Numerical propagation of light beams in refracting / diffracting devices

1. Numerical propagationof light beams inrefracting/diffractingdevices Jean-Yves VINET Observatoire de la Côte d’Azur (Nice, France) J.-Y. Vinet

2. Summary • Needs for optical simulations • General principles of numerical propagation : severalmethods • Someexamples : • Fourier Transform • Hankel Transform • Modal • Monte-Carlo • Advantages/drawbacks J.-Y. Vinet

3. Needs for Optical simulations in GW interferometer design • 1) Sensitivity of a GW interferometerisstronglydependent on the quality of the Fabry-Perotcavities • Efficiency of power recycling • Power in sidebands • 2) Quality of Fabry-Perot’sdepends on the quality of the mirrors • 3) Mirrors are not perfect ! Requirements areneeded for manufacturers • 4) Heatedmirrors change of internal/externalproperties J.-Y. Vinet

4. General principles of Propagation Methods Expandopticalfield on a family of functions of which propagation iswellknown • Plane waves • Bessel waves • Gaussian modes (eg. HG or LG) • Photons J.-Y. Vinet

5. Propagation by Fourier Transform : General principles J.-Y. Vinet

6. Paraxial diffraction theory Maxwell+single frequency Helmholtz : Slowlyvaryingenvelope : Paraxial diffraction equation ( math. analogous to Heat-Fourier and to Schrödinger eq. ) : 2D Fourier Tr. : propagator J.-Y. Vinet 6

7. Propagation by Fourier Transform Diffraction over FT-1 FT Use of Discrete Fourier Transform (in practice : FFT) x=0 x=W x-window 1 2 3 N p=0 p-window Positive frequencies N/2 Negativefrequencies J.-Y. Vinet

8. Mode of a Fabry-Perotcavity E A L B E’ Implicitequation : Mirror operators in xy plane Curvature radius Measuredroughness (Lyon’s surface charts) propagator Optical thickness Propagation J.-Y. Vinet

9. Solution by simple relaxation scheme : With initial guess : Large number of iterations if large finesse and/or large defects Accelerated convergence (a la Aitken): of simple relaxation With optimal choice of ateachiteration Seee.g. : Saha, JOSA A, Vol 14, No 9, 1997 J.-Y. Vinet

10. Black fringe :( ideal TEM00) – (TEM00 reflected by a Virgocavity) 10-8 W/W 2 x perfect 35cm mirors 30 cm J.-Y. Vinet

11. Propagation by Bessel Transform : General principles Suitable for axisymmetricalproblems Fourier Transform : Assume (axial symmetry) : then Bessel Transform J.-Y. Vinet

12. Inverse B transform : Assume negligible for be the zeros of Let Sturm-Liouvilletheorem : the are a complete, orthogonal family on So that with J.-Y. Vinet

13. The first 20 zeros of Example of a samplinggridwith 20 nodes 0. 1 N J.-Y. Vinet

14. Reciprocal F transform : Direct F transform : Direct and inverse Bessel transforms are donewithexplicit matrices J.-Y. Vinet

15. Propagator in the Fourier space over distance Dz : In the Fourier-Bessel space : Aftersampling : Diffraction step by a simple matrixproduct : To becomputed once J.-Y. Vinet

16. Example : propagation of a TEM00 over 3000 m Initial wave Diffraction theory Bessel propagated J.-Y. Vinet

17. Representation of mirrors Axiallysymmetricaldefects : diagonal operator Pure parabolic contribution defects Reflectedfield : J.-Y. Vinet

18. Example : reflectance of a Fabry-Perotcavity A E B Intracavityfield : Matrixoperator Intracavityfield by matrix inversion : Reflectedfield by matrixproduct : With the reflectanceoperator J.-Y. Vinet

19. Modal propagation : generalprinciples The set of all complexfunctions of integrable square modulus has the structure of a Hilbert space, with a scalarproduct An example of a basis of such a HS is the Hermite-Gauss family of optical modes So thatanyoptical amplitude canbeexpanded in a series of HG modes J.-Y. Vinet

20. Propagation of a HG mode of parameter (waist) : Rayleigh parameter : Beamwidth : Curvature radius of the wavefront : Gouy phase J.-Y. Vinet

21. Diffraction of a Gaussianbeam z J.-Y. Vinet

22. HG01 HG22 HG55 HG05 J.-Y. Vinet

23. Representation of mirrors by theirmatrixelements Modal expansion widelyused by Andreas Freise’s « Finesse » package

24. Propagation of light in complex structures by Monte-Carlo photons Principle : sendrandompointlikeparticles fromidentified sources Scattered light « Main beam » Rough mirror surface

25. Reflection of a photon

26. Refraction of a photon

27. Diffusion of a photon Random variable with a PD thatmimics the BRDF of the material Rough surface

28. Diffraction of photons ?

29. Example 1 : Propagation of a beam target source ProbabilityDensity of direction ProbabilityDensity of emission point J.-Y. Vinet

30. Monte-Carlo methods Example 1 : propagation of a TEM00 over 3000 m w0=2cm Initial wave : MC Analytical initial TEM MC propagated Diffraction theory Radial coord. [m] J.-Y. Vinet

31. Example 2 : Management of diffraction by obstacles Emission of photons target screen : Centeredrandomdeviate of standard deviation J.-Y. Vinet

32. Example 2 : diffraction by an edge Screenat 5m Histogram : Monte-Carlo Diffraction theory (Fresnel Integral) transverse distances [m] J.-Y. Vinet

33. Conclusion *FFT propagation : generalpurpose codes (DarkF), suitableeven for short spatial wavelengthdefects of mirrors • Propagation by Bessel transform : suitable for axisymmetrical • problems (eg. heating by axisymmetricalbeams) • Propagation by modal expansion : ideal for nearly • perfect instruments, • smallmisalignments, small ROC errors, etc…. • Photons : mandatory for propagation of scattered light • in complex structures (vacuum tanks, etc…)