The ABCD matrix for parabolic reflectors and its application to astigmatism free four-mirror cavities Outline 2 Motivations Geometrical compensation of ellipticity (with spherical mirrors involved) Symmetry considerations Numerical solutions Compensation of ellipticity with mirror shape (with parabolic mirrors) ABCD matrix for parabolic mirror

Parabolic mirror cavities example(4-mirror cavities) 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Recent Developments 3 Increase of the cavity stacked power more than 670 kW (H. Carstens OL 39(2014)9) Burst mode development (K. Sakaue NIMA 637 (2011) S107-S111) Increase of laser beam power up to [email protected] for passive cavity (B.A. Reagan OL 37(2012)17) Increase interest on (Compton) X/-ray machine with optical cavity X-ray for material science, medical, etc.

-ray machine for photonuclear physics, particle physics, etc. 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Optical considerations for Compton -ray beam production 4 Requirements Polarization switching (P/S) High -ray flux High intensity laser beam Constraints Even number of reflective surfaces

Small waist (~30m)m) High laser-cavity coupling large beam size nearly collimated at the injection Reasonable cavity length (few meters: ~100MHz) No ellipticity (on mirrors) Mechanically stable (mode and beam path) Large laser beam area on optics => avoid Laser Damage Threshold 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL

28/07/2016 Starting point 5 Angle 0 on a spherical mirror => ellipticity (astigmatism) Stability For small waist 2 Consideration for optical cavities: Smaller waist => higher ellipticity (due to ) (if no compensation) Higher Ellipticity => smaller beam spot area on optics Smaller beam spot => higher fluence bb = max a

b | | + bab Smaller waist higher ellipticity higher fluence on optics 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Solutions 6 Ellipticity free cavity Geometrical compensation of ellipticity (e.g. T. Skettrup: J.Opt.A 7(2005)7) Compensation of ellipticity with mirror shape Telescope system (e.g. K. Mnig: NIMA 564(2006)212) Many optical surfaces

Stability to be studied 2 Cylindrical mirrors (4-mirror cavity) Tolerance on fabrication Adjustable ? K. Mnig: NIMA 564(2006)212 2 Parabolic mirrors (4-mirror cavity) Intrinsically not astigmatic 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Geometrical compensation of ellipticity: 7 Studies of spherical mirror cavities 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016

8 Symmetry considerations (with 4 spherical mirrors) tetrahedron configuration (I. Pupeza) planar ring configuration (T. Skettrup: J.Opt.A 7(2005)645) x M2 M3 z M4 M1 Unstable No ellipticity by construction Mechanically highly unstable Polarization effect (only circular polarization) (F. Zomer: Appl. Opt. 48(2009)35) With d1 = d3, d2 = d4

Mechanically unstable polarization effects (High incident angle: 45) Not adjustable 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Numerical solutions 9 Solutions of the equations introduced in T. Skettrup (J.Opt.A 7(2005)7) 4 spherical mirrors Bow Tie Cavity configuration (BTC) 2 spherical + 2 flat mirrors BTC configuration High ellipticity on Mirrors High ellipticity on Mirrors Very low coupling efficiency

(no collimated beam + untypical beam mode) Low coupling efficiency (no collimated beam) Long cavity Long cavity 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Summary of cavities compound of spherical mirrors 10 Always circular beam waist (even using spherical mirrors with non vanishing incident angle) Ring and tetrahedron geometry mechanically unstable Bow Tie Configuration :

4 spherical mirrors coupling issues Difficult to inject through spherical mirror = diverging lens Beam mode long cavity 2 spherical mirrors coupling issues long cavity High ellipticity on mirrors Use of stigmatic mirrors (e.g. parabolic mirrors) in BTC configuration with 2 concave mirrors + 2 flat mirrors 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Compensation of ellipticity with mirror shape :

11 Study of parabolic mirrors cavity 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Ellipticity free cavity 12 (with parabolic mirrors) Circular beam spot optical path pass through the 2 focal points of parabolic mirrors (stigmatic configuration) 2D 3D = ellipticity free configurations

4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 ABCD matrix for parabolic mirror 13 Details in K. Dupraz (Opt. Com. 353(2015)178-183) M. Sieber (Nonlinearity 11 (1998) 16071623) gives for any ellipsoidal surface: With: a) b) Where and the main radii of curvature of the surface and the angle made between the reflection plane and the main curvature .

It remains to calculate the two main radii , and the angle . 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL a) Side view b) Front view 28/07/2016 ABCD matrix for parabolic mirror 14 Details in K. Dupraz (Opt. Com. 353(2015)178-183) Normal vector to the surface: From Geometry Analysis: A point on a parabolic surface is expressed by With . The two first metric tensor are: As the Tensor are diagonal we get the two main radii of curvature:

With 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 2 parabolic mirrors + 2 flat mirrors cavity (design) 15 Details in K. Dupraz (Opt. Com. 353(2015)178-183) Parameters Value L (mm) 541,75 h (mm) 102 R (mm) 250 0 (m)m) 30

4 [ 0.1; 0.1 ] 4 Optically perfect Mechanically Stable Cavity length can be chosen 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL = 28/07/2016 2 parabolic mirrors + 2 flat mirrors cavity (Alignment) 16 Details in K. Dupraz (Opt. Com. 353(2015)178-183) Difficulty to bring the cavity to the working point (many

configurations available) alignment algorithm + observables (constraints on the cavity geometry) Iteration Start with large beam spot size (easy to align manually), then: o Act on the tilts (Tx,Ty) of and on the tilts (Tx,Ty,Tz) and the position (Dx,Dy) of , to reach non elliptic beam mode (and maintaining the same optical plane) o Act on the translation Dz of and simultaneously to reduce the beam spot size 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 17 2 parabolic mirrors + 2 flat mirrors cavity (results) Constraints: Even number of

reflective surfaces Small waist (~30m)m) No ellipticity (stigmatic mode with always ellipticity < 1%) Large laser beam area on optics => to avoid Laser Damage Threshold 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 General summary 18 In the way to 1MW stacked inside cavity already ~700 kW stacked (H. Carstens OL 39(2014)9). New consideration of the ellipticity for small waists in cavity compound of spherical mirrors New study on ellipticity free cavities with parabolic mirrors

Good numerical results are obtained for 2 parabolic mirrors + 2 flat mirrors cavities Experiment assembly in progress 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016 Thank you 19 4th International Conference on Photonics - Berlin - K. Dupraz - CNRS / LAL 28/07/2016