Application of Gaussian pulsed beam decomposition in modeling optical systems with diffraction grating.

A diffraction grating is one of the most commonly used components in ultrafast optical systems such as pulse stretchers and compressors. Hence, modeling the temporal dispersion and spatiotemporal distortions associated with the angular dispersion of a diffraction grating is very crucial for wave optical modeling of such systems. In this paper, the Gaussian pulsed beam decomposition (GPBD) method is extended to handle the propagation of ultrashort pulses, with arbitrary spatial and spectral profiles, through complex ultrashort pulse shaping systems containing diffraction gratings. Although the diffraction efficiencies are not rigorously computed, the GPBD method enables modeling of the large angular dispersion of idealized diffraction gratings without running into an impractically large number of spectral samples as in the case of Fourier-transform-based methods. The application of the extended method is demonstrated by performing the wave optical propagation of an ultrashort pulse through a single diffraction grating and then through a Treacy compressor system. By combining the Treacy compressor with the Martinez grating pair stretcher with internal lenses, the pulse shaping through a complete chirped pulse amplification (CPA) setup is modeled. Finally, the effects of using real dispersive lenses in the Martinez stretcher on the output pulse of the CPA setup are presented. For analysis of the output pulses, methods of computing the spatiotemporal and spatio-spectral amplitudes of the output pulse from the phase correct superposition of individual Gaussian pulsed beams are presented.

Gaussian pulsed beam decomposition for propagation of ultrashort pulses through optical systems.

Many applications of ultrashort laser pulses require manipulation and control of the pulse parameters by propagating them through different optical components before the target. This requires methods of simulating the pulse propagation taking into account all effects of dispersion, diffraction, and system aberrations. In this paper, we propose a method of propagating ultrashort pulses through a real optical system by using the Gaussian pulsed beam decomposition. An input pulse with arbitrary spatial and temporal (spectral) profiles is decomposed into a set of elementary Gaussian pulsed beams in the spatiospectral domain. The final scalar electric field of the ultrashort pulse after propagation is then obtained by performing the phase correct superposition of the electric fields all-Gaussian pulsed beams, which are propagated independently through the optical system. We demonstrate the application of the method by propagating an ultrashort pulse through a focusing aspherical lens with large chromatic aberration and a Bessel-X pulse generating axicon lens.

Spatially truncated Gaussian pulsed beam and its application in modeling diffraction of ultrashort pulses from hard apertures.

A new kind of pulsed beam, which we call a spatially truncated Gaussian pulsed beam, is defined to represent a Gaussian pulsed beam that is diffracted from a semi-infinite hard aperture. The analytical equations for the propagation of the spatially truncated Gaussian pulsed beam through a nonrotationally symmetric paraxial system with second-order dispersion is derived starting from the generalized spatiotemporal Huygens integral. The spatially truncated Gaussian pulsed beam is then combined with the conventional Gaussian pulsed beam decomposition method to enable the modeling of diffraction of a general ultrashort pulse from an arbitrarily shaped hard aperture. The accuracy of the analytical propagation equation derived for the propagation of the truncated Gaussian pulsed beam is evaluated by a numerical comparison with diffraction results obtained using the conventional pulse propagation method based on the Fourier transform algorithm. The application of the modified Gaussian pulsed beam decomposition method is demonstrated by propagating an ultrashort pulse after a circular aperture through a dispersive medium and a focusing aspherical lens with large chromatic aberration.

Propagation of truncated Gaussian beams and their application in modeling sharp-edge diffraction.

The two new kinds of truncated Gaussian beams, known as the half and quarter Gaussian beams, are defined as the product of the fundamental Gaussian beam with the Heaviside unit step function. Using the generalized Collins integral, the exact analytical propagation formulas are derived for the truncated Gaussian beams through paraxial optical systems. Combined with the Gaussian beam decomposition method, the truncated Gaussian beams are used to represent the sharp edges of a field after a hard aperture. The modified Gaussian beam decomposition method presented in this work enables the calculation of the diffraction of a given field from a hard aperture with an arbitrary shape. This solves one of the limitations of the conventional Gaussian beam decomposition method, which is its inability to accurately model the diffraction of fields with sharp edges, especially in the near field. The validity and accuracy of the proposed method are demonstrated using a few exemplary diffraction calculations.

Decomposition of a field with smooth wavefront into a set of Gaussian beams with non-zero curvatures.

Decomposition of a general arbitrary field into a set of Gaussian beams has been one of the challenges in the Gaussian beam decomposition method for field propagation through optical systems. The most commonly used method in this regard is the Gabor expansion, which decomposes initial fields into shifted and rotated Gaussian beams in a plane. Since the Gaussian beams used have zero initial curvatures, the Gabor expansion method does not utilize the ability of the Gaussian beams to represent the quadratic behavior of the local wavefront. In this paper, we describe an alternative method of decomposing an arbitrary field with smooth wavefront into a set of Gaussian beams with non-zero initial curvatures. The individual Gaussian beams are used to represent up to the quadratic term in the Taylor expansion of the local wavefront. This significantly reduces the number of Gaussian beams required for the decomposition of the field with smooth wavefront and gives more accurate decomposition results. The proposed method directly gives the five ray sets representing the parabasal Gaussian beams, which can then be directly used for propagation of the Gaussian beams through optical systems. To demonstrate the application of the method, we have presented results for the decomposition of fields with strongly curved spherical wavefronts, a cone shaped wavefront, and a wavefront with large spherical aberration. The numerical comparison of the input field with the field reconstructed after the decomposition shows very good agreement in both amplitude and phase profiles. We also show results for the far field intensity distributions of the decomposed wavefronts by propagating in free space using the Gaussian beam propagation method.

Ray-mapping approach in double freeform surface design for collimated beam shaping beyond the paraxial approximation.

Numerous applications require the simultaneous redistribution of the irradiance and phase of a laser beam. The beam shape is thereby determined by the respective application. An elegant way to control the irradiance and phase at the same time is from double freeform surfaces. In this work, the numerical design of continuous double freeform surfaces from ray-mapping methods for collimated beam shaping with arbitrary irradiances is considered. These methods consist of the calculation of a proper ray mapping between the source and the target irradiance and the subsequent construction of the freeform surfaces. By combining the law of refraction, the constant optical path length, and the surface continuity condition, a partial differential equation (PDE) for the ray mapping is derived. It is shown that the PDE can be fulfilled in a small-angle approximation by a mapping derived from optimal mass transport with a quadratic cost function. To overcome the restriction to the paraxial regime, we use this mapping as an initial iterate for the simultaneous solution of the Jacobian equation and the ray mapping PDE by a root-finding algorithm. The presented approach enables the efficient calculation of double freeform lenses with small distances between the freeform surfaces for complex target irradiances. This is demonstrated by applying it to the design of a single-lens and a two-lens system.