Gaussian beam diffraction in inhomogeneous and nonlinear media: analytical and numerical solutions by complex geometrical optics

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1 Cent. Eur. J. Phys. 6( DOI: 0.478/s Central European Journal of Physics Gaussian beam diffraction in inhomogeneous and nonlinear media: analytical and numerical solutions by complex geometrical optics Research Article Paweł Berczyński, Yury A. Kravtsov 3, Grzegorz Żegliński 4 Institute of Physics, Szczecin University of Technology, Szczecin 70-30, Poland Institute of Physics, Maritime University of Szczecin, Szczecin , Poland 3 Space Research Institute, Russian Acad. Sci., Moscow, 7 997, Russia 4 Institute of Electronics, Telecommunication and Computer Science, Szczecin University of Technology, 7-6 Szczecin, Poland Received 3 November 007; accepted 3 May 008 Abstract: The method of paraxial complex geometrical optics (CGO is presented, which describes Gaussian beam diffraction in arbitrary smoothly inhomogeneous media, including lens-like waveguides. By way of an example, the known analytical solution for Gaussian beam diffraction in free space is presented. Paraxial CGO reduces the problem of Gaussian beam diffraction in inhomogeneous media to the system of the first order ordinary differential equations, which can be readily solved numerically. As a result, CGO radically simplifies the description of Gaussian beam diffraction in inhomogeneous media as compared to the numerical methods of wave optics. For the paraxial on-axis Gaussian beam propagation in lens-like waveguide, we compare CGO solutions with numerical results for finite differences beam propagation method (FD-BPM. The CGO method is shown to provide 50-times higher rate of calculation then FD-BPM at comparable accuracy. Besides, paraxial eikonal-based complex geometrical optics is generalized for nonlinear Kerr type medium. This paper presents CGO analytical solutions for cylindrically symmetric Gaussian beam in Kerr type nonlinear medium and effective numerical solutions for the self-focusing effect of Gaussian beam with elliptic cross section. Both analytical and numerical solutions are shown to be in a good agreement with previous results, obtained by other methods. Keywords: inhomogeneous media Gaussian beam diffraction complex geometrical optics nonlinear Kerr medium Versita Warsaw and Springer-Verlag Berlin Heidelberg.. Introduction Presented at 9-th International Workshop on Nonlinear Optics Applications, NOA 007, May 7-0, 007, Świnoujście, Poland pawel.berczynski@ps.pl Complex geometrical optics (CGO has two equivalent forms: the ray-based form, which deals with complex rays, that is with trajectories in a complex space, and the eikonal-based form, which uses complex eikonal instead of complex rays [ 3]. A surprising feature of CGO is its 603 Download Date 9/4/5 :5 PM

2 Gaussian beam diffraction in inhomogeneous and nonlinear media: analytical and numerical solutions by complex geometrical optics ability to describe Gaussian beam (GB diffraction in both ray-based and eikonal-based approaches. This paper describes the advantages of the eikonal-based form of CGO for numerical solutions of diffraction problems in inhomogeneous and nonlinear media. Sect. presents the basic equations of paraxial CGO. Analytical solutions for homogeneous space are described in Sect. 3. High efficiency of CGO for numerical solutions of diffraction problems as compared with FD-BPM approach is demonstrated in Sect. 4. Finally Sect. 5 outlines the ability of CGO to describe evolution of the 3D Gaussian beams in nonlinear media of Kerr-type.. Paraxial complex geometrical optics (CGO Like traditional geometrical optics, CGO starts with the presentation of the wave field in the form of Debye expansion in inverse powers of wave number: u (r = m=0 A m (r (ik m exp [ikψ (r], ( where ψ is eikonal and A is amplitude. We consider here a D scalar problem (Cartesian coordinates are x and y and restrict ourselves only by the main term A 0 in Debye expansion (,that is we take A = A 0. Substituting the Debye expansion into the wave equation u (r + k ε (r u (r = 0, ( one can derive the eikonal equation for eikonal ψ: and transport the equation for A A 0 : ( ψ = ε, (3 div(a ψ = 0. (4 All the values are complex in the framework of CGO. As previously mentioned, complex geometrical optics (CGO has two equivalent forms: the ray-based and eikonal-based ones [, 3, 4]. The ray-based form deals with the complex trajectories, obeying the following ray equations dr dτ = p, dp dτ = ε (r (5 similar to those in conventional GO. Here p(τ is the ray momentum and dτ relates to the elementary arc length dσ by dτ = dσ/ ε. The ability of ray-based CGO to describe Gaussian beam diffraction was established approximately 40 years ago [4 6] (see also [, 3]. Development of numerical methods allowed later describe GB propagation and diffraction in inhomogeneous media [7 9]. Eikonal-based form of CGO deals directly with eikonal and is also able to describe Gaussian beam diffraction. According to [0], to describe paraxial beam propagation and diffraction in inhomogeneous D medium it is convenient to use a D curvilinear frame of reference (τ, ξ associated with the central ray: dr c dτ = p c, dp c dτ = ε (r c. (6 Curvilinear coordinates (τ, ξ are connected with Cartesian coordinates (x, y by r = r c (τ + ν (τ ξ, (7 where ν (τ is normal to the central ray, and ξ = ξ = r r c (τ is the distance from the nearest point r c (τ on a central ray. Lame coefficients in this case are h = h = ε c (τ [ Kξ], h = h = 0, h =, (8 ds = h dτ + h dτdξ + h dξ, (9 where ε c (τ = ε (r c ( is permittivity calculated along the central ray and K = is a ray curvature. ε(r ε(r ξ r=r c As a result, in curvilinear coordinates (τ, ξ the eikonal equation takes the form h ( ψ + τ ( ψ = ε (r. (0 ξ In a frame of paraxial approximation one can expand permittivity ε(r = ε (r c + νξ in a Taylor series for a small deviation ξ from the central ray and arrive to the eikonal equation of following form: ( ( ψ ψ + h τ ξ { + ξ ε (r ξ [ ε (r = ε (r c ( ] 3 4 [ ] } ε (r ξ r=r c. It is worth seeking a solution of the eikonal equation as a sum of two terms: ψ = ψ c + φ, ( 604 Download Date 9/4/5 :5 PM

3 Paweł Berczyński, Yury A. Kravtsov, Grzegorz Żegliński where ψ c is the eikonal, calculated along the central ray: dψ c dτ = ε c (τ, (3 and φ is a small deviation from ψ c. In a frame of paraxial approximation deviation, φ can be presented in the quadratic form, like in traditional paraxial optics [7, 0]: φ = B (τ ξ. (4 Here B (τ is a complex phase front curvature changing along the central ray []. Substitution of (4 into the eikonal Eq. ( leads to the following ordinary differential equation of Riccati-type for parameter B (τ [0], similar to that obtained within Luneburg paraxial optics [7]: db dτ + B (τ = α (τ, (5 where α (τ = { [ ε (r ξ ] 3 4ε (r [ ] } ε (r ξ r=r c. (6 The Riccati-type equation for B(τ can be readily solved numerically for arbitrary smoothly inhomogeneous medium. The real (R and imaginary (I parts of parameter B, B = R+iI, determine the wave front curvature κ and the beam widthw: R (τ = εc (τ ε c (τ κ = ; I (τ = ρ (τ kw (τ. (7 The Riccati-type equation for B(τ can be readily solved numerically for arbitrary smoothly inhomogeneous medium. In curvilinear coordinates (τ, ξ the transport Eq. (4 for amplitude A = A (τ takes the following form: [ da ψ h dτ τ + h τ ( h ] ψ + ψ A = 0. (8 τ ξ In accordance with paraxial approximation the above equation takes a much simpler form [0]: dã dτ + B (τ à = 0, (9 where à = ε/4 c A. It admits the following solution: w0 à = Ã0 w, (0 which corresponds to the energy flux conservation through the GB cross-section. As a result the wave field in the frame of paraxial CGO takes the form: w0 u = A (τ exp{ikψ c (τ + ikb ξ } = ε/4 c0 A 0 w ] { [ ]} exp [ ξ εc κ (τ ξ ik ψ w c (τ +. ( (τ Thus, the problem of Gaussian beam diffraction in inhomogeneous medium in fact comes to a solution of the ordinary differential equation, what is the valuable advantage of CGO. Basic equations of paraxial CGO for 3D inhomogeneous media are derived in a consequent way in []. Interrelations between paraxial CGO and other approximated methods of GB beam diffraction description are reviewed in []. 3. Gaussian beam diffraction in homogeneous medium For the case of propagation in homogeneous medium we can go back to Cartesian coordinate system with transformation τ x, ξ y. In such a case parameter α equals zero and Riccati Eq. (5 acquires the form db (x dx + B (x = 0 or d ( = dx. ( B This equation has the following analytical solution [3, 8]: B (x = + x, (3 B(0 where B(0 is an initial value of B(x. For GB with initial width w (0 = w 0 and with plane initial wave front κ(0 = 0, one has B(0 = i/kw 0. Thus, B(x = /(x ikw 0 and ψ(x, y = x + y /. (4 x ikw0 As a result, the wave field in frame of the paraxial CGO has the following form: u(x, y = ( A 0 exp ikx + iky /, (5 + ix/kw 0 x ikw0 This wave field is in total agreement with the diffraction solution of the parabolic wave equation [0, 3]. 605 Download Date 9/4/5 :5 PM

4 Gaussian beam diffraction in inhomogeneous and nonlinear media: analytical and numerical solutions by complex geometrical optics 4. Gaussian beam diffraction in inhomogeneous planar waveguide with parabolic permittivity profile Let us consider Gaussian beam diffraction in a waveguide with permittivity depending on vertical (transverse coordinatey, in the following form: ε (y = ε 0 ε y. (6 For such a case the right-hand side of Riccati Eq. (5 is constant α = / L =const and according to (0, (, with transformation τ x, ξ y, the wave field in a lens-like waveguide is of the form: u (x, y = A 0 w0 [ { exp i k [ w exp y w (x + κy ] arctan [( L L R where A 0 = A(0 is an initial amplitude, [ ( w (x = w 0 L ( x + ] sin L is a beam width and L R (7 ( x ]}] tan, L (8 [ ( κ (x = L ( x ] [ ( L ( x ] tan + tan L L R L L R L (9 is wave front curvature. Besides, L = / ε is a characteristic inhomogeneity scale of the waveguide and L R = kw0 is diffraction length (Rayleigh distance. The same results for parabolic permittivity profile were obtained in a similar way to the CGO analytical procedure in [9]. The CGO solution for GB wave field (7 completely coincides with the results of quasi-optical theory of diffraction [0 ]. The eikonal-based CGO equations for complex phase front curvature (5 and amplitude (9 are ordinary differential equations and can be easily solved numerically by the Runge-Kutta method for arbitrary inhomogeneous waveguides. To assess accuracy of the numerical CGO algorithm, we have compared CGO numerical results for GB width, phase front curvature and amplitude with analytical solutions (7-9 and obtained comparatively high accuracy. CGO results were compared with Finite differences beam propagation method (FD-BPM for narrow GB with the following parameters: A(0 =, w 0 = 3 µm, κ(0 = 0, kl R = 355, L = 3.5L R, λ = µm. (30 The discrepancy between both the methods was estimated according to the formula: δ w = w CGO w BPM w CGO. (3 The FD-BPM algorithm was based on parabolic wave equation in frame of the Crank-Nicholson scheme [4 7]. The results of comparison are presented in the Table. Table. The results of calculations by CGO and FD-BPM methods for the lens-like waveguide. Propagation distance τ Beam width w [µm] FD-BPM kw FD-BPM kw CGO 0 µm % δ w [%] 0 µm % 50 µm % 00 µm % 00 µm % 300 µm % 400 µm % It follows from Table that relative the difference δ w between CGO and FD-BPM results never exceeds %. Additionally we have compared time of calculations, provided by the CGO with that of FD-BPM method for waveguide with parameters, given by Eq. (30. For both methods, we have carried out calculations with numerical step 0.0 µm along both axes xand y, using a computer with an IBM Intel 750 MHz processor. The results are shown in the Table. Thus, eikonal-based CGO have demonstrated 50-times higher rate of calculation as compared with FD-BPM method at accuracy comparable with that of FD-BPM. CGO method can be applied also for description of Gaussian beams of elliptical cross section. The paper [] gives an example of Gaussian beam evolution along 3D spiral trajectory in a waveguide with a parabolic permittivity profile. 5. The nonlinear Kerr type medium In this section the CGO method is applied to a nonlinear medium of Kerr type. The phenomenon of Gaussian beam 606 Download Date 9/4/5 :5 PM

5 Paweł Berczyński, Yury A. Kravtsov, Grzegorz Żegliński Table. Time of calculations by CGO and FD-BPM methods for lenslike waveguide. Propagation distance x Time of numerical Time of numerical calculations t for calculations t for FD-BPM CGO (IBM Intel 750 MHz (IBM Intel 750 MHz x = 400 µm t = 3 min 30s t < min x = 000 µm t = 7 min 57s t < min x = 4000 µm t = 35 min 55s t < min x = 6000 µm t = 54 min 5s t < min Combination of Eqs. (35 and (36 leads to the following equation for beam width evolution w: d w dτ ε NL + Ã0 w0 =. (37 w 3 k0 w3 Introducing dimensionless beam widthf = w/w 0, Eq. (37 can be written in the form: d f dτ = (, (38 f 3 L NL L D self focusing in Kerr-type nonlinear media was first analyzed by Akhmanov, Khokhlov and Sukhorukov [4]. The nonlinear Kerr medium is characterized by nonlinear electrical permittivity of the form: ε = ε 0 + ε NL u. (3 We assume for simplicity that ε 0 = and accept ε NL to be positive: ε NL > 0. Following [4], let us first consider a axially symmetric Gaussian beam: u = Ã0 w 0 ( w exp ξ, (33 w where ξ = ξ + ξ is a distance from the beam axis. According to (3 and (33, the Riccati Eq. (5 takes the following form: db ε NL Ã 0 w dτ + 0 B =. (34 w 4 The above equation is equivalent to the set of two equations for real, R = ReB and imaginary, R = ImB parts of complex parameter B: dr ε NL Ã 0 w dτ + 0 R (τ I (τ = w 4 di + R (τ I (τ = 0. (35 dτ According to (7, parameter R = ReB equals R = w dw dτ. (36 where L D = k 0 w 0 is diffraction (Rayleigh length distance, and L NL = w 0 / ε NL Ã 0 denotes characteristic nonlinear distance. Integrating Eq. (38 one obtains the following: ( df = dτ L NL f + C. (39 L Df For the initial conditions: f(0 =, df (0 dτ = 0 (40 which correspond to the axial symmetric beam of the width w=w 0, and to the plane initial wave front, κ(0 = 0, the integration constant in Eq. (39 equals C =. (4 L D L NL As a result an equation for the beam width evolution in Kerr medium takes the form: ( df = f dτ L NL f + f L D f (4 and acquires the following analytical solution: ( f = + τ L D. (43 L D L NL The above result is identical with those obtained in the frame of classical theory of self-focusing based on a nonlinear parabolic wave Eq. [4]. The above eikonal-based CGO solution illustrates three regimes of Gaussian beam propagation in nonlinear Kerr medium:. If L NL > L D, then diffraction widening prevails over nonlinear refraction; 607 Download Date 9/4/5 :5 PM

6 Gaussian beam diffraction in inhomogeneous and nonlinear media: analytical and numerical solutions by complex geometrical optics 3. If L NL = L D, then f = w = w 0 and stationary self-trapping effect takes place; 3. If L NL < L D, then nonlinear refraction prevails over diffraction widening and beam width decreases, tending to zero. More sophisticated behavior is demonstrated by the Gaussian beam of elliptical cross section. Analysis of such a beam diffraction in 3D smoothly inhomogeneous linear media has preveiously been carried out in our paper []. In this section we generalize linear theory, developed in [], for Gaussian beam of elliptical cross section propagating in nonlinear Kerr medium. Let w0 w 0 u = Ã0 exp ( ikb ξ w w / + ikb ξ/ (44 where, B = ε c κ + i / k 0 w and B = ε c κ + i / k 0 w, be the Gaussian beam wave field. In this case the system of two complex Riccati equations takes the form: db ε NL Ã 0 w0 w 0 dτ + B = w 3w, db ε NL Ã 0 w0 w 0 dτ + B =. (45 w w 3 Equations for beam widths evolution stem from Eqs. (45 in analogy with axially symmetric beam: d w ε NL Ã 0 w0 w 0 + dτ w w =, k0 w3 d w ε NL Ã 0 w0 w 0 + dτ w w =. (46 k0 w3 Introducing dimensionless beam widths f = w /w 0, and f = w /w 0, the above set of equations take the form: d f dτ = ( f L NL f d f dτ = ( f L NL f where L Di = k 0 w i0 and L NLi = w i0 / L D f L D f,, (47 ε NL Ã 0 (i =, are diffraction (Rayleigh length and nonlinear distance corresponding to the large and small axes of elliptical GB cross section. Using identity ( f = [ (f + ff ], Eqs. (47 can be transformed to the form: d f dτ d f dτ ( ( df = dτ L NL f f L D f ( ( df = dτ L NL f f L D f,. (48 Summing Eqs. (48, one obtains the following equation ( d ( f dτ + f df dτ = f f ( f L D f ( df dτ + f L D f L NL L NL. (49 Stationary solution f =, f = takes place, when w = w 0 ; w = w 0 for arbitrary τ. Putting f =, f = into Eq. (49, we obtain the following condition for selftrapping of the Gaussian beam of elliptic cross-section: L D + = +. (50 L D L NL L NL Both L Di and L NLi (i=, depends on corresponding initial beam widths w i0, so the only possible solution of (50 is that: w 0 = w 0. (5 The above condition means that stationary self-trapping is not possible for elliptic beam in a Kerr-medium. This conclusion is in agreement with the results, known from literature [5, 6]. Complex Riccati Eqs. (45 were solved numerically by means of CGO numerical algorithm based on fourth order Runge-Kutta method for the following parameters: ε NL = m V, ε 0 =, A 0 = 0 6 V m. (5 Fig. shows the effect of non-stationary self-trapping for GB of elliptical cross-section. Fig. demonstrates that a very small increase in beam ellipticity (width along large axis was increased about 0.0 comparing to self-trapping value leads to GB collapse. Fig. shows that close to the self-trapping condition, both widths and phase-front curvatures decrease, expiring oscillations. Fig. 3 presents the case when ellipticity is much greater as compared to Fig.. In this case both the widths decrease what in consequence leads to collapse. Wave front curvatures decrease at Fig. 3 significantly faster comparing to the case presented in Fig.. Finally, Fig. 4 corresponds to the case when diffraction widening prevails over nonlinear refraction and as a consequence both the beam widths and wave front curvatures increase. 608 Download Date 9/4/5 :5 PM

7 Paweł Berczyński, Yury A. Kravtsov, Grzegorz Żegliński Figure. Beam widths and phase front curvatures plots in selftrapping regime. Initial conditions: k 0 w 0 = 0.048π, k 0 w 0 = 9.95π, κ 0 /k 0 = 0, κ 0 /k 0 = 0. Figure. Collapse with oscillations close to self-trapping regime. Initial conditions: k 0 w 0 = 0.05π, k 0 w 0 = 9.95π, κ 0 /k 0 = 0, κ 0 /k 0 = 0 and L D /L NL =, L D /L NL = Conclusion The simple and effective method to calculate Gaussian beam wave field, diffracted in arbitrary smoothly inhomogeneous and nonlinear media is presented. The method, based on paraxial complex geometrical optics, reduces the diffraction problem to the ordinary differential equation for complex curvature and GB amplitude. These equations can be readily solved numerically by the Runge- Kutta method. For the paraxial on-axis Gaussian beam propagation in lens-like waveguide, CGO analytical solutions happened to be identical with analytical solutions of parabolic wave equation. For the same waveguide, the difference between CGO and FD-BPM numerical results never exceeds % at a 50-times higher rate of calculation. Another important result is the generalization of the CGO method for nonlinear Kerr medium. CGO allows one to obtain an analytical solution for cylindrically symmetric Gaussian beam in nonlinear Kerr medium. We also performed numerical simulations for the self-focusing effect of the Gaussian beam of elliptic cross section in the Kerr medium. Thereby, the paraxial complex geometrical optics greatly simplifies description of Gaussian beam diffraction as compared to the numerical methods of wave theory reducing radically time of numerical calculations. Acknowledgements Section 5 devoted to nonlinear Kerr medium was prepared in cooperation with Prof. Mietek Lisak and his scientific group in Chalmers University of Technology in Sweden. The authors are indebted to: Prof. E. Winert-Ra czka from Szczecin University of Technology; Prof. M. Karpierz; 609 Download Date 9/4/5 :5 PM

8 Gaussian beam diffraction in inhomogeneous and nonlinear media: analytical and numerical solutions by complex geometrical optics 4 Figure 3. Collapse for much greater beam cross-section ellipticity comparing to the case presented in Fig.. Initial conditions: k 0 w 0 = 0π, k 0 w 0 = 0π, κ 0 /k 0 = 0, κ 0 /k 0 = 0 and L D /L NL = 0.99, L D /L NL =.99. Figure 4. The case when diffraction widening prevails over nonlinear refraction. Initial conditions: k 0 w 0 = 6π, k 0 w 0 = 8π, κ 0 /k 0 = 0, κ 0 /k 0 = 0 and L D /L NL = 0.6, L D /L NL = 0.8. Prof. J. Jasiński from Warsaw University of Technology; and Prof. Z. Jaroszewicz from Institute of Applied Optics in Warsaw for attention to the paper and for valuable advice. This work was supported in part by the Association IPPLM- EURATOM, project P- as well as by the Polish Ministry of Science and Higher Education. References [] Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan, Theory and applications of complex rays, In: E. Wolf (Ed., Progress in Optics 39, (Elsevier, Amsterdam, [] S.J. Chapman, J.M. Lawry, J.R. Ockendon, R.H. Tew, SIAM Rev. 4, 47 (999 [3] Yu.A. Kravtsov, Geometrical Optics in Engineering Physics (Alpha Science International, UK, 005 [4] Yu.A. Kravtsov, Radiophys. Quantum Electron. 0, 79 (967 [5] J.B. Keller, W. Streifer, J. Opt. Soc. Am. 6, 40 (97 [6] G.A. Deschamps, Electron. Lett. 7, 684 (97 [7] R.A. Egorchenkov, Yu.A. Kravtsov, Radiophys. Quantum Electron. 43, 5 (000 [8] R.A. Egorchenkov, Yu.A. Kravtsov, J. Opt. Soc. Am. A 8, 650 (00 [9] R.A. Egorchenkov, Physics of Vibrations 8, (000 [0] V.M. Babich, Eigenfunctions, concentrated in the vicinity of closed geodesics, In: Mathematical Problems of Theory of Waves Propagation 9 (Nauka, Leningrad, 968, 5 (in Russian [] V.M. Babic, V.S. Buldyrev, Asymptotic Methods 60 Download Date 9/4/5 :5 PM

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