Stable optical resonator made of a pair of flat mirrors and a positive lens: numerical analysis (original) (raw)
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A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors
Applied Optics, 1970
A paraxial resonance equation is derived. This gives the mirror separation as a function of the radii of curvature of the mirrors and an integer N which is the number of return transits necessary to form a closed path of rays. Differentiating the paraxial resonance equation gives a formula for the relative mode density as a function of mirror separation. It is shown that the output power from a laser incorporating solid mirrors is inversely proportional to the mode density. In the case of hole coupling, the output power follows the same general profile but dips in power occur at the mirror separations corresponding to the resonance configurations of modes characterized by low values of N. Further confirmation of the paraxial resonance equation is obtained from passive resonators in which conic interference fringes and sudden increases in transmitted intensity are found to occur at the predicted mirror separations for low values of N corresponding to mode-degenerate configurations. The positions of the vertices of the ray traces are found to correspond to the patterns of discrete spots which are obtained in the output of a CO 2 laser incorporating Brewster angle windows and a solid germanium mirror. The laser configurations which give maximum output power are plotted as a cliff of constant height above the gig2 plane of the stability diagram, where g, and g2 are the configuration coordinates. The relative merits of all possible cavity configurations having one mirror in common are shown as a set of equipower contours, and the hyperbolic curves of constant N are also superimposed on the stability diagram. The advantages of simplicity and directness in using the ray model are made clear.
Unstable-stable resonators with toroidal mirrors
Applied Optics, 1981
A resonator with toroidal mirrors is described. This resonator behaves like an off-axis unstable confocal resonator in one transverse dimension and like an on-axis concave-convex stable resonator in the other orthogonal dimension. Some experimental results are reported for a fast flow high power cw-CO 2 laser whose transverse cross section is restricted in the stable-resonator direction. These cavities allow an output laser beam with a fully illuminated cross section which is well suited for focusing. Moreover, the fraction of the available laser power which may be concentrated in the central lobe of the focal plane intensity distribution is 2.5-4.5 times higher than other unstable resonators with similarly restricted modal volumes. Finally the alignment requirements are examined.
Active Laser Resonator Performance: Formation of a Specified Intensity Output
Applied Optics, 2001
We discuss the formation of a specified super-Gaussian intensity distribution of a fundamental mode by means of an intracavity controlled mirror, which is a water-cooled bimorph flexible mirror equipped with four controlling electrodes. Analysis has confirmed the possibility to form fourth-, sixth-, and eighthorder super-Gaussian intensity distributions at the output of the stable resonators of industrial cw CO 2 and YAG:Nd 3ϩ lasers. We present the results of the experimental formation of fourth-order and sixthorder super-Gaussian fundamental modes at the output of a cw CO 2 laser by means of an intracavity flexible mirror. We observed an increase in power up to 12% and an enlargement of the peak value of the far-field intensity by as much as 1.6 times that with a Gaussian TEM 00 mode of the cw CO 2 laser.
Numerical simulation of the w-axicon type optical resonator for coaxial slab CO 2 lasers
Journal of Physics D: Applied Physics, 2001
The numerical simulation of the optical resonator for a coaxial slab CO 2 laser is conducted. The resonator consists of a toric mirror, a w-axicon, and a plane output coupler. To hold the two slabs coaxially, struts exist in the annular gap region. The effects of the struts on the output power and beam quality for the given resonator are discussed. It is found that varying the vertex of the toric mirror can control the oscillation mode between the E 00 fundamental mode and the E 01 first azimuthal mode. It is revealed that the resonator loss of the fundamental mode can be smaller than the first azimuthal mode by shifting the vertex inward, and this results in single-mode oscillation at the E 00 fundamental mode. On the other hand, the maximum output is obtained under conditions where the resonator oscillates at the E 01 mode, because the nodes of the mode coincide with the position of the struts. Loss measurement of each eigenmode well describes the aforementioned phenomena.
IEEE Journal of Quantum Electronics, 1979
We present an analysis of resonator properties for a cavity bounded by a phase conjugate mirror, which is generated by a degenerate four-wave nonlinear optical interaction. Using a ray matrix formalism to describe the conjugate mirror, resonator stability conditions are derived. Longitudinal and transverse mode characteristics are discussed. Results are compared with an experiment where laser oscillation was observed at
Mode Selectivity and Mirror Misalignment Effects in Unstable Laser Resonators
Applied Optics, 1972
It is known that single transverse mode operation of an unstable laser resonator can be achieved using large output coupling and large Fresnel number operation. However, it has not been recognized that there is an upper limit on usable Fresnel number, determined by such effects as mirror alignment stability. We have analyzed a variable-reflectivity unstable resonator using mode solutions obtained by Zucker. The differential loss between the lowest order modes leads to conditions on the Fresnel number and the output coupling if single mode operation is to be realized. The Zucker solution is extended by a perturbation analysis to examine mode distortion caused by mirror misalignment. The on-axis intensity in the far field of the laser output is shown to be a sensitive function of mirror misalignment, and the mirror alignment tolerance permitted varies inversely as the Fresnel number. In practice, this places an upper limit on the usable Fresnel number. The results of this theory are shown to be consistent with laser experiments using unstable resonators.
Recent developments in laser resonator design
Optical and Quantum Electronics, 1991
The problems related to resonators suitable for generation of diffraction-limited beams of high power or energy, and a few of the most significant recent solutions, are reviewed. In particular, this paper is addressed to two promising resonator configurations developed mainly for Nd : YAG (yttrium aluminium garnet) lasers: dynamically stable resonators of minimum misalignment sensitivity for lasers with a strong thermal lensing in the active rod and unstable resonators with variable reflectivity output mirrors of super-Gaussian profile. For both cases experimental data and simple design guidelines are discussed.