The modified propagation equation for TM polarized subwavelength spatial solitons in a nonlinear planar waveguide

Optics

Optik

Optik

The modified propagation equation for TM polarized subwavelength

spatial solitons in a nonlinear planar waveguide

Chi-Feng Chen



, Sien Chi

a

Department of Mechanical Engineering and Institute of Opto-Mechatronics Engineering, National Central University, Jhongli, Taiwan 320, ROC

b

Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu, Taiwan 30050, ROC Received 7 July 2005; accepted 15 November 2005

Abstract

The wave equation of TM polarized subwavelength beam propagations in a nonlinear planar waveguide is derived beyond the paraxial approximation. This modified equation contains more higher-order linear and nonlinear terms than the nonlinear Schro¨dinger equation. The propagation of fundamental subwavelength spatial solitons is numerically studied. It is shown that the effect of the higher nonlinear terms is significant. That is, for the propagation of narrower beam the modified nonlinear Schro¨dinger equation is more suitable than the nonlinear Schro¨dinger equation.

r2006 Elsevier GmbH. All rights reserved.

Keywords: Nonlinear effect; Spatial soliton; Subwavelength spatial soliton; Nonlinear planar waveguide; Nonlinear Schro¨dinger equation

1. Introduction

Spatial solitons which are the balance of the diffrac-tion and the self-focusing have been studied both theoretically and experimentally in a nonlinear planar

waveguide [1,2]. Generally, the nonlinear Schro¨dinger

equation (NLSE) making the paraxial approximation can describe well the propagation of spatial solitons. If the beam width of spatial solitons is as narrow as one wavelength or less, the validity of the paraxial approx-imation becomes questionable. The full-vector nonlinear

Maxwell’s equations were use to avert this problem[3],

but it is very time consuming. In addition, to enhance the validity of the wave equation, the modified NLSEs

are presented [4–7]. The additional terms including a

polarization-dependent correction to the soliton

propa-gation constant were considered, the dynamics of a narrow spatial soliton with an arbitrary polarization were affected

[4]. The solution of a modified NLSE describing the

electric field for TM mode was found, they analyzed the effects of those addition terms on the shapes of bright and

dark solitons of TM mode with a fixed polarization [5].

Beyond the paraxial approximation, a modified NLSE describing the electric field for subwavelength TE solitons was drived and an analytical solution for the soliton was

found[6]. Very narrow solitons in (1+1)-dimensional and

(2+1) dimensional versions of cubic–quintic and full

saturable models were analyzed[7]. For the solitons of TE

and TM polarizations, it was shown that there is always a finite minimum of the soliton’s width, and the solitons cease to exist at a critical value of the propagation constant, at which their width diverges.

In this paper, we will derive the propagation equation in a nonlinear planar waveguide by the iteration

method and the order of magnitude method [6,8,9].

The derived equation contains more higher-order linear

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and nonlinear terms than the NLSE. It is different to the one for the propagation of TE polarized subwavelength

beams[6]. We also numerically study the propagation of

TM polarized subwavelength beams in a nonlinear planar waveguide. It is found that the fundamental spatial soliton is not stable due to these higher-order terms and the effect of the higher nonlinear terms is significant and must be considered.

2. Derivation of the wave equations

We now derive the wave equation which can describe the propagation of TM polarized subwavelength beams in a nonlinear planar waveguide. The electric field E of the light obeys the vector wave equation

r2E
o 2_n2 0 c2 E þ o2 c2_e0PNLþ 1 n2 0e0 r r ð PNLÞ ¼0 (1)

which follows from the Maxwell equations, where e0is

the vacuum permittivity, n0is linear refractive index, o

is the light frequency, c is the velocity of light in vacuum,

and PNLis the third-order nonlinear polarization and

PNL ð Þ_i¼3e0 4 X j;k;l wð3Þ_i;j;k;l
o ¼ ojþok
olEjEkE_l.

Here w(3)(o) is the third-order susceptibility, i, j, k and l

refer to the Cartesian components of the fields. In the following, we will derive the wave equations which describe the propagations of subwavelength beams in a nonlinear planar waveguide.

The electric field of the light can be written as

E x; y; zð Þ ¼½xAx^ ðx; zÞ þ^zAzðx; zÞF ðyÞ exp ik0zð Þ, (2)

where Ax(x, z) and Az(x, z) are slowly varying amplitude

envelopes; F(y) is the normalized linear eigenfunction of the

mode excited in the nonlinear planar waveguide; k0¼n0o/c

is the propagation constant, n0is linear refractive index, o is

the light frequency, and c is the velocity of light in vacuum.

The total refractive index is given by n ¼ n0þn2jEj2,

where n2 is the Kerr coefficient and n2¼3wð3Þxxxx=8n0.

Substituting Eq. (2) into Eq. (1), we obtain

i q qzAxþ 1 2k0 q2 qx2Ax þg Axj j2Ax¼
1 2k0 q2 qz2Ax
g k2₀ q2 qx2 jAxj 2_A x

2g 3 jAzj 2_A x
g 3ðAzÞ 2_A x, ð3aÞ i q qzAzþ 1 2k0 q2 qx2Az þg Aj zj2Az¼
1 2k0 q2 qz2Az
2g 3 jAxj 2_A z
g 3ðAxÞ 2_A z, ð3bÞ

where g ¼ k0n2/n0. For the weakly guided mode,

|Az|5|Ax|. The relation

Az¼

i

k0

qAx

qx isAx

is obtained from r  D ¼ 0, where D is the electric displacement. Therefore, Eq. (3a) can be rewritten as

i q qzAxþ 1 2k0 q2 qx2Axþg Aj xj 2_A x¼
1 2k0 q2 qz2Ax
g k2₀ q2 qx2 jAxj 2_Ax
 " þ2 3 q qxAx         2 Ax
1 3 q qxAx  2 A_x # ð4Þ and Eq. (3b) can be neglected. To normalize Eq. (4), we make the following transformations:

Axðx; zÞ ¼ ffiffiffiffiffiffi P0 p N u Z; xð Þ ¼s ffiffiffiffiffi n0 n2 r u Z; xð Þ, x ¼ w0Z, z ¼ Ldx, ð5Þ

where the parameter

N ¼ n2P0

k2₀w2_n

0

" #1=2

is the order of the spatial soliton, N ¼ 1 for the

fundamental soliton, w0¼wF/1.763 and wF is the

full-width at the half-maximum (FWHM) of the beam, P0is

peak power of the incident beam, the parameter s ¼ 1/

(k0w0) ¼ 0.28(l0/wF), l0¼2p/k0is the wavelength in the

waveguide, and Ld¼k0w2

0 is the diffraction length. Eq.

(4) can be normalized to qu qx¼ i 2 q2 qZ2u þ i uj j 2_u þis2 q 2 qZ2 j ju 2_u
 þ2 3 qu qZ         2 u
1 3 qu qZ  2 u " # þis 2 2 q2 qx2u. ð6Þ

Then we use the iteration method and the order of magnitude method to derive the propagation equation beyond the paraxial approximation. First, a zero order approximation equation is obtained from Eq. (6)

neglected the q2u=qx2 term,

qu qx¼ i 2 q2 qZ2u þ ijuj 2_{u. (7)}

which is the well-known NLSE for spatial soliton propagations. For the first iteration, we differentiate

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Eq. (7) with respect to x and obtain q2 qx¼
1 4 q4 qZ4u
2 q2u qZ2   u j j2
qu qZ  2 u
2qu qZ         2 u
uj j4u. (8) Substituting Eq. (8) into Eq. (6), we obtain the wave equation of the first-order approximation

q qxu ¼ i 2 q2 qZ2u þ i uj j 2_u þis2
1 8 q4 qZ4u þ q2u qZ2   u j j2þ q 2_u qZ2   u2 " þ7 6 qu qZ  2 u_þ11 3 q2u qZ         2 u
1 2j ju 4_u # ð9Þ For the second iteration, we differentiate Eq. (9) with

respect to x and obtain an expression of q2u=qx2 with

higher order terms. Substituting this q2u=qx2 backs into

Eq. (6), we obtain the wave equation of the second-order

approximation with s4higher-order terms, which is the

same as Eq. (9) up to s2terms.

3. Solution and discussion

When the same s2-order terms in Eq. (9) are ignored,

Eq. (9) is approximated to NLSE. Surely, the funda-mental spatial soliton is a solution of one. That is, the beam will maintain its shape unchanged after propagat-ing long distance in a nonlinear planar waveguide if the beam width of spatial soliton is much wider than one wavelength. However, when the beam width of spatial soliton is as narrow as one wavelength or less, the higher-order terms cannot be neglected. To show the effects of higher-order terms and the necessity of the

modified NLSE, we consider wF¼0.75l0. And, Eq. (9)

is solved by the split-step Fourier method with the initial

condition u(0, Z) ¼ sech(Z).Fig. 1shows the peak power

and beam width versus propagation distance with and

without higher order terms, respectively. Here z0 is

soliton period, z0¼ ðp=2ÞLd. One can see that the

changes of the beam width and the peak power are

very obviously. At the distance of 7Z0, the beam width is

about 1.86wFand the peak power is down to 0.61P0at

the distance of 6Z0. The changes are due to the effects of

higher-order terms. Comparing to without higher-order terms, the results are apparently different. In other words, the propagation of narrower beam must describe by the modified NLSE containing higher-order terms.

4. Conclusion

In conclusion, we have derived an accurate wave equation beyond paraxial approximation by the

itera-tive method and the order of magnitude method for the TM polarized subwavelength optical beam propagation in a nonlinear planar waveguide. The derived equation contains higher-order linear and nonlinear terms than the NLSE. We numerically show that the fundamental subwavelength spatial soliton cannot maintain any more

due to these higher-order terms. For wF¼0.75l0, the

changes of the beam width and the peak power are very obviously when the higher-order terms are considered. In other words, for the propagation of narrower beam, the modified NLSE is more suitable than the NLSE.

References

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[3] R.M. Joseph, A. Taflove, Spatial soliton deflection mechanism indicated by FD-TD Maxwell’s equations modeling, IEEE Photon. Technol. Lett. 6 (1994) 1251. [4] A.W. Snyder, D.J. Mitchell, Y. Chen, Spatial solitons of

Maxwell’s equations, Opt. Lett. 19 (1994) 524.

[5] E. Granot, S. Sternklar, Y. Isbi, B. Malomed, A. Lewis, Subwavelength spatial solitons, Opt. Lett. 22 (1997) 1290. [6] C.F. Chen, S. Chi, Subwavelength spatial solitons of TE

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[7] B.V. Gisin, B.A. Malomed, One- and two-dimensional subwavelength solitons in saturable media, J. Opt. Soc. Am. B 18 (2001) 1356.

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[9] C.F. Chen, S. Chi, B. Luo, Femtosecond soliton propaga-tion in an optical fiber, Optik 113 (2001) 267–271.

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Fig. 1. Peak power (dashed curve) and beam width (solid curve) versus propagation distance simulated by the modified NLSE for wF¼0.75l0. Dashed-dotted curve is peak power

simulated by the NLSE.