition rates,�?? IEEE J. Sel. Top. Quantum Electron. 7, 728-744 (2001) [CrossRef]

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