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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 3 — Feb. 2, 2009
  • pp: 1454–1465
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Method for effectively utilizing tunable one-pump fiber parametric wavelength converters as an enabling device for WDM routers

Sheng Cui, Deming Liu, Ying Wang, and Feng Tu  »View Author Affiliations


Optics Express, Vol. 17, Issue 3, pp. 1454-1465 (2009)
http://dx.doi.org/10.1364/OE.17.001454


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Abstract

In this paper a method is proposed to maximize the bandwidth of the WDM router based on one-pump fiber parametric wavelength converters. It is proved that for such converters there exists an optimum signal (idler) frequency at which the output (input) tuning range can be maximized. Analytical expressions of the optimum frequency and the maximal tuning range are deduced. Then a two-stage bidirectional wavelength conversion method is proposed. With this method the bandwidth of the WDM router based on such a converter 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.

© 2009 Optical Society of America

1. Introduction

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

10. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001), Chap. 10.

]

G=(1+κ24g2)sinh2(gL),
(1)

where the parametric gain, g, is defined as

g=[(γP0)2(κ/2)2]1/2.
(2)

The phase mismatch κ is given by

κ=2γP0+Δβ,
(3)

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

Δβ=βs+βi2βp,
(4)

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

8. R. W. McKerracher, J. L. Blows, and C. M de Sterke, “Wavelength conversion bandwidth in fiber based optical parametric amplifiers,” Opt. Express 11, 1002–1007 (2003). [CrossRef] [PubMed]

]:

Δβ=β3(ωpω0)(ωpωs)2+β42(ωpω0)2(ωpωs)2+β412(ωpωs)4
(5)

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

Δβ(ωp)=β3(ωpω0)(ωpωs)2.
(6)

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

ωe1=ωs,
ωe2=(2ω0+ωs)/3.
(7)

The extrema are found to be

Δβ(ωe1)=Δβ(ωs)=0,
Δβ(ωe2)=427β3(ω0ωs)3.
(8)

Fig. 1 shows the schematics of the profiles of Δβ(ωp) when ωs < ω 0 and ωs > ω 0. As seen in Fig. 1, with the same pump power, Δωp is much larger when ωs < ω 0 [13

13. T. Yamamoto and M. Nakazawa, “Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching,” IEEE Photonics Tech. Lett. 9, 327–329 (1997). [CrossRef]

]. Thus ωs < ω 0 is taken as a premise in the following discussion. Considering a cutoff at Δβ(ωp) = 0 , from Eq. (6), the cutoff frequencies are found to be

ωp1=ωsandωp2=ω0.
(9)
Fig. 1. The profile of Δβ(ωp) with different ωs (the gray area represents exponential gain region)

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

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As seen in Fig. 1 Δωpω 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 Δωp 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

ωs=ωsopt=ω03(γP0/β3)1/3.
(10)

With ωs = ωopts, from Eq. (6), the cutoff frequencies for Δβ(ωp) = -4γ P 0 are found to be

ωp3=ω04(γP0/β3)1/3,ωp4=ω0(γP0/β3)1/3.
(11)

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

ω04(γP0/β3)1/3ωpω0.
(12)

Thus the maximal Δωp for a fixed input wavelength equals

Δωpmax=4(γP0/β3)1/3.
(13)

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

ωp1,p2=ωs,
ωp3,p4=17β4(6β3+6β4ω0+β4ωs)±66β32+2β3β4(ω0ωs)β42(ω0ωs)2.
(14)

The solutions ω p3,4 are real when ω s satisfies the following inequality

ω0(1+7)β3/β4ωsω0(17)β3/β4.
(15)

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

Δωp=ωp3ωp4=267β46β32+2β3β4(ω0ωs)β42(ω0ωs)2.
(16)

Eq. (16) shows that Δωp reaches its maximum

Δωpmax=267β3β4,
(17)

when

ωs=ωsopt=ω0β3/β4.
(18)

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

ω0(1+67)β3β4=ωp4ωpωp3=ω0(167)β3β4
(19)

Second considering the cutoff at Δβ(ωp) = -4γ P 0, with ωs = ωopts, the cutoff frequencies are found to be

ωp5,6=ω0β3β4±17β43β32Φ,
ωp7,8=ω0β3β4±17β43β32+Φ,
(20)

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 Δωp, the following inequality must be satisfied,

Φ=9β34336P0β43γ<0.
(21)

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

β4=β4opt=(3β34112P0γ)1/3
(22)

Δω max p reaches its global maximum value

Δωp,globalmax=4(96/7)1/6(γP0β3)1/3.
(23)
Fig. 2. The profile of Δβ(ωp) when Φ < 0, Φ = 0 and Φ > 0 (ωs = ωopts).
Fig. 3. Contours of the cutoff gain

When β 4 < 0, Δβ(ωp) has a M-type profile. The outmost cutoff frequencies are determined by Δβ(ωp) = -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

ωs=ωp±6Φ'±Φ'2(4/3)β4γPpβ4
(24)

where Φ′ = (β 4/2)(ωp - ω 0)2 + β 3(ωp - 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 λopts corresponding to the maximal tuning is found to be 1572.5nm.

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

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

2. M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, “Transparent wavelength conversion in fibre with 24nm pump tuning range,” Electron. Lett. 38, 85–86 (2002). [CrossRef]

], 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 λp reported in Ref. [2

2. M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, “Transparent wavelength conversion in fibre with 24nm pump tuning range,” Electron. Lett. 38, 85–86 (2002). [CrossRef]

]. 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 λs = λopts =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 = βopt 4 = 2.20 × 10-54 s 4 m -1, λs = λopts =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 λs and β 4 a larger part of the curve representing Δβ(ωp) vs. ωp can be retained in the exponential gain region, resulting an increase in Δωp. Note that in this region the gain ripple is always equal to G max/Gc = 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 Gc in the (λs, λi) space given in Ref. [9

9. R. W. McKerracher, J. L. Blows, and C. M de Sterke, “Systematic analysis of wavelength conversion in a fiber optical parametric device with a single, tunable pump,” Opt. Express 12, 2810–2815 (2004). [CrossRef] [PubMed]

] 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 λns = 9nm (λns is the renormalized signal wavelength). From Eq. (10) λopts = 1558.5nm when P 0 = 760mW and the renormalized λns =9.25nm. From Eq. (12) the maximal idler tuning range is from 1540.11nm to 1564.73nm (λni 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 Gc given in Ref. [9]. Converts from an arbitrary λs to any λi 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.

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

8. R. W. McKerracher, J. L. Blows, and C. M de Sterke, “Wavelength conversion bandwidth in fiber based optical parametric amplifiers,” Opt. Express 11, 1002–1007 (2003). [CrossRef] [PubMed]

, 9

9. R. W. McKerracher, J. L. Blows, and C. M de Sterke, “Systematic analysis of wavelength conversion in a fiber optical parametric device with a single, tunable pump,” Opt. Express 12, 2810–2815 (2004). [CrossRef] [PubMed]

]. 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 ωs = 2ωp - ωi 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 ωs in the above section are also applicable to ωi. In other words there also exists an optimum idler frequency ωopti 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

14. G. Kalogerakis and M. E. Marhic, “Methods for full utilization of the bandwidth of fiber optical parametric amplifiers and wavelength converters,” J. Lightwave Tech. 24, 3683–3691 (2006). [CrossRef]

], 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, ωs and ω p1 , are input in the forward direction. Note that the optimal idler (output) wavelength of 1P-FPWC-1 ωopt i1 is determined after the pump power and media fiber are chosen. Thus by tuning ω p1 to ω p1 = (ωs + ω opt i1)/2, a forward converted wave ωi = 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 = 2Δωmax p = 8(γ P 0/β 3)1/3, which is increased by 42/3 (252%) compared to the one-stage ones [8

8. R. W. McKerracher, J. L. Blows, and C. M de Sterke, “Wavelength conversion bandwidth in fiber based optical parametric amplifiers,” Opt. Express 11, 1002–1007 (2003). [CrossRef] [PubMed]

]. 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 ωi2 = 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).
Fig. 7. The bandwidth when β 4 = βopt 4 (a) and β 4 = 1.1βopt 4 (b). The shadowed region represents the guard band against λ 0 variations.

When HNLFs with optimal β 4 are used the bandwidth of the WDM router is increased to Δω = 2Δωp = 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

8. R. W. McKerracher, J. L. Blows, and C. M de Sterke, “Wavelength conversion bandwidth in fiber based optical parametric amplifiers,” Opt. Express 11, 1002–1007 (2003). [CrossRef] [PubMed]

]. 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 = βopt 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 = 0.96βopt 4 and β 4 = 1.2βopt 4.

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 βopt 4 obtained above [9

9. R. W. McKerracher, J. L. Blows, and C. M de Sterke, “Systematic analysis of wavelength conversion in a fiber optical parametric device with a single, tunable pump,” Opt. Express 12, 2810–2815 (2004). [CrossRef] [PubMed]

, 15

15. M. Hirano, T. Nakanishi, T. Okuno, and Masashi Onishi, “Broadband wavelength conversion over 193-nm by HNL-DSF improving higher-order dispersion performance,” in Proc. of ECOC 2005, Glasgow, Scotland, paper TH4.4.4 (2005)

, 16

16. J. Hiroishi, N. Kumano, K. Mukasa, R. Sugizaki, R. Miyabe, S. Matsushita, H. Tobioka, S. Namiki, and T. Yagi, “Dispersion slope controlled HNL-DSF with high γ of 25 W-1km-1 and band conversion experiment using this fiber,” in Proc. of ECOC 2002, Copenhagen, Denmark, paper PD1.5 (2002).

]. 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

16. J. Hiroishi, N. Kumano, K. Mukasa, R. Sugizaki, R. Miyabe, S. Matsushita, H. Tobioka, S. Namiki, and T. Yagi, “Dispersion slope controlled HNL-DSF with high γ of 25 W-1km-1 and band conversion experiment using this fiber,” in Proc. of ECOC 2002, Copenhagen, Denmark, paper PD1.5 (2002).

]. 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 βopt 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

16. J. Hiroishi, N. Kumano, K. Mukasa, R. Sugizaki, R. Miyabe, S. Matsushita, H. Tobioka, S. Namiki, and T. Yagi, “Dispersion slope controlled HNL-DSF with high γ of 25 W-1km-1 and band conversion experiment using this fiber,” in Proc. of ECOC 2002, Copenhagen, Denmark, paper PD1.5 (2002).

]. From Eq. (22) β4opt 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

4. K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, “Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber,” Optics Express 15, 15418–15423 (2007). [CrossRef] [PubMed]

, 5

5. J. H. Lee, T. Nagashima, T. Hasegawa, S. Ohara, N. Sugimoto, T. Tanemura, and K. Kikuchi, “Wavelength conversion of 40-Gbit/s NRZ signal using four-wave mixing in 40-cm-long bismuth oxide based highly-nonlinear optical fiber,” in Proc. OFC 2005, paper PDP23, Anaheim, USA (2005).

, 17-19

17. M. R. E. Lamont, B. T. Kuhlmey, and C. M. D. Sterke, “Multi-order dispersion engineering for optimal four-wave mixing,” Opt. Express 16, 7551–7563 (2008). [CrossRef] [PubMed]

]. 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 4βopt 4 the bandwidth is inversely proportional to β 4 as predicted by Eq. (17). When β 4 becomes a little smaller than βopt 4 a sharp decrease in bandwidth occurs. This is because when β 4 < βopt 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

20. M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B , 15, 2269–2275 (1998) [CrossRef]

,21

21. M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express 12, 136–142 (2003) [CrossRef]

]. To solve this problem one can select and splice appropriate segments of fibers to approximate the required dispersion [22-24

22. K. Inoue, “Arrangement of fiber pieces for a wide wavelength conversion range by fiber four-wave mixing,” Opt. Lett. 19, 1189–1191 (1994). [CrossRef] [PubMed]

] or use very short ultra-highly nonlinear fibers to decrease the variations [4

4. K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, “Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber,” Optics Express 15, 15418–15423 (2007). [CrossRef] [PubMed]

, 5

5. J. H. Lee, T. Nagashima, T. Hasegawa, S. Ohara, N. Sugimoto, T. Tanemura, and K. Kikuchi, “Wavelength conversion of 40-Gbit/s NRZ signal using four-wave mixing in 40-cm-long bismuth oxide based highly-nonlinear optical fiber,” in Proc. OFC 2005, paper PDP23, Anaheim, USA (2005).

, 17-19

17. M. R. E. Lamont, B. T. Kuhlmey, and C. M. D. Sterke, “Multi-order dispersion engineering for optimal four-wave mixing,” Opt. Express 16, 7551–7563 (2008). [CrossRef] [PubMed]

]. 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 βopt 4, in order to avoid the sharp drop of bandwidth at β 4 = βopt 4 in case fiber dispersion changes. Polarization dependent gain (PDG) also has significant impact on the system performance. Schemes such as pump depolarization [25

25. T. Yang, C. Shu, and C. Lin, “Depolarization technique for wavelength conversion using four-wave mixing in a dispersion-flattened photonic crystal fiber,” Optics Express 13, 5409–5415 (2005). [CrossRef] [PubMed]

] and a birefringent fiber pumped at 45° from a principal axis can be used to mitigate the impact [26-28

26. F. Yaman, Q. Lin, and G. P. Agrawal, “A novel design for polarization-independent single-pump fiberoptic parametric amplifier,” IEEE Photonics Tech. Lett. 18, 2335–2337 (2006). [CrossRef]

].

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 Δωp when Φ < 0

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

ωe1=ωs,
ωe2,e3=114β4(9β3+9β4ω0+5β4ωs±327β32+2β3β4(ω0ωs)β42(ω0ωs)2),
(25)

when ωs satisfy the following inequality

ω0(1+27)β3/β4ωsω0(127)β3/β4.
(26)

If ωs is not in this range, Δβ(ωp) exhibits a V-type profile with only one extremum at ω e1 = ωs. From Eqs (14) and (26), noting ω e1 = ω p1,2 = ωs, 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 Δωp = 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(ωp) with different ωs (the gray area represents exponential convert gain region)

Acknowledgments

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

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2.

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5.

J. H. Lee, T. Nagashima, T. Hasegawa, S. Ohara, N. Sugimoto, T. Tanemura, and K. Kikuchi, “Wavelength conversion of 40-Gbit/s NRZ signal using four-wave mixing in 40-cm-long bismuth oxide based highly-nonlinear optical fiber,” in Proc. OFC 2005, paper PDP23, Anaheim, USA (2005).

6.

P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, “Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber,” IEEE Photonics Tech. Lett. 17, 1908–1910 (2005). [CrossRef]

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9.

R. W. McKerracher, J. L. Blows, and C. M de Sterke, “Systematic analysis of wavelength conversion in a fiber optical parametric device with a single, tunable pump,” Opt. Express 12, 2810–2815 (2004). [CrossRef] [PubMed]

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11.

M. E. Marhic, N. Kagi, T. K. Chiang, and L. G. Kazovsky, “Broadband fiber optical parametric amplifiers,” Opt. Lett. 21, 573–575 (1996). [CrossRef] [PubMed]

12.

M. Yu and C. J. McKinstrie, “Modulational instabilities in dispersion-flattened fibers,” Physical Review E , 52, 1072–1080 (1995). [CrossRef]

13.

T. Yamamoto and M. Nakazawa, “Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching,” IEEE Photonics Tech. Lett. 9, 327–329 (1997). [CrossRef]

14.

G. Kalogerakis and M. E. Marhic, “Methods for full utilization of the bandwidth of fiber optical parametric amplifiers and wavelength converters,” J. Lightwave Tech. 24, 3683–3691 (2006). [CrossRef]

15.

M. Hirano, T. Nakanishi, T. Okuno, and Masashi Onishi, “Broadband wavelength conversion over 193-nm by HNL-DSF improving higher-order dispersion performance,” in Proc. of ECOC 2005, Glasgow, Scotland, paper TH4.4.4 (2005)

16.

J. Hiroishi, N. Kumano, K. Mukasa, R. Sugizaki, R. Miyabe, S. Matsushita, H. Tobioka, S. Namiki, and T. Yagi, “Dispersion slope controlled HNL-DSF with high γ of 25 W-1km-1 and band conversion experiment using this fiber,” in Proc. of ECOC 2002, Copenhagen, Denmark, paper PD1.5 (2002).

17.

M. R. E. Lamont, B. T. Kuhlmey, and C. M. D. Sterke, “Multi-order dispersion engineering for optimal four-wave mixing,” Opt. Express 16, 7551–7563 (2008). [CrossRef] [PubMed]

18.

M. D. Pelusi, F. Luan, E. Magi, M. R. E. Lamont, D. J. Moss, B. J. Eggleton, J. S. Sanghera, L. B. Shaw, and I. D. Aggarwal, “High bit rate all-optical signal processing in a fiber photonic wire,” Opt. Express 16, 11506–11512 (2008). [CrossRef] [PubMed]

19.

Dong-II Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. Fu, and B. J. Eggleton, “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires,” Opt. Lett. 33, 660–662 (2008). [CrossRef] [PubMed]

20.

M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B , 15, 2269–2275 (1998) [CrossRef]

21.

M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express 12, 136–142 (2003) [CrossRef]

22.

K. Inoue, “Arrangement of fiber pieces for a wide wavelength conversion range by fiber four-wave mixing,” Opt. Lett. 19, 1189–1191 (1994). [CrossRef] [PubMed]

23.

M. E. Marhic, F. S. Yang, Min-Chen Ho, and L. G. kazovsky, “High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation,” J. Lightwave Tech. 17, 210–215 (1999). [CrossRef]

24.

J. Hansryd and P. A. Andrekson, “Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency,” IEEE Photon. Tech. Lett. 13, 194–196 (2001). [CrossRef]

25.

T. Yang, C. Shu, and C. Lin, “Depolarization technique for wavelength conversion using four-wave mixing in a dispersion-flattened photonic crystal fiber,” Optics Express 13, 5409–5415 (2005). [CrossRef] [PubMed]

26.

F. Yaman, Q. Lin, and G. P. Agrawal, “A novel design for polarization-independent single-pump fiberoptic parametric amplifier,” IEEE Photonics Tech. Lett. 18, 2335–2337 (2006). [CrossRef]

27.

Z. Wang, N. Deng, C. Lin, and C. K. Chan, “Polarization-insensitive widely tunable wavelength conversion based on four-Wave mixing using dispersion-flattened high-nonlinearity photonic crystal fiber with residual birefringence,” in Proc. of ECOC 2006, Cannes, France, Paper We3.P.18 (2006).

28.

A. S. Lenihan and G. M. Carter, “Polarization-insensitive wavelength conversion at 40 Gb/s using birefringent nonlinear fiber,” in Proc. Of CLEO 2007, Baltimore, USA, paper CThAA2 (2007)

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(190.4360) Nonlinear optics : Nonlinear optics, devices
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: October 22, 2008
Revised Manuscript: December 4, 2008
Manuscript Accepted: January 12, 2009
Published: January 26, 2009

Citation
Sheng Cui, Deming Liu, Ying Wang, and Feng Tu, "Method for effectively utilizing tunable one-pump fiber parametric wavelength converters as an enabling device for WDM routers," Opt. Express 17, 1454-1465 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1454


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References

  1. D. J. Blumenthal, "Photonic Packet and All-Optical Label Switching Technologies and Techniques," in Proc. OFC 2002, paper WO3, Anaheim, USA (2002).
  2. M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, "Transparent wavelength conversion in fibre with 24nm pump tuning range," Electron. Lett. 38, 85-86 (2002). [CrossRef]
  3. A. Zhang and M. S. Demokan, "Broadband wavelength converter based on four-wave mixing in a highly nonlinear photonic crystal fiber," Opt. Lett. 30,2375-2376 (2005). [CrossRef] [PubMed]
  4. K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007). [CrossRef] [PubMed]
  5. J. H. Lee, T. Nagashima, T. Hasegawa, S. Ohara, N. Sugimoto, T. Tanemura, and K. Kikuchi, "Wavelength conversion of 40-Gbit/s NRZ signal using four-wave mixing in 40-cm-long bismuth oxide based highly-nonlinear optical fiber," in Proc. OFC 2005, paper PDP23, Anaheim, USA (2005).
  6. P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005). [CrossRef]
  7. K. K. Chow, C. Shu, C. Lin, and A. Bjarklev, "Polarization-Insensitive Widely Tunable Wavelength Converter Based on Four-Wave Mixing in a Dispersion-Flattened Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 624-626 (2005). [CrossRef]
  8. R. W. McKerracher, J. L. Blows and C. M de Sterke, "Wavelength conversion bandwidth in fiber based optical parametric amplifiers," Opt. Express 11, 1002-1007 (2003). [CrossRef] [PubMed]
  9. R. W. McKerracher, J. L. Blows, and C. M de Sterke, "Systematic analysis of wavelength conversion in a fiber optical parametric device with a single, tunable pump," Opt. Express 12, 2810-2815 (2004). [CrossRef] [PubMed]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001), Chap. 10.
  11. M. E. Marhic, N. Kagi, T. K. Chiang, and L. G. Kazovsky, "Broadband fiber optical parametric amplifiers," Opt. Lett. 21, 573-575 (1996). [CrossRef] [PubMed]
  12. M. Yu and C. J. McKinstrie, "Modulational instabilities in dispersion-flattened fibers," Physical Review E,  52, 1072-1080 (1995). [CrossRef]
  13. T. Yamamoto and M. Nakazawa, "Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching," IEEE Photonics Tech. Lett. 9, 327-329 (1997). [CrossRef]
  14. G. Kalogerakis and M. E. Marhic, "Methods for full utilization of the bandwidth of fiber optical parametric amplifiers and wavelength converters," J. Lightwave Tech. 24, 3683-3691 (2006). [CrossRef]
  15. M. Hirano, T. Nakanishi, T. Okuno, and Masashi Onishi, "Broadband wavelength conversion over 193-nm by HNL-DSF improving higher-order dispersion performance," in Proc. of ECOC 2005, Glasgow, Scotland, paper TH4.4.4 (2005)
  16. J. Hiroishi, N. Kumano, K. Mukasa, R. Sugizaki, R. Miyabe, S. Matsushita, H. Tobioka, S. Namiki, and T. Yagi, "Dispersion slope controlled HNL-DSF with high γ of 25 W-1km-1 and band conversion experiment using this fiber," in Proc. of ECOC 2002, Copenhagen, Denmark, paper PD1.5 (2002).
  17. M. R. E. Lamont, B. T. Kuhlmey, and C. M. D. Sterke, "Multi-order dispersion engineering for optimal four-wave mixing," Opt. Express 16, 7551-7563 (2008). [CrossRef] [PubMed]
  18. M. D. Pelusi, F. Luan, E. Magi, M. R. E. Lamont, D. J. Moss, B. J. Eggleton, J. S. Sanghera, L. B. Shaw, and I. D. Aggarwal, "High bit rate all-optical signal processing in a fiber photonic wire," Opt. Express 16, 11506-11512 (2008). [CrossRef] [PubMed]
  19. Dong-II Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. Fu, and B. J. Eggleton, "Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires," Opt. Lett. 33, 660-662 (2008). [CrossRef] [PubMed]
  20. M. Karlsson, "Four-wave mixing in fibers with randomly varying zero-dispersion wavelength," J. Opt. Soc. Am. B,  15, 2269-2275(1998) [CrossRef]
  21. M. Farahmand and M. de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express 12, 136-142 (2003) [CrossRef]
  22. K. Inoue, "Arrangement of fiber pieces for a wide wavelength conversion range by fiber four-wave mixing," Opt. Lett. 19,1189-1191 (1994). [CrossRef] [PubMed]
  23. M. E. Marhic, F. S. Yang, Min-Chen Ho and L. G. kazovsky, "High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation," J. Lightwave Tech. 17, 210-215 (1999). [CrossRef]
  24. J. Hansryd and P. A. Andrekson, "Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency," IEEE Photon. Tech. Lett. 13,194-196 (2001). [CrossRef]
  25. T. Yang, C. Shu and C. Lin, "Depolarization technique for wavelength conversion using four-wave mixing in a dispersion-flattened photonic crystal fiber," Optics Express 13, 5409-5415 (2005). [CrossRef] [PubMed]
  26. F. Yaman, Q. Lin, and G. P. Agrawal, "A novel design for polarization-independent single-pump fiber-optic parametric amplifier," IEEE Photonics Tech. Lett. 18, 2335-2337 (2006). [CrossRef]
  27. Z. Wang, N. Deng, C. Lin, and C. K. Chan, "Polarization-insensitive widely tunable wavelength conversion based on four-Wave mixing using dispersion-flattened high-nonlinearity photonic crystal fiber with residual birefringence," in Proc. of ECOC 2006, Cannes, France, Paper We3.P.18 (2006).
  28. A. S. Lenihan and G. M. Carter, "Polarization-insensitive wavelength conversion at 40 Gb/s using birefringent nonlinear fiber," in Proc. Of CLEO 2007, Baltimore, USA, paper CThAA2 (2007)

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