1. Introduction

Phase-sensitive optical devices have received a great interest in recent years for all-optical signal processing applications in optical communications. Four-wave mixing (FWM) is a well-known nonlinear effect used for the implementation of phase-sensitive amplifiers [1], regenerators [2] or frequency converters [3]. Recently, R. P. Webb et al. have demonstrated phase discrimination and frequency conversion in a semiconductor optical amplifier (SOA) by use of FWM [4]. The principle of operation requires to inject four pump waves together with a signal wave in the SOA. At the output of the SOA, due to FWM, the two quadrature components of the signal are converted to two different idlers at different frequencies. Using this scheme, F. Da Ros et al. demonstrated more recently the conversion of the two complex quadratures of a quadrature phase-shift keying signal to two binary phase-shift keying signals in a highly-nonlinear fiber (HNLF) [5] and in a periodically-poled lithium niobate (PPLN) waveguide [6]. In each of these three demonstrations four pumps were used and power levels and phases of the pump waves had to be carefully adjusted (eight parameters in total). An optimization routine has been used to find the appropriate parameters. To the best of our knowledge, no modeling of this scheme has been proposed yet to obtain analytic expressions for the parameters required in the experiment.

In this article we propose a theoretical study of the mechanism of phase-sensitive frequency conversion (PSFC) in nonlinear optical fibers. We start by writing the complete set of equations of a multiple FWM process and show that, under some assumptions, the system can be reduced to two equations. We also show that only three pumps are needed instead of four as originally proposed. Then we give simple relations to determine the initial values of the power levels and the phases of the pump waves. We show that the principle of the PSFC is similar to a nonlinear interferometer [7] and that its implementation requires careful adjustment of only two parameters (two pump phases) instead of eight as originally proposed. Our approach is therefore very useful for the implementation of PSFC and can easily be extended to the cases of SOAs and PPLN waveguides. Finally, we experimentally demonstrate the three-pump PSFC scheme in a HNLF and validate our theoretical approach.

The article is organized as follows. In Sec. 2 we present the theoretical study of the experiment originally demonstrated by F. Da Ros et al. in a HNLF [5] and give physical interpretation of the mechanism. In Sec. 3, we propose an experimental validation of our theoretical approach.

2. Theoretical study

2.1. Principle

The principle of FWM-based PSFC, as implemented by F. Da Ros et al. in a HNLF [5], is depicted in Fig. 1. Four continuous wave (CW) pumps, equally spaced in frequency, named P1, P3, P5 and P7 in Fig. 1, are launched in a HNLF. A signal wave S6, whose frequency falls between pumps P5 and P7 is injected in the same fiber. Provided the pumps and signal are coherent, depending on the phase ϕ₆ of signal S6 and due to FWM, an idler I2 or I4 can be generated. For example, the idler I4 is generated if ϕ₆ = 0 and the idler I2 is generated if ϕ₆ = π/2, as explained in Fig. 1. This setup enables the simultaneous conversion of the two orthogonal quadratures of the optical signal S6 to different wavelengths.

 

Fig. 1 Principle of phase-sensitive FWM frequency conversion.

Download Full Size | PDF

In order to ensure that the idlers I2 and I4 actually correspond to the two quadratures of the signal S6, the power levels and phases of the four pumps have to be carefully adjusted together with the signal power. In the original report by F. Da Ros et al. [5], a numerical optimization procedure has been carried out using the nonlinear Schrödinger equation in order to find the eight experimental input values for the pumps (four power values and four phase values) and the input power for the signal. This numerical multi-parameter optimization is time consuming and does not guarantee an absolute optimum is found. Indeed, several sets of pumps power levels and phase values could lead to comparable results. Furthermore it does not provide any physical insight on the mechanisms leading to the desired quadrature decomposition. We propose in the following a theoretical study of this experiment and give analytic expressions for the pump power levels and phases.

2.2. Modeling

The PSFC experiment described previously involves seven waves equally-spaced in frequency, as depicted in Fig. 1: four pumps, one signal and two idlers. We start by writing the set of seven coupled equations of a FWM process involving seven equally-spaced waves. The general formula for the slowly-varying amplitude A_n of of the n^th wave of a multiple FWM process involving N waves is given, in the CW regime, by [8]:

The phase mismatch Δβ_{n+p−m,m,p,n} is given by:

From relations (1)–(3) we can derive the seven coupled equations giving the evolution of the waves under the influence of the Kerr effect in the fiber. The complete set of equations is given in Appendix A together with the twenty-two associated phase mismatches. The waves are numbered as in Fig. 1. Solving numerically the whole set of equations allows to theoretically study the PSFC experiment but, in order to determine simple analytic expressions for the initial values of the pump waves, the set of equations needs to be simplified.

Since we only focus on idlers power evolution as a function of the signal phase, we assume the undepleted regime for pumps and signal. We therefore consider that the amplitudes A₁, A₃, A₅, A₇ for the pumps and A₆ for the signal are constant with respect to the z coordinate. Then we just consider the two following evolution equations of the two idlers A₂ and A₄:

The Eqs. (4) and (5) contain all the terms that could affect the idlers A₂ and A₄ during their propagation, i.e. loss, self-phase modulation, cross-phase modulation and FWM. Concerning FWM, many possibilities of resonant coupling occur for each wave (eleven FWM terms for idler A₂ and thirteen terms for idler A₄). Each of them is associated with a given phase mismatch. We now consider that the amplitudes of the idlers remain much lower than the other waves. Thus we neglect all the terms in which A₂ and A₄ appear in Eqs. (4) and (5). By also neglecting the loss of the fiber, we obtain the following equations:

In Eqs. (6) and (7), the remaining FWM terms correspond to all the possible combinations between pumps and signal leading to energy coupling to the idlers. The strength of the coupling depends on the phases mismatches Δβ_k present in the equations. Usually, in FWM experiments in HNLFs, fibers are used near their zero-dispersion wavelength in order to increase the FWM efficiency. This is the case in the article by F. Da Ros et al. [5] and will also be the case in our experiment, presented in the next section. Consequently, we assume that the phase mismatches are close to zero in Eqs. (6) and (7). This leads to the following simplified system of equations:

Each term of Eqs. (8) and (9) contains the amplitude A₆ of the signal and a combination of two pump amplitudes. Finally, we remark that the system (8)–(9) can be considerably reduced by removing the pump A₅ (i.e. setting its amplitude to zero). In this case, we obtain:

The right-hand side of the evolution equations for idlers A₂ and A₄ contains finally the sum of only two terms when only three pumps are involved. To correctly describe the phenomenon of PSFC (i.e. the generation, or not, of the idlers I2 or I4 depending of the phase of the signal), the terms $A_1{A}_7{A}_6^{*}$ and $A_3{A}_6{A}_7^{*}$ must add destructively in Eq. (10) whereas the terms $A_3{A}_7{A}_6^{*}$ and $A_1{A}_6{A}_3^{*}$ must add constructively in Eq. (11) for a given value ϕ₆ of the signal phase. Then the opposite situation must occur for a signal phase of ϕ₆ + π/2, i.e. $A_1{A}_7{A}_6^{*}$ and $A_3{A}_6{A}_7^{*}$ must add constructively in Eq. (10) whereas $A_3{A}_7{A}_6^{*}$ and $A_1{A}_6{A}_3^{*}$ must cancel in Eq. (11).

To meet this requirement, one first condition must be satisfied: the modulus of each term in the right-hand side of Eq. (10) or Eq. (11) must be the same. Let us write the amplitude of the waves A_k as $A_k=\sqrt{P_k}\exp (i{\varphi }_k)$ where P_k is the power of the wave and ϕ_k its phase. To satisfy equal modulus of each term, the conditions on the pump powers must be the following:

The second condition concerns the phase of each term of the right-hand side of Eqs. (10) and (11). As the phase of the signal wave is variable, it will be taken of the form ${\varphi }_6={\varphi }_6^0+\delta {\varphi }_6$, where ${\varphi }_6^0$ is a constant value and δϕ₆ the variable part of ϕ₆. If the idler I4 is expected to be maximum and the idler I2 minimum for δϕ₆ = 0, the terms $A_1{A}_7{A}_6^{*}$ and $A_3{A}_6{A}_7^{*}$ of Eq. (10) must be phase-shifted by π (mod. 2π) and the terms $A_3{A}_7{A}_6^{*}$ and $A_1{A}_6{A}_3^{*}$ of Eq. (11) must have the same phase (mod. 2π) for ${\varphi }_6={\varphi }_6^0$. This leads to the following requirements for the phases:

These equations state that no condition is required for the values of ${\varphi }_6^0$ and ϕ₇. Let us point out that the chosen value of ${\varphi }_6^0$ is the one for which the idler A₄ is efficiently generated while the idler A₂ is zero. The phase ϕ₇ can be arbitrary chosen. Table 1 gives examples of values for the phases of the different waves according to Eqs. (15) and (16).

Table 1. Possible values of the phases of the waves according to Eqs. (15) and (16).

View Table | View all tables in this article

By taking these conditions into account, it is straightforward to solve analytically the system (10)–(11) where all the terms of the right-hand sides are constant in z. We find the following relations for the amplitude A₂(L) and A₄(L) at the output of the fiber of length L:

Eqs. (19) and (20) confirm the energy exchange between idlers I2 and I4 as a function of the signal phase ϕ₆ and give access analytically to the conversion efficiency of both idlers.

2.3. Physical interpretation

To have a better understanding of what physically happens when the phase of the signal evolves from ${\varphi }_6^0$ to ${\varphi }_6^0+\delta {\varphi }_6$ in the fiber we represent the two terms of the right-hand side of Eqs. (10) and (11) as phasors in the complex plane. These phasors have the same modulus, due to condition (12), but can experience different phases. For simplicity, we take the particular initial conditions of line 1 of Table 1, i.e. ${\varphi }_1={\varphi }_6^0={\varphi }_7=0$ and ϕ₃ = π.

For δϕ₆ = 0, Fig. 2(a) represents the phasors $A_1{A}_7{A}_6^{*}$ and $A_3{A}_6{A}_7^{*}$ of Eq. (10) and Fig. 2(b) represents the phasors $A_3{A}_7{A}_6^{*}$ and $A_1{A}_6{A}_3^{*}$ of Eq. (11). For this particular value of δϕ₆, phasors of Fig. 2(a) are opposite while they are equal in Fig. 2(b). This leads to constructive interference for idler A₄ and the cancellation of idler A₂. In Figs. 2(c) and 2(d), for 0 < δϕ₆ < π/2, each phasor experiences the additional phase ±δϕ₆ depending on whether A₆ is conjugated or not in their expression. Figures 2(e) and 2(f) represent the phasors for δϕ₆ = π/2. In this case, the additional phase δϕ₆ of π/2 leads to a situation where phasors $A_3{A}_7{A}_6^{*}$ and $A_1{A}_6{A}_3^{*}$ are in opposition while phasors $A_3{A}_6{A}_7^{*}$ and $A_1{A}_7{A}_6^{*}$ are in phase. In this case, idler A₄ completely vanishes while idler A₂ is efficiently generated. It is easy to understand that for a phase shift δϕ₆ of π the same situation than in Figs. 2(a) and 2(b) occurs. It is also easy to verify that the conditions (15) and (16) for the initial phases ϕ₁ and ϕ₃ ensure that, for any values of ${\varphi }_6^0$ and ϕ₇, the two phasors of Eq. (11) are in phase and the two phasors of Eq. (10) are in opposition for δϕ₆ = 0.

 

Fig. 2 Different terms of Eqs. (10) and (11) represented as phasors in the complex plane for the initial conditions ${\varphi }_1={\varphi }_6^0={\varphi }_7=0$ and ϕ₃ = π.

Download Full Size | PDF

It is worth noting that the PSFC mechanism is similar to a two-wave interferometer: both outputs of the interferometer (I2 and I4) depends on the constructive or destructive sum of two waves. The principle of a phase sensitive FWM interferometer has been proposed recently in a dual-pump FWM process in a HNLF [7]. This principle has also been mentioned to describe the mechanism of four-pump PSFC in a SOA [9]. In complement to these works, our theoretical study of FWM-based PSFC in a nonlinear fiber has shown that only three pumps are necessary instead of four. We have also found the initial conditions for these pumps and have in particular found that the power levels of the three pumps must be equal and that the phases of two pumps (P1 and P3) must satisfy Eqs. (15) and (16), whatever the phase ϕ₇ of the pump P7 and the initial phase ${\varphi }_6^0$ of the signal S6. This approach has allowed us to derive analytic expressions for the idler powers at the end of the fiber.

3. Experimental demonstration

To test the validity of our theoretical study we have realized the experiment of PSFC in a HNLF with CW waves. The setup of the experiment is described in Fig. 3. We start from a tunable laser source (TLS). The tunability of the laser is used to adjust its wavelength near the zero-dispersion wavelength of the HNLF. A phase modulator driven at 20 GHz is used to create a frequency comb with optical frequencies separated by 20 GHz. The comb enters a programmable optical filter (Finisar Waveshaper). This filter selects four lines among the comb corresponding to the three pumps P1, P3 and P7 and the signal S6 as represented in Fig. 3. The programmable optical filter allows to set independently the power level and the phase of each wave. Pumps and signal are then amplified through an erbium-doped fiber amplifier (EDFA) and launched into the HNLF. The generation of idlers I2 and I4 is analyzed with an optical spectrum analyser (OSA). The central wavelength of the comb (wavelength of idler I4) falls at λ₀ = 1547.83 nm. The HNLF has a length L of 500 m, loss of 0.2 dB/km, a nonlinear coefficient γ of 10.8 W⁻¹km⁻¹, a zero-dispersion wavelength of 1548 nm and a dispersion slope of 0.006 ps.nm⁻²km⁻¹. At the central wavelength λ₀, the dispersion coefficients are β₂ = −6.3 × 10⁻³ ps²km⁻¹ and β₃ = 9.7 × 10⁻³ ps³km⁻¹ while β₄ is negligible. These values justify the fact that the phase mismatches Δβ_k L present in Eqs. (6) and (7) are close to zero.

 

Fig. 3 Experimental setup of a phase-sensitive frequency converter.

Download Full Size | PDF

The power levels of the three pumps are set to approximately the same value. Their phases are set in order to have, at the output of the programmable optical filter, values in compliance with the first line of Table 1. The parameters used in the experiment for the pumps and the signal are summarized in Table 2. The fact that the power levels of the three pumps are not perfectly equal does not affect the experiment, as we will see later.

Table 2. Power levels and phases of the three pumps and the signal at the input of the HNLF.

View Table | View all tables in this article

The phase ϕ₆ of the signal can be varied from 0 to π with the help of the programmable optical filter. Figure 4(a) represents the optical spectrum at the output of the HNLF for ϕ₆ = 0. The four more intense waves are P1, P3, S6 and P7 and we clearly show the presence of the idler I4 at 1547.83 nm for this value of the signal phase. Additional waves at 1547.15 nm, 1547.66 nm and 1548.51 nm are due to extra FWM coupling and will be ignored in the following. Figure 4(b) represents the optical spectrum for ϕ₆ = π/2. In this case the idler I4 no longer exists and the idler I2 is generated, as expected.

 

Fig. 4 Optical spectra at the output of the HNLF for (a) δϕ₆ = 0 and (b) δϕ₆ = π/2.

Download Full Size | PDF

In Fig. 5, we have plotted the evolution of the power levels P₂ and P₄ of idlers I2 and I4, respectively as a function of the signal phase ϕ₆. Symbols represent the experimental values while solid lines are the theoretical curves obtained by the analytic expressions (19) and (20). One set of dashed lines (named coupled Eqs.) corresponds to the results of the numerical resolution of the complete set of Eqs. (21)–(27) of Appendix A. The other set of dashed lines (named NLSE) corresponds to the numerical solution obtained by solving the nonlinear Schrödinger equation using a split-step Fourier method. The first comment is that, as expected, P₂ and P₄ evolve in quadrature and a very good agreement is found between experimental results and our analytic solution. The second comment concerns the fact that both numerical solutions (coupled Eqs. and NLSE) coincide very well with the analytic one. This proves that the assumptions used to reduce the complete system of seven equations to the system of two Eqs. (10)–(11) are fully justified.

 

Fig. 5 Evolution of the powers P₂ and P₄ of idlers I2 and I4 as a function of ϕ₆.

Download Full Size | PDF

4. Conclusion

In this paper, we have proposed a theoretical study of the mechanism of PSFC in a HNLF. We have derived analytic expressions for the generated idlers and have given simple relations to determine the required initial conditions on the pumps. We have in particular shown that only three pumps are required and that only the phases of two pumps have to be carefully adjusted in order to obtain the desired effect. In this case, the PSFC mechanism is similar to a FWM-based nonlinear interferometer in which, depending on the phase of the signal, pairs of waves add constructively or destructively. Our approach allows to easily design PSFC devices for all-optical signal-processing applications.