Dependence of the amplification performance of unsaturated degenerate phase-sensitive amplification on wavelength allocation

Kyo Inoue

doi:10.1364/OE.25.029724

1. Introduction

Phase-sensitive amplification (PSA) is a particular type of optical parametric amplification (OPA) based on the second- or third-order nonlinearity [1]. It has a low-noise property (i.e., the quantum limited noise figure of 0 dB) and a phase clamping effect available for phase noise suppression. The PSA operation is achieved when the optical frequencies of incident lights satisfy particular conditions. A typical frequency allocation for PSA is that signal light of f_s and two pump lights of f_p1 and f_p2 are incident, satisfying 2f_s = f_p1 + f_p2. There have been several experimental reports demonstrating the PSA operation employing this frequency arrangement [2–7].

In OPA, phase matching is generally an important factor determining amplification performances. When the propagation phases of co-propagating lights are matched, an parametric interaction efficiently occurs and the signal gain is maximum. It is known that the phase-matching condition is satisfied when incident lights are positioned in the zero-dispersion wavelength region for OPA using an optical fiber as a nonlinear medium. In fact, the previous experiments [3–7] employed signal and pump wavelengths in this wavelength region. However, detailed analysis on the phase matching condition or wavelength dependency is not found in those reports. Xie et al. analyzed PSA gain dependence on the incident light wavelengths in detail [8]. However, signal gain was focused and other PSA properties, such as noise figure and phase clamping effect, were out of the scope of their study.

On the above background, this paper analytically studies dependence of PSA performances, including signal gain, noise figure, and phase-clamping effect, on the wavelength allocation of incident lights. The phase mismatch Δβ is expressed in terms of the frequencies of interacting lights and the pump power, and, using the obtained Δβ, PSA characteristics (gain, noise figure, and phase-clamping effect) are evaluated as a function of the signal wavelength. The results quantitatively show that the PSA performances degrade when the wavelength allocation is detuned from the phase-matched condition.

2. Analytical model

We consider degenerate PSA, where signal light of frequency f_s is amplified by two pump lights of frequencies f_p1 and f_p, satisfying 2f_s = f_p1 + f_p2. Nonlinear coupled equations describing this parametric process are given by [1]

\frac{d E_{p 1}}{d z} = i γ (| E_{p 1} |^{2} + 2 | E_{p 2} |^{2} + 2 | E_{s} |^{2}) E_{p 1} + i γ E_{s}^{2} E_{p2}^{*} e^{i Δ β_{0} z},

\frac{d E_{p 2}}{d z} = i γ (2 | E_{p1} |^{2} + | E_{p 2} |^{2} + 2 | E_{s} |^{2}) E_{p 2} + i γ E_{s}^{2} E_{p1}^{*} e^{i Δ β_{0} z},

\frac{d E_{s}}{d z} = i γ (2 | E_{p 1} |^{2} + 2 | E_{p 2} |^{2} + | E_{s} |^{2}) E_{s} + 2 i γ E_{p 1} E_{p 2} E_{s}^{*} e^{- i Δ β_{0} z},

where E_p1, E_p2, and E_s are the amplitudes of pump-1, pump-2, and the signal, respectively, γ is the nonlinear coefficient, and Δβ₀ = 2β (f_s) – β (f_p1) – β (f_p2) is the linear phase mismatch with β (f) being the propagation constant at frequency f. The propagation loss is neglected, assuming that the fiber length is not long. Equations (1), (2), and (3) are propagation equations for pump-1, pump-2, and the signal, respectively. Under the condition |E_p1|, |E_p2| >> |E_s|, the pump amplitudes can be expressed as E_p1(2)(z) = E_p1(2)(0)exp[iγ(P_p1(2) + 2P_p2(1))z] from Eqs. (1)(2), where P_p1(2) = | E_p1(2) (0)|² denotes the pump power. Hereafter, we assume P_p1 = P_p2 = P₀ for simplicity. Substituting these expressions of the pump lights into Eq. (3), we have

\frac{d E_{s}}{d z} = 4 i γ P_{0} E_{s} + 2 γ P_{0} e^{i φ} e^{6 i γ P_{0} z} E_{s}^{*} e^{- i Δ β_{0} z},

where ϕ = θ_p1 + θ_p2 + π/2 with θ_{p1, p2} being the pump light phase at z = 0. This equation can be analytically solved, by utilizing forms of E_s(z) = A(z)exp(4iγP₀z) and A(z) = B(z)e^–ⁱ^Δ^βz^/2, as

E_{out} = {\cos h (g_{0} L) + i (Δ β / 2 g_{0}) \sin h (g_{0} L)} E_{in} + e^{i φ} \sqrt{1 + {(Δ β / 2 g_{0})}^{2}} \sin h (g_{0} L) E_{in}^{*},

where E_in and E_out are the input and output signal amplitudes, respectively, g₀ = {(2γP₀)² – (Δβ/2)²}^1/2, Δβ = Δβ₀ + 2γP₀ is the nonlinear phase mismatch, and L is the fiber length. Equation (5) is the solution under condition (2γP₀)² – (Δβ/2)² > 0, i.e., the phase mismatch Δβ is small. For (2γP₀)² – (Δβ/2)² < 0, i.e., large phase mismatch, the solution is

E_{out} = {\cos (g_{1} L) + i (Δ β / 2 g_{1}) \sin (g_{1} L)} E_{in} + e^{i φ} \sqrt{{(Δ β / 2 g_{1})}^{2} - 1} \cdot \sin (g_{0} L) E_{in}^{*},

with g₁ = {(Δβ/2)² – (2γP₀)²}^1/2. PSA properties can be evaluated using Eqs. (5) and (6). Note here that |E₁|, |E₂| >> |E_s| is assumed in deriving the above equations, meaning that the present analysis is limited to unsaturated conditions, where pump depletion is neglected, nonlinear phase shift is induced only by the pump lights, and higher-order parametric processes [9] are neglected.

Equations (5) and (6) includes the phase mismatch Δβ, and thus PSA properties depend on Δβ, which is composed of the linear phase mismatch Δβ₀ and the nonlinear phase shift 2γP₀. The linear phase mismatch is determined by the propagation constant at each frequency β(f), thus it depends on the frequency allocation of the pump and signal lights. Here, we expand β(f) around the fiber zero-dispersion frequency as [10]

β_{0} (f) = β_{0} (f_{0}) + (f - f_{0}) {[\frac{d β_{0}}{d f}]}_{f_{0}} + {(f - f_{0})}^{2} \frac{π λ^{4}}{3 c^{2}} {[\frac{d D_{c}}{d λ}]}_{f_{0}},

where f₀ is the zero-dispersion frequency of a fiber; c is the light velocity in the vacuum, λ is the wavelength; D_c is the dispersion parameter; and D_c (f₀) = 0 is applied. Applying this expression, the nonlinear phase matching is expressed as

Δ β = - (f_{s} - f_{0}) {(f_{p 1} - f_{s})}^{2} \frac{2 π λ^{4} D_{cc}}{c^{2}} + 2 γ P_{0},

where D_cc = [dD_c/dλ] at f₀. Using Eqs. (5)(6) with Eq. (8), we can evaluate dependence of PSA properties on the wavelength allocation.

In the next section, we evaluate PSA performance based on the above analytical mode. Before going to the next section, we simplify Eq. (5) for the discussion there:

E_{out0} = A_{0} e^{i ϕ_{0}} E_{in} + B_{0} e^{i φ} E_{in}^{*},

where

A_{0} = \sqrt{\cos h^{2} (g_{0} L) + {(Δ β / 2 g_{0})}^{2} \sin h^{2} (g_{0} L)},

ϕ_{0} = arc \tan [\frac{(Δ β / 2 g_{0}) \sin h (g_{0} L)}{\cos h (g_{0} L)}],

B_{0} = \sqrt{1 + {(Δ β / 2 g_{0})}^{2}} \sin h (g_{0} L) .

We also simplify Eq. (6) as

E_{out1} = A_{1} e^{i ϕ_{1}} E_{in} + B_{1} e^{i φ} E_{in}^{*},

where

A_{1} = \sqrt{\cos^{2} (g_{1} L) + {(Δ β / 2 g_{1})}^{2} \sin^{2} (g_{1} L)},

ϕ_{1} = arc \tan [\frac{(Δ β / 2 g_{1}) \sin (g_{1} L)}{\cos (g_{1} L)}],

B_{1} = \sqrt{{(Δ β / 2 g_{1})}^{2} - 1} \cdot \sin (g_{1} L) .

The above expressions are used in the next section

3. Amplification performance

3.1 Other parametric processes

Based on the previous section, we evaluate dependence of PSA performances on the wavelength allocation. Here, however, we should care about other parametric processes [7, 8]. Under our incident condition, i.e., 2f_s = f₁ + f₂, other parametric processes can occur and affect our evaluation. The primary candidates under gain-unsaturated conditions are (i) an idler light is generated at f_i- = 2f₁ – f_s and the signal light is amplified, and (ii) an idler is generated at f_i+ = 2f₂ – f_s and the signal light is amplified (Fig. 1).

Fig. 1 Frequency allocation concerned in this paper.

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Nonlinear coupled equations for process (i) are

\frac{d E_{p 1}}{d z} = i γ (P_{p 1} + 2 P_{p 2}) E_{1} + 2 i γ E_{s} E_{i -} E_{p1}^{*} e^{- i Δ β_{0 -} z},

\frac{d E_{s}}{d z} = 2 i γ (P_{p1} + P_{p 2}) E_{s} + i γ E_{p1}^{2} E_{i -}^{*} e^{i Δ β_{0 -} z},

\frac{d E_{i -}}{d z} = 2 i γ (P_{p1} + P_{p 2}) E_{i -} + i γ E_{p1}^{2} E_{s}^{*} e^{i Δ β_{0 -} z},

where Δβ_0– = 2β (f_p1) – β (f_s) – β (f_i-). Following the derivation processes similar to those for Eq. (3), the solution is obtained as

E_{s} (L) = {\cos h (g_{- 0} L) + i (Δ β_{-} / 2 g_{- 0}) \sin h (g_{- 0} L)} E_{s} (0)

for small phase mismatch, and

E_{s} (L) = {\cos (g_{- 1} L) + i (Δ β_{-} / 2 g_{- 1}) \sin (g_{- 1} L)} E_{s} (0),

for large phase mismatch, where g_–0 = {(γP₀)² – (Δβ_–/2)²}^1/2, g_–1 = { (Δβ_–/2)² – (γP₀)²}^1/2, and

Δ β_{-} = - (f_{p1} - f_{0}) {(f_{s} - f_{p1})}^{2} \frac{2 π λ^{4} D_{cc}}{c^{2}} - 2 γ P_{0},

On the other hand, nonlinear coupled equations for process (ii) are

\frac{d E_{p 2}}{d z} = i γ (2 P_{p 1} + P_{p 2}) E_{p 1} + 2 i γ E_{s} E_{i +} E_{p2}^{*} e^{- i Δ β_{0 +} z},

\frac{d E_{s}}{d z} = 2 i γ (P_{p1} + P_{p 2}) E_{s} + i γ E_{p2}^{2} E_{i +}^{*} e^{i Δ β_{0 +} z},

\frac{d E_{i +}}{d z} = 2 i γ (P_{p1} + P_{p 2}) E_{i +} + i γ E_{p2}^{2} E_{s}^{*} e^{i Δ β_{0 +} z}

where Δβ₀₊ = 2β (f_p2) – β (f_s) – β (f_i+). The solutions are

E_{s} (L) = {\cos h (g_{+ 0} L) + i (Δ β_{+} / 2 g_{+ 0}) \sin h (g_{+ 0} L)} E_{s} (0)

for small phase mismatch, and

E_{s} (L) = {\cos (g_{+ 1} L) + i (Δ β_{+} / 2 g_{+ 1}) \sin (g_{+ 1} L)} E_{s} (0)

for large phase mismatch, where g₊₀ = {(γP₀)² – (Δβ₊/2)²}^1/2, g₊₁ = { (Δβ₊/2)² – (γP₀)²}^1/2, and

Δ β_{+} = - (2 f_{s} - f_{p 1} - f_{0}) {(f_{s} - f_{p1})}^{2} \frac{2 π λ^{4} D_{cc}}{c^{2}} - 2 γ P_{0},

Equations (20) (26) have no term corresponding to the second term in Eq. (5), because the signal and idler are nondegenerate with no idler incident in these parametric processes. Also noted is that the maximal gain coefficients are g₀(Δβ = 0) = 2γP₀ and g_– (Δβ_– = 0) = g₊ (Δβ₊ = 0) = γP₀, where there is a difference by factor 2. This difference comes from the degeneracy factor in the nonlinear polarization term. These considerations suggest that the signal gain due to parametric processes (i) and (ii) is lower than that of the PSA process of interest in this paper.

We calculated the signal gain due to the above parametric processes around wavelengths in which the PSA will have large gain. The results are shown in Fig. 2. These processes provide signal gains of several dB around the zero-dispersion wavelength, as shown in the figure. In the following subsections where PSA properties are evaluated, we restrict our calculation conditions under which the signal gain due to these parametric processes is sufficiently smaller than the PSA gain, in order to investigate pure PSA characteristics.

Fig. 2 Signal gain of parametric processes other than PSA process, as a function of signal wavelength (λ_s) relative to the fiber zero-dispersion wavelength (λ₀). (a) The frequency difference between the pump and signal lights is fixed at 1 THz, and one pump power is 0.5, 0.4, or 0.3 W. (b) One pump power is 0.5 W, and the signal-pump frequency difference is 0.7, 1.0, or 1.3 THz. In each figure, lines in the shorter and longer wavelength sides are gains due to amplification processes of (i) pump-1 → signal + idler(-), and (ii) pump-2 → signal + idler( + ), respectively. Parameters used in calculations are λ = 1.55 μm, D_cc = 0.02 ps/km-nm-nm, γ = 12 W/km, and L = 200m.

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Another parametric process that may affect the present calculation model is four-wave mixing (FWM) that generates a new light at 2f_p1 – f_p2 or 2f_p2 – f_p1 from the two pump lights. The pump light powers may be depleted through these FWM processes, and the nonlinear phase mismatch Δβ = Δβ₀ + 2γP₀ may change. Wave equations for FWM lights generated at f₁₁₂ = 2f_p1 – f_p2 and f₂₂₁ = 2f_p2 – f_p1 are expressed as

\frac{d E_{112}}{d z} = 2 i γ (P_{p1} + P_{p 2}) E_{112} + i γ E_{p1}^{2} E_{p 2}^{*} e^{i Δ β_{112(0)} z},

\frac{d E_{221}}{d z} = 2 i γ (P_{p1} + P_{p 2}) E_{221} + i γ E_{p2}^{2} E_{p 1}^{*} e^{i Δ β_{221(0)} z},

where Δβ₁₁₂₍₀₎ = 2β (f_p1) – β (f_p2) – β (f₁₁₂) and Δβ₂₂₁₍₀₎ = 2β (f_p2) – β (f_p1) – β (f₂₂₁). Following the derivation processes similar to those for Eq. (3), the solutions are obtained as

E_{112} (L) = e^{i φ} \sqrt{{(Δ β_{112} / 2 g_{112})}^{2} - 1} \cdot \sin (g_{112} L) E_{p2}^{*} (0),

E_{221} (L) = e^{i φ} \sqrt{{(Δ β_{221} / 2 g_{221})}^{2} - 1} \cdot \sin (g_{221} L) E_{p1}^{*} (0),

and their powers are

P_{112} (L) = {{(Δ β_{112} / 2 g_{112})}^{2} - 1} \sin^{2} (g_{112} L) P_{0},

P_{221} (L) = {{(Δ β_{221} / 2 g_{221})}^{2} - 1} \sin^{2} (g_{221} L) P_{0},

where

Δ β_{112} = 24 {(f_{p 1} - f_{s}) - (f_{s} - f_{0})} {(f_{p 1} - f_{s})}^{2} \frac{2 π λ^{4} D_{cc}}{c^{2}} - γ P_{0},

Δ β_{221} = - 24 {(f_{p 1} - f_{s}) + (f_{s} - f_{0})} {(f_{p 1} - f_{s})}^{2} \frac{2 π λ^{4} D_{cc}}{c^{2}} - γ P_{0},

and g₁₁₂ = { (Δβ₁₁₂/2)² – (γP₀)²}^1/2 and g₂₂₁ = { (Δβ₂₂₁/2)² – (γP₀)²}^1/2.

Using the above equations, we calculated the FWM efficiency defined by P₁₁₂(L)/P₀ and P₁₁₂(L)/P₀, the results of which are shown in Fig. 3. Around particular wavelength conditions, i.e., the phase matching conditions for these FWM interactions, the FWM lights with relatively high power levels are generated. The generated power comes from the pump lights, thus the pump depletion occurs in these conditions and the calculation model described in the previous section causes some errors. Assuming that wavelength conditions giving a FWM efficiency higher than 10% cause non-negligible errors, for example, the wavelength ranges for which our model is not applicable are: from –8.9 to –6.6 nm and from 7.1 to 9.2 nm for f_p1 – f_s = 1 THz and P₀ = 0.3~0.5 W; from –11.0 to –9.5 nm and from 9.9 to 11.2 nm for f_p1 – f_s = 1.3 THz and P₀ = 0.5 W; from –7.5 to –2.8 nm and from 3.9 to 8.4 nm for f_p1 – f_s = 0.7 THz and P₀ = 0.5 W. In figures plotting calculation results in the following sections, data in these wavelength ranges are blanked.

Fig. 3 Generation efficiency of FWM lights at f₁₁₂ = 2f_p1 – f_p2 and f₂₂₁ = 2f_p2 – f_p1, defined by P₁₁₂(L)/P₀ and P₁₁₂(L)/P₀ where P₁₁₂(L) and P₁₁₂(L) are the FWM output power at f₁₁₂ and f₂₂₁, respectively, and P₀ is the pump input power. (a) The frequency difference between the pump and signal lights is fixed at 1 THz, and one pump power is 0.5, 0.4, or 0.3 W. (b) One pump power is 0.5 W, and the signal-pump frequency difference is 0.7, 1.0, or 1.3 THz. Parameters used in calculations are λ = 1.55 μm, D_cc = 0.02 ps/km-nm-nm, γ = 12 W/km, and L = 200m.

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3.2 Signal gain

In this subsection, we evaluate the dependence of the signal gain on the wavelength allocation. We first rewrite Eqs. (9) and (13) as

E_{out i} = {A_{i} e^{i (ϕ_{i} + θ_{in})} E_{in} + B_{i} e^{i (φ - θ_{in})}} | E_{in} |, ....... (i = 0, 1)

where θ_in is the signal incident phase. From this equation, the signal gain is expressed as

G_{i} = | A_{i} e^{i (ϕ_{i} + θ_{in})} + B_{i} e^{i (φ - θ_{in})} |^{2} = A_{i}^{2} + B_{i}^{2} + 2 A_{i} B_{i} \cos (ϕ_{i} + 2 θ_{in} - φ),

This expression shows that the signal gain is dependent on the relative phase between the incident pump and signal lights, i.e., 2θ_in – ϕ, as well as the phase mismatch. Thus, the relative phase should be specified when evaluating the gain. Here, we assume two conditions for the relative phase. One is that the relative phase is set at a value providing the maximal gain when the phase matching condition is satisfied. This assumption clarifies performance degradation due to the phase mismatch from the ideal condition in terms of the phase matching and the relative phase. The other assumption is that the relative phase is optimized to provide the maximal gain for a given phase mismatch. In this case, performance degradation due to the phase mismatch is partially compensated by tuning the relative phase.

The signal gain under the former assumption is evaluated as follows. Equation (38) with Δβ = 0 is

G_{i} (Δ β = 0) = A_{i}^{2} + B_{i}^{2} + 2 A_{i} B_{i} \cos (2 θ_{in} - φ) .

This expression suggests that relative phase providing the maximum gain is 2θ_in – ϕ = 0 when the phase matching condition is satisfied. Then, Eq. (38) with this relative phase is expressed as

\begin{array}{l} G_{0} = A_{0}^{2} + B_{0}^{2} + 2 A_{0} B_{0} \cos ϕ_{0} = A_{0}^{2} + B_{0}^{2} + 2 B_{0} Re [A_{0} e^{i ϕ_{0}}] \\ = \cosh^{2} (g_{0} L) + (^{Δ β / 2 g_{0}) 2} \sin h^{2} (g_{0} L) + {1 + (^{Δ β / 2 g_{0}) 2}} \sinh^{2} (g_{0} L) \\ + 2 \sqrt{1 + {(Δ β / 2 g_{0})}^{2}} \sinh (g_{0} L) \cos h (g_{0} L) \\ = \cosh (2 g_{0} L) + 2 (^{Δ β / 2 g_{0}) 2} \sinh^{2} (g_{0} L) + \sqrt{1 + {(Δ β / 2 g_{0})}^{2}} \sinh (2 g_{0} L) \end{array}

for small phase mismatch, and

\begin{array}{l} G_{1} = A_{1}^{2} + B_{1}^{2} + 2 B_{1} Re [A_{1} e^{i ϕ_{0}}] \\ = \cos^{2} (g_{1} L) + (^{Δ β / 2 g_{1}) 2} \sin^{2} (g_{1} L) + {1 + (^{Δ β / 2 g_{1}) 2}} \sin^{2} (g_{1} L) \\ + 2 \sqrt{{(Δ β / 2 g_{1})}^{2} - 1} \sin (g_{1} L) \cos (g_{1} L) \\ = \cos (2 g_{1} L) + 2 (^{Δ β / 2 g_{1}) 2} \sinh^{2} (g_{1} L) + \sqrt{{(Δ β / 2 g_{1})}^{2} - 1} \cdot \sin (2 g_{1} L) \end{array}

for large phase mismatch. Equations (40) and (41) give the signal gain for the relative phase fixed at a value providing the maximal gain under the phase matched condition.

On the other hand, Eq. (38) shows that the signal gain is maximum when the relative phase satisfies φ_i + 2θ_in – ϕ = 0 under phase mismatched conditions. Equation (38) with this relative phase is expressed as

\begin{array}{l} G_{0} = A_{0}^{2} + B_{0}^{2} + 2 A_{0} B_{0} \\ = \cos h (2 g_{0} L) + 2 (^{Δ β / 2 g_{0}) 2} \sin h^{2} (g_{0} L) \\ + 2 {^{\cos h^{2} (g_{0} L) + (^{Δ β / 2 g_{0}) 2} \cos h (2 g_{0} L) + (^{Δ β / 2 g_{0}) 4} \sin h^{2} (g_{0} L)} 1 / 2} \sin h (g_{0} L) \end{array}

for small phase mismatch, and

\begin{array}{l} G_{1} = A_{1}^{2} + B_{1}^{2} + 2 A_{1} B_{1} \\ = \cos (2 g_{1} L) + 2 (^{Δ β / 2 g_{1}) 2} \sin^{2} (g_{1} L) \\ + 2 [^{(^{Δ β / 2 g_{1}) 2} \cos (2 g_{1} L) + (^{Δ β / 2 g_{1}) 4} \sin^{2} (g_{1} L) - \cos^{2} (g_{1} L)] 1 / 2} \sin (g_{1} L) \end{array}

for large phase mismatch. Equations (42) and (43) are the signal gain when the relative phase is optimized for a given phase mismatch.

Using Eqs. (40) – (43) with Δβ given by Eq. (8), we calculated the signal gain as a function of the signal wavelength relative to the fiber zero-dispersion wavelength. The results are shown in Figs. 4 and 5, where the relative phase is fixed at a value optimized for the phase matched condition in Fig. 4 and it is optimized at each wavelength in Fig. 5. The frequency difference between the signal and pump lights were kept constant. The signal gain decreases as the signal wavelength is detuned from the phase matching condition, as shown in the figures. Comparison of Figs. 4 and 5 indicates that the gain reduction due to phase mismatch is partially compensated by optimizing the relative phase.

Fig. 4 Signal gain as a function of signal wavelength (λ_s) relative to the fiber zero-dispersion wavelength (λ₀). The relative incident phase is fixed at a value optimized for the phase matched condition. (a) The frequency difference between the pump and signal lights is fixed at 1 THz, and one pump power is 0.5, 0.4, or 0.4 W. (b) One pump power is 0.5 W, and the signal-pump frequency difference is 0.7, 1.0, or 1.3 THz. Parameters used in calculations are λ = 1.55 μm, D_cc = 0.02 ps/km-nm-nm, γ = 12 W/km, and L = 200m.

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Fig. 5 Signal gain as a function of signal wavelength (λ_s) relative to the fiber zero-dispersion wavelength (λ₀). The relative incident phase is optimized at each wavelength. The parameters used are the same as those in Fig. 4.

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3.3 Noise figure

The noise performance is important for optical amplifiers, especially for PSA that features a low-noise property. The intrinsic noise performance is determined by quantum mechanics. In this subsection, we investigate the dependence of the quantum-limited noise property, i.e., noise figure to be specific, on the wavelength allocation.

The basic item to discuss quantum mechanical properties of light is the annihilation operator (or the field operator). For phase-mismatched gain-unsaturated PSA, the field operator of the signal light at the PSA output can be derived from the Heisenberg equation of motion as [11]

\begin{array}{l} {\hat{a}}_{out} = {\cos h (g_{0} L) + i (Δ β / 2 g_{0}) \sin h (g_{0} L)} {\hat{a}}_{in} + e^{i φ} \sqrt{1 + {(Δ β / 2 g_{0})}^{2}} \sin h (g_{0} L) {\hat{a}}_{in}^{†} \\ = A_{0} e^{i ϕ_{0}} {\hat{a}}_{in} + B_{0} e^{i φ} {\hat{a}}_{in}^{†}, \end{array}

where

{\hat{a}}_{in}

and

{\hat{a}}_{out}

are the signal field operators at the PSA input and output, respectively,

{\hat{a}}_{in}^{†}

is the Hermitian conjugate of

{\hat{a}}_{in}

, i.e., the creation operator, that satisfies

[{\hat{a}}_{in}^{†}, {\hat{a}}_{in}] = 1

, and the same notations as in the previous subsection are used. Equation (44) is for small phase mismatch. The expression for large phase mismatch can be similarly derived as

\begin{array}{l} {\hat{a}}_{out} = {\cos (g_{1} L) + i (Δ β / 2 g_{1}) \sin (g_{1} L)} {\hat{a}}_{in} + e^{i φ} \sqrt{{(Δ β / 2 g_{1})}^{2} - 1} \sin (g_{1} L) {\hat{a}}_{in}^{†} \\ = A_{1} e^{i ϕ_{1}} {\hat{a}}_{in} + B_{1} e^{i φ} {\hat{a}}_{in}^{†} . \end{array}

The noise figure is given by the ratio of the input and output signal-to-noise ratios (SNRs) in terms of the light intensity or the photon number. With the field operator, the photon number operator is expressed as $\hat{n} = {\hat{a}}^{†} \hat{a}$ , and the photon number SNR is given by $< \hat{n} >^{2} / σ^{2}$ with $σ^{2} = < {\hat{n}}^{2} > - < \hat{n} >^{2}$ , where < > denotes quantum mechanical average with respect to an initial state. From Eqs. (44) and (45), the mean photon number of the signal output is obtained as

\begin{array}{l} < {\hat{n}}_{out} > = < α | {\hat{a}}_{out}^{†} {\hat{a}}_{out} | α > \\ = (A_{i}^{2} + B_{i}^{2}) < {\hat{a}}_{in}^{†} {\hat{a}}_{in} > + A_{i} B_{i} {e^{i (φ - ϕ_{i})} < {\hat{a}}_{in}^{† 2} > + e^{- i (φ - ϕ_{i})} < {\hat{a}}_{in}^{2} >} + B_{i}^{2} \\ = [A_{i}^{2} + B_{i}^{2} + A_{i} B_{i} {e^{i (φ - 2 θ_{in} - ϕ_{i})} + e^{- i (φ - 2 θ_{in} - ϕ_{i})}}] n_{in} + B_{i}^{2} \\ = G_{i} n_{in} + B_{i}^{2}, \end{array}

where the signal input is assumed to be a coherent state |α> with the mean photon number n_in and the phase θ_in. The relationship of an eigenstate and an eigenvalue of

{\hat{a}}_{in}

, i.e.,

{\hat{a}}_{in} | α > = \sqrt{n_{in}} \exp (i θ_{in}) | α >

, and its Hermitian conjugate are used in deriving the above equation. The last term B_i² in Eq. (46) represents spontaneously emitted photons, resulting from

[{\hat{a}}_{in}^{†}, {\hat{a}}_{in}] = 1

. On the other hand, the average of the square of the photon number operator is derived as

\begin{array}{l} < {\hat{n}}_{out}^{2} > = < α | (^{{\hat{a}}_{out}^{†} {\hat{a}}_{out}) 2} | α > \\ = < α | [^{(A_{i}^{2} + B_{i}^{2}) {\hat{a}}_{in}^{†} {\hat{a}}_{in} + A_{i} B_{i} {e^{i (φ - ϕ_{i})} {\hat{a}}_{in}^{† 2} + e^{- i (φ - ϕ_{i})} {\hat{a}}_{in}^{2}} + B_{i}^{2}] 2} | α > \\ = n_{in}^{2} G_{i}^{2} + n_{in} [G_{i}^{2} + 2 G_{i} B_{i}^{2} + 4 (^{A_{i} B_{i}) 2} - (^{A_{i} B_{i}) 2} {^{e^{i (φ - 2 θ_{in} - ϕ_{i})} + e^{- i (φ - 2 θ_{in} - ϕ_{i})}} 2}] \\ + B_{i}^{4} + 2 (^{A_{i} B_{i}) 2} . \end{array}

From Eqs. (46) and (47), the photon number variance of the signal output is obtained as

\begin{array}{l} σ_{out}^{2} = < {\hat{n}}_{out}^{2} > - < {\hat{n}}_{out} >^{2} \\ = n_{in} [G_{i}^{2} + 4 (^{A_{i} B_{i}) 2} - (^{A_{i} B_{i}) 2} {^{e^{i (φ - 2 θ_{in} - ϕ_{i})} + e^{- i (φ - 2 θ_{in} - ϕ_{i})}} 2}] + 2 (^{A_{i} B_{i}) 2}, \end{array}

and the photon number SNR is expressed as

\begin{array}{l} (_{SNR) out} = \frac{< {\hat{n}}_{out} >^{2}}{σ_{out}^{2}} \\ = \frac{n_{in}}{1 + [4 {(A_{i} B_{i})}^{2} - {(A_{i} B_{i})}^{2} {e^{i (φ - 2 θ_{in} - ϕ_{i})} + e^{- i (φ - 2 θ_{in} - ϕ_{i})}}^{2}] / G_{i}^{2}}, \end{array}

where the last term in Eq. (48) is ignored, assuming n_in >> 1. In the definition of the noise figure, the SNR at the input is given by that for a coherent state, which is (SNR)_in = n_in. Then, the noise figure is expressed as

NF = \frac{{(SNR)}_{in}}{{(SNR)}_{out}} = 1 + \frac{{(A_{i} B_{i})}^{2}}{G_{i}^{2}} [4 - {e^{i (φ - 2 θ_{in} - ϕ_{i})} + e^{- i (φ - 2 θ_{in} - ϕ_{i})}}^{2}] .

The noise figure is dependent on the relative phase, as shown in Eq. (50). When the relative phase is set at a value providing the maximal gain at the phase matched condition, i.e., 2θ_in – ϕ = 0, the noise figure is

\begin{array}{l} NF = 1 + \frac{B_{0}^{2}}{G_{0}^{2}} [4 A_{0}^{2} - {^{A_{0} e^{- i ϕ_{0}} + A_{0} e^{i ϕ_{0}}} 2}] \\ = 1 + \frac{4}{G_{0}^{2}} \cdot {(\frac{Δ β}{2 g_{0}})}^{2} {1 + (^{Δ β / 2 g_{0}) 2}} \sin h^{4} (g_{0} L) \end{array}

for small phase mismatch, and

\begin{array}{l} NF = 1 + \frac{B_{1}^{2}}{G_{1}^{2}} [4 A_{1}^{2} - {^{A_{1} e^{- i ϕ_{1}} + A_{1} e^{i ϕ_{1}}} 2}] \\ = 1 + \frac{4}{G_{1}^{2}} \cdot {(\frac{Δ β}{2 g_{1}})}^{2} {(^{Δ β / 2 g_{1}) 2} - 1} \sin^{4} (g_{1} L) \end{array}

for large phase mismatch. Using Eqs. (51) and (52), we calculated the noise figure as a function of the signal wavelength relative to the fiber zero-dispersion wavelength when the relative phase is fixed as a value providing the maximal gain at the phase matched condition. The results are shown in Fig. 6. It is observed that the noise figure degrades from the ideal value 0 dB when the wavelength allocation is shifted from the phase matched condition.

Fig. 6 Noise figure NF as a function of signal wavelength (λ_s) relative to the fiber zero-dispersion wavelength (λ₀). The relative incident phase is fixed at a value optimized for the phase matched condition. The parameters used are the same as those in Fig. 3.

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On the other hand, when the relative phase is optimized for a given phase mismatch as 2θ_in + φ_i – ϕ = 0, Eq. (50) becomes

NF = 1 + \frac{{(A_{i} B_{i})}^{2}}{G_{i}^{2}} [4 - {(1 + 1)}^{2}] = 1.

Thus, the noise figure is always 0 dB irrespective of the phase mismatch in this case. This is because the NF degradation shown in Fig. 6 is compensated by optimizing the relative phase for a given phase mismatch.

As shown above, the noise figure is dependent on the incident light phases and the phase mismatch. This can be explained as follows. The noise figure is intuitively evaluated by the ratio of the signal gain to the photon number of spontaneous light, because it is proportional to the optical signal-to-noise ratio at the amplifier output. Here, the signal gain is given by G = A² + B² + 2ABcos(φ + 2θ_in – ϕ), as shown in Eq. (38), and the spontaneous photon number is given by B², as indicated in Eq. (46). Thus, their ratio is {A² + B² + 2ABcos(φ + 2θ_in – ϕ)}/B, where φ depends on phase mismatch, θ_in is the incident signal light phase, and ϕ depends on the incident pump light phases. This consideration suggests that the noise figure depends on the phase mismatch and the incident light phases.

In the last of this subsection, we note that this paper analyzes the quantum-limited noise figure of the PSA process, without taking into account other noise sources, such as Raman noise [12], pump fluctuation, and noise light from an optical amplifier that may be used before PSA. These noises degrade the noise figure more than the results shown in Fig. 6 in practice.

3.4 Output phase

A unique property of PSA is that the signal output phase is clamped at a constant value determined by the pump phases, which is available for phase noise suppression. In this subsection, we examine the dependence of this PSA property on the wavelength allocation. First, we rewrite Eqs. (9)(13) as

E_{out i} = {A_{i} e^{i {θ_{in} + (ϕ_{i} - φ) / 2}} + B_{i} e^{- i {θ_{in} + (ϕ_{i} - φ) / 2}}} | E_{in} | e^{i (ϕ_{i} + φ) / 2},

Here, we assume that the signal incident phase θ_in deviates from a mean value θ_in0 by Δθ, as θ_in = θ_in0 + Δθ, and substitute it into Eq. (54):

E_{out i} = {A_{i} e^{i {θ_{in0} + Δ θ + (ϕ_{i} - φ) / 2}} + B_{i} e^{- i {θ_{in0} + Δ θ + (ϕ_{i} - φ) / 2}}} | E_{in} | e^{i (ϕ_{i} + φ) / 2} .

When the mean signal phase is synchronized so that 2θ_in0 – ϕ = 0, i.e., the relative phase optimized for the phase matched condition, Eq. (55) is rewritten as

\begin{array}{l} E_{out i} = {A_{i} e^{i {Δ θ + ϕ_{i} / 2}} + B_{i} e^{- i {Δ θ + ϕ_{i} / 2}}} | E_{in} | e^{i (ϕ_{i} + φ) / 2} \\ = {(A_{i} + B_{i}) \cos (Δ θ + ϕ_{i} / 2) + i (A_{i} - B_{i}) \sin (Δ θ + ϕ_{i} / 2)} | E_{in} | e^{i (ϕ_{i} + φ) / 2} . \end{array}

From this equation, the output phase is expressed as

\arg [E_{out i}] = \frac{ϕ_{i} + φ}{2} + arc \tan [\frac{A_{i} - B_{i}}{A_{i} + B_{i}} \cdot \tan (Δ θ + ϕ_{i} / 2)] .

Then, the suppression ratio of the phase deviation is evaluated by

R = \frac{1}{Δ θ} {arc \tan [\frac{A_{i} - B_{i}}{A_{i} + B_{i}} \cdot \tan (ϕ_{i} / 2 + Δ θ)] - arc \tan [\frac{A_{i} - B_{i}}{A_{i} + B_{i}} \cdot \tan (ϕ_{i} / 2)]} .

On the other hand, when the mean signal phase is synchronized as 2θ_in0 + φ_i – ϕ = 0, i.e., the relative phase optimized for a given phase mismatch, Eq. (55) is rewritten as

E_{out i} = {A_{i} e^{i Δ θ} + B_{i} e^{- i Δ θ}} | E_{in} | e^{i (ϕ_{i} + φ) / 2},

from which the output phase is expressed as

\arg [E_{out i}] = \frac{ϕ_{i} + φ}{2} + arc \tan [\frac{A_{i} - B_{i}}{A_{i} + B_{i}} \cdot \tan (Δ θ)],

and then the suppression ratio is evaluated as

R = \frac{1}{Δ θ} arc \tan [\frac{A_{i} - B_{i}}{A_{i} + B_{i}} \cdot \tan (Δ θ)] .

Using Eqs. (58) and (61) with Δθ = ± π/4, we calculated the phase noise suppression ratio as a function of the signal wavelength relative to the fiber zero-dispersion wavelength. The results for the relative phase fixed at a value optimized for the phase matched condition are shown in Fig. 7, and those for the relative phase optimized for a given phase mismatch are shown in Fig. 8. The suppression ratio R is different for Δθ = + π/4 and Δθ = –π/4, and the larger one, max[R( + π/4), R(–π/4)], is plotted in the figure. It is observed that the phase-clumping effect is well achieved in a wavelength range where the phase matching is nearly satisfied and signal gain is high (as shown in Figs. 4 and 5), but degrades as the wavelength allocation is largely detuned from the phase matched condition, depending on the pump power and the frequency difference between the signal and pump lights.

Fig. 7 Suppression ration of phase deviation as a function of signal wavelength (λ_s) relative to the fiber zero-dispersion wavelength (λ₀). The relative incident phase is fixed at a value optimized for the phase matched condition. The parameters used are the same as those in Fig. 4.

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Fig. 8 Suppression ration of phase deviation as a function of signal wavelength (λ_s) relative to the fiber zero-dispersion wavelength (λ₀). The relative incident phase is optimized at each wavelength. The parameters used are the same as those in Fig. 4.

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4. Summary

We studied dependence of performance of dual-pump degenerate phase-sensitive amplification (PSA) on incident light wavelengths. Fiber-based PSA under unsaturated-gain conditions was analyzed. Phase mismatch was formalized in terms of incident light frequencies, the fiber zero-dispersion frequency, and the pump light power, based on which signal gain, noise figure, and phase-clamping effect were evaluated as a function of the signal wavelength. The results quantitatively indicated that these PSA performances are degraded as the signal wavelength is detuned from the phase-matched condition.

References and links

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Dependence of the amplification performance of unsaturated degenerate phase-sensitive amplification on wavelength allocation

Abstract

1. Introduction

2. Analytical model

3. Amplification performance

3.1 Other parametric processes

3.2 Signal gain

3.3 Noise figure

3.4 Output phase

4. Summary

References and links

Cited By

Figures (8)

Equations (61)

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