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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 6881–6886
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Simultaneous frequency conversion, regeneration and reshaping of optical signals

C. J. McKinstrie and D. S. Cargill  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 6881-6886 (2012)
http://dx.doi.org/10.1364/OE.20.006881


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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

Parametric devices based on four-wave mixing (FWM) in nonlinear fibers can amplify, frequency convert (FC), phase conjugate, regenerate, sample and switch optical signals in communication systems [1

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

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]

]. This letter focuses on FC by the nondegenerate FWM process called Bragg scattering (BS). In this process, which is illustrated in Fig. 1, a sideband (signal) photon and a pump photon are destroyed, and different sideband (idler) and pump photons are created (πp + πsπq + πr, where πj represents a photon with carrier frequency ωj.) BS is a versatile process. It can provide tunable [3

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

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]

] and low-noise [5

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

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]

] FC for classical signals or single photons [7

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

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]

]. It can also FC waves with similar (nearby) frequencies [9

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]

], or waves with dissimilar (distant) frequencies [10

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

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]

]. Not only does BS transfer power (photon flux) from the input signal to the output idler, it also transfers the quantum state [5

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

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]

]. For example, if the input photon is entangled with another quantum degree of freedom, so also will be the output photon. These features make BS useful for conventional communications and quantum information science. In this letter, it will be shown that BS also has the ability to regenerate (clean-up) noisy signals and reshape (reformat) signals arbitrarily.

Fig. 1 Frequency diagram for nearby (left) and distant (right) Bragg scattering. Long arrows denote pumps (p and q), whereas short arrows denote idler and signal sidebands (r and s). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons. The directions of the arrows are reversible.

2. Analysis

In BS, the conservation of energy and momentum are manifested by the frequency- and wavenumber-matching conditions
ωp+ωs=ωq+ωr,kp+ks=kq+kr,
(1)
respectively, where ωj denotes a carrier frequency and kj = k(ωj) 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.

For a wide range of frequencies, the dispersion function k(ω) can be approximated by the Taylor expansion
k(ω)β0(ωa)+β1(ωa)ω+β2(ωa)ω2/2+β3(ωa)ω3/6+β4(ωa)ω4/24,
(2)
where ωa is a reference frequency, the difference frequency ω is measured relative to the reference frequency and the dispersion coefficient βn = dnk/dωn. It is convenient to let ωa be the average frequency of the waves, in which case ωp = −ωs and ωq = −ωr. Define the wavenumber mismatch δ = kp + ks – (kq + kr). Then it follows from Eq. (2) and the preceding definition that
δ(ωp2ωq2)[β2+β4(ωp2+ωq2)/12].
(3)
For low and moderate frequencies, the effects of fourth-order dispersion can be neglected. In this case, wavenumber matching is achieved by setting β2 = 0. This condition corresponds to wave frequencies that are perfectly symmetric about the zero-dispersion frequency ω0. The group-slowness function dk(ω)/β1 + β3ω2/2 depends quadratically on frequency. Hence, the outer pump co-propagates with the signal (β1p = β1s), whereas the inner pump co-propagates with the idler (β1q = β1r). For high frequencies, the effects of fourth-order dispersion cannot be neglected. However, the waves still co-propagate approximately (β1pβ1s and β1qβ1r) 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]

], |β1rβ1s| was 1.36 ps/m, whereas |β1pβ1s| and |β1qβ1r| were only 0.027 ps/m. These slowness relations enable the reshaping functions described below.

Suppose that the pump and sideband frequencies are chosen so that the matching conditions are satisfied. Then the sideband evolution is governed by the coupled-mode equations
(z+βrt)Ar=i2γ(|Ap|2+|Aq|2)Ar+i2γApAq*As,
(4)
(z+βst)As=i2γ(|Ap|2+|Aq|2)As+i2γAp*AqAr,
(5)
where Aj is a slowly-varying wave (mode) amplitude, z = ∂/∂z is a distance derivative, t = ∂/∂t is a time derivative, βj is an abbreviation for the group slowness β1j 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

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]

].

The effects of nonlinear phase modulation (NPM) are omitted from the following analysis, but will be incorporated numerically. The pumps are not affected by the sidebands, so they convect through the fiber with constant shape. In the low-conversion-efficiency regime, the input signal seeds the growth of a weak idler, whose presence does not affect the signal significantly, so the signal also convects through the fiber with constant shape. Hence, the pump and signal amplitudes can be written in the form
Aj(t,z)=ajfj(tβjz),
(6)
where aj is a complex constant and fj is a complex shape-function. It is convenient to impose the normalization condition |fj(t)|2dt=1, in which case the amplitude aj is the square root of the pulse energy. The idler equation can be written in the form
(z+βrt)Ar(t,z)=i2γAp(tβsz)Aq*(tβrz)As(tβsz),
(7)
where the terms on the right side are specified functions.

Define the retarded time τ = t – βrz. Then the idler equation becomes
zAr(τ,z)=i2γAp(τ+βrzz)As(τ+βrsz)Aq*(τ),
(8)
where βrs = βrβs is the differential slowness (walk-off). By integrating Eq. (8), in which τ is a parameter, one finds that the output idler
Ar(τ,l)=i2γ0lAp(τ+βrsz)As(τ+βrzz)dzAq*(τ).
(9)
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 βrsl (so the waves collide entirely within the fiber), the integral in Eq. (9) does not depend on 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
Ar(t,l)=iγ¯asfq*(tβrl),
(10)
where the strength parameter γ¯=2γapaq*/βrs is proportional to the nonlinearity coefficient and the product of the pump amplitudes, and is inversely proportional to the walk-off. (If βp differs slightly from βs, 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.

3. Examples

Three examples of idler generation and signal reshaping are now considered. Time is measured in units of the (common) pump width σ and distance is measured in units of σ/βr. For these conventions, the group slowness is measured in units of βr and the interaction length βrl/σ 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 βsr = −1 [because in a frame moving with the average slowness, the (apparent) idler slowness is βr – (βr + βs)/2 = βrs/2 and the signal slowness is βs – (βr +βs)/2 = −βrs/2] and γ̄ = 0.325 (which corresponds to an energy-conversion efficiency of 10%). The idler amplitude is measured in units of ias.

In the first example, all three input pulses (both pumps and the signal) are Gaussians, and are timed to collide (overlap completely) in the middle of the fiber. The idler generated by these inputs is illustrated in Fig. 2. As the interaction length increases, so also does the idler amplitude and time delay. Increasing the length beyond a value of about 2 delays the idler, but does not change significantly its peak amplitude or shape. Equation (10) omits the effects of signal depletion, which limits the idler growth by reducing the driving strength, and NPM, which limits the idler growth by detuning the FWM process [15

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]

]. Numerical solutions of Eqs. (4) and (5) show that neither effect changes the idler magnitude significantly (for low conversion efficiencies). However, NPM does impose a moderate chirp on the output idler (not shown).

Fig. 2 (a) Predicted amplitude of the idler generated by a Gaussian signal and Gaussian pumps. The dotted, dashed, dot-dashed and solid curves denote interaction lengths of 0.5, 1.0, 2.0 and 4.0, respectively. (b) Comparison of the analytical predictions (dashed curve) and numerical results for a length of 4.0. The dot-dashed (solid) curve was obtained by solving Eqs. (4) and (5) numerically, without (with) the phase-modulation terms.

In the second example, the signal and its co-propagating pump are Gaussians, whereas the other pump is an 8th-order super-Gaussian. The idler generated by these inputs is illustrated in Fig. 3. As the interaction length increases, the idler amplitude increases and the idler shape becomes more rectangular. Increasing the length beyond a value of about 3 delays the idler, but does not change its shape significantly. Once again, the predictions of perturbation theory are accurate. The output idler is phase-shifted, but unchirped, because the pumps are flat-topped.

Fig. 3 (a) Predicted amplitude of the idler generated by a Gaussian signal, a Gaussian pump and a super-Gaussian pump. The dotted, dashed, dot-dashed and solid curves denote interaction lengths of 0.5, 1.0, 2.0 and 4.0, respectively. (b) Comparison of the analytical predictions (dashed curve) and numerical results for a length of 4.0. The dot-dashed (solid) curve was obtained by solving Eqs. (4) and (5) numerically, without (with) the phase-modulation terms.

Fig. 4 Amplitudes of a noisy Gaussian signal (top) and the regenerated idler (middle and bottom) produced by Gaussian pumps (left), and a Gaussian pump and a super-Gaussian pump (right). Solid curves denote the idler, whereas dashed curves denote the (rescaled) pumps. The top, middle and bottom figures correspond to distances of 0.0, 0.5 and 4.0, respectively. The curves were obtained by solving Eqs. (4) and (5) numerically.

4. Discussion

In summary, BS has the potential to regenerate and reshape trains of noisy input signals. Perturbation theory makes accurate predictions for conversion efficiencies up to 10%, which is high enough to be useful. The effects of signal depletion and nonlinear phase modulation on the idler evolution will be studied in detail elsewhere.

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. 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]

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]

11.

R. Provo, S. G. Murdoch, J. D. Harvey, and D. Méchin, “Bragg scattering in a positive β4 fiber,” Opt. Lett. 35, 3730–3732 (2010). [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]

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]

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]

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. 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]

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

  1. 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]
  2. S. Radic, C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron. E88-C, 859–869 (2005). [CrossRef]
  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, 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, 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, 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, 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, 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, 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, C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006). [CrossRef] [PubMed]
  11. 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]
  12. 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]
  13. 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]
  14. C. J. McKinstrie, H. Kogelnik, L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (2006). [CrossRef] [PubMed]
  15. 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]
  16. 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]
  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]

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