## Simultaneous frequency conversion, regeneration and reshaping of optical signals |

Optics Express, Vol. 20, Issue 7, pp. 6881-6886 (2012)

http://dx.doi.org/10.1364/OE.20.006881

Acrobat PDF (810 KB)

### Abstract

Nondegenerate four-wave mixing in fibers enables the tunable and low-noise frequency conversion of optical signals. This paper shows that four-wave mixing driven by pulsed pumps can also regenerate and reshape optical signal pulses arbitrarily.

© 2012 OSA

## 1. Introduction

1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. **8**, 506–520 (2002). [CrossRef]

2. S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron . **E88-C**, 859–869 (2005). [CrossRef]

*π*+

_{p}*π*→

_{s}*π*+

_{q}*π*, where

_{r}*π*represents a photon with carrier frequency

_{j}*ω*.) BS is a versatile process. It can provide tunable [3

_{j}3. K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. **6**, 1451–1453 (1994). [CrossRef]

4. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. **16**, 551–553 (2004). [CrossRef]

5. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express **13**, 9131–9142 (2005). [CrossRef] [PubMed]

6. A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express **14**, 8989–8994 (2006). [CrossRef] [PubMed]

7. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. **283**, 747–752 (2010). [CrossRef]

8. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. **105**, 093604 (2010). [CrossRef] [PubMed]

9. K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. **8**, 560–568 (2002). [CrossRef]

10. D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express **14**, 8995–8999 (2006). [CrossRef] [PubMed]

12. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. **23**, 109–111 (2011). [CrossRef]

5. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express **13**, 9131–9142 (2005). [CrossRef] [PubMed]

7. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. **283**, 747–752 (2010). [CrossRef]

## 2. Analysis

*ω*denotes a carrier frequency and

_{j}*k*=

_{j}*k*(

*ω*) denotes the associated carrier wavenumber. BS is driven by pump-power-induced nonlinear coupling and suppressed by fiber-dispersion-induced wavenumber mismatch, so it is important to determine the conditions under which BS is wavenumber matched.

_{j}*k*(

*ω*) can be approximated by the Taylor expansion where

*ω*is a reference frequency, the difference frequency

_{a}*ω*is measured relative to the reference frequency and the dispersion coefficient

*β*=

_{n}*d*. It is convenient to let

^{n}k/dω^{n}*ω*be the average frequency of the waves, in which case

_{a}*ω*= −

_{p}*ω*and

_{s}*ω*= −

_{q}*ω*. Define the wavenumber mismatch

_{r}*δ*=

*k*+

_{p}*k*– (

_{s}*k*+

_{q}*k*). Then it follows from Eq. (2) and the preceding definition that For low and moderate frequencies, the effects of fourth-order dispersion can be neglected. In this case, wavenumber matching is achieved by setting

_{r}*β*

_{2}= 0. This condition corresponds to wave frequencies that are perfectly symmetric about the zero-dispersion frequency

*ω*

_{0}. The group-slowness function

*dk*(

*ω*)/

*dω*≈

*β*

_{1}+

*β*

_{3}

*ω*

^{2}/2 depends quadratically on frequency. Hence, the outer pump co-propagates with the signal (

*β*

_{1p}=

*β*

_{1s}), whereas the inner pump co-propagates with the idler (

*β*

_{1q}=

*β*

_{1r}). For high frequencies, the effects of fourth-order dispersion cannot be neglected. However, the waves still co-propagate approximately (

*β*

_{1p}≈

*β*

_{1s}and

*β*

_{1q}≈

*β*

_{1r}) for a wide range of pump frequencies. For example, in a recent experiment [8

8. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. **105**, 093604 (2010). [CrossRef] [PubMed]

*β*

_{1r}−

*β*

_{1s}| was 1.36 ps/m, whereas |

*β*

_{1p}−

*β*

_{1s}| and |

*β*

_{1q}−

*β*

_{1r}| were only 0.027 ps/m. These slowness relations enable the reshaping functions described below.

*A*is a slowly-varying wave (mode) amplitude,

_{j}*∂*=

_{z}*∂/∂z*is a distance derivative,

*∂*=

_{t}*∂/∂t*is a time derivative,

*β*is an abbreviation for the group slowness

_{j}*β*

_{1j}and

*γ*is the Kerr nonlinearity coefficient. The effects of intra-pulse dispersion were neglected, because they are weak for a wide range of relevant system parameters. Equations (4) and (5) apply to scalar FWM, which involves waves with the same polarization. Similar equations apply to vector FWM, which involves waves with different polarizations [13

13. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B **20**, 2425–2433 (2003). [CrossRef]

14. C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express **14**, 8516–8534 (2006). [CrossRef] [PubMed]

*a*is a complex constant and

_{j}*f*is a complex shape-function. It is convenient to impose the normalization condition

_{j}*a*is the square root of the pulse energy. The idler equation can be written in the form where the terms on the right side are specified functions.

_{j}*τ*=

*t –*

*β*. Then the idler equation becomes where

_{r}z*β*=

_{rs}*β*–

_{r}*β*is the differential slowness (walk-off). By integrating Eq. (8), in which

_{s}*τ*is a parameter, one finds that the output idler Equations (6) and (9) imply that the amplitude of the output idler is proportional to the amplitude of the input signal, so information is transferred from the signal to the idler. The shape of the output idler depends in a complicated way on the shapes of the inputs. If the input durations are much shorter than the walk-off time

*β*(so the waves collide entirely within the fiber), the integral in Eq. (9) does not depend on

_{rs}l*τ*and the shape-function of the output idler is the conjugate of the shape-function of pump

*q*(which co-propagates with the idler). The interaction between the signal and pump

*p*(which also co-propagate) is strongest if their shape-functions are conjugates of each other, in which case where the strength parameter

*β*differs slightly from

_{p}*β*, the strength parameter is reduced slightly.) Although the amplitude of the output idler is proportional to the amplitude of the input signal, the shape-function of the output idler is specified by the shape-function of the co-propagating pump (not the signal). This pump shape-function could be the same as, or different from, the signal shape-function. Hence, the output idler can be reshaped arbitrarily relative to the input signal. In particular, the amplitude fluctuations associated with a noisy signal can be removed. Similar results apply to the generation of an output signal by pumps and an input idler.

_{s}## 3. Examples

*σ*and distance is measured in units of

*σ/β*. For these conventions, the group slowness is measured in units of

_{r}*β*and the interaction length

_{r}*β*is the ratio of the idler transit time to the pump duration. The pumps and signal have Gaussian or super-Gaussian (nearly rectangular) shape-functions. Most of the following results were obtained by integrating Eq. (9) numerically, for cases in which

_{r}l/σ*β*= −1 [because in a frame moving with the average slowness, the (apparent) idler slowness is

_{s}/β_{r}*β*– (

_{r}*β*+

_{r}*β*)/2 =

_{s}*β*/2 and the signal slowness is

_{rs}*β*– (

_{s}*β*+

_{r}*β*)/2 = −

_{s}*β*/2] and

_{rs}*γ̄*= 0.325 (which corresponds to an energy-conversion efficiency of 10%). The idler amplitude is measured in units of

*ia*.

_{s}15. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B **10**, 1856–1869 (1993). [CrossRef]

*t*depends on the input signal amplitude at time

*t*′, where

*t*–

*β*≤

_{r}l*t*′ ≤

*t*–

*β*. As the interaction length increases, so also does the number of input signal values on which the output idler value depends. Since these input values are statistically independent, the variance of their sum increases (at most) linearly with distance, so the deviation of their sum increases as the square root of distance. Hence, the idler fluctuations increase less rapidly than the mean amplitude, so the idler profile is smoothed. For short distances the back of the (slow) idler is smoothed more than the front, because it has sampled more (fast) signal values, whereas for long distances the whole pulse is smoothed. It should be emphasized that pulsed FC regenerates the pulse shape, but does not regenerate (constrain) the pulse peak-amplitude. This process is complementary to gain-saturated amplification, which removes peak-amplitude variations from a sequence of pulses, but distorts the pulse shapes [16

_{s}l16. A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. **34**, 1410–1411 (1998). [CrossRef]

17. 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]

## 4. Discussion

*p*and the input signal are delayed by about 2

*σ*relative to the center of the bit slot (and so extend from about 0 to 4

*σ*), whereas pump

*q*and the virtual idler are advanced by 2

*σ*(and so extend from −4

*σ*to 0). After a complete (symmetric) collision, the output signal is advanced by 2

*σ*, whereas the idler is delayed by 2

*σ*. Completeness requires that

*β*= 8

_{rs}l*σ*, whereas noninterference between pumps and sidebands in neighboring bit slots requires that max(8

*σ,β*) ≤

_{rs}l*τ*, where

*τ*is the bit duration. As an example, consider a system operating at 10 Gb/s, for which the bit duration is 100 ps. If the pulses have full-widths-at-half-maximum of 20 ps (

*σ*= 12 ps), the bit duration equals 8.3 pulse widths, and if

*β*= 1.4 ps/m, the collision length is 68 m. Speciality fibers with customizable dispersion and lengths of 1–300 m are available, so a variety of useful experiments can be designed.

_{rs}## References and links

1. | J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. |

2. | S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron . |

3. | K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. |

4. | T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. |

5. | C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express |

6. | A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express |

7. | M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. |

8. | H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. |

9. | K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. |

10. | D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express |

11. | R. Provo, S. G. Murdoch, J. D. Harvey, and D. Méchin, “Bragg scattering in a positive |

12. | H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. |

13. | M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B |

14. | C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express |

15. | C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B |

16. | A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. |

17. | 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. |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 3, 2012

Revised Manuscript: February 29, 2012

Manuscript Accepted: March 1, 2012

Published: March 12, 2012

**Citation**

C. J. McKinstrie and D. S. Cargill, "Simultaneous frequency conversion, regeneration and reshaping of optical signals," Opt. Express **20**, 6881-6886 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-6881

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### References

- J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002). [CrossRef]
- S. Radic, C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron. E88-C, 859–869 (2005). [CrossRef]
- K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994). [CrossRef]
- T. Tanemura, C. S. Goh, K. Kikuchi, S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004). [CrossRef]
- C. J. McKinstrie, J. D. Harvey, S. Radic, M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005). [CrossRef] [PubMed]
- A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006). [CrossRef] [PubMed]
- M. G. Raymer, S. J. van Enk, C. J. McKinstrie, H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010). [CrossRef]
- H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010). [CrossRef] [PubMed]
- K. Uesaka, K. K. Y. Wong, M. E. Marhic, L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002). [CrossRef]
- D. Méchin, R. Provo, J. D. Harvey, C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006). [CrossRef] [PubMed]
- R. Provo, S. G. Murdoch, J. D. Harvey, D. Méchin, “Bragg scattering in a positive β4 fiber,” Opt. Lett. 35, 3730–3732 (2010). [CrossRef] [PubMed]
- H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011). [CrossRef]
- M. E. Marhic, K. K. Y. Wong, L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003). [CrossRef]
- C. J. McKinstrie, H. Kogelnik, L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (2006). [CrossRef] [PubMed]
- C. J. McKinstrie, X. D. Cao, J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993). [CrossRef]
- A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998). [CrossRef]
- 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]

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