Optics Communications 236 (2004) 7–20 www.elsevier.com/locate/optcom Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity S. Konar *, Soumendu Jana Department of Applied Physics, Birla Institute of Technology, Mesra 835215, Ranchi, India Received 24 September 2003; received in revised form 16 February 2004; accepted 5 March 2004 Abstract This paper presents an investigation on linear and nonlinear propagation of sinh-Gaussian pulses in a dispersive medium possessing Kerr nonlinearity. First, the effects of group velocity dispersion and nonlinearity have been treated separately, and then, the dynamic interplay between group velocity dispersion and nonlinearity induced self phase modulation have been discussed. In both normal and anomalous dispersive media, these pulses broaden due to GVD at a much slower rate in comparison to Gaussian pulses. With the increase in the value of sinh factor X0 , the broadening decreases for both chirped and unchirped pulses. It has been found that the self phase modulated spectra are associated with considerable internal structure. For small value of X0 , the number of peaks in the spectrum is less, whereas, for large value of X0 , number of internal peaks is more in comparison to Gaussian pulses. Moreover, number of internal peaks increases with the increase in the value of X0 . When the pulse power is appropriate, they can propagate as antisymmetric solitons in anomalous dispersive media. Linear stability analysis shows that such solitons are stable. The dynamic behavior of these pulses, when magnitude of nonlinearity and dispersion are not same, has been also discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 42.65; 42.65.W; 42.65.T; 42.81.D Keywords: Group velocity dispersion; Self phase modulation; Sinh-Gaussian pulse; Pulse broadening; Chirped pulse; Antisymmetric soliton 1. Introduction Transmission of optical pulses in nonlinear dispersive media is a fascinating area of research * Corresponding author. Tel.: +91-65-1227-6274; fax: +91-651227-5401. E-mail address: skonar@bitmesra.ac.in (S. Konar). due to its technological relevance in pulse compression technique [1,2], optical communication [3–8] and signal processing. It is well known that dispersion alone leads to the pulse broadening, which in a conventional optical communication system would limit transmission capacity [9]. While an unchirped pulse broadens in both normal and anomalous dispersive media, appropriate initial frequency chirp may lead to pulse compression 0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.03.012 8 S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 [8]. Since it limits transmission capacity, the group velocity dispersion (GVD) induced broadening is detrimental in conventional non soliton based optical communication systems. However, GVD induced pulse broadening is useful in soliton based optical communication systems [8] and compressing optical pulses [1,2,10] that lead to generation of pico and femto second pulse. Optical sources emit light pulse whose temporal profile is very close to Gaussian. Naturally, propagation characteristics of such pulses in normal and anomalous dispersive media have been considered both theoretically and experimentally by many workers [4,5,7,11]. Compression of chirped Gaussian pulses also has been investigated [7]. Other pulse shapes, such as hyperbolic secant, super Gaussian have been also considered [8]. While qualitative features of dispersive broadening of Gaussian and ÔsechÕ pulses are identical, due to very sharp tail super Gaussian pulses not only broadens at a faster rate but also distorts in shape. Recently, propagation characteristics of sinhGaussian, cosh-Gaussian (ChG) and Hermite– Gaussian beams in linear media have drawn attention of several workers [12–16]. Analytical reports have focused on the above-mentioned spatial field and propagation characteristics, highlighting potential applications such as improved pump lasers with flat top beam shape for more efficient optical pumping [14–16]. However, propagation of sinh-Gaussian pulses in nonlinear dispersive medium have not received due attention. If such pulses suffer less dispersion induced broadening, then they may be useful in optical communication systems. They may be also useful for the generation of pico and femto second pulses using pulse compression technique that employ GVD induced pulse broadening. Therefore, it is usual that one should be curious to know the dispersive broadening characteristics of sinhGaussian pulses in linear dispersive media. Recently, there has been growing interest in the nonlinear effects of pulse propagation. One important issue which has drawn enormous attention is the interplay between group velocity induced pulse broadening and nonlinearity induced self phase modulation (SPM). Nonlinearity induced SPM can arrest the GVD induced pulse broaden- ing. As a consequence of balance between dispersion and nonlinearity, the pulse can propagate as a soliton [17–19]. They have been shown to form in different nonlinear media such as Kerr [20], cubic quintic [21], power law [22,23], saturating [24], quadrically nonlinear [25], and in photorefractive nonlinear crystals [26]. A wide variety of optical solitons have been theoretically identified and experimentally verified, and their individual characteristics are now well documented [27]. Different types of bright, dark and grey soliton shapes such as sech, Gaussian, super Gaussian, tanh, etc. have been investigated. However, to the best of our knowledge the issue of the possibility of sinhGaussian pulse as antisymmetric optical solitons is not explored yet. Therefore, in this paper we have investigated the linear and nonlinear propagation characteristics of a sinh-Gaussian pulse in dispersive medium possessing Kerr nonlinearity. Both anomalous and normal dispersion have been considered. First, the effects of group velocity dispersion and nonlinearity have been treated separately, and then, the dynamic interplay between group velocity dispersion and nonlinearity induced SPM have been discussed. In addition, it has been shown that the sinh-Gaussian pulse can propagate as an antisymmetric soliton. The organization of the paper is as follows. The mathematical model which describes the propagation of an optical pulse in a nonlinear dispersive medium has been developed in Section 2. The linear broadening of the pulse in the absence of nonlinearity has been discussed in Section 2.1 for both anomalous and normal dispersive media. In Section 2.2, we have discussed nonlinearity induced SPM. The description of the pulse propagation taking into account of the interplay between the nonlinearity induced SPM and GVD has been presented in Section 3. The well known variational formalism has been used in this section and an appropriate Langrangian has been established. Using this Lagrangian and with the help of Legendre transformation, a suitable Hamiltonian has been constructed. The Hamiltonian is then employed to derive a set of ordinary differential equations (ODE), which has been used subsequently to study antisymmetric solitons. Linear stability analysis has been employed to investigate S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 robustness of these solitons. In Section 4, we have presented a brief conclusion. 9 frame of reference which is moving with the group velocity of the pulse the above equation reduces to oq s o2 q 2
jkxx j 2 þ cjqj q ¼ 0; ð5Þ oz 2 oT where T ¼ t
zkx , c ¼ n2 k=n0 , s ¼ 1 depending on the sign of the GVD parameter kxx ; +sign corresponds to normal (kxx  0) and )sign corresponds to anomalous dispersion (kxx  0), respectively. To normalize Eq. (5) we introduce pffiffiffiffithe ffi following transformation s ¼ T =T0 , q ¼ P0 Q. Therefore, i 2. General formalism Consider the propagation of an optical pulse in a dispersive medium possessing Kerr nonlinearity. The electric field ~ Eð~ r; tÞ of the pulse may be assumed to oscillate at the frequency x as 1_ ~ Eðr; tÞ ¼ e qðz; tÞ exp½iðkz
xtÞ þ c  c; 2 ð1Þ where k ¼ ðx=cÞn0 , n0 is the linear refractive index. We consider the case of a sinh-Gaussian pulse for which the incident field at z ¼ 0 may be assumed to be of the form
 ð1 þ iC0 Þt2 qð0; tÞ ¼ A0 exp
sinhðXtÞ; ð2Þ 2T02 where the parameter X is associated with sinh part, T0 may be identified with the half width at 1=e intensity point for a Gaussian pulse. A0 is a constant, the parameter C0 signifies initial frequency chirp. C0  0 is up chirp, whereas C0  0 is down chirp. For C0 ¼ 0, Eq. (2) can be rewritten as  2 !
2 " A0 X0 1 t exp
X0 qð0; tÞ ¼ exp
2 T0 2 2  2 !# 1 t
exp
; ð3Þ þ X0 2 T0 where X0 ¼ XT0 . Above relationship implies that a sinh-Gaussian pulse can be produced in the laboratory by out of phase superposition of two decentered Gaussian pulses whose centers are, respectively, located at t=T0 ¼ X0 and t=T0 ¼
X0 . Making use of the slowly varying envelope approximation, the wave equation that describes the propagation of slowly varying envelope qðz; tÞ of the pulse along z-direction may be written as [8]     oq oq kxx o2 q n2 k 2 i þ þ kx
jqj q ¼ 0; ð4Þ oz ot n0 2 ot2 where kx ¼ ok=ox is the inverse group velocity, kxx ¼ o2 k=ox2 is GVD at the carrier frequency x of the pulse and n2 is the Kerr coefficient. In a i oQ s o2 Q
2 þ cP0 jQj2 Q ¼ 0; oz 2LD os2 ð6Þ where, LD ¼ T02 =jkxx j is the dispersion length. Above equation can be used to study the combined effects of GVD induced broadening and nonlinearity induced SPM on the pulse propagation. Before proceeding further, a comment on the pulse propagation through an optical fiber is in order. Optical fibers are cylindrical dielectric waveguide that can guide light in a direction parallel to its axis. In a fiber optic based communication system, information is transmitted over a fiber by using a coded sequence of optical pulses whose width is determined by the bit rate B of the system. Present day optical fibers offer very low loss 0.2 dB/km and has played a great role in increasing the bandwidth of the telecommunication network to several times. Although the conventional fiber optic based system allow high transmission capacity, the GVD induced pulse broadening in fiber causes the optical pulses in the adjacent bit period to overlap, and thus, GVD limits the transmission capacity for a fixed transmission distance in conventional optical communication systems which rely on low power non-return-to-zero (NRZ) pulse format. This situation can be dramatically improved in a soliton based optical communication system in which low power conventional linear pulses are replaced by high power optical soliton pulses which represent a stable balance between the fiber GVD and its nonlinear intensity dependent refractive index. The nonlinearity in optical fiber is cubic and originates from the Kerr effect. The Kerr coefficient n2 in fiber is extremely small and S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 has a value on the order of 10
22 m2 V
2 . In optical fibers the GVD can be also very small since it can be controlled by proper design of the fiber. This allows the formation of solitons in a single mode fiber (SMF) or in dispersion shifted fiber (DSF) for a reasonable choice of the electric field power (for example a milliwatt of power) in infrared wavelength 1.3–1.5 lm. In standard silica fiber, the zero dispersion wavelength kD is in the vicinity of 1.3 lm. When the wavelength of light is in the region of anomalous dispersion (k  kD ) the fiber supports bright solitons, when it is in the normal dispersion region (k  kD ) the fiber supports dark solitons. The interesting point to note is that, the equation which governs the rules of the propagation of optical pulses in loss less SMF and/or DSF fibers is identical to the NLS Eq. (6). However, due to the guiding property of the fiber, the GVD parameter kxx is modified by the wave guide dispersion that depends on the mode structure in the fiber [19]. In addition, due to finite transverse dimension of the waveguide the Kerr coefficient is also reduced by the waveguide effect by a factor g, where g is the reduction factor of the effective intensity of the peak electric field due to its variation in the fiber crosssection. Numerical value of g  1=2 [19]. Besides optical fibers, there are other guiding structures such as the thin film planar waveguide in which the refractive index depends not only on frequency and intensity of light but on transverse dimension of the waveguide as well. In such devices, diffraction is balanced by the combination of the linear and nonlinear refractive index profiles and the propagation characteristics is again governed by an identical NLS equation with s ¼
1, however, the notable difference is that the time variable in Eq. (6) is replaced by space variable. Solutions of such NLS equations are known as spatial solitons as oppose to the temporal solitons in optical fibers. This is an important space–time analogy between temporal solitons in optical fibers and spatial solitons in planar nonlinear waveguide. Different configurations and novel devices such as optical switches, coupler, and scanners are based on the dynamics existing in the NLS equation. 2.1. GVD induced broadening Before we investigate the combined effects of GVD and SPM on the pulse propagation, it would be worth of investigating the dispersive broadening of the pulse in a linear dispersive medium. Infact, when the pulse power is very low or the dispersion length is very small i.e., cP0 L2D  1, the pulse propagation in nonlinear medium is also dominated by dispersion. Hence, governing equation for a low power pulse and pulses in linear dispersive media are identical. Therefore, appropriate governing equation which can describe pulse propagation in a linear dispersive medium or a very low power pulse propagating in a nonlinear medium can be obtained by dropping the last term in the l.h.s. of Eq. (5), thus, oq s o2 q ð7Þ i
jkxx j 2 ¼ 0: oz 2 oT Before proceeding further it would be worth of defining an arbitrary normalized intensity distribution IN of a sinh-Gaussian pulse at the entry (i.e., z ¼ 0) of the dispersive media as 2 IN ¼ ¼ 1 T0 jqð0; T Þj R1 2 jqð0; T Þj dT
1 2 exp½
s2  sinh2 ðX0 sÞ : pffiffiffi p½expðX20 Þ
1 ð8Þ The normalized intensity distribution is shown in Fig. 1. The root mean square temporal pulse width r of an arbitrary pulse is defined as 0.4 Ω0=0.5 0.3 Ω =2 0 Ω =3 0 IN 10 0.2 0.1 0 −6 −4 −2 0 τ 2 4 6 Fig. 1. Distribution of normalized intensity distribution of sinh-Gaussian pulse with s for different X0 at the input z ¼ 0. 11 S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 r¼ h where i1=2  T 2
hT i2 ; R1 p ð9Þ ððDxÞrms ÞshG ððDxÞrms ÞG 2 31=2 2 2 X20 2 2 e 1þC 1þ2X
1þC
2X 0 0 0 0 5 : ¼4 2 eX0
1 ð1þC02 Þ Rx ð0Þ ¼ 2 hT i ¼
1 R1 T p jqðz; T Þj dT
1 jqðz; T Þj2 dT : With the above definition, the r.m.s. width of the input pulse is obtained as " #1=2 T0 ð1 þ 2X20 Þ expðX20 Þ
1 : ð10Þ rð0Þ ¼ pffiffiffi expðX20 Þ
1 2 The r.m.s.pffiffiwidth of a Gaussian pulse is ffi ðrð0ÞÞG ¼ T0 = 2. The ratio RT ð0Þ of the r.m.s temporal widths of an input sinh-Gaussian pulse rð0Þ to that of an input Gaussian pulse (rð0Þ)G can be written as " #12 rð0Þ ð1 þ 2X20 Þ expðX20 Þ
1 : RT ð0Þ ¼ ¼ ðrð0ÞÞG expðX20 Þ
1 ð13Þ The general solution of Eq. (7) is given by Z 1 1 ^qð0; xÞ qðz; T Þ ¼ 2p
1   i sjkxx jx2 z
ixT dx;  exp ð14Þ 2 where ^qð0; xÞ is the Fourier transform of the incident field at z ¼ 0. By performing the integration in Eq. (14) we may obtain the pulse profile of the sinh-Gaussian pulse at any point z inside the medium, which turns out to be qðn; sÞ ¼ ð11Þ The root mean square spectral width ðDxÞrms can be defined as h  i12 ðDxÞrms ¼ x2
hxi2 ; ð12Þ where r ¼ z2c þ n2 rð0Þ 1=2 " and 1
1 2  qð0; T Þ expðixT Þ dT  : The ratio Rx ð0Þ of the spectral widths of a sinhGaussian pulse ððDxÞrms ÞshG to that of a Gaussian pulse ððDxÞrms ÞG at the entry of the dispersive medium can be written as ð15Þ rðzÞ can be where n ¼ LzD . The broadening factor rð0Þ easily evaluated using Eqs. (9) and (15), which turns out to be exp X20 zcd
exp
X20 zld R1 n x SðxÞ dx R1 hx i ¼
1 SðxÞ dx
1 1=2 ð1
isnð1 þ iC0 ÞÞ
 s2 ð1 þ iC0 Þ þ isnX20  exp
2ð1
isnð1 þ iC0 ÞÞ   X0 s  sinh ; 1
isnð1 þ iC0 Þ 1 þ 2X20 zcd expðX20 zcd Þ
1
2X20 zld exp
X20 zld n Z  SðxÞ ¼  1 exp X20
1 1 þ 2X20 exp X20
1 #1=2 ; ð16Þ where zcd ¼ z2c n2 ; zzc ¼ 1 þ sC0 n: ; zld ¼ 2 z2c þ n2 z2c þ n The broadening factor of Gaussian pulses can be easily obtained by using a Gaussian pulse as incident field in Eq. (2), the result is   r 1=2 : ð17Þ ¼ z2c þ n2 rð0Þ G In order to have an idea about the influence of the sinh parameter X0 on the temporal width of the 12 S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 input pulse, the variation of the ratio RT ð0Þ of the r.m.s temporal widths of an input sinh-Gaussian pulse rð0Þ to that of an input Gaussian pulse for same value of T0 has been displayed in Fig. 2. It is evident that for same T0 , sinh-Gaussian pulses possess larger temporal width in comparison to Gaussian pulses, and this ratio increases with the increase in the value of X0 . Using Eq. (13), the ratio Rx ð0Þ of the spectral widths of the input sinhGaussian pulse to that of an input Gaussian pulse can be estimated. The variation of Rx ð0Þ for different initial chirp C0 has been displayed in Fig. 3. 5 ∑ T (0) 4 3 2 1 0 0 1 2 3 Ω0 Fig. 2. Variation of the ratio RT ð0Þ of the r.m.s. temporal widths of an input sinh-Gaussian pulse to that of a Gaussian pulse. For unchirped pulse the ratio decreases with the increase in X0 . For large initial chirp, Rx ð0Þ increases slowly with the increase in the value of X0 . The influence of X0 on the variation of Rx ð0Þ is insignificant for small initial chirp. In order to investigate the influence of X0 on pulse broadening, we have displayed variation of broadening factor as a function of normalized propagation distance n for values of X0 ranging from 0 to 1.25 in Fig. 4. It is evident from figure that the sinhGaussian pulses broaden monotonically irrespective of the value of X0 . However, the broadening factor decreases with the increase in value of X0 . Thus, it is important to take note of the fact that an unchirped sinh-Gaussian pulse broadens at a much slower rate in comparison to Gaussian pulses. It should be pointed out that similar result is obtained for both normal and anomalous dispersive media. For initially chirped pulses, the magnitude of pulse broadening depends on the relative sign of the GVD parameter kxx and chirp parameter C0 . The behavior of pulse broadening with the propagation distance in an anomalous dispersive medium is depicted in Fig. 5 for C0 ¼ 1 (positive chirp). Initially the pulse width decreases with the increase in propagation distance, attains a minimum value, and with further increase in the propagation distance broadening takes place monotonically. The broken line curve in figure 5 3 C0=1.5 2.5 C =1 σ/σ(0) ∑ω (0) 0 2 Ω0=0.25 4 3 Ω0=0.75 C0=0.5 1.5 Ω =1.25 0 2 C =0 0 1 0 0.5 1 Ω0 1.5 2 Fig. 3. Variation of the ratio Rx ð0Þ of r.m.s. spectral widths of an input sinh-Gaussian pulse to that of a Gaussian pulse with X0 for different initial chirp C0 . 1 0 1 2 ξ 3 4 Fig. 4. Pulse broadening factor for unchirped sinh-Gaussian pulse as a function of normalized propagation distance n. Broken line is for a Gaussian pulse. 13 S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 7 5 Ω0=0.25 Ω =0.25 Ω =1.25 0 3 Ω =0.75 0 5 σ/σ(0) σ/σ(0) 0 6 4 Ω =0.75 0 Ω0=1.25 4 2 3 1 0 0 2 1 2 ξ 3 4 Fig. 5. Pulse broadening factor for upchirped (C0 ¼ 1) sinhGaussian pulse as a function of normalised propagation distance n. The broken line is for a Gaussian pulse. The medium is anomalous dispersive (s ¼
1). For a normal dispersive medium (s ¼ 1), identical behavior is obtained for C0 ¼
1. depicts the pulse broadening for Gaussian pulse. Two points worth mentioning are, first, here too the sinh-Gaussian pulse broadens at a slower rate, and second, the distance at which the minimum pulse width occurs increases with the increase in the value of X0 . For a normal dispersive medium (s ¼ 1), we notice identical behavior for C0 ¼
1 (negative chirp). The qualitative behavior for negative initial chirp in anomalous dispersive medium is shown in Fig. 6, where irrespective of the value of X0 pulses broaden monotonically. 2.2. Self phase modulation The SPM and associated spectral broadening can be investigated using Eq. (6). For simplicity, we consider the SPM in absence of GVD. Thus, neglecting second term in Eq. (6), we get oQ þ cP0 jQj2 Q ¼ 0: oz The solution of above equation is ð18Þ Qðz; T Þ ¼ Qð0; T Þ exp½iUðz; T Þ; ð19Þ i 2 where Uðz; T Þ ¼ jQð0; T Þj d and d ¼ cP0 z is the nonlinear phase shift due to intensity dependent change in refractive index. The instantaneous nonlinear frequency shift arising out due to SPM is given by 1 0 1 2 ξ 3 4 Fig. 6. Broadening factor for down chirped C0 ¼
1 sinhGaussian pulse as a function of propagation distance n. The medium is anomalous dispersive (s ¼
1). Identical behavior is obtained for C0 ¼ 1 in normal dispersive medium (s ¼ 1). Broken line is for a Gaussian pulse. dxðT Þ ¼
oU o ¼
d ðjQð0; T Þj2 Þ: oT oT ð20Þ The intensity spectrum of the self phase modulated pulse is given by c 2 ð21Þ Sðz; xÞ ¼ jQðz; xÞj ; 4p where Qðz; xÞ is the Fourier transform of Qðz; T Þ i.e., Z 1 1 Qðz; xÞ ¼ Qðz; T Þ exp½iðx
x0 ÞT  dT : 2p
1 ð22Þ The temporal variation of frequency chirp due to SPM is shown in Fig. 7. The corresponding variation for a Gaussian pulse is also depicted in the same figure for comparison. In a Gaussian pulse, the leading edge undergoes red shift, whereas the trailing edge undergoes blue shift. However, the behavior is qualitatively different for a sinh-Gaussian pulse. Both the leading and trailing edges of the pulse undergo red and blue shift simultaneously. More specifically, part of the leading edge undergoes red shift while remaining portion of it undergoes blue shift. Similarly, part of the trailing edge undergoes blue shift, whereas, remaining portion suffers red shift. The SPM broadened spectra of unchirped pulse is shown in 14 S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 broadened spectra of Gaussian as well as sinhGaussian pulses are associated with considerable internal structure. Most dominant peaks in a Gaussian pulse are located at the spectral boundaries. On the other hand, for a sinh-Gaussian pulse with small X0 , the most dominant peaks are located in the neighborhood of central region of the pulse spectra. With the increase in the value of X0 , most dominant peaks move towards spectral boundaries. However, dominant peaks are still located slightly away from spectral boundaries in a sinh-Gaussian pulse. For small value of X0 , for example, for X0 ¼ 0:5 the number of peaks in the spectrum is less in comparison to Gaussian pulses, whereas, for large value of X0 i.e., fo X0 ¼ 1:0 or 1.5, the number of internal peaks is more in comparison to a Gaussian pulse. Moreover, number of internal peaks increases with the increase in the value of X0 . Numerically we have 3 Ω0=1.5 2 1 δω Ω0=1 0 −1 −2 −3 −4 −2 0 τ 2 4 Fig. 7. Temporal variation of nonlinear frequency shift. Solid line: Gaussian; broken line: sinh-Gaussian. Fig. 8 for d ¼ 20. The result of a Gaussian pulse, though well known, is incorporated for comparison with the result of sinh-Gaussian pulses. The 2 0.6 INTENSITY INTENSITY 1.5 0.4 1 0.2 0 0.5 −5 0 (ω - ω )T /2π 0 8 3 6 2 1 2 4 1 (c) 0 (ω- ω0)T0/2π 4 0 −4 −1 (b) 0 INTENSITY INTENSITY (a) 0 −2 5 2 −2 0 (ω- ω0)T0/2π 2 4 0 −10 (d) −5 0 5 10 (ω- ω )T /2π 0 0 Fig. 8. The SPM broadened spectra for d ¼ 20. (a) Gaussian pulse. (b), (c) and (d) are for sinh-Gaussian pulses. (b) X0 ¼ 0:5, (c) X0 ¼ 1:0, (d) X0 ¼ 1:5. S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 and No. of Peaks 60 Ω0=1.25 40 Ω0=1 20 Ω =0.75 0 0 20 30 δ 40 50 Fig. 9. Variation of number of peaks with d for different X0 . found that the number of internal peaks increases almost linearly with the increase in the value of d, which is depicted in Fig. 9. Another important point is that, while for small X0 broadening is less in comparison to Gaussian pulses, it is more for large X0 . 3. Nonlinear propagation In this section we describe the dynamic interplay between SPM and GVD induced pulse broadening and investigate the criterion required for antisymmetric solitons. We consider only anomalous dispersive media i.e s ¼
1 and begin with Eq. (6) and introduce P0 ¼ jkxx j=cT02 . Therefore Eq. (6) reduces to i oQ 1 o2 Q þ þ jQj2 Q ¼ 0: on 2 os2 ð23Þ The above NLS equation possess infinitely large number of conserved quantities or integral of motion. Thus, above equation has an infinite number of symmetries, corresponding to the conserved quantities. Two most important of them are energy P also known as the wave power and linear momentum M. These quantities are, respectively, given by Z 1 2 P¼ jQj ds ð24Þ
1 15  Z  i 1 oQ oQ Q ds: ð25Þ
Q M¼ 2
1 os os It is well known that the NLS Eq. (23) is integrable. Zakharov and Shabat [20] have solved the NLS equation using inverse scattering transform. It should be pointed out that even though the NLS equation in the above-mentioned form is integrable, the explicit information that can be obtained from the solution is often rather limited. This situation has prompted an effort to complement the exact analytical solution methods by approximate methods, which sacrifice exactness in order to obtain explicit results and a clear picture of the properties of the solution. One such method is the direct variational method based on trial function [28,29]. The variational formalism has been used successfully and extensively by several authors [28–31] to address different nonlinear optical problems involving nonlinear Schr€ odinger equation and its modified form. This formalism rely on the construction of a field Lagrangian for the pulse with a number of slowly varying free parameters which may describe the pulse amplitude, duration and chirp, and one can increase the number of free parameters for more accurate description of the physical phenomenon. With the help of the field Lagrangian and the prescribed pulse profile, one may obtain a set of ODE for slowly varying free parameters. The system of coupled first order ODE is in general convenient to solve analytically or otherwise numerically. The main advantage of the variational method is its simplicity and capacity to provide clear qualitative picture and good quantitative result. This has motivated the use of this method in the present investigation. The field Lagrangian density L for the propagating pulse, which can reproduce Eq. (23) and its complex conjugate, may be written as      oQ 2 oQ oQ 
jQj4 : L¼i Q ð26Þ þ 
Q on os  on Eq. (26) can be solved using the vanishing of the variation, i.e., Z n d hLidn ¼ 0; ð27Þ 0 R1 where hLi ¼
1 L ds. 16 S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 As outlined earlier, the NLS Eq. (23) is integrable resulting in one soliton, two soliton and N-soliton solutions. The one soliton solution is a pulse which is hyperbolic secant shaped and from this one soliton solution we can get an idea about the initial pulse shape, amplitude and width of the pulse which is propagating through the nonlinear media. However, since Eq. (23) is nonlinear, it may also allow propagating optical pulses which are not hyperbolic secant shaped and that are not obtainable by direct integration. Technically speaking, if a pulse which is not an ideal soliton is injected into the medium then what happens when it propagates through the medium. To address this issue we look for a soliton pulse of the form pffiffiffiffiffiffiffiffiffi Qðn; sÞ ¼ A pðnÞ sinhðX0 ðnÞsÞ 
p2 ðnÞs2 CðnÞs2 þi þ i/ðnÞ ;  exp
2 2 " 2 # 2 1 Xp20 X20 Xp20 e þ2 2 e
1 ; F ðX0 ; pÞ ¼ 2 p and " 2 # 2X2 0 1 2Xp20 2 1=4 p e
4ðe Þ þ 3 : GðX0 ; pÞ ¼ 8 The Lagrangian in (31) does not contain n explicitly, and hence, the Hamiltonian of this system is a constant of motion. We may apply Legendre transformation [32] to the Lagrangian in (31) to generate the Hamiltonian. In particular, for the present case one may define canonical coordinates as C and / with their respective conjugate momenta as ohLi bC ¼ o ð28Þ where A, pðnÞ, CðnÞ and /ðnÞ are amplitude, inverse of the pulse width, chirp and longitudinal phase, respectively, and X0 6¼ 0. For such a form of the soliton as given by Eq. (28), two integral of motions from Eqs. (24) and (25) reduce to ! pffiffiffi 2 p 2 Xp20 P¼ A e
1 ð29Þ 2 and M ¼ 0: ð30Þ The trial function in (28) can be used to evaluate hLi which turns out to be, " ! X2 0 pffiffiffi 2 F ðX0 ; pÞ dC d/ þ e p2
1 hLi ¼ pA 2p2 dn dn !   2 1 2 C2 X20 Xp20 p þ 2 F ðX0 ; pÞ
þ e
1 2 2 p # A2 p
pffiffiffi GðX0 ; pÞ ; ð31Þ 2 where b/ ¼ oC on ohLi o o/ on pffiffiffi A2 F ðX0 ; pÞ ; p 2p2 ð32Þ ! 2 pffiffiffi 2 X20 ¼ pA e p
1 : ð33Þ ¼ With above definitions the Lagrangian can be recasted as hLi ¼ p A4 F 2 X2 þ C 2 bC
0 b/ 4 bC 2 pffiffiffiffi 3=4 5 ðp=2Þ A F G dC d/ bC þ b :
þ 1 pffiffiffiffiffiffi dn dn / 24 b C ð34Þ The required Hamiltonian is defined through Legendre transformation as H ðC; bC ; /; b/ ; nÞ ¼ b/ d/ dC þ bC
hLi dn dn p A4 F 2 X2
C 2 bC þ 0 b/ 4 bC 2 pffiffiffiffi 3=4 5 ðp=2Þ A F G : ð35Þ þ 1 pffiffiffiffiffiffi 24 bC ¼
The above Hamiltonian immediately yields one conservation law and three ODE, they are 17 S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 b/ ¼ Constant of motion; ð36aÞ dX0 ¼
X0 C; dn ð36bÞ dp ¼
Cp; dn ð36cÞ dC A2 Gðp; X0 Þp3 ¼ ðp4
C 2 Þ
pffiffiffi : dn 2F ðX0 ; pÞ ð36dÞ sinh(X0 s) in the pulse profile, the pulse appears as an antisymmetric soliton. Typical permissible antisymmetric soliton profiles are shown in Fig. 11 for different set of ps , X0s and amplitude A. Eqs. (36b)–(36d) have been solved numerically, and a typical behavior of p and X0 with distance of propagation is shown in Fig. 12 when initial values are chosen as stationary value. As expected p and X0 remain constant with the distance of propagation. Robustness of these solitons mentioned earlier is an important issue. In order to examine We now proceed to investigate the existence of antisymmetric solitons. Eqs. (36b)–(36d) have one set of nontrivial stationary point ðps ; X0s ; Cs Þ represented by A2 p ¼ pffiffiffi 4 2 4e 2X2 0s p2 " e
4 e X2 0s p2 þ 2X2 0s p2 2X20s p2 e !1=4 X2 0s p2 3 þ 35
1 # ¼ ps : ð37Þ 0 A=1 p=1.0087 Ω =2 −2 0 −4 The last relationship can be solved to obtain stationary values of p and X0 . Such values of pð¼ ps Þ and X0 ð¼ X0s Þ for which the pulse remains stationary is displayed in Fig. 10 treating A as a parameter. A set of ps and X0s for Cs ¼ 0 admits a possible stable pulse state. Due to the factor A=0.01 p=0.54087 Ω0=2 2 Re(Q) C ¼ 0 ¼ Cs and 2 4 −10 −5 0 5 10 15 τ Fig. 11. Typical amplitude profile Re(Q) of antisymmetric solitons. Values of A, X0s and ps are chosen from curve of Fig. 10. 25 5 20 p, Ω0 A=1.0 4 2 A=0.01 ps 3 A=0.1 15 10 5 1 0.2 0.4 0.6 0.8 1 ξ 0 0 2 4 6 8 10 Ω 0s Fig. 10. Behavior of stationary points (ps , X0s ) with A as a parameter. Fig. 12. Behavior of X0 and p with distance of propagation. Initial values of X0 and p are their stationary value; X0 ¼ X0s ¼ 20:0, p ¼ ps ¼ 7:88 and C ¼ Cs ¼ 0:0. The parameter A ¼ 1:0. Throughout propagation distance C ¼ 0. Dashed line X0 , solid line p. 18 S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 0_1 0 0 d BXC @ 0 ¼ @pA dn ^ ^ X C 0 0 Y 10 _ 1
X0s BXC
ps A@ ^ p A; ^ 0 C 
A2 X0s ps3 x1 k2 k3 X ¼ pffiffiffi
k1 ; 2D 2D 
A2 p 2 3 X2 Y ¼ 4ps3 þ pffiffiffi s X20s k1
ps2 k2
0s x1 k2 k3 4 2D 2D x1 ¼ e ; x2 ¼ e 2X2 0s ps2 ; x3 ¼ e 3X2 0s ps2 p,Ω0 0 0 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 p,Ω 0 (b) 5 0 0 (c) 5 0 0 ð38Þ where, X2 0s ps2 (a) 5 p,Ω0 whether predicted soliton state is stable, we should undertake linear stability analysis around the equilibrium point (ps ; X0s ; Cs ). In order to do that we consider small perturbation from equilibrium ^ and point, and write p ¼ ps þ ^ p, X0 ¼ X0s þ X ^ C ¼ Cs þ C, where it has been assumed that quantities with hat are very small in magnitude in comparison to their respective stationary value. After linearizing Eqs. (36b)–(36d) around stationary point we easily get a set of three equations which can be put in a matrix form as ; ξ Fig. 13. Variation of X0 and p with distance of propagation when only one initial value i.e., value of p is different from stationary value ps i.e., p ¼ ps þ Dp. and X0 ¼ X0s ¼ 5:0. (a) ps ¼ 2:25 and A ¼ 1:0 (b) ps ¼ 1:59 and A ¼ 0:1 (c) ps ¼ 1:30 and A ¼ 0:01. For all cases Cs ¼ 0:0 and Dp ¼ 0:01. Dashed line X0 , solid line p. different from their stationary values X0s and ps is shown in Fig. 14. Both in Figs. 13 and 14 the nature of oscillation appears to be simple harmonic. Fig. 15 shows a phase portrait of p and C. For D ¼ ps2 x1 þ 2X20s x1
ps2 1=4 p,Ω 0 20 0 0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 (b) 20 0 The three eigenvalues (aj ; j ¼ 1; 2; 3) of the 3  3 stability pffiffiffimatrix are given by a1 ¼ 0 and ffi a2;3 ¼  D, where D ¼
ðYps þ X X0s Þ. We have confirmed that the value of D is indeed negative for all the three branches shown in the Fig. 10. Therefore, a2 and a3 are purely imaginary. The fixed point is thus neutraly stable as the real part of a2 and a3 is zero. Thus, in the neighborhood of the fixed point, soliton parameters will oscillate around the steady state value. Variation of p and X0 with distance of propagation when only initial value of p is different from its stationary value ps is shown in Fig. 13. Pulse widths p and X0 oscillate with finite amplitude as the pulse propagates. Variation of p and X0 with the distance of propagation when both initial values X0 and p are (a) p,Ω þ 2X20s : 0 (c) 0 20 0 k3 ¼ 3ps2 p,Ω 1=4 k1 ¼ x2
x2 ; k2 ¼ x2
4x2 þ 3 and 0 0 ξ Fig. 14. Variation of X0 and p with distance of propagation when both initial conditions are different from stationary values X0s and ps . (a) p ¼ 2:26 and A ¼ 1:0 (b) p ¼ 1:60 and A ¼ 0:1 (c) p ¼ 1:31 and A ¼ 0:01. For all cases initial value of C ¼ 0:0 and X0 ¼ 5:2. Dashed line X0 , solid line p. S. Konar, S. Jana / Optics Communications 236 (2004) 7–20 20 C 10 0 −10 −20 7 8 9 10 11 12 p Fig. 15. Typical phase plane dynamics showing variation of p and C. small perturbation around the fixed point trajectories are closed. It is evident from Figs. 13–15 that the soliton state is stable against perturbation. We believe it would be appropriate to point out that a symmetric ChG pulse would possess qualitatively similar behavior [33] necessitating further investigations on these pulses. 4. Conclusion In conclusion, in this paper we have presented an investigation on linear and nonlinear propagation of sinh-Gaussian pulses in a dispersive medium possessing Kerr nonlinearity. First the effects of group velocity dispersion and nonlinearity have been treated separately, and then, the dynamic interplay between group velocity dispersion and nonlinearity induced SPM have been discussed. We have shown that in both normal and anomalous dispersive media these pulses broaden due to GVD at a much slower rate in comparison to Gaussian pulses. With the increase in the value of sinh factor, broadening decreases for both chirped and unchirped pulses. When the influence of nonlinearity is considered alone, we found that SPM broadened spectra are associated with considerable internal structure. For a sinh-Gaussian pulse with small value of X0 , the most dominant peaks are located in the neighborhood of central region of the spectra. With the increase in the va- 19 lue of X0 , most dominant peaks move towards spectral boundaries. However, most dominant peaks are still located slightly away from boundaries. Number of peaks in the spectra increases with the increase in either of the value of X0 or d. For small X0 , SPM induced broadening is less in comparison to Gaussian pulses, whereas for large X0 it is more. We have shown that when the pulse power is appropriate in anomalous dispersive media, these pulses can propagate as antisymmetric solitons. We have undertaken linear stability analysis and found the robust behaviour of these solitons. The dynamic behaviour of these pulses, when magnitude of nonlinearity and dispersion are not same has been also discussed. It has been found that the pulse width oscillates around equilibrium value and the nature of oscillation appears to be simple harmonic. Acknowledgements We thank gratefully two anonymous referees for insightful comments and valuable suggestions. 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