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In this paper a detailed analysis is made of the phase-sensitive amplification (PSA) of optical pulses by degenerate four-wave mixing (phase-conjugation) in fibers. Formulas are derived, which show that the amplification level and phase sensitivity depend strongly on the phase of the signal pulse, but weakly on its chirp, and the difference between its carrier frequency and the average pump frequency. Solitons, which are unchirped, and dispersion-managed solitons, which are weakly chirped, are suitable for in-line and post-transmission PSA. Pseudo-linear pulses, which are strongly chirped, are unsuitable for in-line PSA, but post-transmission dispersion compensation makes them suitable for PSA prior to detection.
©2007 Optical Society of America
1. Introduction
Long-haul optical communication systems require optical amplifiers to compensate for fiber loss. Current systems use erbium-doped and Raman fiber amplifiers. These amplifiers are examples of phase-insensitive amplifiers (PIAs), which produce signal gain that is independent of the signal phase. In principle, one could also use phase-sensitive amplifiers (PSAs) in communication systems. The potential advantages of PSAs include, but are not limited to, noise reduction [1], the reduction of noise-induced frequency, arrival-time [2] and phase [3] fluctuations, dispersion compensation [4], and the suppression of the modulation instability [5]. The properties of systems with PSAs were reviewed in [6].
In a recent paper [7], schemes based on four-wave mixing (FWM) were described, which produce PSA in fibers. The analysis of this paper was based on the single-mode model, in which the pumps and signal are assumed to be monochromatic (continuous waves). However, bits of information are transmitted as pulses: In amplitude-shift-keyed (ASK) systems the presence of a pulse denotes a 1, whereas the absence of a pulse denotes a 0. In differential-phase-shift-keyed (DPSK) systems successive pulses with the same phase denote a 0, whereas successive pulses with opposite phases denote a 1. These pulses could be solitons [8], which propagate without distortion and are unchirped, dispersion-managed (DM) solitons [9], which breathe periodically and have moderate chirps, or pseudo-linear pulses [10], which broaden significantly and have large chirps. In this paper the continuous-mode model is used to analyze the PSA of optical pulses in fibers, and the effects of chirp on this process are studied in detail.
This paper is organized as follows: In Section 2 the properties of FWM in randomly-birefringent fibers (RBFs) and rapidly-spun fibers (RSFs) are reviewed, and the input-output relations that characterize PSA in such fibers are stated. Subsequently, these relations are used to model the PSA of chirped pulses. Section 3 treats the idealized case in which only (linear) dispersion produces chirp, whereas Section 4 treats realistic cases in which chirp is produced by a combination of dispersion and (nonlinear) phase modulation. Although the phase-sensitivity of the amplification processes depends on the magnitude of the chirp, it does not depend on its physical origin. The effects of PSA on the subsequent evolution of the pulses are studied briefly. Finally, in Section 5 the main results of this paper are summarized.
2. Four-wave mixing in fibers
Detailed studies of FWM in RBFs and RSFs were made in [11] and [12], respectively. In this section the properties of degenerate and nondegenerate phase conjugation (PC) are reviewed, because they enable phase-sensitive amplification (PSA) in fibers [7].
Let E denote the electric-field (Jones) vector of a light wave. It is convenient to measure (angular) frequencies relative to a reference frequency ωr and wavenumbers relative to the associated reference wavenumber kr = k(ωr), and to write the field vector as
In a RBF or a RSF, the evolution of the slowly-varying amplitude vector A is governed by the generalized Schrödinger equation (GSE)
where β (ω) = ∑∞ n=2 k (n)(ωr)ωn/n! represents the higher-order terms in the Taylor expansion of the (polarization-independent) dispersion function of the fiber about the reference frequency. There is no convection term in Eq. (2) because t represents the retarded time. The superscripts † and t denote the hermitian conjugate and transpose, respectively. For a RBF, γa = 8γ /9, where γ is the (Kerr) nonlinearity coefficient of the fiber, γb = 0 and Eq. (2) is valid in a frame that rotates randomly with a (virtual) reference wave [11]. For a RSF, γa = 2γ /3, γb =γ /3, and Eq. (2) is valid in a frame that rotates slowly, but deterministically, because of the residual effects of spinning [12].
Degenerate PC involves two strong pump waves, with frequencies ω 3 and ω1, and a weak signal wave, with frequency ω 2 = (ω 3+ω 1)/2. All frequencies are measured relative to the reference frequency. By substituting the ansatz A(t, z) = ∑3 j=1 Aj(z)exp(-iωjt) into Eq. (2), and collecting terms of like frequency, one obtains coupled equations for the wave amplitudes. It is convenient to define A 3 = e 3 B 3 exp(iβ′3 z), A 1 = e 1 B 1 exp(iβ′1 z) and A 2 = e 2 B 2 exp[i(β′3+β′1)z/2], where each ej is a polarization (unit) vector, each Bj is a scalar amplitude and each wavenumber β′j=β′(ωj) includes both dispersive and nonlinear contributions [from self-phase modulation (SPM) and cross-phase modulation (CPM)]. For reference, ex = [1,0]t and ey = [0,1]t are linearly-polarized (LP) unit vectors, whereas e + = [1, i]/21/2 and e - = [1,-i]/21/2 are circularly-polarized (CP) unit vectors. If the signal remains weaker than the pumps, B 3 and B 1 are approximately constant, in which case
where the wavenumber-mismatch coefficient δ =(2β′2-β′3-β′1)/2 and the nonlinear-coupling coefficient κ is proportional to B 3 B 1. Both coefficients depend on the polarizations of the interacting waves.
The properties of degenerate PC driven by parallel LP pumps, or co-rotating CP pumps, are summarized in Table 1. Each column pertains to a particular combination of polarization vectors, which is illustrated in Fig. 1 or 2. The first row describes the magnitude of the coupling coefficient and the second row describes the nonlinear contribution to the mismatch coefficient. Each Pj = |Bj|2 is a wave power. For a parallel LP signal (x or ∥) the γa and γb terms in the GSE both contribute to PC, whereas for a perpendicular LP signal (y or ⊥) only the γb term contributes. In contrast, for a co-rotating CP signal (+) only the γa term contributes, whereas for a counter-rotating CP signal (-) neither term contributes, so the coupling coefficient is zero.
Table 1. Degenerate PC driven by aligned pumps (1 and 3)
The coupling coefficient |κ| should be multiplied by (P 1 P 3)1/2 and the mismatch coefficient δ should be multiplied by (P 1+P 3)/2.
Fig. 2. Polarization diagrams for degenerate PC driven by co-rotating CP pumps. The dashed line denotes a signal that is not coupled to the pumps.
One can write the solution of Eq. (3) in the input-output form
where the transfer functions
and the PC wavenumber k = (δ 2-|κ|2)1/2. The transfer functions satisfy the auxiliary equation |μ|2-|ν|2 = 1, and the phase of ν depends on the pump phases, which can be controlled. Because B 2(z) depends on a combination of B 2(0) and B * 2(0), the amplification process described by Eq. (4) is termed phase sensitive (PS). For example, if μ and ν are real, the inphase (real) signal quadrature is amplified, whereas the out-of-phase (imaginary) quadrature is attenuated. In the quantum-optics literature, Eq. (4) is called a one-mode squeezing transformation [13, 14]. Table 1 implies that the transfer functions depend on the signal polarization: Polarization-independent PS amplification (squeezing) does not occur in RBFs or RSFs if the pumps are aligned.
If the pumps are LP and perpendicular, the parallel component of the signal is coupled to the perpendicular component. By including both components in the derivation of the signal equations, and defining A ∥ = B ∥ exp[i(β′3+β′1+β′∥-β′⊥)z/2] and A ⊥ = B ⊥ exp[i(β′3+β′1-β′∥+β′⊥)z/2], from which the subscript 2 was omitted for simplicity, one finds that
where the wavenumber mismatch δ = (β′∥+β′⊥-β′3-β′1)/2]. One obtains similar equations for pumps that are CP and counter-rotating (∥ and ⊥ are replaced by + and -, respectively).
The properties of degenerate PC driven by perpendicular LP pumps, or counter-rotating CP pumps, are summarized in Table 2. Each column pertains to a particular combination of polarization vectors, which is illustrated in Fig. 3. For perpendicular LP pumps only the γa term in the GSE contributes to degenerate PC, whereas for counter-rotating CP pumps the γa and γb terms both contribute.
Table 2. Degenerate PC driven by orthogonal pumps (1 and 3)
The coupling coefficient |κ| should be multiplied by (P 1 P 3)1/2 and the mismatch coefficient δ should be multiplied by (P 1+P 3)/2.
One can write the solutions of Eqs. (7) and (8) in the input-output form
where the transfer functions were defined in Eqs. (5) and (6). In the quantum-optics literature, Eqs. (9) and (10) are called a two-mode squeezing transformation [13, 14]. If B ⊥(0) = B ∥(0)exp(iψ), Eq. (9) reduces to Eq. (4), with ν replaced by ν exp(-iψ), which describes PS amplification. This condition includes a LP signal that is oriented at 45° to perpendicular LP pumps, a LP signal that is driven by counter-rotating CP pumps or a CP signal that is driven by perpendicular LP pumps. In contrast, if B ⊥(0) = 0 the output powers P ∥(z) and P ⊥(z) do not depend on the phase of B ∥(0), and the amplification process is termed phase-insensitive (PI). For such a process the signal gain G = |μ|2. It follows from the auxiliary equation that the idler gain |ν|2 = G-1. Polarization-independent PS amplification does not occur in RBFs or RSFs if the pumps are orthogonal.
The preceding discussion applies to monochromatic waves. Although continuous-wave (CW) pumps are monochromatic, in communication systems information is transmitted by signal pulses, which have finite frequency bandwidths. Equations (4), (9) and (10) do not apply to pulses, and must be generalized.
First, consider PC driven by aligned pumps and let the reference frequency be the average pump frequency (in which case ω 1+ω 3 =0). Then the frequency-matching condition for FWM implies that the signal component with frequency ω is coupled to the component with frequency -ω. Nondegenerate PC driven by aligned pumps was analyzed, and its polarization properties were illustrated, in [11, 12]. It is characterized by the input-output relation
which is a simple generalization of Eq. (4). The frequency dependences of the transfer functions were omitted from Eq. (11). This omission is justified for the parameters of typical experiments. To convert Eq. (11) from the frequency domain to the time domain, one replaces B 2(ω) and B * 2 (-ω) by B 2(t) and B * 2 (t), respectively.
Second, consider PC driven by orthogonal pumps. Then the frequency-matching condition implies that the (parallel or co-rotating) signal component with frequency ω is coupled to the (perpendicular or counter-rotating) component with frequency -ω. Nondegenerate PC driven by orthogonal pumps was also analyzed, and its polarization properties were also illustrated, in [11, 12]. It is characterized by the input-output relations
which are simple generalizations of Eqs. (9) and (10). To convert Eqs. (12) and (13) to the time domain, one replaces B ∥(ω) and B * ⊥(-ω) by B ∥(t) and B * ∥(t), respectively.
There are two standard frequency-configurations for experiments. In the first configuration, the pump frequencies straddle the zero-dispersion frequency (ZDF) of the fiber and differ by more than 10 Tr/s. (The pump wavelengths differ by more than 13 nm.) In this configuration, the effects of dispersion are strong. However, by tuning the average pump frequency judiciously, relative to the ZDF, one can produce uniform gain over a bandwith that is comparable to the pump separation. Pulses in 10-Gb/s systems have full-widths-at-half-maximum (FWHMs) of about 30 ps, which correspond to frequency bandwidths of about 0.10 Tr/s (wavelength bandwidths of about 0.13 nm). The frequency dependences of the transfer functions can be neglected, because the pulse bandwidths are much smaller than the gain bandwidth. In the second configuration, the pump frequencies are comparable to the ZDF and the difference between them is of order 1 Tr/s (1.3 nm). In this configuration the pulse bandwidths are smaller than, but of the same order as, the pump separation. However, the aforementioned frequency dependences can still be neglected, because the effects of dispersion are weak (near the ZDF).
3. Propagation and phase-sensitive amplification of a weak signal
Pulse propagation in a transmission fiber is also governed by Eq. (2), with γa =8γ/9 and γb =0. Because pulses in 10-Gb/s systems have FWHMs of about 30 ps, the effects of higher-order dispersion can be neglected for distances of order 1 Mm, in which case β (i∂t)≈-βr∂ 2 tt/2, where βr = k (2)(ωr) is the second-order dispersion coefficient, evaluated at the reference frequency (average pump frequency). In this section analyses are made of the propagation and PSA of a signal pulse that is so weak the effects of nonlinearity can be neglected (γa|A|2 ≈ 0). The evolution of such a signal is governed by the linearized Schrödinger equation (LSE)
At the input to the transmission fiber, the signal has the form
where fs is the power (energy flux) of the signal (which is proportional to the photon flux), ωs is the central frequency of the signal, measured relative to the reference frequency, and σs is the (Gaussian) width of the signal. (The root-mean-square width is σs/21/2.) One can solve the LSE by using the Fourier-transform method. We use the definitions
which allow the Parseval relation to be written as
It follows from Eqs. (15) and (16) that
which implies that the (angular) frequency bandwidth of the signal is 1/σs. It follows from Eqs. (14), (17) and (19) that
where the retarded time t′ = t-βrωsz. The frequency-domain formula (20) shows that the signal acquires a frequency-dependent phase shift, which prevents the amplification process from being optimally phased for all frequency components simultaneously. The time-domain formula (21) shows that the signal acquires the time delay τs=βrωsz, the (average) phase shift ϕs = tan-1(βrz/σ2 s)/2-βrω2sz/2 and the chirp cs=βrz/σ2 s.
It follows from Eq. (11), and the text that precedes Eq. (3), that PSA is characterized by the input-output relation
where zi = z and zo are the input and output distances, respectively, and = ν exp[i(β′3+β′1)(zo-zi)]. Because one can control the phase of ν by controlling the pump phases, one can absorb the preceding phase factor into the definition of ν without loss of generality (→ν). By combining Eqs. (20) and (22), one finds that
Define the energy N = ∫|A(ω)|2 dω (which is proportional to the photon number). Then the input energy Ni=π 1/2 fsσs. It follows from Eq. (23) that the output energy
where the reduction and phase factors
respectively. Let μ = |μ|exp(iϕμ) and ν = |ν|exp(iϕν). Then the condition for maximal gain is ϕν=ϕμ+θ , which depends on the chirp [Eq. (26)], whereas the condition for minimal gain is ϕν=ϕμ+θ+π. If these conditions are satisfied, the output energies
For the ideal case in which ωs = 0 and cs = 0, Eq. (27) is equivalent to the single-mode result Po =(|μ|±|ν|)2 Pi, where P is the signal power [7]. In this case, the minimal gain is the reciprocal of the maximal gain. As ρ decreases, the maximal output energy decreases and the minimal output energy increases: The phase sensitivity of the amplification process decreases. Because one can control the central frequency precisely (the inaccuracy in the central frequency is a small fraction of the bandwidth) one can make the approximation ρ ≈ 1/(1+c 2 s)1/4. Hence, the maximal output energy of the signal pulse is limited mainly by its chirp.
The energy-amplification factor No/Ni, normalized to the PI signal gain, is plotted as a function of the chirp in Fig. 4, for pulses whose frequencies equal, and differ from, the average pump frequency (solid and dashed curves, respectively). The mismatch δ = 0 and the gain G = 20 dB. The upper curves represent maximal gain [+ sign in Eq. (27)], whereas the lower curves represent minimal gain [- sign in Eq. (27)]. As stated in the preceding paragraph, moderate frequency offsets do not degrade significantly the phase-sensitivity of the amplification process. For chirps of order 1 PSA works well, whereas for chirps larger than 10 it does not. As the chirp increases, the maximal and minimal gain-factors both tend to 2. This result has a simple explanation: The coherent terms in Eq. (24) depend on the frequency integral of the phase factor exp(iβrω 2 z), in which ω has a range of values, of order 1/σs. If the chirp βr z/σ2 s is small, the phase factor does not vary significantly and PSA is obtained. However, if the chirp is large, the phase factor varies significantly, the integral of the coherent terms is small and PSA is not obtained.
Fig. 4. Energy-amplification factor plotted as a function of the chirp cs. The solid and dashed curves represent the frequency offsets ωs = 0 and ωs = 0.5/πs, respectively, where 1/πs is the frequency bandwidth. The upper curves represent maximal gain, whereas the lower curves represent minimal gain.
Now consider how the signal evolves after it has been amplified. It follows from Eqs. (14) and (23) that
where the total (sum) distance zs = z+z′ and the differential distance zd = z-z′. It follows from Eqs. (20), (21) and (28) that
where the retarded times t′ = t-βrωszs and t′′ = t-βrωszd. If z′ = 0, Eq. (29) describes the signal at the output of the amplifier. Equation (29) shows that the nonconjugated (direct) and conjugated contributions to the signal evolve in different ways for z′ > 0: The former continues to expand as it propagates, whereas the latter contracts. If z′ = z, the latter regains the shape of the input signal [Eq. (15)]. The delay between the centers of the two contributions is 2βrωsz′, which depends only on the distance traveled after the amplifier.
4. Propagation and phase-sensitive amplification of strong signals
In this section the effects of nonlinearity on signal-pulse propagation and PSA are considered briefly. Equation (2) is difficult to solve exactly. One can solve it approximately by assuming that the pulse propagates in a self-similar manner. A common choice of shape function (ansatz) is the chirped Gaussian
where the shape parameters (moments) fs, ϕs,ωs, τs, cs and σs are the pulse power (photon flux), phase, frequency, delay, chirp and width, respectively, and vary with distance. By substituting ansatz (30) in Eq. (2), and proceeding as described in Appendix A, one obtains the moment equations
where the pulse energy es = π1/2 fsσs and the subscript a was omitted from the nonlinearity coefficient. Equations (31)–(36) are based on the simplifying assumptions that fiber loss is compensated by uniformly-distributed amplification and filters are absent. Under these conditions, the energy and frequency are constant. If the frequency is nonzero, the pulse acquires the delay ω s∫z 0 βr(z′)dz′ and phase-shift contribution ω 2 s∫z 0 βr(z′)dz′/2, both of which are proportional to the cumulative dispersion. General moment equations, which model the effects of nonuniformly-distributed amplification and filters, are stated in Appendix A.
It follows from Eq. (30) that
By combining Eqs. (22) and (37), one finds that the output energy of the amplified pulse is determined by Eq. (24), and the (modified) reduction and phase factors
4.1. Solitons
In an idealized system with anomalous dispersion (βr < 0), pulse propagation is governed by Eqs. (33) and (35). When the soliton condition βr/σ2 s = -γes/(2π)1/2σs is satisfied, nonlinearity (SPM) balances dispersion, and a pulse that is unchirped initially maintains its shape as it propagates: It remains unchirped [cs(z) = 0] and its width is constant [σs(z)=σ0]. Equation (36) implies that the pulse acquires the phase shift
[If the frequency is nonzero, the pulse acquires the additional phase shift -βr ω 2 s z/2, as stated after Eq. (36).] Provided that ϕν is properly tuned [as discussed between Eqs. (26) and (27)], the soliton experiences optimal PSA.
In a real system, fiber loss is compensated by EDFAs, which provide lumped PIA at the ends of the fiber sections (spans), or RFAs, which provide PIA that is nonuniformly distributed throughout the spans. When the path-averaged nonlinearity (SPM) balances the path-averaged dispersion, the pulse evolves (breathes) periodically, and is referred to colloquially as a soliton. Path-averaged solitons have properties that are similar to those of idealized (true) solitons [8]. These similarities are strongest in systems with forward- and backward-pumped RFAs, in which the amplification is almost uniformly distributed, and the pulse breathes weakly.
In this subsection and the two that follow, numerical solutions of Eq. (2) are used to study the effects of PSA on the evolution of various pulses. For the reasons stated above, we made the simplifying assumption that loss is compensated by uniformly-distributed PIA (which is equivalent to the assumption that loss and PIA are both lumped at the ends of the spans). In the spans that precede and follow the PSA, the pulse evolves as it would in a lossless fiber. At the PSA, the pulse is attenuated by an amount that equals the span loss (because there is no PIA in the span that contains the PSA), then amplified, according to Eqs. (5), (6) and (11), by an amount that restores its energy. This approach incorporates PSA-induced pulse distortion exactly, and loss-induced pulse evolution approximately.
In Fig. 5 the temporal energy density (power) of the pulse is plotted as a function of time, and the spectral energy density (spectrum) is plotted as a function of frequency, for the parameters βr=-0.15 ps2/Km and γ =1.7/Km-W. For a 10-Gb/s system, the bit period (slot) is 100 ps and a typical pulse FWHM is 30 ps. For a Gaussian pulse (30), which is an approximate solution of the nonlinear Schrödinger equation (NSE), this FWHM corresponds to σs = 18.0 ps and the soliton condition requires that es = 12.3 fJ. For a sech pulse, which is an exact solution of the NSE, σs = 17.0 ps and the soliton condition requires that es = 10.3 fJ. (Moment equations based on the sech ansatz are also derived in Appendix A.) We chose to base Fig. 5 on the sech ansatz, to highlight the effects of PSA (rather than those of an imperfect input condition). The pulse frequency equals the reference frequency (ωs = 0), which is the average frequency of the pumps used to provide PSA. Results are displayed for the distances z = 0, 1 (before and after the PSA) and 5 Mm. The coincidence of the curves shows that the pulse propagates without distorting.
Fig. 5. Pulse power and spectrum at various points in a system with anomalous dispersion. The input pulse frequency equals the average pump frequency. Because the pulse propagates without distorting, the curves are identical.
In Fig. 6 the power is plotted as a function of time, and the power spectrum is plotted as a function of frequency, for a pulse whose frequency does not equal the reference frequency (ωs = 33 Gr/s). The other parameters are the same as those of Fig. 5. Before the PSA, the pulse propagates without distorting, as it should. After the PSA, the pulse spectrum is a symmetric function of frequency: The PSA guides the pulse frequency toward the reference frequency. Although the pulse energies are the same, before and after the PSA, the pulse shapes are different. After the PSA, the soliton effect causes the pulse shape to evolve toward its input shape. The output pulse is delayed relative to the expected arrival of the input pulse, because dispersion converts the PSA-induced frequency shift into a time shift. [The dispersion coefficient and frequency parameter in Eq. (32) are both negative.] These results show that the pulse propagation is robust, in regard to moderate offsets in pulse frequency. The frequency offsets that occur in practice (about 10 Gr/s) are much smaller than the frequency bandwidth of the pulse.
Fig. 6. Pulse power and spectrum in a system with anomalous dispersion. The input pulse frequency does not equal the average pump frequency. Dotted curves represent z = 0 Mm (input), dot-dashed curves represent z = 1 (before the PSA), dashed curves represent z = 1 (after the PSA) and solid curves represent z = 5 (output). Because the pulse propagates without distorting before the PSA, the dotted and dot-dashed curves are identical.
The preceding results show that solitons are suitable for PSA, at any point in the link. Not only does PSA strengthen the solitons, without adding excess noise, it also removes their noise-induced frequency perturbations, which would otherwise be converted into arrival-time perturbations by dispersion [15, 16]. It does not remove noise-induced power perturbations, which are converted into phase perturbations by nonlinearity [17, 18, 19, 20]. Instead, it removes the phase perturbations themselves. This action is beneficial for DPSK systems [21]. Phase-jitter reduction by PSA in a fiber interferometer was demonstrated recently [22].
4.2. Dispersion-managed solitons
In a DM system the dispersion varies periodically with distance. Each period (stage) consists of a half-section of normal fiber (βn > 0) with length ln/2, a section of anomalous fiber (βa < 0) with length la and another normal half-section. If the power is chosen judiciously, the pulse breathes periodically as it propagates, and is referred to colloquially as a DM soliton. Its width is (locally) minimal at the beginning and end of each stage, and at the midpoint of each anomalous section. The properties of DM solitons were reviewed in [9].
Despite their complexity, Eqs. (33), (35) and (36) can be solved analytically, as described in [9] and Appendix B. Let σ0 be the width at the beginning of the first normal half-section, let τs/σ0→τs, σs/σ0→σs, ωsσ0→ωs and βnz/σ2 0→z, and suppose that cs(0) = 0. Then the pulse expansion in the normal half-section is governed by the equation
where the chirp function cn(σ) = |κnσ2-2γnσ-1|1/2, the nonlinearity coefficient γn=γesσ0/(2π)1/2 βn and the constant of integration κn = 1+2γn. Equation (41) defines σs(z) implicitly. The chirp cs(z) = cn[σs(z)] and the phase shift
At the end of the normal half-section the pulse has (maximal) width σn and chirp cn(σn). Now let |βa|z/σ2 0→z and let σa be the (minimal) width at the midpoint of the anomalous section, where the chirp vanishes. Then the pulse contraction in the first half of the anomalous section is governed by the equation
where ca(σ) = |κaσ2+2γaσ-1|1/2, γa=γesσ0/(2π)1/2|βa|, κa=(1-2γaσa)/σ2 a and distance is measured backward from the midpoint. Equation (43) defines σs(z) implicitly. The chirp c(z) = ca[σs(z)] and the phase shift
By requiring the width and chirp to match at the interface between the normal and anomalous sections, one can determine the power required for periodic evolution, as described in Appendix B.
In Fig. 7, which is based on Eqs. (33), (35) and (36), the width, chirp and phase are plotted as functions of distance, and the power is plotted as a function of time, for the parameters βn =1 2.7 ps2/Km, βa = -6.35 ps2/Km, γ = 1.70×10-3/Km-mW, ln = 33.8 Km, la = 70.1 Km, σ0 = 18.0 ps (which corresponds to a full-width-at-half-maximum of 30.0 ps) and es = 16.8 fJ (which corresponds to a peak power of 0.527 mW). These parameters allow the width and chirp to evolve periodically: The former moment is symmetric about the midpoint of the anomalous section, whereas the latter is anti-symmetric. In contrast, the phase exhibits a large anti-symmetric oscillation, upon which is superimposed a small positive drift [19, 20]. For a 10-Gb/s system, the stretching factor should not exceed 1.2, otherwise the pulse tails would cross the edges of the 100-ps bit slot, as implied by Fig. 7(d).
Fig. 7. (a) Width, (b) chirp and (c) phase plotted as functions of distance for a DM system with average dispersion 〈β〉 = -0.15 ps2/Km. (d) Power plotted as a function of time. The solid, dot-dashed and dashed curves represent the beginning of the normal half-section, end of the normal half-section and midpoint of the anomalous section, respectively.
Figure 7 is based on Eqs. (33), (35) and (36). The predictions of these moment equations were verified by numerical solutions of the NSE. Although the pulse shape is not exactly Gaussian, the pulse does breathe periodically under conditions that are almost the same as those described above. In particular, the simulated pulse shapes do not differ significantly from those displayed in Fig. 7(d).
In Fig. 8 the power is plotted as a function of time and the spectrum is plotted as a function of frequency, for a pulse whose frequency equals the reference frequency (ωs = 0). The other parameters are similar to those of Fig. 7, and the PSA is located at the midpoint of a normal section. Results are displayed for the distances z = 0, 1 (before and after the PSA) and 5 Mm. The coincidence of the curves shows that this PSA does not affect the periodic evolution of the pulse.
In Fig. 9 the power is plotted as a function of time and the spectrum is plotted as a function of frequency, for a pulse whose frequency does not equal the reference frequency (ωs = 42 Gr/s). The other parameters and PSA placement are the same as those of Fig. 8. Before the PSA, the pulse evolves periodically, without distorting, as it should. After the PSA, the spectrum is a symmetric function of frequency: The PSA guides the pulse frequency toward the reference frequency. Although the energies are the same, before and after the PSA, the shapes are different. After the PSA, the DM-soliton effect causes the pulse shape to evolve slowly toward its input shape. The output pulse is delayed relative to the expected arrival of the input pulse, because the average-dispersion coefficient and PSA-induced frequency shift are both negative.
Fig. 8. Pulse power and spectrum at various points in a DM system. The input pulse frequency equals the average pump frequency. Because the pulse evolution is periodic, and the data sets pertain to equivalent points in the dispersion map, the curves are identical.
Fig. 9. Pulse power and spectrum in a DM system. The input pulse frequency does not equal the average pump frequency. Dotted curves represent z = 0 Mm (input), dot-dashed curves represent z = 1 (before the PSA), dashed curves represent z = 1 (after the PSA) and solid curves represent z = 5 (output). Because the pulse evolution is periodic before the PSA, and the data sets pertain to equivalent points in the dispersion map, the dotted and dot-dashed curves are identical.
In Fig. 10 the power is plotted as a function of time, and the spectrum is plotted as a function of frequency, for the case in which the pulse frequency is centered (ωs = 0) and the PSA is located at the beginning of a normal section (end of an anomalous section). The other parameters are the same as those of Fig. 8. Because the PSA is located at the beginning of a section (rather than the midpoint), the pulse is broader before the PSA than at the input, by an amount that is consistent with Fig. 7. Equations (22) and (37) imply that the PSA produces a strengthened copy of the incident pulse (which has the same chirp), together with a conjugated copy (which has the opposite chirp). Because the incident pulse is weakly chirped, the central (high-power) region of the (composite) transmitted pulse is unchirped. Transmission through the PSA also distorts the pulse slightly. After the PSA, the pulse evolves in a quasi-periodic manner. The output pulse is narrower than the input pulse.
Fig. 10. Pulse power and spectrum in a DM system. The PSA is located at the beginning of a normal section. Dotted curves represent z = 0 Mm (input), dot-dashed curves represent z ≈ 1 (before the PSA), dashed curves represent z ≈ 1 (after the PSA) and solid curves represent z = 5 (output).
In Fig. 11 the power is plotted as a function of time, and the spectrum is plotted as a function of frequency, for the case in which the pulse frequency is centered and the PSA is located at the end of a normal section (beginning of an anomalous section). The other parameters are the same as those of Fig. 8. Once again, the pulse is broader before the PSA than at the input, by an amount that is consistent with Fig. 7. The PSA distorts the pulse slightly, by removing its chirp. After the PSA, the pulse evolves in a quasi-periodic manner. However, although the FWHM of the output pulse is comparable to that of the input pulse, the output pulse has significant tails, which extend into the neighboring bit slots. In practice, the location of the PSA can be controlled accurately.
Fig. 11. Pulse power and spectrum in a DM system. The PSA is located at the end of a normal section. Dotted curves represent z = 0 Mm (input), dot-dashed curves represent z ≈ 1 (before the PSA), dashed curves represent z ≈ 1 (after the PSA) and solid curves represent z = 5 (output).
The preceding results show that DM solitons are suitable for PSA, at any stage in the link. If the pulse frequency is centered and the PSA is located at the midpoint of a normal section, PSA strengthens the pulse without distorting it. The propagation of DM pulses is robust, in the sense that small offsets in pulse frequency and PSA position do not distort the transmitted pulse significantly.
4.3. Pseudo-linear pulses
Standard- and DM-soliton systems are designed in such a way that nonlinearity (SPM) compensates dispersion, and pulses in the same channel do not interact. In contrast, pseudo-linear systems are designed to allow the pulses to broaden significantly, so that the effects of SPM are minimized. In such systems, the effect of dispersion (aperiodic breathing) is strong, whereas the effects of nonlinearity (CPM and intra-channel FWM) are weak, and are analyzed perturbatively [23, 24]. In this context, one can model the evolution of pulses, before and after the PSA, using the low-power limits of Eqs. (31)–(36), which are consistent with the LSE (14).
It follows from Eqs. (33), (35) and (36) that
By substituting these formulas in Eqs. (38) and (39), one finds that ρ = 1/(1+ζ2)1/2 and θ = (tan-1ζ)/2. Equations (45)–(47) are consistent with Eq. (21), and the formulas for the reduction and phase factors are consistent with Eqs. (25) and (26), respectively. Hence, the results of Section 3 can be applied to pseudo-linear pulses.
In pseudo-linear systems, the dispersion map is designed to minimize the effects of CPM and intrachannel FWM [10]. Although the cumulative dispersion might vanish at one point (or a few points) in the link, at most points in the link the cumulative dispersion is large, as are the pulse width and chirp. The use of PSA at one of these typical points is inappropriate, for two reasons. First, for large chirps, the level and sensitivity of PSA are reduced significantly, almost to those of PIA (Fig. 4), which is easier to implement. Second, even if one were prepared to accept these performance reductions, the PSA would split the input pulse into two output pulses, one of which is the direct image of the input, and the other of which is a conjugated image [Eq. (22)]. Subsequently, the width of the direct image would increase or decrease, consistent with the dispersion map. However, the width of the conjugated image would decrease or increase, in opposition to the intended evolution [Eq. (29)]. At typical points within the link, each pulse overlaps many others, so information is scrambled. At the end of the link dispersion is compensated (the cumulative dispersion is returned to zero), to return the pulse widths to their transmitted values, and unscramble information, prior to detection. If one were to use PSA at a typical point, the conjugated image pulses would still be broad at the detector. Each data pulse would be surrounded by many dispersed pulses, which would hinder the transfer of information. However, one could use PSA after dispersion compensation, to remove phase perturbations from DPSK systems.
5. Summary
In this paper, a detailed study was made of the phase-sensitive amplification (PSA) of optical pulses by degenerate four-wave mixing (FWM) in randomly-birefringent fibers (RBFs) and rapidly-spun fibers (RSFs). PSA occurs when the carrier frequency of the signal pulse (2) is comparable to the average frequency of the pump waves (3 and 1). The polarization properties of PSA are summarized in Tables 1 and 2. Scalar PSA occurs when the pumps and signal are parallel linearly-polarized (LP) waves, or co-rotating circularly-polarized (CP) waves. Vector PSA occurs when the pumps are perpendicular LP waves, and the signal is LP at 45° to the pumps, or CP. It also occurs when the pumps are counter-rotating CP waves and the signal is LP in an arbitrary direction. The nonlinear coupling between the pumps and signal is strongest when the waves are parallel. None of the aforementioned combinations of pump polarizations produces PSA that is signal-polarization independent.
In a previous paper, the single-mode model was used to determine how the power-amplification factor of a continuous-wave signal depends on its phase. In this paper, the continuous-mode model was used to derive formulas (24), (38) and (39), which describe how the energy-amplification factor of a signal pulse depends on its phase, chirp, and the difference (offset) between its carrier frequency and the average pump frequency. In the absence of chirp, each frequency component of the pulse has the same phase, so every component experiences the same amount of amplification (or attenuation). Consequently, the predictions of the continuous-mode model reduce to those of the single-mode model. However, in the presence of chirp, each component has a different phase, so each component experiences a different amount of amplification. Consequently, the extremal values and phase sensitivity of the amplification factor are reduced, relative to those predicted by the single-mode model. Although the amplification factor depends strongly on the pulse phase, it depends only weakly on the chirp and frequency offset. Specifically, if the chirp is less than unity and the offset is less than the inverse width, the amplification factor is comparable to that predicted by the single-mode model.
Solitons, which are unchirped, are suitable for in-line PSA, which produces less noise than in-line PIA. They are also suitable for post-transmission PSA, which reduces (noiseor collision-induced) phase perturbations and improves the performance of differential-phaseshift-keyed systems. Dispersion-managed solitons (pulses) are also suitable for in-line and post-transmission PSA. Not only are there two points in every stage at which the pulses are unchirped, but at other points in the link the pulses are only weakly chirped. Pseudo-linear pulses, which are strongly chirped at most points in the link, are unsuitable for in-line PSA. However, post-transmission dispersion compensation reduces their chirps, and makes them suitable for PSA prior to detection.
Appendix A: Moment equations for optical pulses
In this appendix the moment method is used to derive evolution equations for the main moments (energy, delay, width, phase, frequency and chirp) of optical pulses. Pulse propagation in a communication system with PI amplifiers and filters is governed by the partial differential equation
where A is the (complex) pulse amplitude, μg is the excess gain coefficient (difference between the amplifier gain and fiber loss rates), νf is the filter coefficient, βr is the (second-order) dispersion coefficient of the fiber, evaluated at the reference frequency, γa = 8γ/9 is the nonlinearity coefficient of the fiber, and the subscripts z and t denote the derivatives ∂/∂z and ∂ /∂t, respectively. Henceforth, the subscripts a, f, g and r will be omitted for simplicity. In many systems μ, ν, β and γ vary with distance. Equation (48) is called the Ginzburg–Landau equation (GLE). For the case in which μ = 0 and ν = 0, it is called the nonlinear Schrödinger equation. Both equations are difficult to solve exactly. One can solve them approximately by using the moment method [25] or the variation method [26], which are based on the assumption that the pulse evolves in a self-similar manner.
In the moment method one defines the power-averaged quantities (moments)
where E is the energy, T is the time delay, V is the squared width, Ω is the frequency, C is the chirp and Φz is the phase rate-of-change [27]. By combining Eqs. (48)–(54), one can show that
Equations (55)–(60) are exact. The integral that is common to Eqs. (55), (59) and (60) is proportional to the (squared) bandwidth moment
Further progress requires the selection of a particular pulse shape.
There are two common shape functions (ansätze). For the Gauss ansatz
which is based on a solution of the linearized GLE [28], the moments E=π 1/2 a 2σ = e, T=τ, V=σ2/2, Ω=ω, C=c, B=ω 2+(1+c 2)/2σ2 and Φz=ϕ z+ωτz+cσz/2σ-cz/4. Because this appendix concerns only the signal (and not the pumps), the subscript s was omitted from the shape parameters (e, τ, σ, ω, c and ϕ). By substituting ansatz (62) in Eqs. (55)–(60), one finds that
For the modified-sech ansatz
which is based on a solitary-wave solution of the GLE [28], the moments E=2a 2σ = e, T=τ, V=π2σ2/12, Ω=ω, C=c, B=ω 2+(1+c 2)/3σ2 and Φz=ϕ z+ωτz+cσz/2σ-κcz, where κ = 1-ln2 ≈ 0.31. By substituting ansatz (69) in Eqs. (55)–(60), one finds that
Subsets of Eqs. (63)–(68) and Eqs. (70)–(75) were derived by several researchers, and their predictions were validated by numerical solutions of the GLE. For the case in which μ = 0 and ν = 0, Eqs. (63)–(68) are identical to the variation equations based on the same ansatz [28, 29].
Appendix B: Dispersion-managed solitons
First, consider pulse propagation in a fiber section with normal dispersion (βn > 0). By making the substitutions σ/σ0→σ and βnz/σ2 0→z, where the subscript 0 denotes an initial value, one can rewrite Eqs. (33), (35) and (36) as
respectively, where the subscript z denotes the derivative d/dz and γn=γeσ0/(2π)1/2 βn is the normalized power. Because this appendix concerns only the signal, and not the pumps, the subscript s was omitted from the shape parameters.
It follows from Eqs. (76) and (77) that
By integrating Eq. (79), one finds that
where the potential function
and κn is the constant of integration. Equation (80) is called the potential equation because it is mathematically equivalent to the equation of motion for a particle moving in a potential well (z, σ and c/σ play the roles of time, position and velocity, respectively). The potential Qn is a monotonically-decreasing function of σ for all values of γn. Consequently, if the pulse is unchirped initially, its width will increase without bound.
The potential equation for the normal section can be rewritten as
where the chirp function
The chirp function is so named because c(z) = ±cn[σ(z)]. Suppose that c(0) = 0. Then κn = 1+2γn [because σ (0) = 1]. By using a standard integral [30], one finds that
Equation (84) defines σ as an implicit function of z, or z as an explicit function of σ . It follows from Eqs. (76) and (78) that
By using standard integrals [30], one finds that
Equations (84) and (86) define ϕ (z) parametrically.
Second, consider pulse propagation in a fiber section with anomalous dispersion (βa < 0). By making the substitutions σ/σ0→σ and |βa|z/σ2 0→z, one can rewrite Eqs. (33), (35) and (36) as
respectively, where γa = γeσ0/(2π)1/2|βa| is the normalized power. By combining Eqs. (87) and (88), one obtains a potential equation of the form (80), where the potential function
If c(0) = 0, the constant of integration κa = 1-2γa. The graph of the potential function has a minimum at σ = 1/γa and intersects the ordinate axis at σ = 1 and 1/(2γa-1). The first root is the initial value. If 0 ≤ γa < 0.5 (low power), the second root is negative (unphysical), so the width increases without bound. If γa = 0.5, there is only one root, so the width increases without bound. If 0.5 <γa< 1 (moderate power) the second root is positive (physical), so the width oscillates between its initial value and a larger value. If the soliton condition γa = 1 is satisfied, the initial value is the minimal point of the potential, so the width is constant. If γa>1 (high power), the width oscillates between its initial value and a smaller value.
The potential equation for the anomalous section can be rewritten in the form (82), where the chirp function
Suppose that c(0) = 0. Then κa = 1-2γa. By using standard integrals [30], one finds that
for κa>0, κa = 0 and κa < 0 (and 0.5<γa<1), respectively. Equations (92) define σ as an implicit function of z, or z as an explicit function of σ. It follows from Eqs. (87) and (89) that
By using standard integrals [30], one finds that
for κa > 0, κa = 0 and κa < 0 (and 0.5 <γa < 1), respectively. Equations (92) and (94) define ϕ (z) parametrically.
Now consider pulse propagation in the DM system described in Section 4.2. At the beginning of the first normal half-section, the width σ (0) = 1 and chirp c(0) = 0. Equations (76) and (77) imply that the width and chirp both increase. At the end of the normal half-section (beginning of the anomalous section) the width σ (ln/2)=σn and the chirp c(ln/2) = cn(σn). Equations (87) and (88) imply that the width and chirp start to decrease. (The chirp could increase, but only for unusually-high powers.) The pulse attains its minimal width σa when its chirp vanishes. If the system parameters are chosen judiciously, the minimum point is the midpoint of the anomalous section. With distance measured backwards from this point, the minimum (initial) conditions are σ (0) = σa and ca(σa) = 0, and the matching conditions are σ (la/2) = σn and ca(σn) = cn(σn). In the second half-stage, the width increases and decreases through positive values, whereas the chirp decreases and increases through negative values.
It follows from the matching conditions that
In the (usual) low-power regime (γa < 0.5), κa is positive and the first forms of Eqs. (92) and (94) are relevant. However, κa could be negative in the medium- and high-power regimes. It follows from the minimum conditions that
where σ is an arbitrary width. Let σ = 1, in which case Eq. (95) implies that c 2 a(1) = κa + 2γa-1 is positive. Then Eq. (96) has two negative roots (σa < 1), one negative root, or one negative and one positive root, for κa > 0, κa = 0 and κ < 0, respectively. In the first two cases at least one negative root is accessible, whereas in the third case the result is ambiguous. Let σ=σn, in which case the matching condition implies that c 2 a(σn) is positive. Then Eq. (96) has one negative root and one positive root (σa>σn). Hence, the negative root is accessible: In the anomalous section the minimal width is always accessible, and smaller than 1.
As stated above, the pulse will only evolve periodically if a certain relation exists among the fiber and pulse parameters. It is customary to specify βa, βn, γ and σ0, and convenient to specify σn (which should be shorter than the bit period). For each value of e, γn and γa are determined explicitly by the formulas that follow Eqs. (78) and (89), respectively, ln/2 is determined explicitly by Eq. (84), and the matching and minimum conditions require that
[Equations (97) and (98) are consistent with Eqs. (95) and (96), respectively.] Because these formulas are explicit, la/2 is determined explicitly by Eqs. (92), modified to account for the fact that σa≠1. By repeating this procedure, one can plot la, ln and the path-averaged dispersion 〈β〉as functions of e. [If the lengths are dimensional, 〈β〉 = (βa la+βn ln)/(la+ln).] By inverting the last of these curves, one can plot e as a function of 〈β〉, as is customary.
Acknowledgments
CM and RM acknowledge useful discussions about the moment equations with V. Grigoryan, W. Kath, M. Kozlov and E. Spiller. The research of RM, SR and RJ was partially supported by the National Science Foundation, under contracts DMS-0511091 and ECS-0406379.
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