ABCD Matrix Formalism Research Articles

Thick lens model for self-focusing in Kerr medium

An asymmetric “thick-lens” model for self-focusing in Kerr medium of finite thickness is presented by introducing a complex curvature radius of a Gaussian beam in nonlinear medium and a transfer matrix for the transition from linear to nonlinear medium. With this model the linear ABCD-matrix formalism can be extended to nonlinear optical systems with a clear physical insight. The modulation of the cavity transverse mode of a resonator containing a Kerr medium is characterized by the asymmetry of the “thick lens.” The intracavity small-signal relative spot size variation outside as well as inside the Kerr medium can be calculated efficiently with the “thick-lens”-like matrix formalisms. Criteria for the design and optimization of a Kerr-lens mode-locked laser resonator with a hard or soft aperture are predicted.

Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission

ABCD matrix formalism in the time domain has been newly developed on the basis of laser beam and resonator analyses which were developed under a Gaussian paraxial approximation. We derive matrix elements for amplitude and frequency modulations, group velocity dispersion, optical bandpass filter dispersion, and lumped self-phase modulation under a parabolic approximation. Applications to AM, FM, and stretched-pulse laser mode-locking are described by using these time-domain matrices. An application to pulse transmission in a dispersion-allocated system is also described.

Kerr-lens mode locking without dispersion compensation

We propose and theoretically investigate a novel operating regime of femtosecond Kerr-lens mode-locked solidstate lasers that avoids group-velocity dispersion compensation by use of a nonresonant semiconductor plate in the focused resonator section that provides an overall negative nonlinear refractive index per round trip. The saturable loss of the laser resonator with an effective self-defocusing nonlinearity is derived from a generalized ABCD matrix formalism, and the correspondingly calculated steady-state pulse parameters show that a Kerrlens mode-locked laser with an overall negative nonlinear refractive index generates stable femtosecond pulses without any dispersion compensation.

ABCD matrix formalism of fractional Fourier optics

The use of the ABCD matrix formalism to study fractional Fourier transforms (FRFTs) opens new perspectives on the optical implementation of the FRFTs and on the analysis and synthesis of optical systems. ABCD matrices of basic geometries implementing perfect and imperfect FRFTs are presented. The ABCD matrix method is used to demonstrate the cascading performances of FRFT elementary units and to support the established cascading rules. The matrix formalism is used to show that real and complex optical systems realize FRFTs of complex order.

Complex ray tracing of nonlinear propagation

A new scale transformation of complex ray parameters is introduced in an unified manner to treat the nonlinear propagation in Kerr media. It is shown that the nonlinear propagation, including soliton propagation, can be consistently traced by complex rays specific to the linear propagation with the imposed scale effects of self-shortening, self-focusing and phase self-modification. The solution obtained is explicitly described as the propagation in a free-space modified by three propagation effects, namely, the nonlinear changes of the beam or pulse waist position, waist width and on-axis phase. A field of cylindrical symmetry and arbitrary transverse dimension is analyzed for power levels ranging from the linear propagation to the soliton self-trapping. It is shown that the method can be formulated within a frame of the nonlinear ABCD matrix formalism and specified to comply with the variational analysis of the problem.

Astigmatism and gain guiding in Kerr-lens mode-locked lasers

The effects of astigmatism and the radial variation of the gain on Kerr-lens mode-locking (KLM) are studied with the help of an extended ABCD matrix formalism. A dramatic broadening of the allowed cavity parameter region is predicted that permits KLM in almost the whole stability region. The cavity design for KLM with an aperture differs significantly from the design for KLM without an aperture, because of the saturable diffraction loss.

Geometrical threshold zones and Gaussian modes in lasers with radially varying gain

The effect of a radially varying gain in tightly focused lasers is investigated with a generalized ABCD matrix formalism in a nonparabolic approximation. It is shown that the beam sizes can differ strongly from the theoretical with a constant gain or with radially varying gain in a parabolic approximation. Geometrical threshold zones are defined and studied and, instead of the stability zones, determine the possibility of a laser action for a given cavity design.

Theory of Kerr-lens mode locking: role of self-focusing and radially varying gain

The ABCD-matrix formalism is generalized for the description of self-focusing and radially varying pump power in a nonparabolic approximation. This extended formalism is used for the investigation of the operation and for the optimization of the saturable aperture loss, the power-dependent average gain, and the resonator magnification of Kerr-lens mode-locked lasers with as well as without an internal aperture. The calculated amplitude modulation parameters are compared with the necessary condition for self-starting and for the shortest possible pulse duration limit.

Multirod unstable resonators for high-power solid-state lasers

The properties of positive-branch and negative-branch unstable resonators with variable reflectivity mirrors and several variable internal lenses were investigated both theoretically and experimentally. Design rules for optimized unstable resonators for one or more active elements are derived on the basis of the ABCD matrix formalism. Experiments were performed with a pulsed Nd:YAG system consisting of three 6 in. × 3/8 in. (15.24 cm × 0.95 cm) rods. This system provided a maximum output power of 550 W per rod when a symmetric flat-flat resonator was used. Unstable resonators achieved up to 75% of this maximum value with beam-parameter products between 2 and 10 mm mrad. The beam quality becomes worse as more active elements are used inside the resonator. This deterioration of focusability is caused by spherical aberration in combination with differences of refractive power for r and Φ polarizations.

Matrix formalism for pulse compressors

A matrix formalism for grating and prism compressors is introduced. It is an extension of the standard ABCD matrix formalism for Gaussian optics, but includes the possibility of considering dispersive elements such as gratings and prisms. The full formation on the frequency-dependent phase shift is obtained to compute the time delay dispersion. The formalism simplifies the design of compressors with focusing optics. Information on beam divergence and astigmatism is also acquired. >