1 Polarisation effects in 4 mirrors cavities Introduction Polarisation eigenmodes calculation Numerical illustrations F. Zomer LAL/Orsay Posipol 2008 Hiroshima june

2 2D: bow-tie cavity 3D: tetrahedron cavity L~500mm h~100mm V 0 = 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 L~500mm h~100mm V0V0 V0V0

3 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)

4 y y x z 11 22 11 22 Planar mirror Spherical mirror Planar mirror Spherical mirror Example of a 3D cavity. k1k1 k2k2 p1p1 s1s1 p2p2 s2s2 s2s2 k3k3 p2’p2’ n i is the normal vector of mirror i We have s i =n i ×k i+1 /|| n i ×k i+1 || and p i =k i ×s i /|| k i ×s i ||, p i ’=k i+1 ×s i /|| k i+1 ×s i ||, where k i and k i+1 are the wave vectors incident and reflected by the mirror i. Denoting by R i the reflection matrix of the mirror i N i,i+1 the matrix which describes the change of the basis {s i,p’ i,k i+1 } to the basis {s i+1,p i+1,k i+1 } With  s ≠  p when mirrors are misaligned !!! r s ≠ r p when incidence angle ≠ 0 V0V0

5 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 V 0 is the incident polarisation vector in the s 1,p 1 basis The 2 eigenvalues of J are e i = |e i  exp(i  i ) and  1 ≠   a priori. The 2 eigenvectors are noted e i. One gets  is the round trip phase:  =2  L if the cavity is locked on one phase, e.g. the first one  1 =2 , then  2 =2  2  1 Transmission matrix

6 Experimentally one can lock on the maximum mode coupling, so that the circulating field inside the cavity is computed using a simple algorithm : Numerical study : 2D and 3D L=500mm, h=50mm or 100mm for a given V 0 Only angular misalignment tilts  x,  y = {-1,0,1} mrad or  rad with respect to perfect aligned cavity 3 8 =6561 geometrical configurations (it takes ~2mn on my laptop) Stokes parameters for the eigenvectors and circulating field computed for each configuration  histograming

7 An example of a mirror misalignments configuration : 2D with 3D misalignments Spherical mirror Planar mirror

8 An example of a mirror misalignments configuration : 3D with 3D missalignments Spherical mirror Spherical mirror planar mirror planar mirror

9 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 !!!

10 Stokes parameters for the eigenvectors shown using the Poincaré sphère Numerical examples of eigenvectors for 1mrad misalignment tilts 2D S 3 =0 3 mirror coef. of reflexion considered N layer =16, 18 and 20 S 3 =1 3D   Circular polarisation   Linear polarisation Elliptical polarisation otherwise S1S1 S2S2 S3S3 2 8 entries/plots (  misalignments configurations)

11 2D 3D For 1mrad misalignment tilts and The circulating field is computed for : Then the cavity gain is computed gain = |E circulating | 2 for |E in | 2 =1

12 2D 1mrad tilts 3D Stokes Parameters distributions

13 1mrad tilts X check Low finesse 2D Eigen vectors Cavity gain Stokes parameters Stokes parameters

14 1mrad tilts X-check low finesse 3D Cavity gain Stokes parameters Stokes parameters Stokes parameters

15 1  rad tilts leads to ~10% effect on the gain for the highest finesse N=20 Numerical examples for U or Z 2D & 3D cavities (6reflexions for 1 cavity round-trip) (proposed by KEK) U 2D U 3D Z 2D ‘closed orbits’ are always self retracing  highest sensitivity to misalignments viz bow-tie cavties

16 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 …

17 U 2D L=500.0;h=150.0, ra=1.e-7, S3=1