End correction in the quasi-fast Hankel transform for optical propagation problems

Theory and operational rules for the discrete Hankel transform

Previous definitions of a discrete Hankel transform (DHT) have focused on methods to approximate the continuous Hankel integral transform. In this paper, we propose and evaluate the theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The proposed transform also possesses requisite orthogonality properties which lead to invertibility of the transform. The standard set of shift, modulation, multiplication, and convolution rules are derived. In addition to the theory of the actual manipulated quantities which stand in their own right, this DHT can be used to approximate the continuous forward and inverse Hankel transform in the same manner that the discrete Fourier transform is known to be able to approximate the continuous Fourier transform.

Absorption-free beam generated by a phase-engineered optical element

Phase engineering of an axicon's phase transmission function results in an exponential growth of an on-axis intensity of a diffraction-free or Bessel beam. We show that in a lossy medium obeying Beer's absorption law/exponential intensity decrease, a beam focused by such an optical element propagates with nearly constant intensity while preserving the quasi-diffraction-free property. Absorption-free beams may find applications in illumination, atmospheric propagation, biomedicine/light diffusion in live tissues, and other situations where light absorption hinders beam propagation.

Design and comparison of composite rod crystals for power scaling of diode end-pumped Nd:YAG lasers

A comparison of composite Nd:YAG laser rod crystals with one, two and three doped segments for high-power diode end-pumping is presented. An approach based on an expansion of the heat generation density and temperature distributions into a Fourier-Bessel basis set for solving the stationary heat conduction equation is used for choosing adequate segment lengths and dopant concentrations. A maximum laser output power of 167.5 W at an optical-to-optical efficiency of 53.6% was achieved by longitudinal pumping a crystal with three doped segments with fibre-coupled laser diodes.

Self-similar parabolic beam generation and propagation

The (1+1) -dimensional and (2+1) -dimensional amplified nonlinear Schrödinger equations incorporating diffraction, Kerr nonlinearity, and gain are solved analytically and numerically. An asymptotic solution is found corresponding to self-similar propagation of a beam with parabolic amplitude and phase profiles. While the (1+1) -dimensional solution is directly analogous to parabolic pulse propagation in nonlinear dispersive media, the existence of self-similar propagation in (2+1) dimensions is a nontrivial question, given that spatial solitons are unstable in bulk media with nonsaturating nonlinearities. We show that self-similar parabolic beams are possible in such media with gain and a negative nonlinear index.

Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields

The method originally proposed by Yu et al. [Opt. Lett. 23, 409 (1998)] for evaluating the zero-order Hankel transform is generalized to high-order Hankel transforms. Since the method preserves the discrete form of the Parseval theorem, it is particularly suitable for field propagation. A general algorithm for propagating an input field through axially symmetric systems using the generalized method is given. The advantages and the disadvantages of the method with respect to other typical methods are discussed.

Frequency-dependent graded reflectivity mirror: characterization and laser implementation

We present a theoretical and experimental investigation of a new family of variable reflectivity mirrors, based on frustrated total internal reflection and interference effects. These mirrors are sensitive to frequency in the sense that their transverse reflectivity distribution changes significantly as a function of the frequency of light. The mirror reported here shows total power reflectivity changes of 50% within 8.0 GHz. The mirror was tested as the output coupler of an unstable resonator in a Nd:YAG laser working in the free-running regime. This configuration was compared with the standard stable multimode resonator configuration. The beam divergence was reduced by more than 1 order of magnitude and the output power was reduced by only 10%. The laser resonator mode competition that is due to the frequency-dependent mirror reflectivity distribution is discussed.

Transverse-mode selection in apertured super-Gaussian resonators: an experimental and numerical investigation for a pulsed CO(2) Doppler lidar transmitter

A representative prototype of a high-energy, long-pulse, and narrow-bandwidth pulsed CO(2) laser suitable for a spaceborne Doppler wind lidar application has been developed. We obtained 10 J of output energy at greater than 8% efficiency in long, narrow-bandwidth, single-longitudinal, and transverse-mode pulses. We used a positive branch unstable resonator with a fourth-order super-Gaussian mirror as the output coupler. Experiments were carried out to assess the effect of intracavity hard apertures of different diameters that induce diffractive perturbation of the theoretical field and reduce the transverse-mode selectivity of the cavity. An upper limit to the choice of the mirror soft radius has been found, which allows optimization of the trade-off between laser efficiency and beam quality. We determined experimentally that a value of 0.75-0.8 for the ratio between the exp(-1) diameter of the beam intensity and the laser clear aperture gave a single-transverse-mode operation without significant loss of efficiency.

Transverse pattern modifications in a stable apertured laser resonator

The effects of hard apertures on the transversal mode distribution and the output near and far field distributions are investigated both theoretically and experimentally. Wavefronts propagating in different directions inside a CO(2) laser cavity were measured at the same plane and found to have different transversal amplitude profiles. The influence of intracavity aperture's location and diameter on the near field distribution and far field beam quality were measured and calculated. Quantitative figures of merit are introduced to evaluate the departure of the transversal mode patterns from ideal Gaussians.

Optical implementation of integral transforms with Bessel function kernels

Integral transforms involving Bessel function kernels are useful in analyzing effects of circularly symmetric optical systems on arbitrary inputs. Methods for performing the integral transforms optically are divided into two categories. The first category involves input data available in Cartesian (x, y) format and uses the close connection between the desired integral transform and the two-dimensional Fourier transform in Cartesian coordinates. The second category involves input data in polar (r, theta) format and uses methods such as change of variables to perform the integral transform as a correlation integral. Experimental results obtained with optical implementation for these two categories are presented.