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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 16 — Aug. 2, 2010
  • pp: 17548–17554
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Nonlinear similariton tunneling effect in the birefringent fiber

Chaoqing Dai, Yueyue Wang, and Jiefang Zhang  »View Author Affiliations


Optics Express, Vol. 18, Issue 16, pp. 17548-17554 (2010)
http://dx.doi.org/10.1364/OE.18.017548


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Abstract

We derive analytical bright and dark similaritons of the generalized coupled nonlinear Schrödinger equations with distributed coefficients. An exact balance condition between the dispersion, nonlinearity and the gain/loss has been obtained. Under this condition, we discuss the nonlinear similariton tunneling effect.

© 2010 Optical Society of America

1. Introduction

The possibility of the propagation of solitons in optical fibers was theoretically predicted by Hasegawa and Tappert [1

1. A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. 23, 142–170 (1973). [CrossRef]

] and was experimentally demonstrated by Mollenauer et al. [2

2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]

]. As we all know, the propagation of a picosecond optical pulse in a monomode optical fiber (not including optical fiber loss) is described by the nonlinear Schrödinger equation (NLSE) [1

1. A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. 23, 142–170 (1973). [CrossRef]

]

iΦZ+εΦTT+σΦ2Φ=0.
(1)

Depending upon the relative sign between the group velocity dispersion (GVD) and the self phase modulation (SPM) originating from nonlinearity, the NLSE admits two distinct types of solitons, namely, bright (εσ > 0) and dark (εσ < 0) solitons. Generally, bright solitons are well localized structures of light while dark solitons appear as localized intensity dips on a finite carrier wave background and are more robust than bright solitons.

The propagation of light pulses in birefringent fibers, multimode fibers, fiber arrays and also incoherent beam propagation in photorefractive media are more generally modelled by a system of coupled nonlinear Schrödinger (CNLS) type equations. Single mode fibers are actually bimodal because of the presence of birefringence. This birefringence creates two principal transmission axes within the fiber known as the fast and slow axes. When two or more optical fields with different frequencies co-propagate in a fiber, they can interplay through the cross-phase modulation (XPM) mechanism. To describe a two-channel wavelength-division multiplexing (WDM) soliton system with dispersion compensation and lumped amplification, we consider the generalized CNLS equations with distributed coefficients [3

3. G. P. Agrawal, Nonlinear Fiber Optics, (New York: Academic Press, 1993).

,4

4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010). [CrossRef]

]

iΨ1z12β(z)Ψ1tt+γ(z)(Ψ12+Ψ22)Ψ1=ig(z)Ψ1,
iΨ2z12β(z)Ψ2tt+γ(z)(Ψ22+Ψ12)Ψ2=ig(z)Ψ2,
(2)

where Ψ1(z, t) and Ψ2(z, t) denote two orthogonal components of an electric field, z is the coordinate along the propagation direction of the carrier wave, and t is the retarded time. The second term represents GVD, the third and fourth terms are SPM and XPM for representing the nonlinear effect, and the term in the right stands for the amplification (g > 0) or the attenuation (g < 0). Bright soliton solutions of Eq. (2) have been investigated [4

4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010). [CrossRef]

]. However, to our knowledge, similariton solutions of the CNLS equations with variable coefficients have been hardly discussed. Similariton [5–7

5. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005). [CrossRef]

], which maintains its overall shape but with their parameters, such as amplitudes and widths changing with the modulation of system parameters, has an increasingly vital role in optical fiber communication system.

In this paper, by employing the similarity transformation, Eq. (2) can be transformed into the standard NLSE [Eq. (1)]. Then by making the reverse transformation variables and functions, we obtain the exact bright and dark multi-similariton solutions for Eq. (2). As an application, we will discuss the nonlinear similariton tunneling effect, which is predicted as early as in 1978 by Newell [8

8. A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978). [CrossRef]

]. It was found that in certain circumstances, depending on the ratio of soliton amplitude to barrier height, the soliton can tunnel through the barrier in a lossless manner [8–10

8. A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978). [CrossRef]

]. The review of soliton tunneling principles and research as it currently stands in [11

11. R. C. Yang and X. L. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759–17767 (2008). [CrossRef] [PubMed]

].

2. General similarity transformations

To connect Eq. (2) with Eq. (1), we begin by using the technique of variable transformation

Ψ1(z,t)=ψ(z,t)cosθ,Ψ2(z,t)=ψ(z,t)sinθeiϑ,
(3)

then construct the similarity transformation [5–7

5. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005). [CrossRef]

]

ψ(z,t)=ρ(z)eiϕ(z,t)Φ(Z,T),
(4)

where both θ and ϑ are constants, the amplitude ρ(z) and the phase ϕ (z, t) are real functions, Φ(Z, T) is a complex function, the effective propagation distance Z(z) and the similarity variable T(z, t) are both to be determined.

The substitution of Eqs. (3) with (4) into Eq. (2) leads to Eq. (1), and after some algebra one gets the following expressions for the pulse:

ρ=ρ0α12exp[0zg(s)ds],ϕ=s0α2t2d0αt+d02αD(z)2,
(5)
Z=αD(z)(2εw02),T=[t(d0+s0t0)D(z)t0]w(z),w(z)=w0α,
(6)

Note that the existence of such similariton solutions should satisfy the following necessary and sufficient condition between the gain or loss profile and the parameters

g(z)=γβzβγz2γ(z)β(z)s0α(z)β(z)2,
(7)

which degenerates into the restriction condition for soliton (s 0 = 0) expressed as Eq. (8) in [4

4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010). [CrossRef]

]. Thus our results are more general than them in [4

4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010). [CrossRef]

].

3. Bright and dark similaritons

The solution of Eq. (2) can be obtained from the solution of NLSE [Eq. (1)] by exploiting a one-to-one correspondence. Employing the transformations [Eqs. (3), (4)] and Darboux transformation (DT) [12

12. L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004). [CrossRef]

], one can obtain bright multi-similaritons for Eq. (2)

{Ψ1Ψ2}=ρ(z)eiϕ(z,t)[Ψ0+22εσΣm=1n(λm+λm*)φ1,m(λm)φ2,m*(λm)Am]{cosθsinθeiϑ},
(8)

with

φj,m+1(λm+1)=(λm+1+λm*)φj,m(λm+1)BmAm(λm+λm*)φj,m(λm),
Am=φ1,m(λm)2+φ2,m(λm)2,Bm=φ1,m(λm+1)φ1,m*(λm)+φ2,m(λm+1)φ2,m*(λm),
(9)

where DT times m = 1, …, n, j = 1, 2, complex spectral parameters λm=12(ηm+iξm) , λ * m is the complex conjugate of λm, T,Z,ρ(z) and ϕ (z, t) satisfy Eqs. (5) and (6). (φ 1,1(λ 1),φ 2,1(λ 1))T is the eigenfunction corresponding to λ 1 for Ψ0 and φj,1=exp(δj2+iκj2) , with

δj=ηjT2εηjξjZδj0,κj=ξjTε(ηj2ξj2)Zκj0.
(10)

Inserting the zero seeding solution of Eq. (2) as Ψ0 = 0 into Eq. (8), one can obtain one similariton solution for Eq. (2). Using that one similariton solution as the seed solution in Eq. (8), we can obtain two-similaritons. Thus in recursion, one can generate up to n-similaritons. Here we present bright one- and two-similariton solutions in explicit forms. The one similariton read

{Ψ1Ψ2}=2εση1ρ(z)sechδ1exp{i[ϕ(z,t)+κ1(Z,T)]}{cosθsinθeiϑ},
(11)

where δ 1 and κ 1 are given by Eq. (10). The analytical bright similariton pairs read

Ψ1=2εσcosθρ(z)eiϕ(z,t)G1F1,
Ψ2=2εσsinθρ(z)eiϕ(z,t)+ϑG1F1,
(12)

where G1=a1coshδ2eiκ1+a2coshδ1eiκ2+ia3(sinhδ2eiκ1sinhδ1eiκ2) , F 1 = b 1 cosh(δ 1 + δ 2) + b 2 cosh(δ 1δ 2) + b 3 cos (κ 2κ 1), aj=ηj2[ηj2η3j2+(ξ1ξ2)2] , bj=14{[η1+(1)jη2]2+(ξ1ξ2)2} , a 3 = η 1 η 2(ξ 1ξ 2) and b 3 = −η 1 η 2,j = 1,2. δ j and κ j are given by Eq. (10).

Since a one-to-one correspondence exists between Eq. (2) and the NLSE [Eq. (1)], one can derive dark (gray) multi-similaritons for Eq. (2). For simplicity, here we only present dark (gray) one- and two-similariton solutions in explicit forms. The exact gray similariton pairs read

{Ψ1Ψ2}=μ2εσρ(z)ei[ϕ(z,t)+φ(Z,T)](1+G2F2){cosθsinθeiϑ},
(13)

where G2=4μ(ω1+ω22μ)4iλ1+λ2η1+η2ρ , F2=4μ2+(λ1+λ2η1+η2)2ρ , ρ = (ω 1μ) (ω 2μ), ωj = (ξjj) [ξj + j tanh(δj)]/μ, δj = ηj[TT j0 − 2(Ω + ξj)εZ],, φ(Z,T)=2ε(μ2+Ω22)ZΩTφ0 , λj = ξj + iηj and μ = ∣λj∣, j = 1,2. And the exact dark one similariton read

{Ψ1Ψ2}=2εσξ1iη1μcosθρ(z)[ξ1+iη1tanh(δ1)]ei[ϕ(z,t)+φ(Z,T)]{cosθsinθeiϑ}.
(14)

4. Nonlinear similariton tunneling effect

To investigate the unique properties of the optical similaritons propagating through dispersion and nonlinear barriers, we consider two particular examples. The first one is a dispersion barrier (DB) or dispersion well (DW) [9

9. V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001). [CrossRef]

]

β(z)=1+h1sech2[k(zz01)],γ(z)=γ0,
(15)

and the second one is a nonlinear barrier (NB) or nonlinear well (NW) [10

10. J. F. Wang, L. Li, and S. T. Jia, “Nonlinear tunneling of optical similaritons in nonlinear waveguides,” J. Opt. Soc. Am. B 25, 1254–1260 (2008). [CrossRef]

]

β(z)=β0,γ(z)=1+h2sech2[k(zz02)],
(16)
Fig. 1. Evolutions of nonlinear tunneling of bright similariton (s 0 = 0.03) and soliton (s 0 = 0) in the (a) and (b) dispersion barrier with h 1 = 2, (c) dispersion well with h 1 = −0.5 and (d) nonlinear barrier with h 2 = 2. Other parameters: k = 2,z 01 = z 02 = 7, σ = ε = t 0 = d 0 = δ 10 = w 0 = 1, θ = η 1 = ξ 1 = 0.5, γ 0 = β 0 = −1.
Fig. 2. The nonlinear tunneling of separating and interacting bright similariton pairs in the dispersion barrier with (a) and (b) h 1 = 2,z 01 = 7, (c) h 1 = 2,z 01 = 3 and (d) dispersion well with h 1 = −0.9, k = 0.6,z 01 = 5.2. The parameters are (a)η 1 = ξ 2 = 1.5, η 2 = ξ 1 = 1.4, θ 10 = −θ 20 = 2,ϕ 10 = ϕ 20 = 1, and (b)–(d)η 1 = η 2 = 1,ξ 1 = −ξ 2 = 1.5, θ 10 = −θ 20 = 5, ϕ 10 = ϕ 20 = 0. Other parameters are the same as that in Fig. 1.

Figure 1 displays the dynamical evolution scenarios I = ∣Ψ12 of the nonlinear tunneling of the single bright similariton with s 0 ≠ 0 or soliton with s 0 = 0 [Eq. (11)] through both the DB or DW [Eq. (15)] and the NB or NW [Eq. (16)]. As β(z) > 0, we assume h 1 > −1, where h 1 > 0 indicates the DB, and −1 < h 1 < 0 represents the DW. Similarly, h 2 > 0 and −1 < h 2 < 0 denote the NB and NW, respectively. As shown in Figs. 1(a) and 1(b), it can be seen that when the similariton and soliton pass through the DB, the pulses are amplified and form the peaks, then attenuates and recovers original shape, respectively. When the similaritons pass through the DW and the NB, the pulses’ amplitudes diminish. The pulses form the dips, then attenuates and increases their amplitudes, as shown in Figs. 1(c) and 1(d), respectively. When the similariton goes through the NW, the dynamical behavior is similar to the case in Fig. 1(a) except increasing its amplitudes after passing across the well, and we omit it. From Figs. 1(a) and 1(d), the DB and NB have different effects on the propagation of the similaritons, i.e., one forms a peak, the other a dip.

In the high bit rate and long-distance optical communication systems, there are always multiple pulses [3

3. G. P. Agrawal, Nonlinear Fiber Optics, (New York: Academic Press, 1993).

]. Therefore, it is necessary to study the multi-similaritons transmission. From Fig. 2(a), similariton pairs pass through the DB and do not interact with each other along the optical fibers. That is what we want in the ultralarge capacity transmission systems and can enhance the capacity of the systems. Although not strictly valid, a two-similariton solution may be interpreted as the nonlinear superposition of two interacting one-similariton solutions approaching each other from infinity. Two interacting similaritons are exhibited in Fig. 2(b). From Figs. 2(c) and 2(d), one can make use of the barrier and well to regulate the influence of interaction by controlling their heights, widths and locations. If the locations of the barrier or well coincide with that of interaction, the barrier in Fig. 2(c) increases the level of interaction, while the well in Fig. 2(d) lessens the degree of interaction.

Fig. 3. Evolutions of nonlinear tunneling of dark similaritons in the (a) and (d) nonlinear barrier, and (b) and (c) dispersion barrier. The parameters are σ = −ε = γ 0 = β 0 = 1 with (a)η 1 = ξ 1 = 0.5,Ω = 1, (b)η 1 = ξ 2 = 1.5, η 2 = ξ 1 = 1.4,T 10 = −T 20 = 1, and (c) and (d) η 1 = −ξ 2 = 1.5, η 2 = ξ 1 = 1.4,T 10 = −T 20 = 10,z 02 = 8. Other parameters are the same as that in Fig. 1.

From Fig. 3(a) and 3(b), when the optical wave passes through the NB and DB, the dark similaritons forms a channel in the background near z = z 02 and a peak near z = z 01. The peak and channel originating from the NB and DB also influence the level of interaction, as shown in Figs. 3(c) and 3(d). If the locations of the barriers coincides with that of interaction, the DB in Fig. 3(c) can partially counteract the influence of interaction, and the shape in the location of interaction becomes from concave to convex. While the NB in Fig. 3(d) increases the degree of interaction and form a deeper dip in the location of interaction.

β(z)=reg0z+h1sech2[k(zz01)],γ(z)=γ0.
(17)

Figure 4 illustrates the evolution of the bright similariton through the dispersion barrier for the decaying or increasing parameter g 0. When g 0 = 0.055, the amplitude of the wave is almost unchanged. When g 0 > 0.055 and g 0 < 0.055, the amplitudes gradually increases and decreases along z, respectively. This implies that the parameter g 0 can control the amplitude of similariton after passing through the barrier.

The dynamic behaviors of component Ψ2 are similar to that of Ψ1, and here we omit it.

5. Conclusions

In summary, we have analytically obtained bright and dark similaritons for the generalized CNLSE with distributed coefficients in the birefringent fiber. Under the parameter condition, we discuss the nonlinear similariton tunneling effect. The results show that the effect of dispersion barrier and well on the similaritons equals that of nonlinear well and barrier, respectively. One can make use of the barrier and well to regulate the influence of interaction between similaritons. The decaying or increasing parameter g 0 can control the amplitude of similariton after passing through the barrier. We expect that these results for similariton tunneling would inaugurate a new and exciting area in the application of optical similaritons.

Fig. 4. Bright similariton through the dispersion barrier [Eq. (17)] for the different system parameter g 0: (a) g 0 = 0.1, (b) g 0 = 0.055 and (c) g 0 = −0.1 with r = 1. Other parameters are the same as that in Fig. 1.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 10672147), and the Program for Innovative Research Team of Young Teachers in Zhejiang A&F University (Grant No. 2009RC01).

References and links

1.

A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. 23, 142–170 (1973). [CrossRef]

2.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]

3.

G. P. Agrawal, Nonlinear Fiber Optics, (New York: Academic Press, 1993).

4.

C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010). [CrossRef]

5.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005). [CrossRef]

6.

V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters,” J. Opt. Soc. Am. B 23, 2541–2550 (2006). [CrossRef]

7.

C. Q. Dai, Y. Y. Wang, and J. F. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. 35, 1437–1439 (2010). [CrossRef] [PubMed]

8.

A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978). [CrossRef]

9.

V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001). [CrossRef]

10.

J. F. Wang, L. Li, and S. T. Jia, “Nonlinear tunneling of optical similaritons in nonlinear waveguides,” J. Opt. Soc. Am. B 25, 1254–1260 (2008). [CrossRef]

11.

R. C. Yang and X. L. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759–17767 (2008). [CrossRef] [PubMed]

12.

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004). [CrossRef]

OCIS Codes
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(190.4370) Nonlinear optics : Nonlinear optics, fibers

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 17, 2010
Revised Manuscript: July 22, 2010
Manuscript Accepted: July 24, 2010
Published: July 30, 2010

Citation
Chaoqing Dai, Yueyue Wang, and Jiefang Zhang, "Nonlinear similariton tunneling effect in the birefringent fiber," Opt. Express 18, 17548-17554 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17548


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References

  1. A. Hasegawa, and F. Tappet, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion,” Appl. Phys. Lett. 23, 142–170 (1973). [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics, (New York: Academic Press, 1993).
  4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010). [CrossRef]
  5. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71, 056619 (2005). [CrossRef]
  6. V. I. Kruglov, and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters,” J. Opt. Soc. Am. B 23, 2541–2550 (2006). [CrossRef]
  7. C. Q. Dai, Y. Y. Wang, and J. F. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. 35, 1437–1439 (2010). [CrossRef] [PubMed]
  8. A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978). [CrossRef]
  9. V. N. Serkin, and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001). [CrossRef]
  10. J. F. Wang, L. Li, and S. T. Jia, “Nonlinear tunneling of optical similaritons in nonlinear waveguides,” J. Opt. Soc. Am. B 25, 1254–1260 (2008). [CrossRef]
  11. R. C. Yang, and X. L. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759–17767 (2008). [CrossRef] [PubMed]
  12. L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004). [CrossRef]

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