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Numerical propagation of light beams in refracting/diffracting devices Jean-Yves VINET Observatoire de la Côte d’Azur (Nice, France) KASHIWA-October-2011 J.-Y. Vinet 1 Summary •Needs for optical simulations •General principles of numerical propagation : several methods •Some examples : •Fourier Transform •Hankel Transform •Modal •Monte-Carlo •Advantages/drawbacks KASHIWA-October-2011 2 J.-Y. Vinet Needs for Optical simulations in GW interferometer design 1) Sensitivity of a GW interferometer is strongly dependent on the quality of the Fabry-Perot cavities -Efficiency of power recycling -Power in sidebands 2) Quality of Fabry-Perot’s depends on the quality of the mirrors 3) Mirrors are not perfect ! Requirements are needed for manufacturers 4) Heated mirrors change of internal/external properties KASHIWA-October-2011 J.-Y. Vinet General principles of Propagation Methods Expand optical field on a family of functions of which propagation is well known •Plane waves •Bessel waves •Gaussian modes (eg. HG or LG) •Photons KASHIWA-October2011 J.-Y. Vinet 4 Propagation by Fourier Transform : General principles KASHIWA-October-2011 J.-Y. Vinet 5 Paraxial diffraction theory Maxwell+single frequency Helmholtz : Slowly varying envelope : 2x 2y 2z 2 / c 2 E ( , x, y, z ) 0 E(, x, y, z) exp ikz F (, x, y, z) and z F kF 2 k c Paraxial diffraction equation ( math. analogous to Heat-Fourier and to Schrödinger eq. ) : 2x 2y 2ik z F ( , x, y, z ) 0 2D Fourier Tr. : f ( p, q) f ( x, y )eipx eiqy dxdy R2 2ik z p 2 q 2 F ( , p, q, z ) 0 p2 q2 F (, p, q, z1 ) exp i ( z1 z0 ) F (, p, q, z0 ) 4 propagator KASHIWA-October-2011 J.-Y. Vinet 6 6 Propagation by Fourier Transform Diffraction over z A( x, y, z0 ) A( x, y, z0 z ) FT-1 FT p2 q2 exp i z 2 k A( p, q, z0 ) A( p, q, z0 z) Use of Discrete Fourier Transform (in practice : FFT) x=0 x=W x-window 1 p=0 2 3 x W / N N p pmax N / W p-window Positive frequencies KASHIWA-October-2011 N/2 J.-Y. Vinet p 2 / W Negative frequencies 7 Mode of a Fabry-Perot cavity E A L B E’ M1 Implicit equation : M2 E T1 A ( R1 PL R2 PL ) E xx 2 y y 2 Ra ra exp 2ik2ik ) 2) R r exp f ( x, yf) a ( x(,a y 1, 2 2 a 2 Mirror operators in xy plane a 2 a a Curvature radius Ta ta exp ikha ( x, y) Optical thickness Propagation KASHIWA-October-2011 (a 1, 2) a Measured roughness (Lyon’s surface charts) propagator PL X F -1 P.F X J.-Y. Vinet 8 Solution by simple relaxation scheme : En T1 A C En1 With initial guess : E0 C M1 P M 2 P t1 TEM 00 1 r1r2 Large number of iterations if large finesse and/or large defects Accelerated convergence (a la Aitken): En n1En1 n1 (T1 A C En1 ) En' of simple relaxation With optimal choice of n , n at each iteration See e.g. : Saha, JOSA A, Vol 14, No 9, 1997 KASHIWA-October-2011 J.-Y. Vinet 9 Black fringe :( ideal TEM00) – (TEM00 reflected by a Virgo cavity) 10-8 W/W 2 x perfect 35cm mirors 30 cm KASHIWA-October-2011 J.-Y. Vinet 10 Propagation by Bessel Transform : General principles Suitable for axisymmetrical problems Fourier Transform : 1 f ( p, q ) dxdy exp i ( px qy ) f ( x, y ) 2 Assume (axial symmetry) : f ( x, y ) f r x 2 y 2 then x r cos , y r sin , f ( ) p cos , q sin 1 rdrd exp i r cos( ) f (r ) J 0 ( r ) f (r )rdr 2 Bessel Transform KASHIWA-October-2011 J.-Y. Vinet 11 Inverse B transform : f (r ) J 0 ( r ) f ( ) d Assume Let f (r ) r a negligible for , 1,..., be the zeros of J1 (r ) (r ) J 0 ( r / a) are a complete, orthogonal family on 0, a Sturm-Liouville theorem : the a (r ) (r )rdr p , 0 So that f (r ) 1 KASHIWA-October-2011 f (r ) p a2 2 p J 0 ( ) 2 a with f f (r ) (r )rdr 0 J.-Y. Vinet 12 J1 ( x ) The first 20 zeros of J1 ( x) x Example of a sampling grid with 20 nodes 0. 1 xi i a / N KASHIWA-October-2011 J.-Y. Vinet a N 13 f (r ) 1 r a / N f (r ) p f f (r ) 1 f 2 (r ) 2 2 J0 p 1 a J 0 ( ) N Reciprocal F transform : f H( ) f 1 with H( ) Direct F transform : 2J0 2 2 N a J 0 ( ) 2a J 0 N N2 J 02 ( ) 2 () f H f 1 f () with H f f ( ) with a Direct and inverse Bessel transforms are done with explicit matrices KASHIWA-October-2011 J.-Y. Vinet 14 Propagator in the Fourier space over distance z : z 2 2 P( p, q, z ) exp i p q 4 p2 q2 2 In the Fourier-Bessel space : After sampling : a z 2 P (z ) exp i 2 4 a Diffraction step by a simple matrix product : 1 1 () ( z z ) P (z) with P (z) H( ) P (z) H To be computed once KASHIWA-October-2011 J.-Y. Vinet 15 Example : propagation of a TEM00 over 3000 m Initial wave Diffraction theory Bessel propagated KASHIWA-October-2011 J.-Y. Vinet 16 Representation of mirrors Axially symmetrical defects : diagonal operator 4i M r exp r2 f (r ) 2 Rc Pure parabolic contribution a with r N defects Reflected field : ' M KASHIWA-October-2011 J.-Y. Vinet 17 Example : reflectance of a Fabry-Perot cavity A E L B Intracavity field : M1 M2 E t1 A e2ikL M 1P( L) M 2 P( L) E C Matrix operator 1 Intracavity field by matrix inversion : E Id e Reflected field by matrix product : B R A With the reflectance operator R M t1 PM 2 P Id e KASHIWA-October-2011 J.-Y. Vinet † 1 C t1 A 2ikL 1 C t1 2 ikL 18 Modal propagation : general principles The set of all complex functions ( x, y ) of integrable square modulus has the structure of a Hilbert space, with a scalar product , * dxdy ( x, y)( x, y) 2 An example of a basis of such a HS is the Hermite-Gauss family of optical modes x2 y 2 2 x 2 y 2 x y nm ( x, y) nm H n 2 H m 2 exp exp i 2 w w w R So that any optical amplitude can be expanded in a series of HG modes A( x, y ) Anm nm ( x, y ) m,n KASHIWA-October-2011 J.-Y. Vinet 19 Propagation of a HG mode of parameter (waist) w0 : 2P lm ( x, y, z ) w( z )2 1 r2 r2 exp ik iGlm ( z ) mn 2 2 m !n ! 2 R( z ) w( z ) x y H l 2 H m 2 w w Rayleigh parameter : Beam width : b w02 / w( z ) w0 1 ( z / b) 2 Curvature radius of the wavefront : Gouy phase KASHIWA-October-2011 b2 R( z ) z z Glm ( z ) (l m 1) arctan( z / b) J.-Y. Vinet 20 Diffraction of a Gaussian beam w( z ) w0 1 ( z / b) 2 w0 z 0 KASHIWA-October-2011 J.-Y. Vinet z b2 R( z ) z z 21 HG01 HG22 HG55 HG05 KASHIWA-October-2011 J.-Y. Vinet 22 Representation of mirrors by their matrix elements M abcd ab , M cd M abcd ab , M cd Modal expansion widely used by Andreas Freise’s « Finesse » package Propagation of light in complex structures by Monte-Carlo photons Principle : send random pointlike particles from identified sources Scattered light « Main beam » Rough mirror surface Reflection of a photon k n k' k ' k 2(k .n ) n Refraction of a photon k n 1 k' N k' k k .n N 2 1 (k .n ) 2 n Diffusion of a photon ' Rough surface Random variable with a PD that mimics the BRDF of the material Diffraction of photons ? Example 1 : Propagation of a beam target source Probability Density of direction 2 dP 1 2 lm ( , ) d Probability Density of emission point dP 2 lm ( x, y ) dS KASHIWA-October-2011 lm ( p, q) p 2 sin cos , q 2 sin sin lm ( , ) J.-Y. Vinet 29 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] KASHIWA-October-2011 J.-Y. Vinet 30 Example 2 : Management of diffraction by obstacles Emission of photons x target screen p.x , p k : Centered random deviate of standard deviation KASHIWA-October-2011 J.-Y. Vinet 4x * arctan 31 Example 2 : diffraction by an edge Screen at 5m Histogram : Monte-Carlo Diffraction theory (Fresnel Integral) transverse distances [m] KASHIWA-October-2011 J.-Y. Vinet 32 Conclusion *FFT propagation : general purpose codes (DarkF), suitable even for short spatial wavelength defects of mirrors •Propagation by Bessel transform : suitable for axisymmetrical problems (eg. heating by axisymmetrical beams) •Propagation by modal expansion : ideal for nearly perfect instruments, small misalignments, small ROC errors, etc…. •Photons : mandatory for propagation of scattered light in complex structures (vacuum tanks, etc…)