Considerations about self-filtering unstable resonators (original) (raw)
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More about the self-filtering unstable resonators
Optics Communications vol. 209, pp. 383-389 (2002)
We present a comparison between the Self-Filtering Unstable Resonator (SFUR) and the Generalised Self Filtering Unstable Resonator (GSFUR) based on some unpublished relationships among GSFUR parameters. Calculations show that, when the resonator magnification M is less than 14, depending on some geometrical parameters, the SFUR may give a larger mode volume (and then a larger output energy) than GSFUR, independently of the cavity length. This unexpected result can help to choose the ‘‘best’’ resonator between SFUR and GSFUR in order to exploit as much as possible the active medium with a rapid steady-state establishment of the transverse mode structure
Experimental study of a generalized self-filtering unstable resonator applied to an XeCl laser
IEEE Journal of Quantum Electronics (ISSN 0018-9197), vol. 24, Nov. 1988, p. 2284-2287., 1988
A diffraction-limited laser beam of 3.5 mJ with a pulse width of 11 ns and a brightness of 4.8 x 10^13 W/sq cm/Sr has been obtained by applying a generalized self-filtering unstable resonator (SFUR) scheme to a short-pulse XeCl laser. In accordance with the theory, the generalized SFUR maintains the excellent properties of the SFUR, such as high transverse-mode discrimination, fast establishment of steady-state condition, and diffraction-limited output beam, but offers the possibility of having higher magnification and higher mode volumes with shorter resonator lengths. It is possible to demonstrate that with the generalized SFUR, one can increase, by a factor more than two, the mode volume with respect to the confocal case by leaving the cavity length unaltered. These properties make the nonfocal SFUR particularly suitable for short-pulse excimer lasers.
Numerical characterizations of unstable optical resonators and evaluation of the geometry effects
Optics & Laser Technology, 2007
Unstable resonators have been widely used in high-power gas lasers as well as solid-state lasers. The phase and the spatial distribution of intensity of these lasers are very important in some applications such as material processing. In this paper, unstable resonators with three different geometries have been characterized numerically and the results have been compared and evaluated. Based on the Fresnel-Kirchhoff integral, the two-dimensional phase and intensity have been calculated for three different positive branch unstable resonators. The results show that the resonators with rectangular geometry have the best performance for near-field as well as far-field intensity, which is more suitable for material processing. The calculations also show that the maximum output power can be extracted from the rectangular resonator with spherical surface, while the circular resonator with spherical surface has minimum output power. The results also show that the laser has a higher divergence for the cylindrical resonator in compare with those for the circular and rectangular resonators. r
Optics Communications, 2008
Field characteristics and beam quality of confocal positive branch unstable resonator under some practical imperfections has been investigated. Based on Fresnel-Kirchhoff integral, the near and far field intensity of the resonator was calculated numerically. Using the second order moment method, the beam quality factor, M 2 of the resonator was also calculated. Variety of practical distorting affects, mainly misalignment, off-axis, and thermal deflection of the mirrors, as well as gain non-uniformity were considered in these calculations. The results show that the beam quality factor is more sensitive to the off-axis and misalignment distortions while the non-uniform gain has significant effect only for unusual conditions. The effects of geometrical parameters, i.e. Fresnel number as well as magnification factor on the beam quality were also investigated.
An alternative method to specify the degree of resonator stability
Pramana, 2007
We present an alternative method to specify the stability of real stable resonators. We introduce the degree of optical stability or the S parameter, which specify the stability of resonators in a numerical scale ranging from 0 to 100%. The value of zero corresponds to marginally stable resonator and S < 0 corresponds to unstable resonator. Also, three definitions of the S parameter are provided: in terms of A&D, B&ZR0 and g 1 g 2 . It may be noticed from the present formalism that the maximum degree of stability with S = 1 automatically corresponds to g 1 g 2 = 1/2. We also describe the method to measure the S parameter from the output beam characteristics and B parameter. A possible correlation between the S parameter and the misalignment tolerance is also discussed.
Unstable Resonators with Variable Reflectivity Mirrors Geometrical Optics Spherical cavity
In an unstable resonator, the transverse electric field profile is magnified by a factor M on each round trip. For a confocal resonator with spherical mirrors, if the field reflected off the output mirror (shorter radius of curvature) is E(x,y) then the field incident on the output mirror after one round trip is E(x/M,y/M)/M (The longer radius mirror is a total reflector). This represents a scale change in both dimensions and a reduction in the field strength by one over M. (The intensity is reduced by 1/M 2). If the output coupler is assumed to have a reflectivity profile, R(x, y), then the field reflection coefficient is and the transverse resonator modes are defined by the eigenvalue equation where γ is the complex eigenvalue and E(x,y) is the field incident on the output coupler. The lowest order eigenmode can be found by the Fox-Lee method, starting with a uniform field of strength E 0. The field profile after n passes through the resonator is Note that If M > 1 and ρ(x,y) is continuous, then as the factor multiplying the field on each round trip converges to ρ(0,0)/M, which is the eigenvalue of the lowest order mode, γ 0. The cavity feedback, defined as , is then , where R 0 = R(0,0). Thus the feedback depends only on the center reflectivity and the magnification of the resonator and not on the details of the variable reflectivity profile. For a reflectivity profile which is symmetric and goes to zero at infinity, e.g. the lowest order cavity mode is The intracavity mode profile depends only on the resonator magnification and the shape of the reflectivity profile. It is independent of the length of the resonator and of the magnitude of the reflectivity. This is true for an arbitrary reflectivity distribution. The output mode profile, on the other hand, will in general depend on the magnitude of the reflectivity. If the only nonzero coefficients above are c 10 = 1/a x 2 and c 01 = 1/a y 2 then the reflectivity and the lowest order cavity mode are Gaussian, i.e. A closed form solution for the field is possible with variable reflectivities containing non-integer exponents of the coordinates as well. For example, with a reflectivity profile which is super-Gaussian. Long
Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element
IEEE Journal of Quantum Electronics, 2001
A graded-reflectivity-mirror (GRM) unstable resonator with low output coupling is described, where a custom-made optical phase element is used inside the resonator to provide maximally flat output. The phase element removes the dip in the output beam by pre-compensating the internal Gaussian mode. An experiment is performed with a flashlamp-pumped Nd:YAG laser. The resonator's magnification () and the GRM's central reflectivity (0) are 2.3 and 0.7, respectively. The large dip in the center of the output is removed using the custom-made phase element. This resonator has the advantage over a conventional GRM unstable resonator of being suitable for lower-gain laser media. The gain required to overcome fundamental mode cavity losses for maximally flat output is decreased from 22.3 (for a conventional GRM resonator) to 4.3 (for the resonator containing the phase element). This reduction in required gain comes with essentially no loss to the resonator modal discrimination.
Unstable resonators with a distributed focusing gain
Applied Optics, 1994
The geometrical optics approximation is used to form a model of axisymmetric unstable resonators having distributed focus, gain, and loss. A tapered reflectivity feedback mirror is included. The rate equations for propagation through the focusing gain medium are derived. A unique grid is found for propagation without interpolation along eigenrays in each direction. Numerical examples show the effects of distributed gain and focus on the axial and transverse intensity distributions.