Propagation of Lorentz Gaussian Beams in Strongly Nonlocal Nonlinear Media

Commun. Theor. Phys. 6 04 4 45 Vol. 6, No., February, 04 Propagation of Lorentz Gaussian Beams in Strongly Nonlocal Nonlinear Media A. Keshavarz and G. Honarasa Department of Physics, Faculty of Science, Shiraz University of Technology, Shiraz, Iran Received April 5, 03; revised manuscript received November, 03 Abstract In this paper the propagation of Lorentz Gaussian beams in strongly nonlinear nonlocal media is investigated by the ABCD matrix method. For this purpose, an expression for field distribution during propagation is derived and based on it, the propagation of Lorentz Gaussian beams is simulated in this media. Then, the evolutions of beam width and curvature radius during propagation are discussed. PACS numbers: 4.5.Bs Key words: nonlocal nonlinear media, Lorentz Gaussian beam, ABCD matrix Introduction The Lorentz Gaussian beam, as a more generalized case of Lorentz beam, has been introduced by Gawhary and Severini. [] The Lorentz Gaussian beam can describe the radiation emitted by single-mode diode lasers. [ 3] The beam propagation factors and nonparaxial propagation of a Lorentz Gaussian beam have been investigated. [4 6] The Wigner distribution function of Lorentz Gauss beams through a paraxial ABCD optical system has also been derived by Zhou and Chen. [7] The propagation of Lorentz Gauss beams through a few optical systems and different media such as turbulent atmosphere and uniaxial crystals have been investigated. [8 ] In nonlocal nonlinear media NNM, the refractive index at a particular point depends on the beam intensity at all other material points. [] When the beam width is much shorter than the width of the material response function, the media are called strongly nonlocal nonlinear media SNNM. [3] This property is observed in several optical materials such as lead glasses [4] and nematic liquid crystals. [5 6] The propagation of various optical beams such as Hermite Gaussian, Laguerre Gaussian, Hermite Laguerre Gaussian, Ince Gaussian and four-petal Gaussian beams in nonlocal nonlinear media have been studied. [7 ] According to our knowledge, the propagation of Lorentz Gaussian beams in SNNM has not been reported elsewhere. In this paper, the evolutions of intensity distribution, the beam width and the curvature radius of a Lorentz Gaussian beam during propagation in SNNM are studied based on Collins formula and ABCD method. The paper is organized as follows: In Sec., the theory of our formalism is exposed to study the propagation of Lorentz Gaussian beams in SNNM. Propagation properties of Lorentz Gaussian beams during the propagation in SNNM are investigated in Sec. 3. Conclusion of the paper is also presented in Sec. 4. Theory The Lorentz Gaussian beam in the Cartesian coordinates at z = 0 plane is introduced as: [] with Ex, y, 0 = Ex, 0Ey, 0, Ej, 0 = j + ω0j exp j ω0, where j = x or y, is the parameter related to the beam width of Lorentz part in j-direction and ω 0 is the waist of the Gaussian part. The Lorentz distribution can be expressed in terms of Hermite Gaussian functions: [] j + ω 0j = π ω0j m=0 exp j ω0j σ m H m j, 3 where N is the number of terms in the expansion and σ m is given by m { e / m σ m = m m! erfc n + n n!m n! = [ e / n ]} erfc + n n 3!!, 4 π n = where erfc and H m are the complementary error function and the m-th order Hermite polynomial, respectively. With the help of Eq. 4, Ej, 0 can be obtained as follows Ej, 0 = where π ω0j m=0 u j σ m H m j exp = ω 0 + ω0j. j u j, 5 E-mail: keshavarz@sutech.ac.ir E-mail: honarasa@sutech.ac.ir c 03 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

4 Communications in Theoretical Physics Vol. 6 Propagation of the optical beams through an optical system, which is parameterized by an ABCD transfer matrix can be described by the generalized Collins formula: [3] Ej, z = iλb expikz [ ik ] Ej 0, 0exp B Aj 0 jj 0 + Dj dj 0, 6 where Ej 0, 0 is the field distribution at entrance plane z = 0, λ is the wavelength of the beam and k is the wave number. Here A, B, and D are the elements of the transfer ABCD matrix. By using the integral formula π e αp q H m pdp = mhm [ q / ], 7 α α α the following analytical expression for propagation of Lorentz Gaussian beams through an optical ABCD system can be derived as: kπ Ej, z = i4α j B exp j k ω0j 4α j B + ikj D N B + ikz σ m mhm ikjω 0j α m=0 j Bα j α, 8 j / where α j is defined by α j = ω 0 + ika B. 9 The propagation of beams in nonlocal nonlinear media is governed by the nonlocal nonlinear Schrödinger equation where t ψ + ik ψ z + k n n 0 ψ = 0, 0 t = x + y is the transverse Laplacian operator, k is the wave number in the media without nonlinearity, n 0 is the linear refractive index of the media, n = n R r r ψ r, z d r is the nonlinear perturbation of refractive index, n is the nonlinear index coefficient and R is the normalized symmetrical real spatial response function of the media. In the case of strong nonlocality, by the use of Snyder and Mitchell model, Eq. 0 takes the form [4 5] tψ + ik ψ z k γ p 0 x + y ψ = 0, where γ is the material constant relating to the response function and p 0 is the input power at z = 0. As ABCD matrix is widely used to describe the beam propagation through the paraxial optical systems and the beam propagation equation in SNNM can be linearized, it is straight forward to show that the beam propagation in SNNM can be described by following ABCD matrix [] p0 /p c z cos p0 /p c z 0 z sin p0 /p c p0 /p c z z sin p0 /p c cos p0 /p c where = πω0 / is Rayleigh distance, p c = z0 γ is the critical power of Gaussian beam and p 0 is the input power. By inserting the elements of transfer matrix in Eq. 8 the analytical expressions for propagation of Lorentz Gaussian beams in SNNM can be written in the form ikπ p 0 /p c / [ Ej, z = 4α j sin j k ω0j exp p 0/p c p 0 /p c z/ 4α j z0 sin p 0 /p c z/ ikj p0 /p c m=0 σ m α j mhm ikj p0 /p c α j α j / sin p 0 /p c z/, z ] cot p0 /p c + ikz. 3 Figure shows the transverse field distribution of a typical Lorentz Gaussian beam with ω 0 = 3, ω 0x =, and ω 0y = 6 during propagation through the SNNM for different values of input powers. In Fig., the behavior of a Lorentz Gaussian beam is simulated in x z and y z plane. These analytical simulations are based on Eq. 3. The results show that when the Lorentz Gaussian beam propagates through SNNM, the intensity distribution varies periodically. The period distance is z = π / p 0 /p c, which can be found from Eq.. Therefore, the evaluation period with a smaller input power is longer than that with a larger input power.

No. Communications in Theoretical Physics 43 Fig. Transverse field distribution of a Lorentz Gaussian beam with ω 0 = 3, ω 0x =, and ω 0y = 6 through the SNNM at different propagation distances: z/ = 0 first column, z/ = second column, z/ = 4 third column, and z/ = 6 fourth column. The different rows represent various values of the input powers: p 0/p c = 0.7 first row, p 0/p c =.0 second row, and p 0/p c =.3 third row. 3 Propagation Properties of Lorentz Gaus sian Beams In this section, the evolution of the beam width and the mean curvature of Lorentz Gaussian beams in SNNM based on ABCD matrix are investigated. The second-order moment of the intensity certainly gives good eminent information on the beam width. The second-order moment beam width at the waist plane w can be expressed as [6] w = 4 x Ex, y, 0 dx Ex, y, 0 dx, 4 Fig. Transverse distribution of a Lorentz Gaussian beam with ω 0 = 3, ω 0x =, and ω 0y = 6 through the SNNM in x-z and y-z planes for p 0 = p c. and the beam width at the observation plane w after propagation through an optical ABDC system is related to w as follows with w = A w + ABV + B U, 5 U = λ / xex, y, 0 dx π Ex, y, 0 dx V = 4 x [ φx, y x ] ψ x, ydx, 6,

44 Communications in Theoretical Physics Vol. 6 where φx, y and ψx, y are the phase and the amplitude of the beam, respectively. At the waist plane φx, y = 0 and so V = 0. The radius of curvature for a general beam at observation plane is given by R = w w z. 7 Figure 3 displays the evolution of the beam width and the curvature radius of the Lorentz Gaussian beam during propagation through the SNNM. For simplicity, only the beam width in x-direction is plotted. It can be found from Fig. 3a that the beam width varies periodically and its period is shorter in larger input powers. Figure 3b shows that the curvature radius is infinite at entrance plane and it varies periodically during propagation. The positive negative curvature radius represents the cophasal surface is concave convex and induces the focusing defocusing of the beam. Fig. 3 The beam width and the curvature radius of a Lorentz Gaussian beam in x-direction with ω 0 = 3 and ω 0x = through the SNNM for p 0/p c = 0.7 solid line, p 0/p c =.0 dashed line, and p 0/p c =.3 dot-dashed line. Fig. 4 Transverse distribution of a Lorentz Gaussian beam with ω 0 = 3, ω 0x =, and ω 0y = 6 through the SNNM in x-z and y-z planes for a p 0 = p LG cx and b p 0 = p LG cy. Fig. 5 The beam width and the curvature radius of a Lorentz Gaussian beam in x-direction with ω 0 = 3 and ω 0x = through the SNNM for p 0/p LG cx = 0.7 solid line, p 0/p LG cx =.0 dashed line, and p 0/p LG cx =.3 dot-dashed line.

No. Communications in Theoretical Physics 45 By inserting A and B from Eq. into Eq. 5, we can obtain the critical power of the Lorentz Gaussian beam in j-direction as follows ω 4 p LG 0 cj = p / jex, y, 0 dj c 4, j Ex, y, 0 dj j = x or y. 8 When the input power is equal to Eq. 8, the beam width of the Lorentz Gaussian beam in j-direction keeps invariant during propagation through the SNNM. In general case, when ω 0x ω 0y, the value of the critical power of the Lorentz Gaussian beam is different for x and y-direction. In Fig. 4, the propagation of a Lorentz Gaussian beam through SNNM is simulated in x z and y-z plane for a p 0 = p LG cx and b p 0 = p LG cy. The evolution of the beam width and the curvature radius of the Lorentz Gaussian beam in x-direction during propagation through the SNNM are plotted in Fig. 5 for different value of input powers. It is seen that when p 0 = p LG cx the beam width of the Lorentz Gaussian beam is constant during propagation. Otherwise the beam width oscillates periodically along the propagation length. 4 Conclusion In conclusion, a closed-form expression for propagation of the Lorentz Gaussian beams through the SNNM has been derived by using Collins formula, and then based on it the propagation of these beams in SNNM is simulated. The results show that the intensity distribution and beam width of the Lorentz Gaussian beam vary periodically during propagation. This is due to the propagation nature of the beam, which is in competition with nonlinearity of the medium. References [] O.E. Gawhary and S. Severini, J. Opt. A, Pure Appl. Opt. 8 006 409. [] A. Naqwi and F. Durst, Appl. Opt. 9 990 780. [3] J. Yang, T. Chen, G. Ding, and X. Yuan, Proc. SPIE 684 008 6840A. [4] G. Zhou, Appl. Phys. B 96 009 49. [5] G. Zhou, J. Opt. Soc. Am. B 6 009 4. [6] H. Yu, et al., Optik 00 455. [7] G. Zhou and R. Chen, Appl. Phys. B 07 0 83. [8] G. Zhou, J. Opt. Soc. Am. A 5 008 594. [9] W. Du, C. Zhao, and Y. Cai, Opt. Lasers Eng. 49 0 5. [0] G. Zhou and X. Chu, Opt. Express 8 00 76. [] C. Zhao and Y. Cai, J. Mod. Opt. 57 00 375. [] Z. Yang, D. Lu, W. Hu, Y. Zheng, X. Gao, and Q. Guo, Phys. Lett. A 374 00 4007. [3] W. Krolikowski, O. Bang, J.J. Rasmussen, and J. Wyller, Phys. Rev. E 64 00 066. [4] C. Rotschild, O. Cohen, O, Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95 005 3904; C. Rotschild, M. Segev, Z. Xu, Y.V. Kartashov, L. Torner, and O. Cohen, Opt. Lett. 3 006 33. [5] M. Peccianti, K.A. Brzdakiewicz, and G. Assanto, Opt. Lett. 7 00 460. [6] C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 9 004 390. [7] D. Deng, X. Zhao, Q. Guo, and S. Lan, J. Opt. Soc. Am. B 4 007 537. [8] D. Deng and Q. Guo, J. Opt. A: Pure Appl. Opt. 0 008 0350. [9] D. Deng, Q. Guo, and W. Hu, J. Phys. B: At. Mol. Opt. Phys. 4 008 540. [0] D. Deng and Q. Guo, J. Phys. B: At. Mol. Opt. Phys. 4 008 4540. [] Z. Yang, D. Lu, D. Deng, S. Li, W. Hu, and Q. Guo, Opt. Commun. 83 00 595. [] P.P. Schmidt, J. Phys. B, At. Mol. Phys. 9 976 33. [3] S.A. Collins, J. Opt. Soc. Am. 60 970 68. [4] A.W. Snyder and D.J. Mitchell, Science 76 997 538. [5] Q. Guo, B. Luo, F.H. Yi, S. Chi, and Y.Q. Xie, Phys. Rev. E 69 004 0660. [6] P.A. Belanger, Opt. Lett. 6 99 96.