![abcd matrix abcd matrix](http://kilyos.ee.bilkent.edu.tr/~microwave/programs/magnetic/hybrid/ringtheory_files/image020.gif)
to compute the electric field after reflection on a parabolic surface but no explicit and easy-to-use expression is provided.
![abcd matrix abcd matrix](https://image3.slideserve.com/6080995/abcd-matrix-example-11-l.jpg)
The wave-front matching method is indeed employed in Ref. To our knowledge, the general ABCD matrix for a parabolic reflector, valid for any angle of incidence, has not yet been published. The paraxial approximation is assumed to hold so that a 4×4 ABCD matrix formalism, can be used. The feasibility of this kind of cavity is studied in this paper by means of simulations. In bow-tie configuration the cavity length and the waist size can be tuned independently of the beam shape for a given incidence angle on the mirrors. However they are difficult to align and general astigmatism can anyhow be introduced by angular misalignments. Parabolic reflectors are a natural choice since once perfectly aligned they do not induce astigmatism.
#Abcd matrix free
This type of application requires an impulse regime with high average and peak power (astigmatism free beam-mode), the maintaining of polarization, a small beam-waist size (few tens of microns to match the electron-beam size) and a few meter optical path length (to match the repetition rate of the electron bunches, typically ranging from a few tens megahertz to few hundreds megahertz). To the best of our knowledge this has never been done.
![abcd matrix abcd matrix](http://d2vlcm61l7u1fs.cloudfront.net/media%2F7a2%2F7a2eace1-d094-42ae-b208-4d928015f8d6%2FphpARRcxj.png)
The purpose of this paper is to study the alternative possibility of using parabolic mirrors instead of spherical mirrors in the context of laser Compton backscattering off electrons. For a given fluence damage threshold the resonator geometry must be tuned to minimize the fluence impinging on the mirrors in order to maximize the stacked power. But its main effect is to increase the beam fluence on the mirror surfaces as a consequence of the ellipticity.
![abcd matrix abcd matrix](https://image3.slideserve.com/5399078/abcd-matrix-cont3-l.jpg)
To some extent, beam astigmatism can be an issue for some of the above-mentioned applications. However, the non-vanishing incident angles on the reflective spherical surfaces induce astigmatism and a large ellipticity as the beam waist decreases. They also permit quasi-independent tuning of the cavity-mode beam-waist size and round-trip length. Such cavities can provide stable Gaussian modes with at least one small beam-waist. we focus on four-mirror cavities in planar geometries. In these applications one must consider four-mirror cavities made of at least two concave reflective surfaces, or new optical resonator developments. There is also no flexibility to vary the cavity-mode beam-waist size and the cavity round-trip length independently which can also be an issue in some cases. However, in these configurations the odd number of reflective surfaces acts as a linear polarizer filter which can be incompatible with some applications. Three-mirror cavities either consisting of three concave or two concave and one flat reflective surfaces can thus be considered. These remarks rule out two-mirror cavities. The two-mirror cavities made of one concave reflector do not exhibit a very small beam-waist between the two mirrors, the small beam-waist is instead located on the flat mirror and thus cannot be used to some applications. As two-mirror cavities made of two concave reflectors have to be in a concentric configuration to exhibit a very small beam-waist they are mechanically very unstable. The number of reflective surfaces inside the resonator must be minimized to reduce the round-trip losses and the cavity mode must be stable while exhibiting at least one very small beam-waist. However, since the mode properties of these resonators solely depend on their geometries, specific optical designs must be supplied to fulfill the requirements of these applications. Fabry–Perot cavities are nowadays used to stack high average- and peak-power pulsed laser-beams in various applications: quasi-monochromatic X/ γ ray beam production, ,, high harmonic generation, polarized positron production, nuclear waste management, γ – γ collisions.