Derivation of Mueller matrix from Zemax

2018.03.10 Tetsu Anan

Definition of Stokes Q, U, & V

View towards the sun, that is, from the direction of the observer• Signs of Stokes U & V change, when light is reflected• Coordinates in Zemax doesn’t change, when light is reflected

=> We should change the sign of U & V in Zemax manually

Ichimoto et al. 2008Stenflo 1994

Jones vector• “Polarization Ray Trace” procedure in Zemax can calculate

polarization state of light passing (X, Y) field & pupil coordinates, in Jones description, at each surface

• Jones vector J

! = #$%&'()#*%&'(+

,

where the electric field vector of light E

, = Re #$%&' /01()#*%&' /01(+

• Interaction with a medium!2 = 3! ω: 2 � 2 complex matrix

Jones vector cannot describe depolarization

Mueller matrix• Mueller matrix M derived from ω

! = # $⨂$∗ #'(,

where ⨂ is tensor product, � is complex conjugation,

• However, the definition of V in the output is opposite to our definition. Then

! = )# $⨂$∗ #'(), where * =1 0 0 00 1 0 000

00

10

0−1

Stenflo 1994 ‘Solar Magnetic Fields’ chap. 2

T =

1 0 0 11 0 0 −10 1 1 00 −i i 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

T −1 =12

1 1 0 00 0 1 i0 0 1 −i1 −1 0 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

, and

Method

Step 1

1. Edit Lens Data2. Open the ‘Polarization Ray Trace’

Step 2• Edit ‘Settings’ in ‘Polarization Ray Trace’

– Generate a polarized light source (JX, JY, X, Y-Phase)– Specify the ray path (HX, HY, PX, PY) and wavelength

No check

Step 3• Read ‘E field before (after) coating ’ of S & P pol. at the interestring surface

from output of the ‘Polarization Ray Trace’• An example of the output regarding with a surface

(surface 3, silver coated mirror)Tracing ray to surface 3:

Path length through air (tau): 0.0000000E+00Internal absorption per mm (alpha): 0.0000000E+00Internal Transmittance of ray (IT): 1.000000000000Propagation Phase Factors (pc,ps): 1.000000000000 -0.000000000000Coordinates on surface (x,y,z): 0.0000000E+00 0.0000000E+00 0.0000000E+00Direction cosines of incident ray (l1,m1,n1): 0.000000000000 -0.707106781187 0.707106781187Cosine of angle of incident ray : 0.707106781187 (45.000000 deg)Cosine of angle of exit ray : 0.707106781187 (45.000000 deg)Direction cosines of exit ray (l2,m2,n2): -0.000000000000 0.707106781187 0.707106781187Direction cosines of normal (ln,mn,nn): 0.000000000000 0.000000000000 -1.000000000000Direction cosines of S vector (sx,sy,sz): -1.000000000000 -0.000000000000 -0.000000000000Direction cosines of P vector (px,py,pz): 0.000000000000 -0.707106781187 -0.707106781187E field before coating (xyz) (Exr,Eyr,Ezr): 0.056728078820 -0.228864797933 -0.228864797933

(Exi,Eyi,Ezi): -0.892626419659 -0.218222144309 -0.218222144309E field before coating (s&p) (Esr,Epr): -0.056728078820 0.323663701187

(Esi,Epi): 0.892626419659 0.308612716092Ray intensity before coating (I1): 1.000000000000Coating : SILVERIntensity Reflection coefficients (Rs,Rp): 0.964552361746 0.930361258549Intensity Transmission coefficients (Ts,Tp): 0.035447638254 0.069638741451Intensity Absorption coefficients (As,Ap): 0.000000000000 0.000000000000Diattenuation (D): 0.018043621002Field Amplitude Reflection S pol (rsr,rsi): -0.894113163503 0.406342235802Field Amplitude Reflection P pol (rpr,rpi): -0.634324336553 0.726631883835Field Amplitude Transmission S pol (tsr,tsi): 0.047476359022 0.182191200633Field Amplitude Transmission P pol (tpr,tpi): 0.118627893187 0.235724764097Field Reflection Phase (Prs,Prp): 2.715032677668 2.288472701745Field Reflection Retardance (P-S) (Sr): -0.426559975922 (-24.440086 deg)Field Reflection Retardance (P-S+pi) (Sr): 2.715032677668 (155.559914 deg)

Field Transmission Phase (Pts,Ptp): 1.315880054209 1.104554111076Field Transmission Retardance (P-S) (St): -0.211325943133 (-12.108085 deg)Field Transmission Retardance (P-S+pi) (St): 2.930266710457 (167.891915 deg)Ray Amplitude Reflection S pol (rsr,rsi): -0.894113163503 0.406342235802Ray Amplitude Reflection P pol (rpr,rpi): -0.634324336553 0.726631883835Ray Amplitude Transmission S pol (tsr,tsi): 0.047476359022 0.182191200633Ray Amplitude Transmission P pol (tpr,tpi): 0.118627893187 0.235724764097Ray Reflection Phase (Prs,Prp): 2.715032677668 2.288472701745Ray Reflection Retardance (P-S) (Sr): -0.426559975922 (-24.440086 deg)Ray Reflection Retardance (P-S+pi) (Sr): 2.715032677668 (155.559914 deg)Ray Transmission Phase (Pts,Ptp): 1.315880054209 1.104554111076Ray Transmission Retardance (P-S) (St): -0.211325943133 (-12.108085 deg)Ray Transmission Retardance (P-S+pi) (St): 2.930266710457 (167.891915 deg)Electric field after coating (Esr,Epr): -0.311990493087 -0.429555601791

(Esi,Epi): -0.821160046288 0.039423808535Ray intensity after coating (I2): 0.957714141106Direction cosines of new S vector (sx,sy,sz): -1.000000000000 -0.000000000000 -0.000000000000Direction cosines of new P vector (px,py,pz): 0.000000000000 -0.707106781187 0.707106781187E field after (Exr,Eyr,Ezr): 0.311990493087 0.303741678923 -0.303741678923

(Exi,Eyi,Ezi): 0.821160046288 -0.027876842356 0.027876842356X, Y, and Z direction Amplitude (Ax, Ay, Az): 0.878431493855 0.305018238561 0.305018238561X, Y, and Z direction Phase (Px, Py, Pz): 1.207702869709 -0.091521732210 3.050070921379Phase difference between X and Y (Pxy): 1.299224601919 (74.440086 deg)Major, Minor semi axis XY ellipse (EM, Em): 0.882706442588 0.292416400814Angle of XY polarization ellipse (Ap): -3.037222448442 (-174.020028 deg)Ray intensity out (I2): 0.957714141106

Polarization property of the surfacein Jones description

• 2 sets of E field before/after coating with different polarized light source!",$!%,$ = '(( '()

')( '))!",*!%,*

!′",$!′%,$ = '(( '()

')( '))!′",*!′%,*

where E(S,P),(a,b) is E field (after, before) in (S, P) pol.– If you consider rays pass some points on a pupil, you need to derive average ω

matrix after calculation of the ω for each ray• Sampling points on a pupil should be in symmetry around the optical axis

Þ ω00, ω01, ω10, ω11Þ Mueller matrix� positive Q-direction is S-direction� no depolarization� zemax2mm.pro

Example

Exercise 1

• Folding mirror– Silver coating

Exercise 1

Zemax1.000 0.016 0.000 0.0000.016 1.000 0.000 0.0000.000 0.000 −0.924 −0.3830.000 0.000 0.383 −0.924

• λ: 587.6 nm• Silver coating

– n=0.150160 − 3.4727 i• A ray pass center on pupil • Normalized by (0,0)

Stenflo 19931.000 0.016 0.000 0.0000.016 1.000 0.000 0.0000.000 0.000 −0.924 −0.3830.000 0.000 0.383 −0.924

• n=0.150160 − 3.4727 i• Incidence angle 45�• Normalized by (0,0)

Summary• I present a principle, a method, and an example of derivation of the

Mueller matrix from Zemax– We should manually change the sign of U & V in case of

reflector– Definition of V in Mueller matrix derived with a method

described in Stenflo 1993 is opposite to our definition– It does not have depolarization effect

• I developed a IDL function named ‘zemax2mm.pro’– It is succeed in derivation of Mueller matrix of silver coated

folding mirror – It does not have depolarization effect

Coating

• COATING.DAT���– Coating Materials: e.g. AIR, AG

• Wavelength, Index (n), Extinction (k)– Coating Definitions:

• Coating name: e.g. HEAR1, 3 layer(s)– Material, Thickness, Absolute, Loop, Taper– e.g. MGF2, 0.25, 0, 0

ZRO2, 0,50, 0, 0CEF3, 0.25, 0, 0

or– S or P Polarization: T, tr, ti. R, rr, ri, A, TIR

or– Wavelength, Rs, Rp, Ts, Tp, Ars, Arp

Absolute: �����- 0���������- 1���� μmLoop: ������� ������Taper:

����� η=n+ik