Simultaneous frequency conversion, regeneration and reshaping of optical signals

C. J. McKinstrie; D. S. Cargill

doi:10.1364/OE.20.006881

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 [1,2]. 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 (π_p + π_s → π_q + π_r, where π_j represents a photon with carrier frequency ω_j.) BS is a versatile process. It can provide tunable [3,4] and low-noise [5,6] FC for classical signals or single photons [7,8]. It can also FC waves with similar (nearby) frequencies [9], or waves with dissimilar (distant) frequencies [10–12]. Not only does BS transfer power (photon flux) from the input signal to the output idler, it also transfers the quantum state [5, 7]. 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.

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2. Analysis

In BS, the conservation of energy and momentum are manifested by the frequency- and wavenumber-matching conditions

\begin{matrix} ω_{p} + ω_{s} = ω_{q} + ω_{r}, & k_{p} + k_{s} = k_{q} + k_{r}, \end{matrix}

respectively, where ω_j denotes a carrier frequency and k_j = k(ω_j) 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

\begin{array}{l} k (ω) & \approx & β_{0} (ω_{a}) + β_{1} (ω_{a}) ω + β_{2} (ω_{a}) ω^{2} / 2 \\ + β_{3} (ω_{a}) ω^{3} / 6 + β_{4} (ω_{a}) ω^{4} / 24, \end{array}

where ω_a is a reference frequency, the difference frequency ω is measured relative to the reference frequency and the dispersion coefficient β_n = dⁿk/dωⁿ. It is convenient to let ω_a be the average frequency of the waves, in which case ω_p = −ω_s and ω_q = −ω_r. Define the wavenumber mismatch δ = k_p + k_s – (k_q + k_r). Then it follows from Eq. (2) and the preceding definition that

δ \approx (ω_{p}^{2} - ω_{q}^{2}) [β_{2} + β_{4} (ω_{p}^{2} + ω_{q}^{2}) / 12] .

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)/dω ≈ β₁ + β₃ω²/2 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], |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.

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

\begin{matrix} (\partial_{z} + β_{r} \partial_{t}) A_{r} & = & i 2 γ ({| A_{p} |}^{2} + {| A_{q} |}^{2}) A_{r} + i 2 γ A_{p} A_{q}^{*} A_{s}, \end{matrix}

\begin{matrix} (\partial_{z} + β_{s} \partial_{t}) A_{s} & = & i 2 γ ({| A_{p} |}^{2} + {| A_{q} |}^{2}) A_{s} + i 2 γ A_{p}^{*} A_{q} A_{r}, \end{matrix}

where A_j is a slowly-varying wave (mode) amplitude, ∂_z = ∂/∂z is a distance derivative, ∂_t = ∂/∂t is a time derivative, β_j is an abbreviation for the group slowness 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, 14].

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

A_{j} (t, z) = a_{j} f_{j} (t - β_{j} z),

where a_j is a complex constant and f_j is a complex shape-function. It is convenient to impose the normalization condition

\int_{- \infty}^{\infty} {| f_{j} (t) |}^{2} d t = 1

, in which case the amplitude a_j is the square root of the pulse energy. The idler equation can be written in the form

(\partial_{z} + β_{r} \partial_{t}) A_{r} (t, z) = i 2 γ A_{p} (t - β_{s} z) A_{q}^{*} (t - β_{r} z) A_{s} (t - β_{s} z),

where the terms on the right side are specified functions.

Define the retarded time τ = t – β_rz. Then the idler equation becomes

\partial_{z} A_{r} (τ, z) = i 2 γ A_{p} (τ + β_{r z} z) A_{s} (τ + β_{r s} z) A_{q}^{*} (τ),

where β_rs = β_r – β_s is the differential slowness (walk-off). By integrating Eq. (8), in which τ is a parameter, one finds that the output idler

A_{r} (τ, l) = i 2 γ \int_{0}^{l} A_{p} (τ + β_{r s} z) A_{s} (τ + β_{r z} z) d z A_{q}^{*} (τ) .

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_rsl (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

A_{r} (t, l) = i \bar{γ} a_{s} f_{q}^{*} (t - β_{r} l),

where the strength parameter

\bar{γ} = 2 γ a_{p} a_{q}^{*} / β_{r s}

is proportional to the nonlinearity coefficient and the product of the pump amplitudes, and is inversely proportional to the walk-off. (If β_p differs slightly from β_s, 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/β_r. For these conventions, the group slowness is measured in units of β_r and the interaction length β_rl/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 β_s/β_r = −1 [because in a frame moving with the average slowness, the (apparent) idler slowness is β_r – (β_r + β_s)/2 = β_rs/2 and the signal slowness is β_s – (β_r +β_s)/2 = −β_rs/2] and γ̄ = 0.325 (which corresponds to an energy-conversion efficiency of 10%). The idler amplitude is measured in units of ia_s.

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]. 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.

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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.

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In the previous examples, the input signals were perfect noiseless Gaussians. However, real signals are usually degraded by noise, as illustrated in Fig. 4 (top row). It is natural to ask what happens to such signals when they are frequency converted. In the third example, the signals are noisy Gaussians and the pumps are noiseless Gaussians or super-Gaussians. The idlers generated by these inputs are also illustrated in Fig. 4 (middle and bottom rows). In each case, the output idler amplitude at time t depends on the input signal amplitude at time t′, where t – β_rl ≤ t′ ≤ t – β_sl. 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, 17].

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.

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4. Discussion

To regenerate or reshape trains of signals, the fiber (nonlinear medium) should be long enough that each pump–sideband collision is complete, but short enough that the signal (idler) from one collision does not intersect the idler (signal) from the next collision. Suppose that input pump p and the input signal are delayed by about 2σ relative to the center of the bit slot (and so extend from about 0 to 4σ), whereas pump q and the virtual idler are advanced by 2σ (and so extend from −4σ to 0). After a complete (symmetric) collision, the output signal is advanced by 2σ, whereas the idler is delayed by 2σ. Completeness requires that β_rsl = 8σ, whereas noninterference between pumps and sidebands in neighboring bit slots requires that max(8σ,β_rsl) ≤ 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_rs = 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.

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

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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]

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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]

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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]

Simultaneous frequency conversion, regeneration and reshaping of optical signals

Abstract

1. Introduction

2. Analysis

3. Examples

4. Discussion

References and links

Cited By

Figures (4)

Equations (10)

Optics Express