Pulsed beams in a nonlinear nematic fiber

Physica D 141 (2000) 333–343

Pulsed beams in a nonlinear nematic fiber J.A. Reyes1 , R.F. Rodr´ıguez∗,2 Instituto de F´ısica, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 Mexico, D.F., Mexico Received 14 July 1999; received in revised form 2 December 1999; accepted 20 December 1999 Communicated by E. Ott

Abstract A model for a cylindrical fiber whose cladding is an initially homogeneous nematic liquid crystal is presented. We construct a narrow wavepacket of transverse magnetic modes propagating through the fiber taking into account the dispersion, self-focusing and diffraction in the nematic. Then a nonlinear Schrödinger equation for the time evolution of its envelope is derived. The speed, time- and length-scales and nonlinear index of refraction of the optical solitons are estimated by using typical experimental values for some of the parameters. © 2000 Elsevier Science B.V. All rights reserved. PACS: 77.84.N; 61.30.G; 42.65 Keywords: Nonlinear Schrödinger equation; Pulsed beam; Nematic fibers; Optical solitons

1. Introduction During the last decade, the interest in the nonlinear optics of liquid crystals has increased enormously due to the possibility of producing gigantic optical nonlinearities and strong nonlinear effects. The latter are 6–10 orders of magnitude larger than the corresponding ones for doped glasses, and can be achieved by using lasers with moderate intensity (kW/cm2 ) [1,2,24–26]. Recent experiments using continuous beams have shown the presence of steady spatial patterns for cylindrical [3] and planar [4] geometries. The basic mechanism which governs these time independent patterns is the balance between the nonlinear refraction (self-focusing) and the spatial diffraction of the nematic. A study of these experiments using separation of scales [5,6] shows that the field amplitude at the center of a gaussian beam (inner solution), follows a nonlocal, nonlinear Schrödinger (NLS) equation which is able to describe the undulation and filamentation observed in the experiments. ∗

Corresponding author. E-mail address: [email protected] (R.F. Rodr´ıguez) 1 Fellow of SNI, Mexico. 2 Fellow of SNI, Mexico. 0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 0 ) 0 0 0 4 4 - 0

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A different situation is obtained when the propagation of wavepackets, instead of continuous beams is considered. In this case there exists the possibility of propagation of stable solitary waves (optical solitons) through the liquid crystal, when the equilibrium between dispersion and self-focusing is reached. This possibility for a planar waveguide in a specific distorted configuration has been previously considered [7] for a planar geometry. However, the sluggish nonlinear optical response of the liquid crystal, as well as the lossy and nonlocal aspects of the director dynamics were entirely neglected. To some extent these last features were considered recently in a more general formulation which, however, does not take into account neither the confinement of the liquid crystal in a specific geometry [8] or the absorption effects. The main objective of the present article is to generalize our previous work in [7], for a more realistic cylindrical geometry and taking into account the nonlocal features of the re-orientation dynamics. The case of a nematic cylindrical waveguide has been described before numerically [11,12] and analytically [13], but without specifying the continuous or pulsed nature of the beam. For this purpose of incorporating this last feature into our model, in Section 2, the coupled time evolution equation for both, the transverse magnetic modes (TM) and for the orientational configuration are derived in an explicit retarded form and in terms of a coupling parameter q. In Section 3, these general equations are solved to linear order in q for the final stationary orientational configuration. These solutions are then used in Section 4 to construct the propagation equation of a wavepacket of TM modes. It is shown that the envelope of the wavepacket obeys an NLS equation which takes into account self-focusing, dispersion and diffraction in the nematic. This equation has soliton-like solutions for which their speed, time-and length-scale and nonlinear index of refraction are estimated using experimental values for some of the parameters [15]. It should be stressed that although in our previous works we have almost entirely neglected all effects associated with absorption, the dissipation due to the re-orientation of the nematic may be considered explicitly [8]. However, in the present model and as a first approximation, again we ignore all effects due to absorption; their full incorporation into our formalism is feasible and is presently under way [9].

2. Coupled dynamics Let us consider a cylindrical waveguide with an isotropic core of radius a, dielectric constant c and a quiescent nematic liquid crystal cladding of radius b satisfying planar axial boundary hard-anchoring conditions n(r ˆ = a, z) = n(r ˆ = b, z) = eˆz ,

(1)

as depicted in Fig. 1. The nematic director is to be written in terms of the angle θ as follows: n(r, ˆ z) = sin θ eˆr + cos θ eˆz ,

(2)

Fig. 1. Schematics of a laser beam propagating through a nematic liquid crystal cylindrical waveguide. The TM modes are shown explicitly.

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where eˆr and eˆz are the unit cylindrical vectors. If the re-orientation process is isothermal, the equilibrium orientational configurations are determined by minimizing the corresponding total Helmholtz free energy functional [10] Z 1 E r , t) · E(E E r , t)] ˆ 2 − D(E dV [K(∇ · n) ˆ 2 + K(∇ × n) F= 2  Z t Z 1 2 2 i E E ∗ (Er , t 0 ) ˆ − E (Er , t) · dt 0 ∇ × H dV K(∇ · n) ˆ + K(∇ × n) = 2  Z t a 0 ∗ 0 E E dt ∇ × H (Er , t ) , (3) +E (Er , t) · where K ≡ K1 = K2 = K3 is the elastic constant in the equal elastic constant approximation and the asterisk E r , ω) =  ↔ (Er , ω) · E(E E r , ω) with denotes complex conjugation. Here, we have used the constitutive relation D(E  ↔ (Er , ω) = 0 [⊥ (ω)I ↔ + a (ω)nˆ n], ˆ where 0 is the permittivity of the vacuum, and which leads to the retarded E given by relation between EE and D Z t ↔−1 E r , t 00 ) = EEi (Er , t) + EEa (Er , t), E r , t) = dt”  (Er , t − t 00 ) · D(E (4) E(E where EEi and EEa are electric fields defined by the following nonlocal and retarded relations: Z Z t E rE0 , t 0 ) 1 ∇ × H( , dt 0 dt 00 EEi (Er , t) = 0 ⊥ (t 00 − t 0 ) Z Z t 1 a 00 E rE0 , t 0 ). dt 00 (t − t 0 )nˆ nˆ · ∇ × H( dt 0 EEa (Er , t) = 0 ⊥ k

(5) (6)

E r , t) in terms of H(E E r , t) by using Ampère–Maxwell’s law without In Eqs. (3), (5) and (6), we have substituted D(E sources. For the specific geometry, Eq. (3) takes the form "     Z ∂θ ∂θ 2 ∂θ 2 sin θ ∂θ − r sin θ + cos θ + sin θ + K cos θ F = r dr K r ∂z ∂r ∂z ∂r V    Z t Z t Z t Z t a∗ 0 ∂Hφ a∗ 0 1 ∂(rHφ ) i∗ 0 ∂Hφ i∗ 1 0 ∂(rHφ ) − Er dt dt dt dt − Ez + Er − Ez . (7) ∂z r ∂r ∂z r ∂r If we now minimize Eq. (7) with respect to θ , we find the following Euler–Lagrange equation:      Z t Z t ∂θ sin θ cos θ 1 ∂ ∂ 2θ 2 cos 2θ a∗ 0 ∂xHφ a∗ 0 ∂Hφ x − Ez + Er + −q dt dt x ∂r ∂x x ∂x ∂ζ ∂ζ 2 x2   Z t Z t ∂H ∂xH sin 2θ φ φ + Eza∗ = 0, (8) dt 0 dt 0 −xEra∗ + 2 ∂ζ ∂x x √ where ζ ≡ z/a, x ≡ r/a, Hφ ≡ Hφ /(c0 E0 ) with c = 1/ µ0 0 , where µ0 is the magnetic permeability of free space. Eia ≡ Eia /E0 , with i = r, z, are dimensionless variables and q 2 ≡ 0 E02 a 2 /K is a dimensionless parameter which is equal to the ratio between the electric field energy density and the elastic energy density of the nematic; that is to say, it is a measure of the coupling between the nematic and the optical field. It is important to emphasize at this point that we have not considered a time dependent equation for θ and we only use the final stationary state defined by (8). This assumption is motivated by the large difference between the time-scales of re-orientation and

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of time variations of the optical field. Since we have also ignored all effects due to absorption, as a first adiabatic approximation we will only consider the stationary final reoriented state for the liquid crystal. Since only the TM components are coupled with the re-orientation [16], we assume that the optical field is a TM whose electric and magnetic components are Er , Ez and Hφ , as shown in Fig. 1. Thus, Hφ is governed in general by the nonlinear, nonlocal and retarded equation obtained by substituting Eqs. (5) and (6) in Faraday’s law, namely, Z 2 2 2 2 0 a 2 ∂ 2 Hφ 0 ((∂ Hφ /∂ζ ) + (∂ Hφ /∂x ))(t − t ) = − dt c2 ∂t 2 ⊥ (rE0 , t 0 )   Z ∂Hφ 1 ∂ a 0 ∂2 dt 0 + sin θ cos θ xHφ (t − t 0 ) (t ) − sin2 θ + ∂t ∂ζ ⊥ k ∂ζ x ∂x   Z 2 ∂Hφ ∂ 0 a 0 2 1 ∂ (t ) − sin θ cos θ (9) dt + cos θ xHφ (t − t 0 ). − ∂t ∂x ⊥ k ∂ζ x ∂x Notice that Eqs. (8) and (9) provide a complete set of general coupled dynamics of the nematic and electromagnetic field for our system. Next we shall solve them iteratively in the weakly nonlinear regime.

3. Linear and weakly nonlinear dynamics The solution of Eq. (8) to zeroth order in q and satisfying the axial boundary conditions defined above, is θ (0) = 0. Substitution of this solution into Eq. (9) and taking a monochromatic beam of frequency ω, we obtain a (0) linear equation for the zeroth order field U ≡ Hφ which is given by     ω 2  ⊥ ∂ ∂U ⊥ ∂ 2U − ⊥ k a x = 0. (10) U − k 2 − c x ∂x ∂x x2 ∂ζ Solving this equation by the method of separation of variables, its propagating solution is given by   !1/2    2 2 2 β a k ω0  a − k U (r, ω0 ) = exp(−iβaζ ) A1 I1 x ⊥ c  !1/2    2 2 2 β a k ω0  , a − k +A2 K1 x ⊥ c

(11)

where A1 and A2 are arbitrary constants to be determined by using the boundary conditions (1). Here I1 (x) and K1 (x) are, respectively, the modified Bessel and Neumann functions of order 1. On the other hand, Hφc (r, z) in the isotropic dielectric core governed by the following equation [17]: Z 2 c ((∂ 2 Hφc /∂z2 ) + (∂ 2 Hφc /∂x 2 ))(t − t 0 )0 1 ∂ Hφ 0 , (12) = − dt c (t 0 ) c2 ∂t 2 and its monochromatic solution, which remains finite at the origin, is !   ω 2 1/2 0 c 2 2 a Hφ = exp(−iβaζ )B1 J1 x β a − c , c

(13)

where J1 (x) is the Bessel function of order 1 and B1 is also an undetermined constant. To find the constants A1 , A2 and B1 , it is necessary to impose the following boundary conditions over Hφ and its derivative at the boundary [18]:

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Hφc |x=1 = U |x=1 , c 1 dU 1 dHφ = . c dx ⊥ dx x=1

337

(14) (15)

x=1

Now, we shall restrict our analysis to describe beams in the optical spectrum (λ = 0.4–0.6 ␮m) and we shall only consider numerical values of some of the fiber parameters which are typical of experimental situations where the problem of selection of modes arises [15]. Then we choose for the core and cladding radii, a ≈ 10 ␮m and b ≈ 100 ␮m, respectively. Since a, b  λ, we are in the geometrical optics limit and we can approximate the Bessel and Neumann functions in Eq. (11) by their corresponding asymptotic expressions. Moreover, since the liquid crystal is located at the cladding which is thicker than the core, we can assume the cladding as an infinite medium. Thus, A1 vanishes and Eq. (11) reduces to r π exp(−iβaζ − γ ax), (16) U (x, ω0 ) = A2 2γ ax where   2   1/2 ω0 2 β − . γ = k ⊥ c Thus, substitution of Eqs. (13) and (16) into Eqs. (14) and (15), leads to the following transcendental equation for the allowed values of β: J10 (a(β 2 − c (ω0 /c)2 )1/2 ) =a J1 (a(β 2 − c (ω0 /c)2 )1/2 )



πk 2⊥



β 2 − ⊥ (ω0 /c)2 β 2 − c (ω/c)2

1/2 ,

(17)

whose solution is shown in Fig. 2 for the same parameter values as in [15], namely, c = 2.1374, ⊥ = 2.1228 and k = 2.2611. To obtain the weakly nonlinear equations for θ and Hφ , we perform another iteration to find their next nonvanishing order corrections in q. For this purpose we first insert Eq. (16) into Eq. (8) to obtain

Fig. 2. Graphic solution of the transcendental equation (17). The right-and left-hand sides are plotted as a function of βa and the intersections yield the allowed values of βa. The right-hand side is the continuous line.

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Fig. 3. Orientational profile giving θ vs. the radial distance r for the allowed values of βa.

0=x

∂ 2θ ∂x 2

+

∂θ ∂ 2θ sin 2θ +x 2 − θx x ∂ζ "

# q 2 a A22 [4x 2 ((βa)2 a − ⊥ k ((ω0 /c)a)2 ) + ⊥ ] exp(−2γ x) 2βa cos 2θ + 2 sin 2θ , − π⊥ k 4x ⊥ [(k /⊥ )((βa)2 − ⊥ ((ω0 /c)a)2 )]1/2

(18)

and look for a solution of the form θ = θ (0) + q 2 |A(ζ, t)U (x, ω)|2 θ (1) (r) + · · · = q 2 |A(ζ, t)U (x, ω)|2 θ (1) (r) + · · · , where A(ζ, t) is a slowly varying function of its arguments. Hence the equation for θ (1) takes the form x

2a βaA22 θ (1) ∂θ (1) ∂ 2 θ (1) − − + exp(−2γ x) = 0, 2 θx x π⊥ k ∂x

(19)

and its solution satisfying the hard-anchoring homotropic boundary conditions (1), θ (x = 1) = θ (x = b/a) = 0 may be written in terms of the exponential integral function; however, the resulting complicated equation can be approximated using the asymptotic expressions of these functions with the following result: θ (1) =

βaa J12 [(c (ω0 a/c)2 − β 2 a 2 )1/2 ] 2 {(a − b2 ) exp(γ a(1 − x)) π⊥ k x(a 2 − b2 ) +(b2 − x 2 a 2 ) + exp(γ (a − b))a 2 (1 − x 2 )}.

(20)

θ (1)

as a function of x is given in Fig. 3. If we now insert this expression into Eq. (9) and expand the The plot of result up to first-order in q, we arrive at an equation of the form ˆ φ ) = 0, ˆ L(β, ω, x)Hφ + q 2 N(H where the linear and nonlinear operators Lˆ and Nˆ are defined, respectively, by      ∂ ∂2 ω0 2 1 2 2 2 ˆ a − (βa) + x⊥ + x ⊥ 2 , −⊥ + x k ⊥ L≡ 2 c ∂x x ⊥ k ∂x ! 2 (1) (x) dθ dU |A(ζ, t)U (x, ω)|  a (1) (1) + Ux A(ζ, t)U (x, ω). iβa U θ (x) + 3xθ (x) Nˆ ≡ x⊥ k dx dx

(21)

(22)

(23)

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4. Wavepacket of TM modes The explicit Fourier representation of a monochromatic field as the one considered in last section, is given by Hφ (x, ζ, ω) = 2πδ(ω − ω0 , ζ ) exp(iβ(ω0 )aζ )Uφ (x, ω0 ) + c.c.,

(24)

where c.c. denotes complex conjugate and δ Dirac’s delta function. Eq. (24) suggests that a narrow wavepacket centered around the frequency ω0 , may be expressed in the form ¯ − ω0 , ζ ) exp(iβ(ω0 )aζ )Uφ (x, ω0 ) + c.c., Hφ (x, ζ, ω) = A(ω

(25)

¯ − ω0 , ζ ) characterizes the distribution of frequencies around ω0 . We assume that this where the function A(ω distribution has a small dispersion q = (ω − ω0 )/ω0 . Thus, if the amplitude Hφ (x, ζ, ω) is expanded in a Taylor series around ω = ω0 and the inverse Fourier transform of Hφ (x, ζ, ω) is taken, we arrive at ∞

Hφ (x, ζ, t) =

1 X 1 dn Uφ (x, ω0 ) 2π n! dωn

Z

¯ − ω0 , ζ ) (ω − ω0 )n A(ω

n=0

×exp(−i(ω − ω0 )t) d(ω − ω0 ) exp(iβ(ω0 )aζ − iω0 t) + c.c. By using the identity [19]  n Z 1 ∂ ¯ − ω0 , ζ ) exp(−i(ω − ω0 )t) d(ω − ω0 ) A(x, t) = (ω − ω0 )n A(ω i ∂t 2π with n = 0, 1, 2, . . . , Eq. (26) can be written in the more compact form   ∂ A(4, T ) + c.c., Hφ (x, ζ, t) = exp(iβ(ω0 )aζ − iω0 t)Uφ x, ω0 + iq ∂T

(26)

(27)

(28)

¯ − ω0 , ζ ) and is a slowly varying function of the variables 4 ≡ qζ where A(4, T ) is the Fourier transform of A(ω and T ≡ qt. Due to the coupling between the re-orientation and the optical field, it is to be expected that when a monochromatic TM mode propagates along the cell, higher harmonics may be generated. Therefore, we assume that the solution of Eq. (21) can be written as the superposition   ∂ A(4, T ) + q 2 U (1) + q 3 U (2) + c.c. + · · · (29) Hφ (x, ζ, t) = qUφ x, ω0 + iq ∂T The superindices identify the first, second, etc. harmonics. Note that the presence of the powers of q implies that the contribution of the higher-order harmonics are smaller than the dominant term which is itself a small amplitude narrow wavepacket. To follow the dynamics of the envelope A(4, T ), we substitute Eq. (29) into Eq. (21) and identify the Fourier variables iβa = iβ0 a + q(∂/∂41 ) + q 2 (∂/∂42 ) and −iω = −iω0 + q(∂/∂T ), in consistency with a narrow wavepacket, and where Z = q41 = q 2 42 are the spatial scales associated with upper harmonics contributions. Expanding the resulting expression in series up to third-order in q, that is to say,   ∂ ∂ 2 ∂ ˆ +q , −iω0 + q (30) Hφ (x, ζ, t) − q 2 Nˆ (Hφ (x, ζ, t)) = 0, L iβ0 a + q ∂41 ∂42 ∂T and grouping contributions of the same order in q, we find the following expressions: ˆ 0 a, −iω0 , x)Uφ (x, ω0 )A = 0, q : L(iβ

(31)

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∂Uφ (x, ω0 ) ∂ ∂ + Uφ (x, ω0 )Lˆ 2 (iβ0 a, −iω0 ) ∂ω ∂T ∂T ∂ ˆ (1) , A = LU +Lˆ 1 (iβ0 a, −iω0 )Uφ (x, ω0 ) ∂41

ˆ 0 a, −iω0 ) q 2 : iL(iβ

 2 2 2  ˆ (2) = − 1 L(iβ ˆ 0 a, −iω0 ) ∂ Uφ (x, ω0 ) ∂ + i ∂Uφ (x, ω0 ) Lˆ 2 (iβ0 a, −iω0 ) ∂ q 3 : LU 2 ∂ω ∂ω2 ∂T 2 ∂T 2  Uφ (x, ω0 ) ˆ ∂2 ∂2 ∂ + L22 (iβ0 a, −iω0 ) 2 + Lˆ 1 (iβ0 a, −iω0 ) +Lˆ 1 (iβ0 a, −iω0 ) ∂41 ∂T 2 ∂42 ∂T !! 2 2 ˆ L11 (iβ0 a, −iω0 ) ∂ ∂ + A − Nˆ (Uφ (x, ω0 )A), +Lˆ 12 (iβ0 a, −iω0 ) ∂41 ∂T 2 ∂421

(32)

(33)

ˆ 0 a, −iω0 ) with respect to its first or second argument. where Lˆ i (iβ0 a, −iω0 ), i = 1, 2, denotes the derivative of L(iβ ˆ 0 a, −iω0 )Uφ (x, ω0 ) = 0. Taking the first and second Note, Eq. (31) reproduces the usual dispersion relation L(iβ derivatives of Eq. (31) with respect to ω, we obtain the expression that will allow us to simplify Eqs. (32) and (33). The first derivative leads to   ˆ 0 a, −iω0 ) ∂Uφ (x, ω0 ) = iUφ (x, ω0 ) Lˆ 2 (iβ0 a, −iω0 ) − a dβ Lˆ 1 (iβ0 a, −iω0 ) , (34) L(iβ ∂ω dω and substitution of this expression into Eq. (32) yields   dβ ∂ ∂ ˆ (1) . ˆ + A = LU L1 (iβ0 a, −iω0 )Uφ (x, ω0 ) a dω ∂T ∂41

(35)

This expression is a linear inhomogeneous equation U1 , whose solution is assured to exist by imposing the so-called ˆ (r, ω0 ) = 0 and U (r, ω0 ) → 0 as r → ∞. In our case alternative Fredholm condition [20], which is fulfilled if LU this condition reads explicitly Z ∞ ˆ (1) dx = 0, ˆ (1) , Uφ i = Uφ LU (36) hLU 0

ˆ 1 , U i 6= 0, implies that ((a dβ/dω)∂/∂T + ∂/∂41 )A = 0, which expresses the fact that up to and since hLU second-order in q, the envelope A travels with the group velocity dβ/dω. The second derivative of Eq. (31) is given by 2 ˆ 0 a, −iω0 ) ∂ Uφ (x, ω0 ) + 2iLˆ 2 (iβ0 a, −iω0 ) ∂Uφ (x, ω0 ) + Uφ (x, ω0 )Lˆ 2,2 (iβ0 a, −iω0 ) 0 = −L(iβ ∂ω ∂ω2 2 dβ ∂Uφ (x, ω0 ) ˆ d β L1 (iβ0 a, −iω0 ) −ia 2 Uφ (x, ω0 )Lˆ 1 (iβ0 a, −iω0 ) − 2ia dω ∂ω dω   dβ 2 dβ Uφ (x, ω0 )Lˆ 1,1 (iβ0 a, −iω0 ). +2a Uφ (x, ω0 )Lˆ 1,2 (iβ0 a, −iω0 ) − a dω dω

Therefore, substitution of ((a dβ/dω)∂/∂T + ∂/∂Z1 )A = 0 and Eq. (37), in Eq. (33) yields ! " dUφ (x, ω0 ) a A|A|2 dθ (1) (x) 3 (1) ˆ + 3xθ (1) (x)Uφ2 (x, ω0 ) Uφ (x, ω0 ) θ (x) + x LU2 = −iβ x⊥ k dx dx !# Uφ (x, ω0 ) ∂A d2 β ∂ 2A 2 . + ia 2 +Lˆ 1 (iβ0 a, −iω0 ) 2 ∂42 dω ∂T 2

(37)

(38)

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ˆ (2) , Uφ i = 0, we find Using again the alternative Fredholm condition hLU in¯ 2 A|A|2 + 2

∂A d2 β ∂ 2 A + ia 2 = 0, ∂42 dω ∂T 2

(39)

where the dimensionless refractive index n¯ 2 ≡ Kn2 /0 a 2 is given by n¯ 2 =

a 1 [hUφ3 (x, ω0 )((θ (1) (x)/x) + (dθ (1) (x)/dx)), Uφ (x, ω0 )i k hUφ (x, ω0 ), Uφ (x, ω0 )i

+3hθ (1) (x)Uφ2 (x, ω0 )(dUφ (x, ω0 )/dx), Uφ (x, ω0 )i] a 4 1 (c ω02 − β 2 c2 )1/2 exp(−γ b + 2γ a) = a2 βa 3 J1 4 c −a e−3γ b + a exp(γ (a − 4b)) + b exp(−γ (4a − b)) − b e−3γ a . × πk2 b(a 2 − b2 )⊥ (−e−2γ b + e−2γ a )

(40)

5. Results Using the above derived expressions, we now estimate the values of some of the properties of the wavepacket, such as the nonlinear contribution n2 to the refractive index, its coefficient d2 β/dω2 , the soliton typical length-and time-scales and its speed. For the same typical values of the dielectric permitivities given in the previous section, from (40) we get the set of values of n2 corresponding to the allowed values of βa given by the solution of (17) shown in Fig. 2, given in Table 1. 2 = Note that the last two values of n2 are several orders of magnitude larger than its value for glass, nSiO 2 2 1.2 × 10−28 (km/V ). This shows the existence of the giant optical nonlinearity expected for a liquid crystal [1,24]. Another physical quantity which can be estimated from the above results is the coefficient (d2 β/dω2 ) of the wavepacket given by the third term of Eq. (39). Using n0 = 1 + n00 +

g01 ω12 − ω2

+

g02 ω22 − ω2

,

(41)

where n00 = 0.4136, ω1 = 8.9 × 1015 rad/s, ω2 = 6.68 × 1015 rad/s, g01 = 4.8 × 1030 (rad/s)2 and g02 = 1.66 × 1030 (rad/s)2 for 5CB from [21], we find that (d2 kn0 /dω2 )5CB ≈ 1.1 × 10−4 ps2 /km. Thus, the width of a picosecond pulse traveling in 5CB in the linear regime is doubled in a distance of 0.1 m; while for glass (SiO2 ), (d2 kn0 /dω2 )SiO2 = 1.8 ps2 /km, it is doubled in a distance of 0.5 km. This is consistent with the fact that liquids are considerably more dispersive than solids. Table 1 βa

n2 (km/V)2

228.993 229.192 229.352 229.471 229.552 229.59

8.222 × 10−29 9.341 × 10−28 5.191 × 10−27 2.835 × 10−26 1.718 × 10−25 2.902 × 10−24

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Note that Eq. (39) can be rewritten as the NLS equation ∂A ∂4

+i

∂ 2A − i|A|2 A = 0 ∂ T¯ 2

(42)

¯ ≡ 42 /Z0 and T¯ ≡ T /T0 , where A0 = c0 E0 is the amplitude of the optical by using the dimensionless variables 4 pulse. Z0 , T0 are the soliton length-and time-scales given by Z0 =

2⊥ K , aA20 a2

(43)

T02 =

2⊥ K d2 β . a2 A20 a dω2

(44)

As is well known, the NLS equation admits soliton type solutions given in [22]   dk Z0 ¯ ¯ exp[ik(ω0 )Z0 Z¯ − iω0 T0 T¯ ]. A = 2A0 sech T − Z dω T0

(45)

We now estimate the length-and time-scales of this pulse given by Eqs. (43) and (44). For a 500 mW laser at λ = 0.5 ␮m, with a beam waist of 10 ␮m, the field amplitude is A20 = 1.9 × 106 V/m. Then by using the material values given above, the spatial and temporal scales for the pulse turn out to be Z0 = 4.2 × 10−5 m and T0 = 0.21 × 10−11 s. From Eq. (45), we find that the soliton propagates with the speed v¯ = v/c s n 1 λ0 dω Z0 = , (46) v¯ = dk T0 c A0 2πn2 d2 β/dω2 which for the chosen values of the parameters yields v¯ nem = 0.1, which is one order of magnitude smaller than the speed of light c in vacuum, and roughly has the same value as for glass, v¯ SiO2 = 2.5 × 10−1 . The difference between v¯ nem and v¯ SiO2 comes from the product n2 d2 (kn0 )/dω2 in Eq. (46), which measures the balance between nonlinearity and dispersion.

6. Summary Summarizing, we have derived an NLS equation for the amplitude of a wavepacket of TM modes propagating through a cylindrical nematic waveguide which takes into account the self-focusing, dispersion and diffraction in the nematic. For the soliton-like solution of this equation, we estimated the nonlinear refractive index n2 , the coefficient (d2 kn0 /dω2 ), the time T0 and length Z0 scales of the soliton and its speed v. ¯ We found that n2 may be several orders of magnitude larger than its typical value for glass (SiO2 ); on the other hand, d2 β/dω2 is much smaller than for glass. However, the speed v¯ of the soliton v¯ is comparable to its speed in glass, which shows that the more intense nonlinearity is compensated by a larger dispersion. Our model also illustrates another feature of liquid crystal cored optical fibers, namely, how the anisotropic nature of its physical properties allows to separate between different modes. We have shown that only the TM modes couple to the re-orientation dynamics and that only the higher-order modes are self-focused. This is a consequence of the fact that n2 and v¯ differ for the different modes. These features might be useful in mode selection liquid crystal clad-tapered fibers, as has been shown experimentally in [15].

J.A. Reyes, R.F. Rodr´ıguez / Physica D 141 (2000) 333–343

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It should be mentioned that our NLS equation, which contains time explicitly, might be an adequate model to describe the propagation of a pulsed laser beam in a liquid crystalline waveguide. However, an important limitation is that our model does not take into account the possibility of absorption by the fluid; the full consideration of absorption requires a generalization of our formalism that is currently in progress [9]. Actually, the neglect of these effects is responsible for the existence of caustics for the ray trajectories found in the previous works [13,14], and which are spurious when absorption is considered. These effects should also modify other dynamical properties such as response times or the speed propagation of the soliton and would remove the instantaneous character of the Kerr nonlinearity implicit in our description. The effect of dissipation in the re-orientation was considered to some extent through a coupled relaxation equation for the order parameter [8], which yields a finite penetration length. Finally, it should also be pointed out that we have ignored all the effects due to hydrodynamic flow (backflow) which would also affect the re-orientational dynamics and the response times [23].

Acknowledgements We acknowledge partial financial support from DGAPA-UNAM IN105797 and from FENOMEC through grant CONACYT 400316-5-G25427E, Mexico. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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