Four-wave mixing in a fiber loop mirror constructed from two polarization-maintaining dispersion-shifted fibers

1 October 2000

Optics Communications 184 (2000) 277±282

www.elsevier.com/locate/optcom

Four-wave mixing in a ®ber loop mirror constructed from two polarization-maintaining dispersion-shifted ®bers Y.Z. He *, H.L. An, X.Z. Lin, H.D. Liu Mesoscopic Physics National Laboratory, Department of Physics, Peking University, Beijing 100871, People's Republic of China Received 27 April 2000; received in revised form 22 June 2000; accepted 24 July 2000

Abstract A ®ber loop mirror constructed from two polarization-maintaining dispersion-shifted ®bers is proposed and fourwave mixing (FWM) in this loop is theoretically investigated. Our results show that this ®ber loop can inherently suppress the pump and signal waves even without optical ®ltering in continuous-wave or long-pulse cases. The difference between one-pump (nearly degenerate) and two-pump (nondegenerate) FWMs is also presented. Ó 2000 Published by Elsevier Science B.V. Keywords: Nonlinear optical loop mirror; Four-wave mixing; Dispersion-shifted ®ber; Polarization maintaining

The nonlinear optical loop mirror (NOLM) has found its application in many ®elds such as wavelength conversion [1], time demultiplexing [2], and signal regeneration [3]. It has been developed into many kinds of basic forms, among which the dispersion-imbalanced ®ber loop mirror constructed from two optical ®bers through a 3-dB ®ber coupler is a very important type. It has been pointed out by Steele [4] that the symmetry of the two counter-propagating waves can be broken due to the di€erent dispersion characteristics of the two optical ®bers combined with the nonlinear e€ects, leading to the transmission of the laser beam from the loop. This feature has been introduced for pulse compression or pulse self-switching [5] and four-wave mixing (FWM) [6] experimentally.

*

Corresponding author. Fax: +86-10-6275-1615. E-mail address: [email protected] (Y.Z. He).

In this paper, we propose a speci®c ®ber loop mirror constructed from two polarization-maintaining dispersion-shifted ®bers (DSF) through a 3-dB ®ber coupler for FWM, and the ®ber loop will be referred to as TDFLM hereafter. The two DSFs have di€erent zero-dispersion wavelengths (ZDW). It should be noted that the TDFLM is di€erent from the optical parametric loop mirror (PALM) that is composed of one DSF and one short high-dispersion optical ®ber [6]. The FWM can only happen in the DSF in a PALM, while in the TDFLM, the FWM can happen in both the optical ®bers. The FWM feature in the TDFLM will be discussed below. One-pump FWM is ®rst discussed to set up the basic equations of the resulting idler and signal waves, and two-pump case will be discussed thereafter. The basic con®guration of our TDFLM is shown in Fig. 1. Ap0 and As0 denote the amplitudes of the initial input pump and signal waves, respectively. Each of them will be split into two

0030-4018/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 0 9 1 7 - 2

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Y.Z. He et al. / Optics Communications 184 (2000) 277±282

Act ˆ

A2p0 As0 …c K1 ÿ c1 K2 † sinh g1 L1 sinh g2 L2 ; 4g1 g2 2 …1b†

Ast ˆ 0;

…1c†

"

4 c1 c2 Ap0 As0 sinh g1 L1 sinh g2 L2 Asr ˆ i 4g1 g2   iK1 sinh g1 L1 ‡ coshg1 L1 ‡ 2g1 Fig. 1. Basic con®guration of the dispersion-imbalanced ®ber loop mirror.

beams by the 3-dB ®ber coupler: Ap0 is split into Ap1 and Ap2 , As0 is split into As1 and As2 . Ast and Act denote the amplitudes of the transmitted signal and idler waves, respectively; Asr and Acr denote the amplitudes of the re¯ected signal and idler waves, respectively. Parameters for DSF1 are as follows: L1 is the optical ®ber length, c1 , the nonlinear coecient, k01 , the ZDW, and Dk1 , the dispersion slope near k01 . L2 ; c2 ; k02 ; Dk2 are the corresponding parameters for DSF2. We assume that all the laser beams launched into the TDFLM are continuous-wave (CW) beams or long pulses such that the pump-signal walk-o€ can be neglected. It should be noted that the counter-propagating cross-phase modulation (CP XPM) by the pump waves, which is the feature of an NOLM, is considered. Analysis shows, however, that all laser beams su€er identical phase delays introduced by the CP XPM, allowing us to neglect the CP XPM. Some basic considerations are made in the following discussion: the two ZDWs are all at 1550 nm wave band, nonlinear interaction occurs in the ZDW regions, the pump is undepleted, and all the energy loss is neglected. The pump wave will be re¯ected back from the input port as discussed in Ref. [7]. The resulting idler and signal waves can be expressed as Acr ˆ ÿ ÿ

c2 A2p0 As0 sinh g2 L2 coshg1 L1 2g2 c1 A2p0 As0 sinh g1 L1 cosh g2 L2 ; 2g1

 

 # iK2 sinh g2 L2 As0 ; coshg2 L2 ‡ 2g2

where Ki ˆ Dki ‡ ci P0 …i ˆ 1; 2), Dki is the 2 phase mismatch caused by dispersion, P0 ˆ Ap0 is the input pump intensity and gi is the parametric gain in the DSFs. Dki and gi can be expressed as Dki  ÿ


2 ÿ
2pc ÿ Dki kp ÿ ks kp ÿ k0i ; 2 k

 s 2  2 1 Ki ci P 0 ÿ ; gi ˆ 2 2

i ˆ 1; 2;

…2†

…3†

where kp and ks are the pump and signal wavelengths, respectively. Eq. (1) illustrates that the transmitted signal is zero while the idler wave can be transmitted. This feature enables the TDFLM to inherently suppress the pump and signal waves even without optical ®ltering. In the FWM process, the phase-matching conditions in the two DSFs are di€erent due to the di€erent dispersions of the two optical ®bers, and we can satisfy the phase-matching condition in only one of the DSFs by setting the pump wavelength to be the ZDW of that DSF [8]. When kp is set to be k01 , the transmitted idler can be given by 1 Dk2 sinh g2 L2 : Act ˆ ÿ c1 A2p0 As0 L1 g2 4

…1a†

…1d†

…4†

When kp is set to be k02 , the transmitted idler can be given by

Y.Z. He et al. / Optics Communications 184 (2000) 277±282

1 Dk1 Act ˆ c2 A2p0 As0 L2 sinh g1 L1 : g1 4

…5†

When kp is set to be away from k01 and k02 , for example kp is set to be between k01 and k02 , the phase-matching condition cannot be met in both the DSFs, and FWM conversion eciency is impaired. Fig. 2 shows the di€erence in the conversion eciency pro®les (de®ned as g ˆ jAct j2 =jAs0 j2 ) versus the signal wavelength in the three cases. The eciency pro®le when kp is set to be between k01 and k02 decreases fast with increasing pump-signal wavelength separation as shown in Fig. 2(b). Comparing the three cases, it can be seen that kp should be set to be k01 or k02 to get a broadband transmission pro®le. It is found that the two pro-

Fig. 2. Conversion eciency of transmitted idler wave versus signal wavelength: (a) kp is set to be k01 or k02 , (b) kp is set to be between k01 and k02 . The parameters are P0 ˆ 40 mW, L1 ˆ L2 ˆ 2 km, Dk1 ˆ Dk2 ˆ 0:085 ps nmÿ2 kmÿ1 , k01 ˆ 1550 nm, k02 ˆ 1548 nm, c1 ˆ c2 ˆ 2:2 Wÿ1 kmÿ1 .

279

®les when kp is set to be k01 or k02 have a complementary relationship to some extent, meaning that the device can convert more signal wavelengths into idler waves that can be e€ectively transmitted by shifting kp from k01 to k02 (or from k02 to k01 ). The di€erence between the PALM and TDFLM can be given by assuming that the nonlinear coecient c of one of the DSF is zero so that the TDFLM is equivalent to the PALM. In the simulation, di€erent levels of pump power are set as 40, 200 and 400 mW, respectively. In a practical case, the stimulated Brillouin scattering (SBS) process possibly prevents a high-power laser beam (such as a power higher than 200 mW) which is useful to achieve a high conversion eciency from passing through a 4-km-long ®ber (our setting). However, the SBS e€ect can be e€ectively suppressed by frequency-sweeping spread-spectrum technique as shown in Ref. [9], where the pump laser is frequency-modulated. We choose kp ˆ k01 , while k01 > k02 and c2 can be set to be zero. The di€erence of the transmission pro®les is shown in Fig. 3. There is little di€erence between the TDFLM and PALM for the pump power of 40 mW. It can be seen that the higher the level of the pump power, the greater is the di€erence between PALM and TDFLM. This di€erence is apparent when the pump-signal wavelength separation is small, and diminishes when the pump-signal wavelength separation increases. All this can be explained by the fact that the FWM e€ect is still signi®cant in DSF2 even when kp is set away from the ZDW of that DSF when the pump-signal wavelength separation is small enough, especially in the region ÿ2c2 P0 < Dk2 < 0 where modulation instability [10] occurs in DSF2. Whichever ZDW kp is set to be k01 or k02 , the FWM contribution in the DSF whose ZDW is away from kp , cannot be neglected when the pump-signal wavelength separation is small. In the region where pump-signal wavelength separation is large enough, the TDFLM is similar to the PALM. Fig. 4 shows the eciency pro®le of the re¯ected idler wave (de®ned as g ˆ jAcr j2 =jAs0 j2 ). Comparing Fig. 2 with Fig. 4, it can be found that there is no idler transmission at ks ˆ kp ˆ k0i (with no wavelength shift), where the totally degenerate

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Y.Z. He et al. / Optics Communications 184 (2000) 277±282

Fig. 4. Conversion eciency of the re¯ected idler wave versus signal wavelength. Parameters are the same as set in Fig. 2.

for spectral inversion that can be introduced for dispersion compensation. The diculty lies in the separation of the idler wave from the initial input signal waves and has been overcome in Refs. [11,12]. The problem can also be solved by using two pump waves in our construction. For the twopump FWM in the TDFLM, the result can still be expressed by Eq. (1) with the exception Dki and gi become Dki  ÿ


ÿ
ÿ
pc ÿ Dki kp1 ÿ ks kp2 ÿ ks kp1 ‡ kp2 ÿ 2k0i ; 2 k …6†

 s 2  2 1 Ki ci P 0 r ÿ gi ˆ ; 2 2

Fig. 3. Conversion eciency of transmitted idler wave versus signal wavelength: (a) P0 ˆ 40 mW, (b) P0 ˆ 200 mW, and (c) P0 ˆ 400 mW. Other parameters are the same as set in Fig. 2.

FWM happens. This can be explained by the fact that Dki (i ˆ 1,2) in Eqs. (4) and (5) are zero when the totally degenerate FWM happens. FWM with no wavelength shift, however, is very important

i ˆ 1; 2;

…7†

where P0 ˆ P1 ‡ P2 is the total intensity of the two 1=2 pump waves, r ˆ 2…P1 P2 † =P0 , kp1 and kp2 are the two pump wavelengths, respectively. We set the two pump wavelengths to be symmetrical with respect to the ZDW of DSF1 and ks ˆ k01 . Under this condition, Dk1 ˆ 0 and Dk2 6ˆ 0 can be achieved, ensuring the transmission of the idler wave as shown in Eq. (4). Fig. 5(a) shows that both the transmission and the re¯ection pro®les of the idler wave are quasi-trigonometric functions of the wavelength separation of the two pump waves. Thus, we can get a maximum transmission of the idler wave by adjusting the wavelength separation of the two pump waves.

Y.Z. He et al. / Optics Communications 184 (2000) 277±282

281

Fig. 6. Conversion eciency of transmitted idler wave versus signal wavelength: the parameters set are the same as in Fig. 5(b) with the exception that P1 ˆ P2 ˆ 200 mW.

Fig. 5. Two-pump FWM operation pro®le: the parameters set are the same as in Fig. 2 except that P1 ˆ P2 ˆ 20 mW: (a) conversion eciency of transmitted idler wave versus pump wavelength separation and (b) conversion eciency of transmitted idler wave versus signal wavelength.

Fig. 5(b) shows the eciency pro®le of the transmitted idler wave with respect to a certain wavelength separation of the two pump waves marked as ``A'' in Fig. 5(a). The idler wave at ks ˆ k01 can be e€ectively transmitted. Similar discussion is applicable to DSF2. It should be noted that the total pump power is set to be 40 mW …P1 ˆ P2 † under which there is little di€erence between the TDFLM and PALM as discussed in Fig. 3. It is understandable that similar results in Fig. 5 could be achieved by the PALM under this condition in Ref. [11]. The di€erence of the TDFLM and PALM is apparent when the pump power is raised much higher as shown in Fig. 6, the principle of which is the same as the one-pump case. For both the one- and two-pump FWMs, the pump and signal can be e€ectively suppressed in

CW or long-pulse cases in theory. In short-pulse operation, the pump-signal walk-o€ and the pulse self-switching e€ect [5], which leads to the transmission of the signal and pump waves, should be considered. The suppression ratio of the input pulses will be less than that in CW or long-pulse cases. The suppression ratio will also be limited by the accuracy of the 3-dB ®ber coupler as has been pointed out in Ref. [6]. In conclusion, we have theoretically investigated the FWM behavior in the TDFLM. For CW or long-pulse FWM, the ®ber loop can inherently separate the idler wave from the pump and signal waves. With one pump, more signal wavelengths can be converted into idler waves that can be effectively transmitted by shifting the pump wavelength between the two ZDWs of the DSFs. With two pumps, we can convert two signal wavelengths (ks ˆ k01 ; k02 ) into phase conjugate idler waves with no wavelength shift and ®lter out the idler waves successfully. The results will be most useful to wavelength conversion and dispersion compensation via FWM in DSF.

References [1] K.A. Rauschenbach, K.L. Hall, J.C. Livas, G. Raybon, All-optical pulse width and wavelength conversion at 10 Gb/s using a nonlinear optical loop mirror, IEEE Photon. Technol. Lett. 6 (1994) 1130±1132.

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[2] H.K. Lee, K.H. Kim, S.Y. Park, E.H. Lee, A walk-o€ balanced nonlinear ®ber loop mirror switch, IEEE Photon. Technol. Lett. 7 (1995) 1441±1443. [3] M. Jinno, All optical signal regularizing/regeneration using a nonlinear ®ber Saganc interferometer switch with signalclock walk-o€, J. Lightwave Technol. 12 (1994) 1648±1659. [4] A.L. Steele, Pulse compression by an optical ®bre loop mirror constructed from two di€erent ®bres, Electron. Lett. 29 (1993) 1972±1974. [5] W.S. Wong, S. Namiki, M. Margalit, H.A. Haus, E.P. Ippen, Self-switching of optical pulses in dispersion-imbalanced nonlinear loop mirror, Opt. Lett. 22 (1997) 1150± 1152. [6] K. Mori, T. Morioka, M. Garuwatari, Optical parametric loop mirror, Opt. Lett. 20 (1995) 1424±1426. [7] E.A. Swanson, J.D. Moores, A ®ber frequency shifter with broad bandwidth high conversion eciency pump and pump ASE cancellation, and rapid tunability for WDM

[8] [9]

[10] [11] [12]

optical network, IEEE Photon. Technol. Lett. 6 (1994) 1341±1343. K. Kikuchi, C. Lorattanasane, Design of highly ecient four-wave mixing devices using optical ®bers, IEEE Photon. Technol. Lett. 6 (1994) 992±994. A. Hirose, Y. Takushima, T. Okoshi, Suppression of stimulated Brillouin scattering and Brillouin crosstalk by frequency-sweeping spread-spectrum scheme, J. Opt. Commun. 12 (1991) 82±85. P.O. Hedekvist, M. Karlsson, P.A. Andrekson, Fiber fourwave mixing demultiplexing with inherent parametric ampli®cation, J. Lightwave Technol. 15 (1997) 2051±2058. K. Mori, T. Morioka, M. Saruwatari, Wavelength-shiftfree spectral inversion with an optical parametric loop mirror, Opt. Lett. 21 (1996) 110±112. K. Inoue, Spectral inversion with no wavelength shift based on four-wave mixing with orthogonal pump beams, Opt. Lett. 22 (1997) 1772±1774.