Complex geometrical optics of Kerr type nonlinear media Paweł Berczyński and Yury A. Kravtsov 1) Institute of Physics, West Pomeranian University of Technology, Szczecin , Poland 2) Institute of Physics, Maritime University of Szczecin, Szczecin , Poland SEMINAR AT THE INSTITUTE OF PHYSICS WEST POMERANIAN UNIVERSITY OF TECHNOLOGY Szczecin, 26-th February (2010)

It has been shown recently that Eikonal-based form deals directly with complex eikonal and is able to describe Gaussian beam diffraction for arbitrary smoothly inhomogeneous media. Now CGO is generalized for the case of Gaussian beam (GB) diffraction and self-focusing in nonlinear media of Kerr type, including nonlinear graded index fiber.

3. CGO for linear media of cylindrical symmetry 3.1. Riccati equation for complex parameter B For axially symmetric GB in axially symmetric medium CGO solution has the form is the linear wave number for GB propagating in the z direction Complex parameter: and is a distance from the axis z is complex-valued eikonal The eikonal equation in coordinates takes the form: Paraxial approximation can be expanded in Taylor series in in the vicinity of symmetry axis z

Substitution: into eikonal equation leads to the ordinary differential equation of Riccati type for complex parameter B: where Substituting into eikonal, one obtains Gaussian beam in the form: Above solution reflects the general feature of CGO, which in fact deals with the Gaussian beams.

3.2. The equation for GB complex amplitude In frame of paraxial approximation the transport equation for axially symmetric beam in coordinates takes the form: For eikonal we obtain that As a result the transport equation reduces to the ordinary differential equation in the form: The above equation for GB complex amplitude, as well as the Riccati equation for complex curvature B are the basic CGO equations. Thus, CGO reduces the problem of GB diffraction to the domain of ordinary differential equation. Having calculated the complex parameter B from Riccati equation one can readily determine complex amplitude A:

3.3. The equation for GB width evolution The Riccati equation is equivalent to the set of two equations for the real and imaginary parts of the complex parameter B: Substituting and into above equations, one obtains the known relation between the beam width and the wave front curvature [Kogelnik (1965)]: Substituting above relation into the first equation of the system, we obtain the ordinary differential equation of the second order for GB width evolution The absolute value of complex amplitude: leads to energy flux conservation principle in GB cross-section

4. Generalization of CGO for nonlinear media of Kerr type In nonlinear media of Kerr type the permittivity depends on the beam intensity where coefficient is assumed to be positive:. Substituting Gaussian wave field into above equation one obtains Thus, in the framework of CGO method the nonlinear medium of Kerr type can be formally treated as a smoothly inhomogeneous medium, whose profile is additionally modulated by GB parameters and. The alpha parameter can be given as: As a result the Riccati equation, generalized for nonlinear media of Kerr type has form: where describes the linear refraction and accounts for self-focusing process in nonlinear medium.

4.1. CGO solution for nonlinear medium of Kerr type without contribution of linear refraction When the contribution of linear term is negligibly small: the Riccati equation has the form: and the equation for the beam width has the form (where ): where is diffraction length and is the nonlinear scale. Following the papers of Akhmanov, Sukhorukov and Khokhlov (1967,1968,1976) we prove that: where is the total beam power and is the critical power.

. As a result, equation for GB width evolution takes the form: The above CGO result is in total agreement with the solution of nonlinear parabolic equation (NLS) [Akhmanov, Sukhorukov and Khokhlov (1967), (1968), (1976)]. Thus, the PCGO method reproduces the classical results of nonlinear optics but in a more simple and illustrative way [P.Berczynski, Yu.A.Kravtsov, A.P.Sukhorukov, Physica D, 239, p , (2010)]. The three partial cases deserve to be distinguished in the above solution: 1. Under-critical power:, where the beam width increases. 2. Critical power:, Gaussian soliton. It was shown by Desaix, Anderson, Lisak, [Phys. Rev. A. 40(5), , (1989)] that Gaussian soliton can approximate exact soliton solution (hyperbolic secant) with relative error not exceeding 6%. 3. Over-critical power:, the beam width decreases to zero at a finite propagation distance and GB amplitude increases to infinity: collapse phenomenon.

Fig. 1. Evolution of the squared relative beam width in nonlinear medium of Kerr type for the beam power: P=1.5Pcrit (curve 1), P=Pcrit (curve 2), P=0.5Pcrit (curve 3) It follows from CGO solution for nonlinear medium of Kerr type that the self-focusing distance is given by:

4.2. Influence of the initial phase front curvature on the beam evolution in nonlinear medium of Kerr type The first integral of eq. for GB width takes the form The initial condition for at presents the squared initial wave front curvature: As a result:, where Taking advantage of differential relation one obtains:

As in the previous case, it is worth analyzing the following partial cases: 1)A. Sub-critical regime: a)For a divergent beam (positive initial curvature) : diffraction widening prevails over self-focusing effect and GB width at once increases (curve 1 in Fig. 2). For a convergent beam, corresponding to, the beam width initially decreases reaching minimum value 1)Next, diffraction widening dominates and the beam width starts increasing (curve 2 in Fig. 2). Fig. 2. Presented for and (trace1) (trace2)

B. Critical regime: In this case diffraction divergence and self-focusing effects compensate each other and depends only on the value of. As a result, the beam width w changes as a linear function of distance z: a) In above solution factor leads to a growth of the GB width, whereas for one obtains Gaussian soliton with w=w(0). b) For the beam is focused at a distance.

C. Over-critical regime: Fig. 3. Evolution of for and (trace1), (trace 2) One can notice in Fig. 3 that the positive value of the initial wave front curvature can eliminate collapse in over critical regime. In fact one can distinguish the characteristic beam power for above which GB is always focused regardless of the sign and value of the initial wave front curvature: But when GB power is greater than the critical power and smaller than the characteristic power, a positive value of the initial wave front curvature eliminates the collapse effect whereas a negative value enhances it.

C.1. When the total power is equal to characteristic power, the beam width squared is a linear function of z: For the beam width increases and for the beam is focused at a distance C.2. When the GB total power is greater than the characteristic power, the GB is focused at a distance: Fig. 4: illustrates the case

5. Gaussian beam propagation in nonlinear inhomogeneous medium of cylindrical symmetry The CGO method is applied for the beam propagation in a nonlinear and inhomogeneous medium with electrical permittivity of the form The case when and corresponds to graded-index nonlinear optical fiber, with distance from the fiber axis. The Riccati equation takes now the form:, where leads to Integration above equation with the initial conditions:

Taking advantage of differential relation we have and differentiating it we obtain the equation: which has this time a simple physical interpretation in the form of a harmonic oscillator with a constant force, where, This analog can let us easily find the solution of above equation in the form

6. Conclusions 1. CGO is developed to describe the diffraction and self-focusing of axially symmetric Gaussian beams in nonlinear media of Kerr type. 2. This paraxial method reduces the GB diffraction problem to the ordinary differential equations for complex curvature of the wave front, amplitude and for GB width evolution. 3. This method supplies results, which happen to be identical with the solutions of nonlinear parabolic equation (NLS) for a nonlinear medium of Kerr type. 4. CGO allows easily to include into analysis the initial curvature of the wave front and to study its influence on GB dynamics.

5. It is shown for nonlinear Kerr medium that in an over-critical regime one can distinguish the characteristic beam power for above which GB is always focused regardless of the sign and value of the initial wave front curvature. It is also shown that, when GB power is greater than the critical power and smaller than the above mentioned characteristic power, a positive value of the initial wave front curvature eliminates the collapse effect whereas a negative value enhances it. 6. CGO method is also applied for the case of inhomogeneous and nonlinear medium. The solution for GB diffraction in a graded-index Kerr nonlinear optical fiber is obtained and influence of the initial curvature of the wave front is taken into account. 7. Thereby, the complex geometrical optics greatly simplifies description of Gaussian beam diffraction and self-focusing as compared with the traditional methods of nonlinear optics based-on nonlinear parabolic equation.