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Four-wave mixing in the form of Bragg scattering (BS) has been predicted to enable quantum noise-less frequency conversion by analytic quantum approaches. Using a semi-classical description of quantum noise that accounts for loss and stimulated and spontaneous Raman scattering, which are not currently described in existing quantum approaches, we quantify the impacts of these effects on the conversion efficiency and on the quantum noise properties of BS in terms of an induced noise figure (NF). We give an approximate closed-form expression for the BS conversion efficiency that includes loss and stimulated Raman scattering, and we derive explicit expressions for the Raman-induced NF from the semi-classical approach used here. We find that Raman scattering induces a NF in the BS process that is comparable to the 3-dB NF associated with linear amplifiers.
© 2017 Optical Society of America
1. Introduction
Parametric processes offer a range of important potential applications in future all-optical communication systems: low-noise phase-insensitive and phase-sensitive amplification [1,2]; the ability to regenerate noisy optical signals [3,4]; and alteration of the temporal mode profile as well as the frequency of optical quantum states, while preserving other properties of the quantum state of light [5, 6]. The third-order nonlinearity of optical fibers has been shown to enable tunable frequency conversion by four-wave mixing (FWM) in a special configuration known as Bragg scattering (BS) [7]. In BS, one uses two pumps p and q, which interact with a signal s and an idler i fulfilling energy conservation ωp + ωs = ωq + ωi, where ωj is the angular frequency of the electric field for j = {p, q, s, i}, see Fig. 1. Even though BS is driven by two strong pumps, the process has been predicted by analytic quantum approaches to be free of additional quantum noise [8–11] as opposed to phase-insensitive parametric amplifiers or Raman and Erbium-doped fiber amplifiers, where amplified spontaneous emission induce a 3-dB noise figure (NF) in the high gain limit. These predictions are based on FWM being effectively phase-matched and the only influence on the wave components. However, material and waveguide dispersion of the third-order nonlinear device in which BS is realized may induce a significant phase mismatch and thereby reduce the conversion to take place in a confined frequency window. Also, amorphous materials such as silica and chalcogenide (As2S3) have broad Raman response spectra that typically extend beyond this limit of phase matching [12]. Finally, attenuation in the nonlinear medium reduces the efficiency of BS and therefore limits the achievable output. Raman scattering can be avoided on crystalline nonlinear platforms, such as devices based on silicon, but such integrated devices have significantly lower nonlinearity relative to linear loss and two-photon absorption compared to silica fibers [13,14].
Fig. 1 Sketches of selected frequency configurations performing down-conversion from signal (s) to idler (i); the two pumps, p and q, need not have equal magnitudes. δ is the frequency separation between the two side-bands in either of (a)–(d), and Δ is the frequency separation between the zero-dispersion frequency and the closest wave component on either side. By changing ωs ↔ ωi and ωp ↔ ωq, up-conversion is achieved
BS has been studied extensively for both classical [15,16] and quantum signals [10,17,18], and the effects of stimulated and spontaneous Raman scattering (SRS and SpRS) have been studied in parametric amplifiers [1,19–21] and for photon-pair generation by degenerate FWM [22,23], but little attention has been given to their impacts on frequency conversion using BS, which is mainly due to the difficulty of implementing Raman scattering in quantum descriptions of FWM.
In this paper, we use a semi-classical approach [21] to produce realistic predictions for the conversion efficiency (CE) and noise properties of BS in silica-based highly nonlinear fibers. The semi-classical approach simulates coherent states by classical field ensembles that display correct Gaussian statistics. One ensemble for each wave component is then propagated through the classical equations describing FWM including dispersion, loss, and Raman scattering. During propagation in the fiber, fluctuations are added to the ensemble, which account for spontaneous emission, and thus distort the performance of the FWM process. Note that such semi-classical approaches can predict the quantum noise properties of classical signals but they do not capture the quantum nature of single-photon states.
In Fig. 1, four possible frequency configurations of BS are shown: in Figs. 1(a) and 1(b), the signal and idler are placed on the Stokes (S) and anti-Stokes (aS) side of the two pumps, respectively; Figs. 1(c) and 1(d) offer alternative configurations that have the same FWM properties but different Raman interaction. In all four setups, δ is the frequency separation between the two components of each side-bands, and Δ is the separation between the zero-dispersion frequency, ω0, and the closest wave component on either side of ω0. It is implied that the waves are placed symmetrically around ω0 to obtain phase matching independent on the dispersion slope of the fiber. In this work, we investigate only the two setups in Figs. 1(a) and 1(b), henceforth referred to as Case (a) and Case (b).
This paper is structured as follows: in Sec. 2, the classical propagation equations that describe FWM, SRS and loss in a fiber are derived and the approach for semi-classical modeling of quantum noise is outlined. In Sec. 3, we investigate the effects of fiber attenuation using realistic physical parameters. Finally in Sec. 4, we quantify the combined effects of FWM, SRS, and SpRS, and we show that Raman scattering has a significant impact on both the CE and noise properties of BS in a typical highly nonlinear silica fiber.
2. Theory
2.1. Propagation equations of FWM and Raman scattering
Equations that describe the propagation of an electromagnetic field through nonlinear optical fibers including the effects of both FWM and Raman scattering can be derived directly from Maxwell’s equations; using the approach of [24], a general propagation equation in a single mode nonlinear fiber for the amplitude of a continuous-wave electric field, En, at frequency ωn is [25]
where βn is the propagation constant at frequency ωn, and r = (x, y, z) is the position vector where z is the longitudinal coordinate in the fiber and x and y are the transverse coordinates. In the derivation of Eq. (1), the total electric field was expanded in a set of continuous-wave components where Fm(x, y) is a three-component vector holding the transverse field distribution functions of the three spatial components of the electric field, and Nm is a normalization constant defined as [25] where Hn is the same for the magnetic field as Fn(x, y) is for the electric field. For a linearly polarized field, Fm(x, y) = Fm(x, y)x̂, where x̂ is a unit vector pointing in the transverse direction x in the fiber, is calculated to be where is the effective refractive index at frequency ωm. The time-independent third order nonlinear induced polarization in Eq. (1) is defined in terms of the total third order nonlinear induced polarization as The total nonlinear induced polarization is written as [26] where the vertical dots denote the third order tensor product. The third order response function including both FWM and Raman scattering is [25], under the assumption that only one term of the third order susceptibility, , is non-zero, where χ(3) is the total third order susceptibility, fR = 0.18 is the Raman fraction, and hR(t) is the Raman response function.In this paper, we consider four frequency components that are connected through energy conservation ω1 + ω4 = ω2 + ω3 and they interact through both FWM and Raman scattering. Inserting Eq. (7) into Eq. (6), collecting terms in Eq. (6) that oscillate at ωj where j = {1, 2, 3, 4} [27], then using Eq. (5) to insert into Eq. (1), and finally using Eq. (4), the propagation equation for the field amplitude at ω1 is derived,
where is the nonlinear refractive index, Aeff is the usual effective area of the fundamental mode in the nonlinear fiber [26], Δβ is the phase mismatch elaborated below (note that in the equations of ω2 and ω3, the sign in front of Δβ is opposite), and h̃R(Ωnm) is the Fourier transform of hR(t) evaluated at Ωnm = ωn − ωm. The linear loss term with coefficient α has been added and it is assumed common for all frequencies.Equation (8) and the corresponding equations of ω2 through ω4 are applied to Case (a) in Fig. 1 by setting ω1 = ωi, ω2 = ωs, ω3 = ωp, and ω4 = ωq, and similarly for Case (b). Note that other FWM processes, depending on the values of δ and Δ and the dispersion properties of the fiber, comprising one or both of the pumps may take place simultaneously with Cases (a) and (b). For example, when the pumps are situated in the anomalous dispersion regime, the signal or idler may stimulate parametric amplification, which both affects the CE of the BS process and induce additional noise. In the special case of the amplification process being properly phase matched, the noise properties of the combined processes may be dominated by the parametric amplification noise contribution. Therefore, it must be emphasized that this paper studies only the noise properties of the BS processes of Cases (a) and (b) and not the contributions from multiple simultaneous FWM processes.
The time-domain Raman response function, hR(t), is often approximated by a single damped oscillator function [28], but it turns out that such a simple description is not accurate for SpRS close to the pumps. Therefore, we use an extended response function,
where the coefficients bj, ωj, ηj, and Γj are determined in [29] and valid for a silica-core fiber [30], and Θ(t) is the heaviside step function for which Θ(t) = 0 for t < 0 and Θ(t) = 1 for t ≥ 0. To implement Eq. (9) in Eq. (8), the Fourier transform is applied and the resulting imaginary part is used while the real part is disregarded for simplicity. According to [31], a significant effect of the real part of the Raman susceptibility is only observed when using a strong pulsed pump; the comparatively weak continuous-wave pumps used in this paper justify neglecting the real part of the Raman susceptibility.By expanding the wave-number of each wave component in a Taylor series around the zero-dispersion frequency, the phase-mismatch Δβ = β2 + β3 − β1 − β4 is written in terms of δ and Δ from Fig. 1 as
Because of the symmetry of the BS configuration around the zero-dispersion frequency, all odd dispersion terms cancel, which means β4 is the lowest one that is used in the calculations. If we consider only the FWM terms, it is instructive to express the CE in terms of the phase mismatch. We assume the two strong pumps to have constant amplitudes such that where Pp and Pq are the constant pump powers and γj = n2ωj/(cAeff). By solving the resulting equations for the signal and idler, neglecting loss and Raman scattering, one easily finds where is the effective nonlinear strength, g2 = ηiηs + (κ/2)2 is the phase-mismatched conversion coefficient, and where the first ‘+’ (second ‘−’) corresponds to Case (a) and Fig. 1(c), and the first ‘−’ (second ‘+’) corresponds to Case (b) and Fig. 1(d). Maximum conversion from signal to idler is achieved for κ = 0, which gives CE = ηi/ηs, which further is the same as full conversion in photon numbers. This condition can be met experimentally by adjusting the difference in pump powers to counter-balance the phase mismatch if the values of δ and Δ are not too large relative to the fourth-order dispersion. Alternatively, a broad bandwidth of phase matching can be achieved through dispersion engineering, as demonstrated recently in a dispersion shifted fiber [32], or special phase matching properties across small [33] or large [34] bandwidths can be achieved using higher order modes. If κ ≈ 0 is valid, one may, with a few simplifying assumptions, derive an approximate analytic expression for the CE where Raman scattering and loss are included. In the pump equations, energy exchange terms, cross-phase modulation terms of the signal and idler, the effect of loss on the phase modulation terms of the pumps, and the Raman interaction between the pumps are disregarded. In the signal and idler equations, only the small phase modulation terms of the signal and idler are disregarded. The CE from signal to idler becomes (after some calculations) where μ2 = ηiηs − (fi − fs)2/4 is the phase matched conversion coefficient, zeff = (1 − exp[−αz])/α is the effective position in the fiber, and fi = iγi fR (h̃R(Ωpi)Pp + h̃R(Ωqi)Pq) and fs = iγs fR (h̃R(Ωps)Pp + h̃R(Ωqs)Pq) are the Raman contributions to the signal and idler, respectively. In Eq. (16), it was assumed that αz ≪ 1, which is valid for typical lengths of highly nonlinear fibers.2.2. Spontaneous Raman scattering
SpRS has been studied earlier [35,36], and a model for the accumulation of amplified spontaneous emission (ASE) seeded by the Raman process at frequency ωm on the S side of a given pump at frequency ωn of power Pn is
where B0 is the frequency bandwidth of the wave component at frequency ωm, is the Raman gain coefficient, and is the phonon equilibrium number at a pump-signal frequency separation of Ωnm, where ħ is Plancks reduced constant, kB is Boltzmanns constant, and T is the temperature [37]. On the aS side, the corresponding expression is The -term in Eq. (17) means that SpRS on the S side does not require any phonons present but on the aS side the -term gives SpRS a proportional dependence on the number of phonons. We define the rates of SpRS for the S and aS processes as and , respectively, and they are plotted in Fig. 2, which shows these quantities at room (solid blue) and at liquid nitrogen (dashed red) temperature versus frequency shift Ωnm from one of the pumps, p or q. In contrast to SRS, the rate of SpRS is asymmetric around the pump, which is important to consider when small signals are situated simultaneously on both sides of the pump, e.g. as they may be in FWM. Lowering the temperature reduces the rate of SpRS significantly on the aS side as expected, whereas on the S side a significant reduction occurs only close to the pump.Fig. 2 Rate of SpRS versus real frequency shift νij = Ωnm/2π from a pump p or q for temperatures T = 300 K (solid blue curve) and T = 77 K (dashed red curve). The inset shows the Raman gain coefficient for silica core fibers (solid black curve) and the phonon equilibrium numbers for the same temperatures as in the main plot (with same line styles).
2.3. Simulating quantum noise classically
The quantum noise properties of parametric processes [9] can be predicted by semi-classical methods [38]. One approach is to create field ensembles for simulating quantum coherent states at the input of the fiber and then propagate each element of the ensemble through classical equations (as derived in the previous section) in parallel; while propagating through the fiber, fluctuations must be added to the ensembles at every numerical step to account for SpRS and loss [21]. The inherent fluctuations of a quantum field is represented in the ensemble, and the signal-to-noise ratio (SNR) is evaluated as
where Aens is the complex valued classical field ensemble and Var(·) denotes the ensemble variance. The induced NF is then defined as NF = SNRin/SNRout. The induced NF for the BS process is defined as SNRs,in/SNRi,out, i.e. the SNR of the signal at the input relative to the SNR of the idler at the output. A coherent state is simulated classically by defining the field amplitude ensemble as where δx and δp are the quadrature fluctuation variables that follow Gaussian distributions with the properties 〈δx〉 = 〈δp〉 = 0 and 〈δx2〉 = 〈δp2〉 = ħωB0/4 [39], and is the mean field power. The variables x0 and p0 are the quadrature mean values, and the last term, ħωB0/2, is the explicit inclusion of the vacuum state energy, which is an artifact of the semi-classical model. The magnitude, however, is so small that no significant error in the mean field power can be observed. To obtain reliable statistics, we use ensembles of 5 · 104 elements in all simulations.By propagating the ensemble of Eq. (20) through the classical equations derived in the previous section, the quantum noise associated with the FWM process is captured [38]. When the field ensemble is subject to loss through the fiber, fluctuations must be added to each quadrature of all field elements to ensure that the mean field power of the field ensemble converge toward the value of ħωB0/2, the vacuum energy explicitly included in the ensemble, instead of zero. The loss fluctuation δaloss follows a Gaussian distribution with the statistical properties [21]
where Δz is an infinitesimal piece of fiber in which the attenuation takes place, e.g. the numerical step size. The variance of the loss fluctuation in Eq. (22) ensures that the mean power of the ensemble that undergoes a large loss converges towards ħωB0/2. SpRS is included by adding another fluctuation to each quadrature of all field elements in the ensemble. Because the rates of SpRS on the S and aS sides are unequal, different fluctuation terms must be assigned to them: on the S side of a wave component n of power Pn, the fluctuation δaRaman,S must be added to each quadrature of a field m with the properties (calculated in the Appendix) where and depend on the frequency shift as explained above and Δz is assumed small. On the aS side of wave component n, the corresponding fluctuation, δaRaman,aS, has the properties The effect of adding fluctuations during propagation is visualized for the loss process in Fig. 3: in diagram (i), a coherent state ensemble of Eq. (20) is visualized in phase space and in diagram (ii), the ensemble is attenuated (blue→red) through the classical Eq. (8) in the absence of FWM and Raman scattering. As described in the classical Eq. (8), loss is a linear and phase-insensitive process, so all elements of the ensemble are translated directly towards the origin of phase space; this process is unphysical because the coherent state is squeezed in all directions simultaneously, thus breaking the uncertainty principle. Diagram (iii) shows that the effect of adding the loss fluctuation of Eqs. (21)–(22) is to increase size of the red ensemble in phase space. Thus, the addition of the fluctuation ensures that the shape of the coherent state is conserved when subject to loss and at the same time it induces a NF equal to the loss as expected of a passive device.Fig. 3 (i) A coherent state ensemble visualized in a phase-space diagram, (ii) how the ensemble is affected by loss in the classical equations, and (iii) the effect of adding loss fluctuations.
The ensemble approach described here has the advantage that it includes both amplitude and phase noise from SpRS in the FWM process. Raman amplifiers are usually described in power or photon number equations and the noise properties are derived from a statistical approach to the photon number mean and variance, which thus excludes phase noise. However, note that the semi-classical modeling presented here inherently works for continuous variable states only; the discrete nature of photon number states is not captured by the model.
3. Fiber attenuation
In the context of realizing quantum state preserving frequency conversion, it is of interest to investigate how loss and the addition of loss fluctuations at the different interacting waves affect the CE and NF of BS, so we exclude Raman scattering and assume κ = 0 by one of the approaches discussed above.
Figure 4 shows the results of solving Eq. (8) (and the three corresponding ones of ω2–ω4) through a fiber of length L = 4 km and the addition of loss fluctuations during propagation with α = 1 dB/km for all wave components, which implies that the results do not depend on which of the setups in Figs. 1(a)–1(d) is considered. The results depend only slightly on the values of δ and Δ through the explicit frequency dependence on the right hand side of Eq. (8). This frequency dependence is so small in highly nonlinear fibers that it is usually neglected [26]. The CE versus position in the fiber is plotted in Fig. 4(i) (blue dots) together with the analytic result of Eq. (15) (solid black), where an excellent agreement is found. The red-dashed line compares the simulated CE to the approximation of Eq. (16), which gives the same as if loss on the pumps is neglected entirely; for small αz the approximation is seen to be reasonable. The importance of the solid-red line, the loss factor, is seen in Fig. 4(ii) in which the loss-induced NF is plotted: the blue dotted curve shows the simulated NF versus position in the fiber where each local minimum corresponds to each local maximum in CE in Fig. 4(i). The solid-red line thus marks the expected loss-induced degradation of the idler SNR relative to the input signal SNR that equals exp(αz). The impact of fiber attenuation can be reduced by increasing the pump powers, which gives a shorter conversion distance since the conversion coefficient is μ2 = ηiηs ∝ PpPq. Thus, the accumulated signal and idler loss becomes smaller.
Fig. 4 (i) CE versus fiber length with optimal phase matching and without Raman scattering. The red lines visually illustrate the effects of attenuation on the CE. (ii) The same for the conversion NF. In the simulation, α = 1 dB/km, Pp = Pq = 0.2 W, γs = 9.89 (W km)−1, β4 = 0 ps4/km, δ/2π = Δ/2π = 1 THz, and Δz = 20 m.
To investigate how the addition of loss fluctuations at the different wave components affects the overall CE and noise properties of BS, we set the loss coefficient to zero in the pumps, signal, and idler in turn and repeated the simulation above (not shown graphically). If the loss in the signal and idler is excluded while keeping the loss in both pumps, the CE oscillates between 0 and 1 as if no loss was present, but the oscillation is still slowed by the loss in the pumps. The NF follows the CE, thus oscillating between infinity and 0. Hence, the presence of loss in the pumps does not hinder efficient and noise-less frequency conversion. Turning on the loss in either of the signal or idler gives indistinguishable results, the characteristics of which are intermediate between turning the loss on or off in both. Consequently, loss in the signal affects the BS process to the same degree as loss in the idler due to their coupling through FWM. If only the loss in the pumps is excluded, the slowing of the CE and NF oscillation disappears, which means that the point of maximum conversion is moved to a shorter fiber length but still limited by the loss in the signal and idler, and the solution of Eq. (16) is very accurate. However, the maximum CE in Fig. 4(i) and the minimum NF in Fig. 4(ii) do not increase and decrease, respectively, relative to the solid red lines. This confirms that loss in the pumps does not hinder efficient and noise-less frequency conversion using BS; it only determines the fiber length of maximum conversion.
4. Stimulated and spontaneous Raman scattering
FWM and Raman scattering have a complicated interplay because they depend differently on the frequency shifts Δ and δ: FWM requires phase matching, while SRS, which is anti-symmetric around the pumps, has a complicated material response in the frequency domain. Furthermore, as shown in Sec. 2.2, SpRS is asymmetric around the pumps. To isolate the effects of SRS and SpRS in the FWM process, we disregard loss henceforth and assume again κ = 0.
Figure 5 shows the results of solving Eq. (8) (and the three corresponding ones of ω2–ω4) through a fiber of length L = 4 km and adding Raman fluctuations at T = 300 K during propagation in Cases (a) (top) and (b) (bottom); Fig. 5(i) shows the CE versus position in the fiber, and the simulated (dotted blue) and the analytic (solid black) results agree very well. After ∼ 2 km the two curves start differing significantly because it was assumed in the analytic expression that the pumps do not exchange energy. The actual energy exchange between the pumps, which is caused by SRS, taking place in the simulation has two impacts on the conversion efficiency due to SRS transferring energy to the lower frequency pump, which is p in these cases: firstly, the difference in pump power causes a phase mismatch cf. the definition of κ; secondly, the conversion coefficient μ2 ≈ ηiηi ∝ PpPq becomes smaller because the product of two functions that have a constant sum is largest when values of the functions are equal.
Fig. 5 (i) CE versus fiber length for Case (a) (top) and (b) (bottom); both simulation (dotted blue) and analytic result of Eq. (15) are shown; the legend applies to both plots, and the thick red line is the Raman amplification term. (ii) The same for the NF; the analytic Raman NF (solid black) is Eq. (27) and Eq. (28) for the top and bottom plots, respectively; the dashed green line is the Raman NF at 0 K. The parameters are the same as in Fig. 4, but α = 0 dB/km and T = 300 K.
In Case (a), where the signal and idler are on the S side of the pumps, the CE grows through the fiber, essentially due to Raman amplification. That is, a CE higher than unity means that the signal and idler have been amplified by SRS such that the output idler has a higher output power than the signal input. The red curve denotes the Raman amplification term. The NF, shown in Fig. 5(ii), oscillates in phase with the CE, but the lowest achievable point increases according to the S side Raman-induced NF (calculated in the Appendix), valid for Case (a),
where is the Raman gain. The dashed green line of Fig. 5(ii) (top plot) marks the Raman-induced NF at 0 K, so the region between that curve and the solid black curve is the NF induced by the existence of thermally excited phonons.In Case (b), where the signal and idler are on the aS side of the pumps, the CE drops off exponentially because energy flows towards the pumps through SRS. The NF on the aS side is different from that on the S side for two reasons: firstly, since the aS process requires the presence of phonons, aS SpRS cannot occur at 0 K (as discussed above), and secondly, Raman depletion removes photons from a wave component so the SNR must change accordingly (much like the effect of loss). Hence, the aS Raman-induced NF (see the Appendix), valid for Case (b), is
where . The first term on the right hand side, the Raman depletion term, is in contrast to the first term in Eq. (27) growing through the fiber. The dashed green line of Fig. 5(ii) (bottom plot) represents the minimum Raman NF of the aS side as caused by Raman depletion, and the region between that curve and the analytic curve is the entire SpRS contribution.Having established that Eq. (15) is a good description of phase matched BS in the presence of Raman scattering and that the S and aS NFs of Eqs. (27) and (28) accurately predicts the NF of BS at the points of optimal conversion, we can use these analytic expressions to analyze the frequency and temperature dependencies of the BS CE and NF. Figure 6(i) and 6(ii) shows the CE of Eq. (15) versus Δ of Cases (a) and (b) of Fig. 1, respectively, at the fiber length of the first optimal conversion point, L = π/(2μ); κ was assumed to be zero and Pp = Pq = 0.2 W, δ = 1 THz, α = 0 dB/km, and γs = 9.89 (W m)−1 was used. The CE is in both cases essentially showing the signal and idler average Raman amplification, Case (a), and depletion, Case (b), that they receive from the two pumps; for these realistic parameter values, stimulated Raman scattering is a significant effect (in both cases up to a factor ×2.5) that must be taken into account in silica fibers. Note that the frequency separation between the pump and the side band that are closest together is 2Δ.
Fig. 6 (i) and (ii) CE of Eq. (15) of Cases (a) and (b) of Fig. 1, respectively; (iii) and (iv) NFs of Eqs. (27) and (28), respectively, color scales are in dB.
Figures 6(iii) and 6(iv) show the NF of Cases (a) and (b), respectively, i.e. Eqs. (27) and (28) versus Δ and the temperature, T; the color scalings on the two plots are equal. For small Δ/2π-values < 3 THz, the two NFs are similar for all values of T, which is expected from Fig. 2 since the magnitudes of the rates of SpRS on the S and aS sides are comparable close to the pump; the aS side SpRS contribution is smaller but Raman depletion also contribute to the aS NF. The NF-values at low temperatures for small Δ (lower left corner) are in both cases approximately 0.4 dB; for small Δ-values at room temperature (upper left corner) the NF is approximately 2.7 dB on the S side but 3.1 dB on the aS side. For larger Δ-values (right side), the Raman depletion term becomes dominant on the aS side so the NF grows beyond 4 dB for all value of T. On the S side, where the NF is not affected by any Raman depletion term, the NF increases with Δ as expected from Fig. 2. Given these numbers, the S side seems advantageous with lower NF but an important note on the difference in origin between the S and aS NFs should be taken. The S NF is solely induced by SpRS, which increases the power variance of a signal ensemble due to the random phase of the spontaneous decay; the aS NF is composed of both SpRS and the effect of Raman depletion. The rate of SpRS is always smaller on the aS side compared to the S side so the power variance increases less there; in the total NF of 3.1 dB at 300 K in Fig. 6(iv), SpRS is only responsible for 45% of the total NF in linear units. This number is found by calculating the ratio of the temperature dependent part of Eq. (28) (the second term) relative to the total NFaS.
At room temperature, the minimum achievable S and aS NFs are 2.1 dB and 2.6 dB, respectively, which underlines that Raman scattering has a significant impact on BS that is comparable to the 3-dB NF induced by linear amplifiers such as phase-insensitive parametric amplifiers, Raman and Erbium-doped fiber amplifiers. Specific, realistic parameters where chosen in all simulations conducted in this paper but it has been verified in subsequent simulations that the presented results do not change notably by changing the γj and the pump powers by an order of magnitude. Changing δ significantly leads to a different Raman interaction, especially between the pump and the side bands that are separated the most.
5. Conclusion
We presented a model for calculating the quantum noise properties of parametric frequency conversion in form of Bragg Scattering; dispersion, loss, and stimulated and spontaneous Raman scattering were included. Closed-form analytic expressions for the conversion efficiency and Raman noise contribution were derived.
It was shown that loss in the signal and idler reduces the conversion efficiency and induces a noise floor equal to the common loss factor, and that loss in the pumps reduces the rate at which the energy oscillates between the signal and idler. Further elaboration showed that loss in the pumps does not hinder efficient and noise-less frequency conversion. On the other hand, loss in either of the signal and idler induces a degradation in the idler signal-to-noise ratio.
Both stimulated and spontaneous Raman scattering were shown to have significant impacts on Bragg scattering for a choice of realistic parameters for a highly nonlinear silica fiber: stimulated Raman scattering affects the conversion efficiency as would be expected from Raman amplifiers, the longer wavelength components receive energy from the shorter wavelength components. Spontaneous Raman scattering, which is asymmetric around the pumps, induces a minimum noise figure of 2.1 dB on the Stokes side of the pumps, Case (a), for one full conversion from signal to idler at room temperature. On the anti-Stokes sides of the pumps, Case (b), a minimum noise figure of 2.6 dB caused by both spontaneous Raman scattering and Raman depletion is predicted. Lowering the temperature reduces spontaneous Raman scattering on the Stokes side of the pumps but does not remove it; on the anti-Stokes side, spontaneous Raman scattering is removed completely when lowering the temperature but the effect of Raman depletion is still present.
Our theoretical predictions confirm that the presence of Raman scattering in silica fiber-based four-wave mixing in form of Bragg scattering contaminates the quantum noise-less frequency conversion to a significant degree that is comparable to the 3-dB noise figure induced by linear amplifiers such as phase-insensitive parametric amplifiers, Raman and Erbium-doped fiber amplifiers.
Appendix: Variance of the Raman fluctuations and the Raman noise figure
Given an ensemble of electromagnetic fields defined as in Eq. (20), we may apply a constant Raman gain G in power from one pump to one signal, which leaves the SNR unchanged (signal and noise are amplified equally). The pump also provide spontaneous emission at the frequency of the signal ensemble, and this contribution changes the SNR. In this semi-classical model, we add fluctuation variables after having applied the gain and then calculate the corresponding change in the SNR. The output field ensemble of a Raman amplifier in the linear gain regime is
where δaj is the fluctuation in quadrature j of the spontaneous emission associated with the physical process that provided the gain. Defining the SNR as Eq. (19) and the Raman NF as the ratio SNRin/SNRout, we get where a large photon number was assumed and it was furthermore assumed that the statistics of the fluctuations in both quadratures are equal, so . To evaluate the NF the variance of δa, which is equal to the second order moment 〈δa2〉 because δa has zero mean value, must be determined.Classical photon-number equations describing the Raman interaction between two waves at different wavelengths are shown in textbooks on nonlinear optics [40] and it is custom to approximate the shorter wavelength component (the anti-Stokes component or pump in the context of Raman amplifiers) by a constant but here we also need to consider the opposite case of the longer wavelength component (the Stokes component) being much stronger than the shorter wavelength component and hence approximate that by a constant. Likewise, the mean output power associated with Eq. (29) is
where Eq. (31) describes the weak Stokes component that receives Raman gain G from the strong anti-Stokes component, and Eq. (32) describes the distinct case of the weak anti-Stokes component that depletes by a factor D by giving energy to the strong Stokes component. The second term of each equation is the amplification/depletion of the vacuum fluctuations explicitly included in Eq. (29) and they are artifacts of the semi-classical modeling; and should be chosen to counter-balance these terms as well to include spontaneous emission. If and are chosen to be and they are inserted into Eqs. (31) and (32), then one gets the equivalent of what may be derived from the classical equations. The Raman NF of a signal on either side of a strong pump is thus easily calculated using Eq. (30) to be The arrows indicate the limits in case of large Raman interaction, i.e. G ≫ 1 and D ≪ 1. In the limit of low temperature, nT ≈ 0, these formulas give the expected results of a 3-dB NF on the S side and a NF equal to the depletion on the aS side.Considering Case (a) and Case (b) of Fig. 1, in which there are two pumps and two small signals, the results Eqs. (35)–(36) do not immediately apply. However, if one regards the two pumps as one wave component with power P = Pp + Pq and the signal and idler as one small signal, one has an artificial two-component system that may be described by the results of this appendix. The frequency separation between the two components is Ωpi = ωp − ωi in Case (a) and Ωip in Case (b).
Acknowledgments
The authors thank Colin J. McKinstrie for valuable discussions.
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