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Polarisation effects in 4 mirrors cavities •Introduction •Polarisation eigenmodes calculation •Numerical illustrations F. Zomer LAL/Orsay Posipol 2008 Hiroshima 16-19 june 1 3D: tetrahedron cavity 2D: bow-tie cavity V0 h~100mm h~100mm L~500mm L~500mm V0 V0 = the electric vector of the incident laser beam, What is the degree of polarisation inside the resonator ? Answer: ~the same if the cavity is perfectly aligned different is the cavity is misaligned numerical estimation of the polarisation effects is case of unavoidable mirrors missalignments 2 Calculations (with Matlab) • First step : optical axis calculation – ‘fundamental closed orbit’ determined using iteratively Fermat’s Principal Matlab numerical precision reached • Second step – For a given set of mirror misalignments • The reflection coefficients of each mirror are computed as a function of the number of layers (SiO2/Ta2O5) – From the first step the incidence angles and the mirror normal directions are determined – The multilayer formula of Hetch’s book (Optics) are then used assuming perfect lambda/4 thicknesses when the cavity is aligned. • Third step – The Jones matrix for a round trip is computed following Gyro laser and non planar laser standard techniques (paraxial approximation) 3 Planar mirror y V0 x y P1 k1 p2 S1 Planar mirror p1 k2 P2 s1 s2 z p2’ k3 s2 Spherical mirror S2 Spherical mirror ni is the normal vector of mirror i We have si=ni×ki+1/|| ni×ki+1|| and pi=ki×si/|| ki×si||, pi’=ki+1×si/|| ki+1×si||, where ki and ki+1 are the wave vectors incident and reflected by the mirror i. Example of a 3D cavity. Denoting by • Ri the reflection matrix of the mirror i • Ni,i+1 the matrix which describes the change of the basis {si,p’i,ki+1} to the basis {si+1,pi+1,ki+1} | rs | eis R 0 Er , s Ei , s , such R i p E E | rp | e r, p' i, p 0 With s≠p when mirrors are misaligned !!! rs ≠ rp when incidence angle ≠ 0 si si+1 p'i si+1 Ni ,i 1 s p p' p i i+1 i i+1 4 J R1 N41R4 N34 R3 N23 R2 N12 Taking the mirror 1 basis as the reference basis one gets the Jones Matrix for a round trip And the electric field circulating inside the cavity where V0 is the incident polarisation vector in the s1,p1 basis Ecirculating n J T1V0 n 0 Transmission matrix The 2 eigenvalues of J are ei = |ei|exp(ifi) and f1≠f2 a priori. The 2 eigenvectors are noted ei . One gets Ecirculating 1 1 e eif1 ei 1 U 0 0 1 1 e2 eif2 ei s 1 e1 s 1 e 2 , U p ' e1 p ' e 2 1 1 with the normalised eignevectors e i = ei ei U 1T V 1 0 is the round trip phase: =2pn L if the cavity is locked on one phase, e.g. the first one f1=2p, then f2=2p f2f1 5 Experimentally one can lock on the maximum mode coupling, so that the circulating field inside the cavity is computed using a simple algorithm : If e1 T1V0 e 2 T1V0 : Ecirc 1 1 e 1 U 0 U 1T V 1 0 1 1 e2 ei (f2 f1 ) If e 2 T1V0 e1 T1V0 : Ecirc 1 1 e e i (f2 f1 ) 1 U 0 0 U 1T V 1 0 1 1 e2 0 Numerical study : 2D and 3D •L=500mm, h=50mm or 100mm for a given V0 •Only angular misalignment tilts dqx,dqy = {-1,0,1} mrad or mrad with respect to perfect aligned cavity •38=6561 geometrical configurations (it takes ~2mn on my laptop) •Stokes parameters for the eigenvectors and circulating field computed for each configuration histograming 6 An example of a mirror misalignments configuration : 2D with 3D misalignments Planar mirror Spherical mirror Planar mirror Spherical mirror 7 An example of a mirror misalignments configuration : 3D with 3D missalignments planar mirror Spherical mirror planar mirror Spherical mirror 8 Results are the following: For the eigen polarisation •2D cavity : eigenvectors are linear for low mirror reflectivity and elliptical at high reflect. •3D cavity : eigenvectors are circular for any mirror reflectivities Eigenvectors unstables for 2D cavity at high finesse eigen polarisation state unstable For the circulating field •In 2D the finesse acts as a bifurcation parameter for the polarisation state of the circulating field The vector coupling between incident and circulating beam is unstable the circulating power is unstable •In 3D the circulating field is always circular at high finesse because only one of the two eigenstates resonates !!! 9 Numerical examples of eigenvectors for 1mrad misalignment tilts Stokes parameters for the eigenvectors shown using the Poincaré sphère S3 3D 28 entries/plots (misalignments configurations) S3=1 S1 S2 q0,p Circular polarisation qp/2 Linear polarisation Elliptical polarisation otherwise 3 mirror coef. of reflexion considered Nlayer=16, 18 and 20 2D S3=0 10 The circulating field is computed for : For 1mrad misalignment tilts and V0 = 3D 1 2 S3,in 1 i 2 Then the cavity gain is computed gain = |Ecirculating|2 for |Ein|2=1 2D 11 Stokes Parameters distributions 1 2 V0 = i 2 S3,in 1 3D 1mrad tilts 2D 12 X check Low finesse 2D Eigen vectors V0 = Cavity gain 1 2 1 2 S 2,in 1 1mrad tilts Stokes parameters Stokes parameters 13 X-check low finesse 3D Cavity gain V0 = 1 2 1 2 Stokes parameters S 2,in 1 1mrad tilts Stokes parameters Stokes parameters 14 Numerical examples for U or Z 2D & 3D cavities (6reflexions for 1 cavity round-trip) Z 2D (proposed by KEK) 1 2 V0 = i 2 S3,in 1 U 2D 1mrad tilts leads to ~10% effect on the gain for the highest finesse N=20 ‘closed orbits’ are always self retracing highest sensitivity to misalignments viz bow-tie cavties U 3D 15 Summary • Simple numerical estimate of the effects of mirror misalignments on the polarisation modes of 4 mirrors cavity – 2D cavity • Instability of the polarisation of the eigen modes Instability of the polarisation mode matching between the incident and circulating fields power instability growing with the cavity finesse – 3D cavity • Eigen modes allways circular • Power stable – Z or U type cavities (4 mirrors & 6 reflexions) behave like 2D bow-tie cavities with highest sensitivity to misalignments • Most likely because the optical axis is self retracing • Experimental verification requested … 16 U 2D L=500.0;h=150.0, ra=1.e-7, S3=1 17