New approach to the analysis of ultrashort pulse diffraction (Invited Paper)

Progress in optics led to the point when scientists and engineers operate the pulses of the electromagnetic energy containing only a few (in fact, sometime less than one, see examples in [1]) oscillations of the field instead of monochrome or quasimonochrome waves. Even omitting physical problems of interaction between field and matter, including those associated with essential nonlinearity due to extremely high amplitudes of the field, one encounters a lot of problems caused by extraordinary large homogeneous spectral bandwidth of ultrashort pulses. Linear optics of ultrashort pulses essentially differs from that of monochrome waves; hence, the development of appropriate methods for description of such pulses propagation through an optical system is necessary. It is common to our perception to deal with monochrome wave transformation, not wide-spectrum signal one, because in the traditional optics there is the set of solutions for the typical problems. The purpose of this presentation is to show that the equivalent set may be developed also for the optics of ultrashort pulses. As a result, the qualitative study of the diffraction and interference ofultrashort pulses becomes as simple as the ray optics. There are different methods of the analysis of diffraction and interference phenomena. The first possible approach is based on Huygens representation of the diffraction phenomena. Traditional procedure involves Fourier transform of the initial pulse process to get a set of monochrome waves. Then one analyzes the diffraction of each of those waves and finally the resulting monochrome diffraction fields sum back to form a scattered pulse. Rather widely used [2, 3, 4], this method implies reconsideration of developed solutions of classical diffraction problems to take into account some second-order components. Thrown away in the conventional theory as contributing only small phase correction for rather law frequencies, they become significant for high frequency components of the field. Another method suggests decomposition of the initial pulse into a set of wavelets [5]. Being convenient to those researchers who operate with an equal ease with this class of functions and with monochrome waves, this method does not possess the clearness of first one. Moreover, some of them does not meet the causality principle. At the same time, there is another group of methods based on one of principal results of general theory of diffraction [6]. Their formulation is closer to Young interpretation of diffraction and considers the interaction of waves scattered by the edge of aperture and through-passing one [7]. In acoustics and electro-engineering, the method based on the formalism of Green function for the wave equation [8, 9] is used. The fact that this method performs well for linear system analysis in electronics supported us in our attempt. In optics, it was successfully used by J. Connes [10] and L. Mertz [1 1], for example. Both of them studied the propagation of pulses through some most simple devices. Moreover, their analysis was based only on intuitive concepts that led to some mistakes, as it becomes clear today. Nevertheless, the general conclusion on the possibility of using the Dirac transform for the description of the waveform conversion in the process of propagation through the linear optical system became obvious. The technique based on 8-wave ideas proposed in our previous publications [12, 13, 14]. Those papers demonstrated the simplicity of interpretation of spatial and temporal form of the diffracted wave transformation using the pulse approach. The present paper extends this concept and demonstrates complete agreement of its results with classical theory for monochrome waves; the method opens a way to the development of specific software for the analysis of propagation of ultrashort pulses. All methods mentioned above give identical results. To our knowledge, the S -wave approach was not used directly for the solution of diffraction problems in the Kirchhoff approximation. However, it seems to be handy for the description of short pulses propagation. Moreover, it was shown in [12, 14] that the method can clarify some features of diffracted wave that can give a key to a fine experiment realization