1. Introduction

Optical amplifiers are used to boost the signal power which decays due to distributed propagation loss as well as lumped losses in lightwave systems. Commonly used amplifiers, such as erbium-doped-fiber amplifiers (EDFAs), semiconductor optical amplifiers, or Raman amplifiers, belong to the category of phase-insensitive amplifiers (PIAs), i.e., their gain does not depend on the optical phase of the input signal. On the other hand, fiber-optic parametric amplifiers (FOPAs) utilizing the χ ⁽³⁾ nonlinear Kerr effect, can be used as either phase-insensitive or phase-sensitive amplifiers, depending on the configuration of the FOPA. The phase-sensitive amplifier (PSA) only amplifies the in-phase component of the signal, while attenuating the quadrature component [1, 2, 3]. The most well-studied implementation of fiber PSA is a frequency-degenerate FOPA based on nonlinear fiber Sagnac interferometer, first realized in the visible region [4]. Based on this configuration, the first fiber-based telecom-band PSA was demonstrated with phase-sensitive gain of 10 dB [5]. This configuration was also used to achieve nearly-noiseless amplification [6, 7, 8, 9]. In addition to improving the noise figure, the PSAs also perform phase regeneration, which enabled the demonstrations of long-term soliton storage [10] and, more recently, regeneration of differential phase-shift-keying signals [11]. The frequency-degenerate, i.e., interferometer-based, PSAs, however, have several significant drawbacks. First of all, they are inherently single-channel devices not compatible with wavelength-division multiplexing (WDM) operation. Second, they are highly susceptible to guided acoustic-wave Brillouin scattering (GAWBS) noise [12] producing phase fluctuations that are converted by the interferometer into amplitude noise. Without complicated GAWBS cancellation schemes [6, 13], the signal’s spectral content up to 2 GHz is severely degraded. Finally, to provide phase-sensitive amplification, the pump should have exactly the same frequency as the signal, and the relative phase between the pump and the signal must be carefully controlled using either pump injection-locking [14] or optical phase-locking loop [15].

An alternative implementation of fiber PSA is based on frequency nondegenerate FOPA, where two pump photons with frequency ω_p produce one signal photon at frequency ω_s and one idler photon at frequency ω_i, so that ω_s +ω_i = 2ω_p. If only the pump and the signal are present at the input of the nonlinear fiber, the nondegenerate FOPA operates as a PIA. In order for this FOPA to be used as a PSA, the idler must be excited at the input of the fiber together with the pump and the signal (a two-pump FOPA can also be used in a similar manner [16]). Such a process has been previously used in the visible region to demonstrate squeezing of classical noise [17] and amplification of signal-idler sidebands produced by acousto-optic modulation of the pump [18]. In a recent experiment, a telecom band fiber PSA was demonstrated by use of the double-sideband modulation format [19]. Data at 2.5 Gb/s was transmitted over 60 km of dispersion-compensated fiber with bit-error rate performance better than that obtainable with a FOPA of the same gain in a PIA configuration [20]. In both these experiments the phase-coherent sidebands were created by high-speed electro-optic modulation of a continuous-wave (CW) light, which was carried along with the sidebands in the transmission line. This CW carrier was then extracted and used as pump in the PSA after boosting its power with a conventional EDFA. The creation of phase-coherent sidebands with modulation of the pump, however, is limited by electronics and does not permit full utilization of the wide parametric-amplification bandwidth capable of supporting many WDM channels.

In this paper, we report on the first PSA based on frequency nondegenerate optical parametric amplification wherein the input phase-coherent signal-idler pair is prepared by all-optical means, which is potentially compatible with wideband WDM operation. Preliminary results from the scheme described in this paper were presented in an invited talk at LEOS 2003 [21]. We use three sections of fibers in a straight-line configuration. The first is a dispersion-shifted fiber (DSF) that acts as a FOPA-based PIA. In our configuration, the signal, pump, and idler have different frequencies. Through the parametric process in the PIA section, the generated idler automatically acquires a certain phase relationship with the pump and signal. Then, the pump, signal, and the generated idler from the PIA section enter a single-mode fiber (SMF) whose dispersion produces wavelength-dependent phase shifts changing the relative phase between the three waves. Even though the phase is modified by the SMF, the new relative phase is stable and no phase-locking control is needed. After the SMF, the pump, signal, and idler enter a third fiber section, which is a second DSF that acts as a FOPA-based PSA. Depending on the signal wavelength, the SMF-induced phase shift leads to either amplification or de-amplification, demonstrating the phase-sensitive behavior of the frequency nondegenerate FOPA.

It may appear at first that a nondegenerate PSA would be wasteful of bandwidth, as it requires twice the optical bandwidth compared to a corresponding degenerate PSA. This is, however, not true since a nondegenerate PSA amplifies simultaneously both the sum of the amplitude quadratures of the signal and idler fields and the difference of their phase quadratures [22]. Thus, a nondegenerate PSA transmits twice as much information as a degenerate PSA, while occupying twice as much bandwidth. The amplitude information can be easily recovered by directly detecting either the signal or the idler fields only. However, if both quadratures are encoded (i.e., the signal spectrum is not symmetric), then a more elaborate detection scheme would be required, e.g., homodyne detection of the signal field alone.

While our experiments only use a short length of SMF to demonstrate that a PIA is a convenient way to prepare the phase-coherent signal-idler input for the PSA, we note that in a practical application the remaining pump from the PIA can be co-transmitted with the signal and idler to a remote downstream PSA as a pilot tone to which a local pump can be phase-locked. Since all three beams propagate in the same fiber, their relative phase experiences only very slow drift due to temperature-caused dispersion variations, and can be easily adjusted for maximum gain. Moreover, multiple phase-matched signal-idler pairs can be potentially simultaneously generated by the PIA from a WDM signal input. In this paper, we do not explore the noise-figure characteristics of the PSA because the PIA stage in this configuration sets a limit on the noise-figure of the system. In practical applications, however, the PIA needs to be used for generation of the signal-idler combination only once at the beginning of the communication link, hence its noise figure would contribute a very little fraction of the total link noise [22].

The structure of this paper is as follows. In section 2 we apply a general theoretical model of a frequency nondegenerate FOPA to the special case of a FOPA-based PSA to predict the PSA gain characteristics. The details of the parametric-gain derivation are provided in the appendix. In section 3, we present a series of experiments confirming the theoretical predictions, and summarize the paper in section 4.

2. Theory

In a FOPA, gain is a consequence of the four-wave mixing (FWM) process in which two photons from the pump are converted into a pair of signal and idler photons. The basic equations describing this process, in terms of optical powers and phases, are [23]:

where P_p, P_s and P_i are the optical powers for pump, signal and idler, respectively. α is the linear loss coefficient of the gain medium fiber. γ=2 π n ₂/λ A_eff is the nonlinear coefficient with n ₂ and A_eff as the fiber’s nonlinear index and effective mode area, respectively. Δβ = β_s + β_i - 2β_p is the linear phase mismatch per unit length between the signal, idler and pump with propagation constants β_s, β_i, and β_p. The relative phase difference is

where ϕ_s, ϕ_i, and ϕ_p are the phases of the signal, idler and pump, respectively.

When the pump, signal, and idler waves are present at the input of the fiber, Eqs. (1)–(4) show that the power flow depends on the relative phase between the three waves, i.e., it can be transferred either from the pump to the signal and idler (when θ = π/2, parametric amplification) or from the signal and idler to the pump (when θ = -π/2, parametric de-amplification). This gives us the possibility to create an optical amplifier whose gain depends on the phase of the input signal, i.e., a PSA.

Figure 1 schematically describes our PSA model, which is comprised of three segments of fibers. In the first segment, DSF1 is configured as a PIA, in which signal is amplified and the idler is generated. The relative phase relationship between the three waves at the output of the PIA, θ ^Out_PIA depends on the gain and the phase-matching characteristics of DSF1. The second segment consists of a piece of SMF which acts as a wavelength-dependent phase shifter, where the phase relationship between the pump, signal and idler is changed. For pump, signal and idler angular frequencies (ω_p, ω_s, and ω_i = 2ω_p - ω_s, respectively) satisfying ∣ ω_s - ω_p∣ = ∣ω_i - ω_p∣ <<ω_p, one can expand Δβ ≅ β ₂(ω_s - ω_p)², where β ₂ is the group-velocity-dispersion coefficient of the SMF. After propagating through the SMF, the relative phase at the input of the PSA-DSF2 segment becomes:

 

Fig. 1. Schematic of the phase-sensitive fiber parametric amplification model; p: pump, s: signal, i: idler.

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where L _SMF is the length of the SMF section and the various superscripts distinguish θ(ω) at various locations along the three-segment fiber line. Note that in the derivation of Eq. (6), the nonlinear phase shift inside the SMF is neglected since it is much smaller than the linear phase shift. Because of its large dispersion, even a few meters of SMF can cause tremendous changes in θ(ω), varying as a function of wavelength. After conversion from frequency to wavelength, Eq. (6) becomes:

where D is the dispersion coefficient of the SMF at the pump wavelength and c is the vacuum speed of light.

To theoretically investigate the phase-sensitive performance of our model (Fig. 1), Eqs. (1) to (7) were solved using parameters that match the experimental conditions to be described in the next section. The nonlinear coefficient γ, the dispersion slope, and the loss coefficient α of DSF1 and DSF2 (pieces of highly-nonlinear fiber with small effective-area core) were ≃ 9 W^-1km^-1, 0.018 ps/nm²/km, and 0.75 dB/km, respectively. The length and zero-dispersion wavelength (λ ₀) of each segment were 1020 m with λ ₀ = 1558 nm for DSF1 and 500 m with λ ₀ = 1556 nm for DSF2. The SMF used in the calculation was chosen to be 10 m long. By using a continuous wave (CW) pump light at 1560 nm and a CW signal light with wavelength tunable from 1538 to 1582 nm we calculated as a function of the signal’s wavelength both the sinusoidal-function values of the relative phase at the input of DSF2, i.e., sin(θ ^In_PSA(λ)), and the gain of the PSA, as shown in Fig. 2. Three steps are followed in the calculation. Firstly, at each signal wavelength the gain G _PIA(λ) and the relative phase θ ^Out_PIA(λ) at the output of the PIA section (DSF1) were calculated using Eqs. (1)–(4). Secondly, changes in the relative phase provided by the SMF, i.e., θ ^SMF(λ), were calculated using $\left(\lambda \right)\simeq -2\mathit{\pi c}{\left(\frac{{\lambda }_p-\lambda }{\lambda }\right)}^{2}D{L}_{\mathrm{SMF}}$. Then using θ ^In_PSA(λ) = θ ^Out_PIA(λ)+ θ ^SMF(λ) [see Eq. (7)] the relative phase at the input of the PSA section was obtained. The last step was to obtain the phase-sensitive signal gain inside the PSA by importing the amplitudes and the relative phases of the pump, signal, and idler into the calculations using Eqs. (1)–(4) for the DSF2.

 

Fig. 2. (a): Plot of the sine values of the relative phase at the input of DSF2; (b): Plot of the calculated gain of the PSA (DSF2).

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Figure 2 clearly shows that the gain spectrum follows θ ^In_PSA(λ) with a quasi-periodically-varying profile that contains peaks corresponding to maximum amplification and valleys corresponding to maximum attenuation (de-amplification). For example, once θ ^In_PSA(λ) changes from - π/2 (module 2π) at wavelength λ _min,k to +π/2 at wavelength λ _max,k and back to - π/2 at λ _min,k+1, the signal gain will change from a minimum point (at λ _min,k+1) to a maximum point (at λ _max,k) and to another minimum point (at λ _min,k+1), respectively. Thus, a 2π change in θ ^In_PSA(λ) from one wavelength to another will generate either one peak and two valleys when starting from a valley location or one valley and two peaks when starting from a peak location.

To study the gain characteristics of a FOPA with and without an idler at its input, an analytical solution for the gain of a FOPA-based PSA can be obtained. By using the procedure described in appendix A, we obtain:

where, due to the fact that only the PSA section (DSF2) is considered, θ = ϕ_s + ϕ_i - 2ϕ_p, κ = Δβ + 2γP_p is the net phase mismatch, g = [(γP_p)² - (κ/2)²]^1/2 is the parametric-gain coefficient, and ${\eta }^2=\frac{P_i\left(0\right)}{P_s\left(0\right)}$ is the power ratio between the idler and the signal at the input of the PSA.

The following two cases should be considered in the solution for the gain in Eq. (8):

Equations (9) and (10) show that when the idler is present at the input together with the signal and the pump, the maximum gain of the signal is larger by 6 dB compared with the case when the idler is not present at the input. Physically, this is because when the idler is present, the effective input to the amplifier is a coherent sum of the signal and idler (giving a factor of 2 in the fields, or factor of 4 in the powers). For the same pump power as in the PIA case (no input idler), this effective input is transformed into a 4-times higher output signal, leading to 6-dB gain increase when we consider the power gain of the signal mode alone. Of course, this holds only at large gains [limiting case in Eq. (10)], and only for the signal and idler fields that align perfectly in phase. When the PSA gain is not large, the 6-dB advantage is reduced somewhat, as is the case in the experiment below [see discussion after Fig. 6(b)].

3. Experiment

The experimental setup of the nondegenerate PSA is shown in Fig. 3. Two spools of DSFs were used to create the PIA and PSA sections. The parameters of the DSFs are the same as given in the theoretical model. The CW pump light at 1559.8 nm was phase-modulated to suppress the stimulated Brillouin scattering in the DSFs, strongly amplified by an EDFA, filtered by a 1-nm optical bandpass filter to remove the out-of-band amplified spontaneous emission (ASE) coming from the EDFA, and then coupled into the PIA stage via a 90/10 coupler. A wavelength-tunable laser (TLS) was used as the signal source. Between the PIA and PSA stages, a piece of SMF was used to create the wavelength-dependent phase shift, followed by a 95/5 splitter whose 95% output was connected to the PSA and the 5% output was used to monitor the PSA input. A 0.2 nm optical notch filter (an isolator followed by a fiber Bragg grating) centered at the pump wavelength (1559.8 nm) was connected after the PSA to remove the high-power pump before the signal was detected with an optical spectrum analyzer.

Figure 4(a) shows the optical spectra of the parametrically amplified spontaneous emission, known as parametric fluorescence (PF), at the input (black curve) and output (blue curve) of the PSA segment (500-m-long DSF) when only the pump was present at the input of the PIA and the length of the SMF was chosen to be 10 m. The pump powers into the PIA and PSA stages were 265 mW and 180 mW, respectively. This figure shows that, unlike the well-known smooth PF curve generated by a PIA (black curve in the figure), the PF at the output of the PSA changes quasi-periodically, with peaks and valleys unevenly distributed in the wavelength domain. The phase-sensitive gain performance of the PSA can be evaluated on a logarithmic scale (without introducing an input signal) by subtracting the input PF from the output PF, as shown by the black curve in Fig. 4(b). The quasi-periodic dependence of the PF gain on wavelength demonstrates the phase-sensitive performance of the amplifier, which is matched pretty well by the theoretical calculation of the PSA gain shown by the blue curve in Fig. 4(b). However, there is a noticeable mismatch at the edges of the optical spectra shown. The cause of this can be attributed to the following: (1) the actual dispersion slope of the SMF around 1550 nm varies, whereas in the simulation it is kept constant; (2) the signal experiences a nonlinear polarization rotation due to the strong pump, which is not taken into account in the simulation; (3) the Raman effect accompanying the parametric process is not included in the simulation, and (4) fluctuations in the zero-dispersion wavelength of the fiber can cause some gain to appear in wavelength regions where the theory for uniform zero-dispersion wavelength does not predict any gain [24, 25].

 

Fig. 4. (a) Measured input (black) and output (blue) PF spectra of DSF2 with 10-m-long SMF. (b) Measured (black) and calculated (blue) PF gain with 10-m-long SMF. (c) and (d) show the measurements and calculations [similar to (a) and (b), respectively] for a 15-m-long SMF.

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To further verify the phase dependence of the gain, we also measured the PF and the gain spectra with 15-m-long SMF, as plotted in Figs. 4(c) and (d), respectively. These figures show that the PF and gain profiles change due to the change of the relative phase spectrum at the input of the DSF2, which is a function of the SMF’s length. When the input phase relationship changes, the gain property of the DSF2 changes [as predicted by Eqs. (1)–(7)], resulting in shifts in the positions of the peaks and valleys in the spectra.

 

Fig. 5. Calculated (black curve) and measured gain (diamonds) and attenuation (squares) of the PSA segment for (a) 1000-m-long PIA and 500-m-long PSA and (b) 500-m-long PIA and 1000-m-long PSA.

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Next, we investigated the parametric amplification/deamplification performance of the non-degenerate PSA with an incoming signal, whose wavelength was tuned to the peaks and valleys of the PF-gain spectra. The SMF between the PIA and PSA stages was 10 m long. Figure 5(a) shows the calculated gain spectra of the PSA together with the measured signal gain at the wavelengths of the peaks and valleys. One can note that the measured PSA gain (attenuation) is larger (smaller) on the Stokes side (signal whose wavelength is longer than that of the pump) and smaller (larger) on the anti-Stokes side of the pump (signal whose wavelength is shorter than that of the pump). We believe that the main reason for this is the Raman-assisted gain (loss) seen by a signal on the Stokes (anti-Stokes) side. This effect is not included in our theoretical model. Otherwise, the experimental results match reasonably well with the theoretical predictions.

The parametric gain/attenuation characteristics of the nondegenerate PSA with an incoming signal were also measured when the shorter DSF (DSF2) was used as the PIA segment and the longer DSF (DSF1) as the PSA segment, as shown in Fig. 5(b). Similar differences between the experimental data and the theoretical simulation are seen in Fig. 5(b), as in Fig. 5(a), owing to the Raman effect. Additionally, because a longer DSF is used in the PSA stage, the gain of the PSA is larger for the same amount of pump power, as is expected for any kind of FOPA in general.

Finally, to investigate the differences between the gain characteristics of a FOPA that acts as a PIA and a FOPA that acts as a PSA, we measured the gain of the third fiber segment with and without the idler at its input. By using the 500-m-long DSF with the same pump power (180 mW) as in the previous experiments, we measured, without any idler input, the gain spectra of a FOPA-PIA [shown as triangles in Fig. 6(a)] with the signal wavelength tuned to the peak wavelengths in Fig. 5(a). The squares in Fig. 6(a) are the measured gain with an idler at DSF’s input, whereas the circles represent differences between the squares and the triangles at each wavelength. Figure 6(b) shows the data when the same procedure was repeated with a 1000-m-long DSF used as the PSA gain medium. In both cases, the results show that when the wavelength detuning between the signal and idler is small (< 10 nm), the gain of the PSA is larger than that of the PIA with a maximum difference of about 5.4 dB. This result supports the theoretical prediction of ~6 dB difference given by Eqs. (9) and (10). We note that when the wavelength detuning increases, the gain difference decreases. This can be due to the following reasons: (1) the phase-matching condition inside the PSA is degraded when the wavelength detuning from the pump increases, (2) the power ratio between the idler and the signal, η, is reduced at the input of the PSA due to gain reduction of the PIA stage when the phase-matching condition degrades.

 

Fig. 6. Measured gain with input idler (squares), without input idler (triangles), and the difference of gains with and without input idler (circles). (a) and (b) are for measurement results with 500-m-long and 1000-m-long DSFs, respectively.

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4. Conclusion

We have demonstrated, for the first time, a phase-sensitive amplifier based on the frequency-nondegenerate parametric amplification process in optical fibers wherein the input is prepared by all-optical means. This is achieved by using a phase-insensitive parametric amplifier to create an input signal-idler pair which is then sent into a second, phase-sensitive, parametric amplifier. We have presented a model describing the two fiber-optic parametric amplifier sections separated by a wavelength-dependent phase shifter. The wavelength-dependent gain profile of the phase-sensitive amplifier is calculated and confirmed in a series of experiments. In agreement with the theory, we have observed that, when the idler is present at the input together with the pump and the signal, not only is the amplification sensitive to the phase of the incoming signal, but also the maximum gain of the signal is approximately four-fold greater than that in the case when the idler is not present at the input. Even though the separation between the phase-insensitive and phase-sensitive amplifiers in our experiments is only 10–15 m, the demonstrated technique can be used to generate an automatically phase-matched signal-idler pair at the transmitter for launching into a phase-sensitively amplified long-distance communication line, thereby facilitating the practical realization of the phase-sensitive amplification technology. One must note that the total dispersion in the long-fiber line would need to be compensated to less than the equivalent dispersion of 10–15 m of SMF. This requirement, however, is not significantly more stringent than the dispersion map accuracy requirement of 100 m of SMF in 40Gb/s systems wherein it is easily handled by tunable dispersion compensators.