Propagation of Hermite-cosh-Gaussian laser beams in n-InSb S.D. Patil * , M.V. Takale, M.B. Dongare Division of Nonlinear Optics and Holography Laboratory, Department of Physics, Shivaji University, Vidyanagar, Kolhapur, Maharashtra 416004, India article info Article history: Received 4 March 2008 Received in revised form 1 May 2008 Accepted 15 May 2008 Keywords: Hermite-cosh-Gaussian beams n-InSb Parabolic wave equation Self-focusing/defocusing abstract The propagation of Hermite-cosh-Gaussian (HChG) laser beams in n-InSb are investigated for 0, 1 and 2 mode indices. The field distribution in the medium is expressed in terms of beam-width parameter f and decentred parameter b. The differential equations for f-parameter are established by parabolic wave equation approach under paraxial approximation. Analytical solutions are obtained under the condition R n < R d , where, R n is the self-focusing length and R d is the diffraction length. The behaviour of f-parameter with the dimensionless distance of propagation g for various b values is examined by numerical esti- mates. The results are presented graphically. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction The interaction of lasers with semiconductors has been a fasci- nating field of research for several decades [1–9]. Semiconductors provide a compact and less expensive medium to model nonlinear phenomena encountered in laser produced plasmas. The observa- tion of self-focusing in semiconductors is of great relevance to the practical applications and possibilities of optical limiting de- vices [10]. In equilibrium, the temperature of free carriers is the same as that of the crystal so that the net energy exchange be- tween the carriers and lattice of the crystal is zero. When an elec- tric field is applied, the free carriers gain energies (from the field), which causes the temperature to be higher than that of the crystal in the steady state. For moderate values of electric field, the in- crease in temperature of the carriers is proportional to the square of electric field. The change in temperature of the carriers leads to corresponding change in the effective mass of the carriers. This ef- fect is important for laser self-focusing in semiconductors [11]. InSb has proved to be a very promising material because it shows nonparabolic energy bands [2,6]. Here the nonlinearity arises from the dependence of effective electronic mass on the electronic tem- perature [6,7]. Recently, a new laser beam called Hermite-cosh-Gaussian (HChG) beam that is one of the solutions of paraxial wave equation has been studied extensively, including parameters, characteriza- tion and propagation properties [12–32] and commented that such HChG beams can be generated in the laboratory by superposition of two decentred Hermite-Gaussian beams as cosh-Gaussian ones [15]. In this paper, we present the propagation of Hermite-cosh- Gaussian laser beams in n-type Indium Antimonide (n-InSb) for 0, 1 and 2 mode indices. We assume negligible absorption in the nonlinear medium. The advantage of this formalism is that similar propagation can be obtained for the various nonlinear media. In Section 2, the field distribution of HChG beams propagating along z-axis and nonlinear dielectric constant for nonparabolic semicon- ductors (e.g. n-InSb) are presented. In Section 3, the differential equations for beam-width parameter are established by parabolic wave equation approach under paraxial approximation. Results and a discussion are given in Section 4, supported by numerical analysis. A brief conclusion is added in Section 5. The present study has been carried out under Wentzel–Kramers–Brillouin (WKB) approximation and paraxial approximation through parabolic wave equation approach [6,7]. 2. Theoretical considerations 2.1. Field distribution of HChG beams We employ the HChG-laser beam propagating along z-axis with the field distribution in the following form [12,15] Eðr; zÞ¼ E 0 2f H m ffiffiffiffiffi 2r p f x 0 ! exp b 2 4 ! exp  r f x 0 þ b 2   2 " # ( þ exp  r f x 0  b 2   2 " #) ð1Þ where m is the mode index associated with the Hermite polyno- mial H m , x 0 is the waist-width of Gaussian amplitude distribution, b is the decentred parameter, r is the radial coordinate, E 0 is the 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.05.045 * Corresponding author. Tel.: +91 0231 2609224; fax: +91 0231 2692333. E-mail address: sdpatil_phy@rediffmail.com (S.D. Patil). Optics Communications 281 (2008) 4776–4779 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/optcom