uides lled with nonlinear Kerr-type media. ... II. Basic Equations. We consider nonlinear electromagnetic wave propaga- ... r ~H = j!0 r~E .... John Wiley &. Sons ...
Full Wave Vector Maxwell Equation Simulation of Nonlinear Self-Focusing E ects in Three Spatial Dimensions Sergey V. Polstyanko, Romanus Dyczij-Edlinger, Jin-Fa Lee ECE Dept., Worcester Polytechnic Institute, 100 Institute Rd., Worcester, MA 01609, U.S.A.
Abstract| In our e ort to meet an increasing demand for more accurate and realistic nonlinear optics simulations, we have developed a nonlinear hybrid vector nite element method (NL-HVFEM) to study different phenomena due to wave propagation in waveguides lled with nonlinear Kerr-type media. Contrary to the most existing scalar models, the NL-HVFEM approach is based upon the vector Helmholtz equation and thus can predict the vectorial properties of elds in nonlinear media. In this paper, we describe the NLHVFEM approach and apply it to study nonlinear selffocusing e ects in nonlinear waveguides. The numerical results of the evolution of the beam self-focusing are also included. Also, we summarized conditions under which electromagnetic beam can produce its own dielectric waveguide and propagate without spreading. I. Introduction
The investigation of nonlinear wave propagation has been attracting attention of many researchers over the last 30 years. Many interesting phenomena have come out of it. A number of theoretical and experimental papers have been written about such nonlinear e ects such as solitons formation and propagation, self-focusing [1]{[4]. This paper is devoted to the problem of the behavior of light beam in a self-focusing nonlinear medium [3]{[5]. The phenomenon itself was originally described by Askarjan [6] and became of a great interest because of its importance for practical applications. Self-focusing describes in general focusing of the beam by itself as a result of the nonlinear refractive index induced by the beam. The e ect may occur in materials whose dielectric constant increases with eld intensity, but which are homogeneous in the absence of electromagnetic eld. The self-focusing is often responsible for the optical damage created in solid by a high power laser beam. Consequently, many attempts have been made to describe it theoretically, but unfortunately, self-focusing is a class of nonlinear wave Manuscript received March 19, 1996. This work was sponsored by NSF through RIA93-09587. Dr. Dyczij-Edlinger is currently with System Technologies, Motorola Inc., 1301 E. Algonquin Rd., Schaumburg, IL60196.
phenomena for which plane wave analysis fails to give essential qualitative information. The fact that the wave amplitude is not independent of transverse dimensions gives rise to the enhancement of inhomogeneities which is characteristic of the e ect. Thus in many cases numerical methods are needed to take into account the transverse e ects of optical beams. One of the most interesting problems, from the physical point of view, is the beam propagation in nonlinear medium with a speci ed initial intensity distribution. Under the assumption made in quasi-optics, the propagation of a stationary light beam can be studied within the framework of parabolic equation. Dynamical focusing solutions of the nonlinear wave equation were originally found numerically by Kelley [4] for a nonsaturable, nonlinear refractive index. To reduce the number of independent variables in the problem, Kelley sought timeperiodic, axially symmetric solutions. To solve the resulting equation numerically with known initial distribution, he ignored the second derivative of the slowly varying amplitude E~ with respect to longitudinal distance z . Unfortunately, this approximation is valid only in the prefocal region where the beam diameter is still relatively large but it fails when E~ changes rapidly within a wavelength, for example near the self-focus. Published numerical results only con rm the physically clear conclusion that the beam is initially narrowed down, but does not determine the subsequent evolution of the phenomenon [7], [8]. Several attempts have been made to modify and extend the numerical calculations done by Kelley through the focal region by assuming a saturable nonlinear re ective index [9], [8], [10]. Unfortunately, this assumption was not physically reasonable for liquids and solids. Thus the question of the self-focusing remains open and a more accurate numerical solution is needed for its theoretical explanation. In our e ort to resolve this issue, we have developed a nonlinear hybrid vector nite element method (NLHVFEM) [11] for modeling beam propagation in nonlinear medium. The proposed method is a combination of the edge elements [12] and nite di erence method. The former is applied to the waveguide cross section and the latter to the propagation direction itself. Since the approach presented herein is based on the use of vector nite element methods, it is applicable to any arbitrary nonlinear waveguides.
II. Basic Equations
III. NL-HVFEM Formulation
We consider nonlinear electromagnetic wave propagation in an isotropic medium, which is uniform in the z direction with an arbitrary shape in the transverse (xy) plane, having intensity-dependent relative permittivity r , such that
By applying the Galerkin principle to Eq. (6), one comes up with the following di erential matrix equation
r = L + f (jE j2 )
(1)
where L = n20 denotes the linear relative permittivity, is the nonlinear coecient and f can be an arbitrary function of the intensity jE j2 such that f (0) = 0. When f (jE j2 ) = jE j2 , Eq. (1) corresponds to the Kerr-type nonlinearity. Assuming exp (j!t) dependence of the elds, the time-harmonic form of Maxwell's equations is given by
r H~ = j!0 r E~ r E~ = ?j!0 H~
(2)
This assumption, concerning time variation, is valid for long duration pulses. To study the spatiotemporal behavior of short pulses, which are obtainable with modern mode-locked laser technology, the present approach is not valid and a nonlinear time domain method, such as the nonlinear nite di erence time domain (NL-FDTD), has to be adopted [13], [14]. Equations (2) can be reduced to a single vector wave equation in term of E~ . The result is
r r E~ = k02 r E~
(3)
where k0 is the wavenumber in free-space. Using the conventional approximation of a slowly varying envelope and assuming also that the axis of the propagation is z , we can rewrite E~ as
E~ = E~(x; y; z ) exp j (!t ? z )
(4)
where E~ represents slowly varying amplitude of the original eld E~ . Furthermore, we de ne the transverse operator r as
r = x^@x + y^@y and
E~ = E~ + z^Ez r = r + z^@z
where
@ e = [F ]e [M ] @z
2 [M ] = 4 2 [F ] = 4 2 e = 4
where E~ ; Ez are the transverse and the z components of E~, respectively. Finally, based on Eqs. (3)-(5), the vector Helmholtz wave equation becomes r r E~ ? z^r2 Ez + z^r @z E~ ? @z2 E~ ? k02 r E~ = 0 ?j z^r E~ + @z r Ez ? j r Ez + 2 E~
(6)
3
[M ] [M z ] [N ] [Mz ] 0 0 5 [ N ] 0 0 3 [F ] [F z ] 0 [Fz ] [Fzz ] 0 5 0 0 [N ]
3 e ez 5 g
(8)
and e ; ez are column vectors of coecients for the transverse and z components, respectively, and g denotes g = @e @z
(9)
Moreover, the element matrices in Eq. (7) are given by
Z
[M ] : ?2j fW~ E~ gd
Z
[M z ] : ? fW~ r Ez gd
[Mz ] : [N ] : [F ] :
Z
Z
Z
fE~ r Wz gd
fW~ E~ gd
(10)
f(r W~ ) (r E~ ) + 2 W~ E~
?k02 WZ~ r E~ gd
[F z ] : ?j fW~ r Ez gd
Z
[Fz ] : j fr Wz E~ gd
Z
[Fzz ] : fr Wz r Ez ? k02 Wz r Ez gd
(5)
(7)
Finally, application of the nite di erence method, based on the backward di erence scheme, to Eq. (7) yields ([M ] ? [F ]z )ei+1 = [M ]ei i = 0; 1; 2::: (11) where ei corresponds to e at z = iz , e0 is the initial state and z is the di erence step along the z axis. Moreover, we would like to point out that in the iterative process for each step i, once a solution vector ei is found, we update not only the relative permittivity r , but also the propagation constant . Equation (11) is then used in a step-by-step calculation to obtain the electric eld pro le corresponding to the initial eld distribution at z = 0.
IV. Numerical Results
The NL-HVFEM approach has been applied to study self-focusing e ects of an axially symmetrical optical beam. The initial intensity pro le of the beam is a Gaussian distribution with constant phase front [4], [5], [10]. Namely, 2 Ex (r; 0) = E0 exp (? 2ra2 )
(12)
2 1=2 Pcr = (1:220 ) (1=02r ) 16 0
(13)
where a, the initial waist, has been taken to be a = 0:4m. Note that all dimensions and parameters of nonlinear medium were chosen simply as a representative case. It would be trivial to tailor the problem to other dimensions and materials. The simulations were conducted for approximately 6000 triangular elements for the waveguide cross section and with z = 0 =8. We would like to study here conditions under which an initial Gaussian beam can produce its own dielectric waveguide and propagate without spreading. Such selftrapping appears to be possible only for high intensity of an input beam when an input power is greater than a critical one. To come up with an expression for a critical power we ran our simulations for di erent parameters characterizing nonlinear medium, such as L and as well as for di erent frequency points. Based on our numerical results (see Fig. 1), the critical power can be approximated fairly well by
where 0 is the wavelength in vacuum. To obtain the corresponding result in cgs units we divide (13) by 40 and result in the formula numerically found by previous authors [3], [5]
0)2 c Pcr = (1:22 64
(14)
where c denotes speed of light in vacuum. Furthermore, the input power of the light beam is de ned to be
Z1 E02 exp (?r2 =a2) 2rdr Pin = ( L0 )1=2 0 0 = ( L 0 )1=2 E02 a2 (15) 0
Finally, from Eqs. (13) - (15) the critical eld amplitude can be derived as (16) Ecr = 14:a22p 0 It is also interesting to notice that the par-axial-ray analysis of Talanov [15], [8] predicts a much lower critical power Ppar = 0:273Pcr . Our numerical simulations show that for
the input power P such that Ppar < P < Pcr , the intensity of the beam rises initially to a semi-focus and then falls to zero as if the beam is di racting. Shown in Fig.2 are on-axis intensity pro les of an input Gaussian pulse for di erent values of input power and various type on nonlinear media. We note that longitudinal distance is given in units of 0 . The solutions have been obtained in suciently large regions including the entire self-focusing process and the obtained pictures of the phenomenon differ greatly from those obtained previously through the use of scalar par-axial approximations. Also, if we de ne the self-focusing length as the distance in which the intensity of an optical beam reaches the maximum, then from Fig.2 we conclude that increase the input power decreases the self-focusing length. The characteristic pro les of the amplitude E (r; z ) for di erent z are shown in Fig.3 for a nonlinear medium with r = 2 and = 0:01. In addition to the self-trapping behavior of an optical beam, we also observed a self-limiting e ect in our simulation results. Unlike the par-axial scalar models, the current simulation predicts the peak intensity of the beam oscillates in the self-focusing region (see Fig. 2). Moreover, as the input beam propagates in the z direction, in the presence of TM modes, the z component also grows. This growth in the z component, which represents the energy transfer from the transverse components to the longitudinal component, eventually limits the self-focusing process. This self-limiting e ect was rst observed and explained by Ziolkowski and Judkins [13] using the NLFDTD method for ultrashort optical pulses in a nonlinear Kerr medium. References [1] E. M. Wright, G. I. Stegeman, C. T. Seaton, J. V. Moloney, and A. D. Boardman. "Multisoliton Emission from a Nonlinear Waveguide". Phys. Rev. A, 34:pp. 4442{4444, 1986. [2] M. D. Feit and J. A. Fleck. "Light Propagation in GradedIndex Optical Fibers". Appl. Opt., 17:pp. 3990{3998, 1978. [3] R. Y. Chiao, E. Garmire, and C. H. Townes. "Self-Trapping of Optical Beams". Phys. Rev. Lett., 13(15):pp. 479{482, 1964. [4] P. L. Kelley. "Self-Focusing of Optical Beams". Phys. Rev. Lett., 15(26):1005{1008, 1965. Erratum 16, pp. 384 (1966). [5] Y. R. Shen. The principles of nonlinear optics. John Wiley & Sons, Inc., 1984. [6] G. A. Askarjan. . Soviet Phys. JETP, 15:pp. 1088, 1962. [7] V. E. Zakharov, V. V. Sobolev, and V. C. Synakh. "Behavior of Light Beams in Nonlinear Media". Soviet Physics JETP, 33(1):pp. 77{81, 1971. [8] W. G. Wagner, H. A. Haus, and J. H. Marburger. "Large-Scale Self-Trapping of Optical Beams in the Paraxial Ray Approximation". Phys. Rev., 175(1):pp. 256{266, 1968. [9] E. L. Dawes and J. H. Marburger. "Computer Studies in SelfFocusing". Phys. Rev., 179(3):pp. 862{869, 1969. [10] J. H. Marburger and E. L. Dawes. "Dinamical Formation of a Small-Scale Filament". Phys. Rev. Lett., 21(8):pp. 556{558, 1968. [11] S. Polstyanko and J.-F. Lee. "Full Vectorial Analysis of Nonlinear Slab Waveguide Using the Nonlinear Hybrid Vector Finite Element Method". Optics Letters, 21:pp. 98{100, 1996.
5.5
1.6
1.3
3.5
I(z) / I(0)
I(z) / I(0)
1.2
4
1.0 0.8 0.6
3
50.0
100.0 Z
150.0
4 epsilon
5
6
(a) Critical power Pcr vs r for = 0:01 and f = 2GHz. 5
theory results
4.5
Pcr (x 0.001)
4
100.0
150.0
250.0
Pin / Pcr = 1.10
16.0
35.0
14.0
30.0
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25.0 20.0 15.0
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3.5
4.0
3
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2.5
10.0 5.0
200.0
400.0
600.0 Z
800.0
1000.0
(b)
2 1.5
0.0 0.0
1200.0
70.0
100.0
200.0
300.0 Z
1
500.0
600.0
Pin / Pcr = 1.72
60.0
0.5
400.0
120.0
Pin / Pcr = 1.72 100.0
4
6 8 alpha (x 0.001)
10
12
(b) Critical power Pcr vs for r = 2:0 and f = 2GHz.
40.0 30.0
20.0
10.0 0.0 0.0
8
20.0
40.0
60.0 Z
80.0
100.0
2 1 1
1.2
1.4
1.6
1.8 2 2.2 Freq. (GHz)
2.4
2.6
2.8
3
(c) Critical power Pcr vs f for r = 2:0 and = 0:01.
60.0
80.0
100.0
120.0
140.0
12.0 Zo = 0 Zo = 50 Zo = 100
6.0
Zo = 0 Zo = 50 Zo = 100
10.0
5.0 8.0 Et(R,Zo)
[12] J. Lee, D. Sun, Z. Cendes. "Full-Wave Analysis of Dielectric Waveguides Using Tangential Vector Finite Elements". IEEE Trans. Microwave Theory Tech., 39(8):pp. 1262{1271, 1991. [13] R. W. Ziolkowski and J. B. Judkins. "Full-Wave Vector Maxwell Equation Modeling of the Self-Focusing of Ultrashort Optical Pulses in a Nonlinear Kerr Medium Exhibiting a Finite Response Time". J. Opt. Soc. Am. B, 10(2):pp. 186{198, 1993. [14] R. W. Ziolkowski and J. B. Judkins. "Applications of the Nonlinear Finite Di erence Time Domain (NL-FDTD) Method to Pulse Propagation in Nonlinear Media: Self-Focusing and Linear-Nonlinear Interfaces". Radio Science, 28:pp. 901{911, 1993. [15] V. I. Talanov. "Self Focusing of Wave Beams in Nonlinear Media". JETP Lett., 2:pp. 138{141, 1965.
7.0
4.0 3.0
6.0
4.0 2.0 2.0
1.0 0.0 -1.5
-1.0
-0.5
0.0 R
0.5
1.0
0.0 -1.5
1.5
(a)
1.2
-1.0
-0.5
0.0 R
0.5
2.5
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0.2
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0.6
1.5
Zo = 50 Zo = 100
1.0
0.0 -0.8
1.0
3.0
Zo = 50 Zo = 100
Ez(R,Zo)
Fig. 1. Comparison between theoretical and computed results for critical power for di erent nonlinear media.
40.0
Fig. 2. Normalized on-axis intensity I (z)=I (0) versus axial distance in units of the wavelength 0 for f = 2GHz. Curves are parameterized by Pin =Pcr . The simulations were made for nonlinear media characterized by = 0:01; L = 2 (left column) and L = 5 (right column) for 3 di erent regions : (a) defocusing and semi-focusing; (b) self-focusing; and, (c) strong self-focusing.
Et(R,Zo)
3
20.0
Z
Ez(R,Zo)
4
0.0 0.0
120.0
(c)
7
5
60.0
40.0
theory results
6
80.0
20.0
10 9
I(z) / I(0)
2
I(z) / I(0)
50.0
0
Pcr (x 0.001)
200.0
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Pin / Pcr = 1.10
7
I(z) / I(0)
3
I(z) / I(0)
2
50.0
Z
(a)
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1
1.0
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1.1
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2
1.2
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Pin / Pcr = 0.191 Pin / Pcr = 0.925
1.4
1.4
4.5 Pcr (x 0.001)
1.5 Pin / Pcr = 0.191 Pin / Pcr = 0.925
theory results
5
0.8
(b)
0.0 -0.8
-0.6
-0.4
-0.2
0.0 R
0.2
0.4
0.6
Fig. 3. Numerical simulations for nonlinear medium with = 0:01 and L = 2 for f = 1:5GHz. Curves are parameterized by z=0 for Pin =Pcr = 1:243 (left column) and Pin =Pcr = 1:720 (right column). (a) Radial distribution of the total electric eld; and, (b) Radial distribution of the Ez component.
0.8
