1. Introduction

A WDM router enabled by all-optical wavelength converters (AOWCs) is one of the key devices for high-capacity and flexible all-optical networks [1]. Among various AOWCs, a one-pump fiber parametric wavelength converter (1P-FPWC) based on degenerated four-wave mixing (DFWM) is an attractive candidate with inherited advantages such as operation speed above 100Gb/s, simple structure and broadband tunability [2-7]. An important requirement for an AOWC to be used in a WDM router is that it should be able to realize conversion between any pair of channels (wavelengths) within a band [1, 8, 9], so that a virtual or real link can be established between any pair of users in a WDM network. It is proposed that a tunable 1P-FPWC can be used to construct a WDM router and the bandwidth, within which conversion between any two channels has exponential gain, is represented by the side length of the square fitting inside the contours of the cutoff gain in the (λ_s, λ_i) space (λ_s,i are the respective wavelengths of signal and idler), and the maximal bandwidth can be obtained by maximizing the side length the square [8, 9]. But the bandwidth of the WDM router obtained in this way is not maximal, leaving a large part of exponential gain region not utilized. In this paper a novel method is proposed to maximize the bandwidth of the WDM router based on 1P-FPWCs. The effects of third and fourth order dispersion on the tuning range of a 1P-FPWC with a fixed input wavelength are analyzed. Then it is proved that there exists an optimum signal (idler) wavelength at which the output (input) wavelength range can be maximized. Analytic expressions of the optimum frequency and bandwidth are deduced. Based on these results, a novel two-stage bidirectional wavelength conversion method is proposed. Using this method the bandwidth of the WDM router can be significantly improved compared to the one-stage converters [8, 9] by 252% if ordinary highly nonlinear fibers (HNLFs) are used or 390% if HNLFs with optimal fourth order dispersion are used.

2. Optimization theory

To maximize the bandwidth of a WDM router based on 1P-FPWCs, we are first going to discuss how to maximize the tuning range of a 1P-FPWC with a fixed input wavelength. For 1P-FPWCs, the wavelength conversion gain G can be derived analytically when pump depletion and fiber loss can be neglected and is given by [10]

where the parametric gain, g, is defined as

The phase mismatch κ is given by

where P ₀ is pump power, γ is the fiber nonlinear coefficient and the linear phase mismatch Δβ is given by

where β_s,i,p are the respective propagation constants of signal, idler and pump. For a tunable pump it is more convenient to expand β_s,i,p in a Taylor series about the zero dispersion frequency ω ₀ than ω_p. Considering up to the fourth order dispersion, Δβ can be rewritten as [8]:

where β_m = (d^mβ/ dω^m)ω=ω₀. Eqs. (1–3) show that exponential gain occurs while g is real. This is true when -4γ P ₀ ≤ Δβ ≤ 0. When Δβ = -2γ P ₀, G = G _max = sinh²(γ P ₀ L). When Δβ = -4γ P ₀,0, G = G_c = (γ P ₀ L)². From Eq. (5) Δβ is a fourth order function of ω_p. Thus the real solutions of equations Δβ(ω_p) = 0, -4γ P ₀ are called cutoff frequencies and G_c is called cutoff gain [8]. The allowable tuning range of ω_p is equal to the cutoff frequency difference Δω_p. Obviously the tuning range of 1P-FPWC with a fixed input wavelength is equal to 2Δω_p, because ω_i = 2ω_p - ω_s. Next we will discuss how to maximize Δω_p. Note that for a fiber parametric amplifier signal tuning range with a fixed pump wavelength (amplification bandwidth) should be maximized instead. A well-known conclusion is that for a given pump power there exists an optimal pump wavelength at which the signal tuning range can be maximized [11,12].

When β ₄ is very small or the tuning range is limited, the effects of β ₄ can be neglected and Eq. (5) is simplified as

Here Δβ(ω_p) is a third order function of ω_p with a S-type profile and two extrema occurring at

The extrema are found to be

Fig. 1 shows the schematics of the profiles of Δβ(ω_p) when ω_s < ω ₀ and ω_s > ω ₀. As seen in Fig. 1, with the same pump power, Δω_p is much larger when ω_s < ω ₀ [13]. Thus ω_s < ω ₀ is taken as a premise in the following discussion. Considering a cutoff at Δβ(ω_p) = 0 , from Eq. (6), the cutoff frequencies are found to be

 

Fig. 1. The profile of Δβ(ω_p) with different ω_s (the gray area represents exponential gain region)

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Table 1. The corresponding ranges of ω_s in Fig. 1

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As seen in Fig. 1 Δω_p ≈ ω _p2 - ω _p1 = ω ₀ - ω _s when Δβ(ω _e2)≥-4γP ₀. Because Δβ(ω _e2) decreases with decreasing ω _s and when Δβ(ω _e2) becomes smaller than -4γ P ₀, the exponential gain region is broken into several narrower segments. Thus Δω_p reaches its maximum when Δβ(ω _e2) = -4γ P ₀. Solving Δβ(ω _e2) = -4γ P ₀ the optimum ω _s is found to be

With ω_s = ω^opt_s, from Eq. (6), the cutoff frequencies for Δβ(ω_p) = -4γ P ₀ are found to be

As seen from Fig. 1(b) ω_p should be in the range of

Thus the maximal Δω_p for a fixed input wavelength equals

When β ₄ ≠ 0, Δβ is a fourth order function of ω_p. Considering the cutoff at Δβ(ω_p) = 0, cutoff frequencies are found to be

The solutions ω _p3,4 are real when ω _s satisfies the following inequality

When β ₄ > 0, Δβ (ω_p) has a W-type profile as seen in Fig. 2. So if inequality (15) is not satisfied Δβ(ω_p) ≥ 0. Thus Δω_p = 0. Otherwise the outmost cutoff frequencies are determined by Δβ(ω_p) = 0 and Δω_p is found to be (see Appendix)

Eq. (16) shows that Δω_p reaches its maximum

when

Substituting Eq. (18) in Eq. (14) one can find that ω_p should be in the range of

Second considering the cutoff at Δβ(ω_p) = -4γ P ₀, with ω_s = ω^opt_s, the cutoff frequencies are found to be

where Φ = 9β ⁴ ₃ -336P ₀ β ³ ₄. If Φ ≥ 0, ω _p5,6,7,8 are real and fall inside the range given by Eq. (19). In this case the exponential gain region is broken into several narrower segments as seen in Fig. 2(c). Thus to obtain the maximal Δω_p, the following inequality must be satisfied,

Thus when β ₄ takes the minimum (or optimum) value

Δω ^max _p reaches its global maximum value

 

Fig. 2. The profile of Δβ(ω_p) when Φ < 0, Φ = 0 and Φ > 0 (ω_s = ω^opt_s).

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Fig. 3. Contours of the cutoff gain

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When β ₄ < 0, Δβ(ω_p) has a M-type profile. The outmost cutoff frequencies are determined by Δβ(ω_p) = -4γ P ₀. Because this equation is an inhomogeneous one, simple analytical expressions of the outmost cutoff frequencies cannot be obtained. Nevertheless we can get the inexplicit expression as follows

where Φ′ = (β ₄/2)(ω_p - ω ₀)² + β ₃(ω_p - ω ₀). With Eqs. (24) and (14), we can easily draw the contours of the cutoff gain. From these contours, we can find the optimum signal frequency numerically. For example, Assuming λ ₀ = 1550nm, γ = 0.02W^-1m^-1, P ₀ = 0.5W, β ₄ = -2.58 × 10^-55 s ⁴ m ^-1, β ₃ = 5.04 × 10^-41 s ³ m ^-1, we can get the contours as plotted in Fig. 3. The red and blue contours are obtained respectively from Eq. (14) and (24). Converts inside the gray region have exponential gain. Thus the optimum signal wavelength λ^opt_s corresponding to the maximal tuning is found to be 1572.5nm.

 

Fig. 4. Comparison between Ref. [2] and optimization results in different cases

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Table 2. The parameters used in Fig. 4.

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In order to get a physical insight into the above theoretical results and demonstrate the effectiveness of the optimization method, we are going to optimize a tunable 1P-FPWC similar to that experimentally studied in Ref. [2], which reported a transparent wavelength conversion with 24nm pump tuning range. The media fiber is a 115m-long HNLF with nonlinear coefficient γ = 10W ^-1 km ^-1, λ ₀ = 1562.0nm and dispersion slope S=0.03 ps/nm ²/km (β ₃ = 5.03 × 10^-41 s ³/m). The other parameters are listed in Table 2. The blue lines in Fig. 4 (a) and (b) show the changes of Δβ and G against pump wavelength λ_p reported in Ref. [2]. The optimization result assuming β ₄ = 0 is denoted as result1. To make the increase of pump tuning range more explicit λ ₀ is assumed to be 1535.5nm in order to align the tuning range before and after optimization. From Eq. (10), by setting λ_s = λ^opt_s =1559.9nm, the tuning range increases 10nm with the same pump power and fiber length. Moreover the average gain increases 1.5dB, while the gain ripples drop by more than 2dB. The optimization result with optimal β ₄ is denoted as result2. Here λ ₀ is assumed to be 1533.6nm to align the tuning range with that of result1. From Eqs. (18) and (22) by setting β ₄ = β^opt ₄ = 2.20 × 10^-54 s ⁴ m ^-1, λ_s = λ^opt_s =1562.7nm, the tuning range increases by another 18.5nm and retains the same gain ripple as result1 after optimization.

The physical explanation for the optimization results is obvious from Fig. 4(a). By carefully choosing λ_s and β ₄ a larger part of the curve representing Δβ(ω_p) vs. ω_p can be retained in the exponential gain region, resulting an increase in Δω_p. Note that in this region the gain ripple is always equal to G _max/G_c = sinh² (γ P ₀ L)/(γP ₀ L)², which keeps the same as long as γ P ₀ L is not changed.

3. A method to maximize the bandwidth of the WDM routers based on 1P-FPWCs

Figure 5(a) shows the measured (colored) contour plots of the cutoff gain G^c in the (λ_s, λ_i) space given in Ref. [9] and the area inside the contours represents the exponential gain region. The media fiber is a 1.5km-long dispersion shifted fiber (DSF) with γ = 2.2W ^-1 km ^-1, λ ₀ = 1549.25nm and β ₃ = 1.2 × 10^-40 s ³/m. As seen from Fig. 5 (a) the maximal tuning range occurs near λⁿ_s = 9nm (λⁿ_s is the renormalized signal wavelength). From Eq. (10) λ^opt_s = 1558.5nm when P ₀ = 760mW and the renormalized λⁿ_s =9.25nm. From Eq. (12) the maximal idler tuning range is from 1540.11nm to 1564.73nm (λⁿ_i is from −9.8nm to 16.6nm). As seen in Fig. 5 (a) the theoretical predications agree very well with the experimental data.

 

Fig. 5. (a): Measured contours of the cutoff gain G_c given in Ref. [9]. Converts from an arbitrary λ_s to any λ_i within the contours has exponential convert gain. (b): Theoretical contours of the cutoff gain. The side lengths of the black and red squares represent the maximal bandwidths of the one- and two-stage 1P-FPWC, respectively. Shadowed region represents the guard band against λ₀ variations.

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Because the maximal tuning range occurs only at a fixed input wavelength, it is proposed that the bandwidth of a WDM router based on a tunable 1P-FPWC is equal to the side length of the black square fitting within the contours as seen in Fig. 5 (b) and the maximal bandwidth can be obtained by maximizing the side length the square [8, 9]. The bandwidth obtained in this way is given by Δω = 2(4γP ₀/β ₃)^1/3 when β ₄ = 0. Obviously this method leaves a large part of the exponential gain region not utilized. Substituting ω_s = 2ω_p - ω_i into Eq. (5), one can find that the equation remains unchanged except the subscript s is changed into i. Therefore the equations and conclusions derived for ω_s in the above section are also applicable to ω_i. In other words there also exists an optimum idler frequency ω^opt_i at which the input wavelength range can be maximized. Therefore a novel two-stage bidirectional conversion method is proposed, as seen in Fig. 6. Because wavelength conversion (FWM) only occurs in the direction of the pump [14], one can actually use a single fiber with two counter-propagating pumps to implement 1P-FPWC-1 and 1P-FPWC-2 in the same fiber. Signal and pump waves, ω_s and ω _p1 , are input in the forward direction. Note that the optimal idler (output) wavelength of 1P-FPWC-1 ω^opt _i1 is determined after the pump power and media fiber are chosen. Thus by tuning ω _p1 to ω _p1 = (ω_s + ω ^opt _i1)/2, a forward converted wave ω_i = ω ^opt _i1 can be obtained and the input bandwidth of 1P-FPWC-1 is maximized. ω ^opt _i1 is then reflected by the FBG and acts as the signal wave in 1P-FPWC-2. If the pump power is the same, ω ^opt _i1 = ω ^opt _s2. Thus output bandwidth of 1P-FPWC-2 is maximized. So the bandwidth of the WDM router can be presented by the side length of the red square in Fig. 5 (b). From Eq. (13), the bandwidth Δω = 2Δω^max _p = 8(γ _{P 0}/β ₃)^1/3, which is increased by 4^2/3 (252%) compared to the one-stage ones [8]. Note that the FBG with reflection frequency ω ^opt _i1 will not limit the bandwidth, because the backward pump will not be blocked except when ω _p2 = ω ^opt _i1. But ω _p2 = ω ^opt _i1 the expected converted wavelength is ω_i2 = 2ω _p2 - ω _s2 = 2ω ^opt _i1 - ω ^opt _i1 = ω ^opt _i1. Noting that ω ^opt _i1 has already been obtained in the first-stage, so one can still get the expected

 

Fig. 6. Schematic the WDM router based on 1P-FPWCs (FBG: fiber Bragg grating, TF: Tunable filter).

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Fig. 7. The bandwidth when β ₄ = β^opt ₄ (a) and β ₄ = 1.1β^opt ₄ (b). The shadowed region represents the guard band against λ ₀ variations.

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When HNLFs with optimal β ₄ are used the bandwidth of the WDM router is increased to Δω = 2Δω_p = 8(96/7)^1/6(γ P ₀/ β ₃)^1/3. The bandwidth is 390% larger compared with the one-stage one without fourth order dispersion [8]. Note that to get the same bandwidth the pump power need increase by about 64 times for the latter. The bandwidth is represented by the side length of the red square in Fig. 7(a). The parameters used are the same as Fig. 5 except β ₄ = β^opt ₄ = 1.492 × 10^-53 s ⁴/m. Compared to Fig. 5(b) the bandwidth of the WDM router is increased from 26.4nm (−9.8nm to 16.6nm) to 38.5nm (−8.7nm to 29.8nm).

 

Fig. 8. The variation of maximal bandwidth against the fourth order dispersion. The insets show the bandwidth when β ₄ = 0.96β^opt ₄ and β ₄ = 1.2β^opt ₄.

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It is noteworthy that for ordinary HNLFs (DSFs) the value of β ₄ is close to 1~2 (2~3) orders of magnitude smaller than the β^opt ₄ obtained above [9, 15, 16]. Thus in such fibers one can assume β ₄ = 0 because the fourth order dispersion has very little effect on the bandwidth. In this case the optimization method with β ₄ = 0 should be adopted. The method involving optimization of β ₄ is supported by some special kinds of HNLFs, like the ones demonstrated in Ref. [16]. The W-type index profile HNLF has a higher γ (25.1W ^-1 km ^-1) and smaller β ₃ (β ₃ = 2.15 × 10^-41 s ³/m, S =0.0 13ps/nm ²/km). Using it β^opt ₄ is decreased to 5.70 × 10^-55 s ⁴/m when P ₀ = 760mW. This value is stated as a typical value for such HNLFs [16]. From Eq. (22) β₄^opt is proportional to (β ⁴ ₃/γ)^1/3. So other novel kinds of HNLFs like chalcogenide fiber and bismuth-oxide fiber with much higher γ may also support the second method [4, 5, 17-19]. Fig. 8 shows the changes of the maximal bandwidth against β ₄ obtained by numerical method. The bandwidth (in nm) is normalized by that of the one-stage one without fourth order dispersion effects. As we can see, when β ₄ ≥ β^opt ₄ the bandwidth is inversely proportional to β ₄ as predicted by Eq. (17). When β ₄ becomes a little smaller than β^opt ₄ a sharp decrease in bandwidth occurs. This is because when β ₄ < β_{opt 4} the exponential gain region is broken into several discontinuous parts as seen in Fig. 2(c) and the insets of Fig. 8.

4. Discussions and conclusions

In practical applications there are several problems that should be addressed. First, EDFAs are often used to boost the pumps. In this case the pump tuning range is limited. From Eqs. (12) and (19), the required pump frequencies can be moved along the frequency axis by moving ω ₀. Thus one can make the degree of the overlap between the pump frequencies and the bandwidth of EDFAs as large as possible by choosing fibers with appropriate ω ₀, so that this limitation can be reduced. Another problem is that the fiber fabrication process inevitably results in undesirable variation of ω ₀ along the fiber, which will then cause a reduction of conversion gain and bandwidth [20,21]. To solve this problem one can select and splice appropriate segments of fibers to approximate the required dispersion [22-24] or use very short ultra-highly nonlinear fibers to decrease the variations [4, 5, 17-19]. To be more robust, a guard band should be left as shown by the shadowed region in Fig. 5(b) and Fig. 7(b). Note in Fig. 7 (b) β ₄ is set a little larger than β^opt ₄, in order to avoid the sharp drop of bandwidth at β ₄ = β^opt ₄ in case fiber dispersion changes. Polarization dependent gain (PDG) also has significant impact on the system performance. Schemes such as pump depolarization [25] and a birefringent fiber pumped at 45° from a principal axis can be used to mitigate the impact [26-28].

In summary, we have proved that for a 1P-FPWC there exists an optimum signal (idler) frequency at which the input (output) bandwidth can be maximized. Analytical expressions of the optimum frequency and bandwidth are deduced. Based on these results, a novel two-stage bidirectional conversion method is proposed to maximize the bandwidth of the WDM router based on 1P-FPWCs. Using this method bandwidth of the WDM router can be significantly improved compared to the one-stage ones by 252% if ordinary highly nonlinear fibers are used or 390% if fibers with optimal fourth order dispersion are used.

Appendix: The analytic expression of Δω_p when Φ < 0

By solving dΔβ(ω_p)/dω_p = 0, the three extrema can be found occurs at

(25)$${\omega }_{e1}={\omega }_s,$$
$${\omega }_{e2,e3}=\frac{1}{14{\beta }_4}\left(-9{\beta }_3+9{\beta }_4{\omega }_0+5{\beta }_4{\omega }_s\pm \sqrt{3}\sqrt{27{\beta }_3^2+2{\beta }_3{\beta }_4\left({\omega }_0-{\omega }_s\right)-{\beta }_4^2{\left({\omega }_0-{\omega }_s\right)}^2}\right),$$

when ω_s satisfy the following inequality

(26)$${\omega }_0-\left(1+2\sqrt{7}\right){\beta }_3/{\beta }_4\le {\omega }_s\le {\omega }_0-\left(1-2\sqrt{7}\right){\beta }_3/{\beta }_4.$$

If ω_s is not in this range, Δβ(ω_p) exhibits a V-type profile with only one extremum at ω _e1 = ω_s. From Eqs (14) and (26), noting ω _e1 = ω _p1,2 = ω_s, one can get the results as shown in Table 3 and the schematics of the profiles can be drawn as Fig. 9. As seen in Fig. 9, the tuning range is always equal to Δω_p = ω _p3 - ω _p4.

Table 3. The corresponding ranges of a>s in Fig. 9

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Fig. 9. The profiles of Δβ(ω_p) with different ω_s (the gray area represents exponential convert gain region)

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Acknowledgments

The authors acknowledge the support of the hi-tech research and development program of China (No. 2007AA01Z229).

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