Crosspolarization and Depolarization in Ellipsometry at Inner Boundaries • Introduction and Motivation • In optics we never measure the fields! • Decoherence • Temporal and Spatial • Depolarization • Temporal • Depolarization Kurt Hingerl & Razvigor Ossikovski • Spatial University Linz & Ecole Polytechnique • Summary [email protected] Opt. Lett. 41, 219, (2016), Opt. Lett. 41, 4044, (2016), Paul Dirac „Quantum Mechanics“: ~ 1935 “each photon then interferes only with itself. Interference between different photons never occurs.” Albert Einstein to his friend Michael Besso 1954: “All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.”

R. Feynman QED- the strange theory matter and light, 1985 When a photon comes down, it interacts with electrons throughout the glass, not just on the surface. The photon and electrons do some kind of dance, the net result of which is the same as if the photon hit only on the surface.

W.E. Lamb 1995, Appl. Phys B “Photons cannot be localized in any meaningful manner, and they do not behave at all like particles, whether described by a wave function or not." Roy Glauber „Nobel lecture: Quantum Mechanics“: 2005 “….if we get a click in a detector, we know that at that very moment the photon is just there….” What does depolarization mean?

Partial state of produced by the interaction of polarized light and an optical element (depolarizer).

Case 1 Case 2

Two phase system: semi-infinite material & air Totally polarized Depolarization E r 22s  sin ( ) cos(  )  sin (  ) Ei r  pp  a : tan  ei  Ersrs 2  2  sin ( ) cos(  )  sin (  )  Ei s a Case 1: For nondepolarising system fully equivalent Mueller matrix formalism Jones formalism

Sin Sout M

Sout  M Sin  Ex  v    s MMMM s v out  J v in   00 11 12 13 14  E     y  s MMMM s 11 21 22 23 24 . s MMMM  s 22 31 32 33 34  out in s MMMM s  E  r r   E  33out 41 42 43 44  in  x    pp ps  x   out   r r   in   Ey   sp ss   Ey  Degree of polarization (DOP)

sss222 DOP123(0 DOP 1) Common states of polarization s 0 Degree of Depolarization () approximately ssss222 0 123 s 0 Totally polarized DOP = 1 Partially depolarizing case:

Sin Sout M 0 < DOP < 1

Partially polarized light Sout  M Sin Stokes vector JJ11 12 rrppps rrrr ppsspsss   J rss  sII0 I xy  EExx EE yy rr rr 1 JJ21 22 sp ss sp ss     sIIQ   EE EE isotropic material S 1  xy  xx yy sIIU   EE EE r 0 2 45 45 xy yx  pp    0 rss sII3 V GD iEEEE xy yx

2 2 2 2 2 2  1 2 1 2 * * * *   2 J pp  J ss  J sp  J ps 2 J pp  J ss  J sp  J ps ReJ ps J pp  J ss J sp   ImJ ps J pp  J ss J sp    2 2 2 2 2 2  1 2 1 2 * * * *  2 J pp  J ss  J ps  J sp 2 J ss  J pp  J sp  J ps ReJ ps J pp  J sp J ss Im J ps J pp  J sp J ss  M    * * * * * * * *  ReJ pp J sp  J ps J ss ReJ pp J sp  J ps J ss ReJ pp J ss  J ps J sp Im J pp J ss  J ps J sp     ImJ * J  J * J ImJ * J  J * J ImJ * J  J * J ReJ * J  J * J   pp sp ps ss pp sp ps ss pp ss ps sp pp ss ps sp 

Field amplitudes Ex, Ey are defined statistically: The Stokes vector components are linear combinations of the second moments of their joint probability distributions, which are directly related to the experimentally measurable intensities. E- field components oscillate ~ 10^(16)/s Experimental observations- Depolarization occurs:

Surface scattering Incidence angle distribution

Sample Sample

Wavelength variation Backside reflection

Thin film Sample Substrate Thickness inhomogeneity

substrate Müller Matrix Ellipsometer Optical set up

 M M M M   11 12 13 14   M 21 M 22 M 23 M 24  .  M M M M   31 32 33 34     M 41 M 42 M 43 M 44 

1: Glan-Taylor 2: Quarterwave at 510nm ferroelectic cell 3: Zero order Quarter at 633nm 4: Halfwave at 510nm ferroelectric liquid crystal cell Decomposition of the Mueller matrix

Experimental data

Coherence matrix Lu Chipman Decomposition: Measured matrix decomposed in

MΔ, MR and MD

Depolarization X Retardance (Phase) X Diattanuation (preferred absorption of one linear polarization state) (not unique‐ what is the correct decomposition?)

Non‐depolarizing Depolarizing Medium medium

Decoherence We never measure in optics the fields, we always measure 2nd moments:  ~ 1015 10 16s 1 ; fastest Detector ~ ps; at least 10 6 oszillations measured 2 2 11T /2 T /2 Inc0 lim Etdti () ExampleIK : lim EtEtdt12 () ()  T  TTT  T /2 i T /2  K    Ert101(,) E cos( kr12 t12 ) Ert 202 (,) E cos( kr  t ) 2 1 T /2   IKlim EtEtdt12 () () T    T T /2 112 2    K E01  EEE02 01 02 cos( kkr12 (21 ))  22   EE01 02 22     KE00 Ecos( k12 k r ()21 ) If phase is not well defined e.g.  0,2 => Statistical Optics! Temporal and Spatial Decoherence

via Wiener Khitchin theorem (FT) 2 Photons (2 interfering beams) => N photons (N interfering beams)  • Finite coherence length of your light source lµm100 c  • Monochromator resolution ~   ~3nm ~1013 s 1 • 1 Watt ~ 1019 photons/s, they all interfere, rather complicated beating pattern („purely?“ statistical?): N=100, 1 ps, 1 µs, 1 s Spectroscopic ellipsometry for a thick overlayer:

Defined Incremental decoherence phase ()q r1  ()q rn  ()qqq ()() ()qq () trt (1)i rr 01 12 10 ()qqq () () 001 rrr11012

 a

2d in n11cos c  o e 

 s

 500nm lnmµm~ 500 100 in Air! ( goes in, too) c    3nm Decoherence 1 22 2cos()EE00 krkr12()  2 21 S D

• Lets assume we have the most perfect sample in the world, with a „most“ homogeneous thickness: SiO2 on Si  lµm100 • Finite coherence length of your light source c    • Monochromator resolution ~ ~3nm ~1013 s 1 • 1 Watt ~ 1019 photons/s, they all interfere, rather complicated beating pattern („purely?“ statistical?)

• If 2‐1 is purely stochastic [, the measured intensity is just the sum of the intensities, all coherence lost! , Retardation: Ewald Oseen , Defined phase , r Incremental decoherence Ept () ,…..,  rad c ,  a ()qq () () q J 00rr 01  Jrtrtr()qqqqqqq () () ()()  ()1  r ()2 2d 2d 2d 1 1 01 12 10 12 01   o 2 2 2  n1 cos 1  n1  n0 sin 0   ()q () q () qqqqq () () ()() () q () qq () c c c J2 r 2 trrrt 01 12 10 12 10  r 1 rr 10 12 ()qqqqqqqq () () () () () () () 2 Jrrrrrrr33 2101211012    (,sp ) ( q ) ( q ) ( q ) ( q ) (1)i  Jrrrrii11012   s

Correlation functions in time ‐‐ ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (,sp ) 2 IDn rEt01( ) rEt ( ) rEt 2 ( 2 ) .... rEt  ( n )

 22 2  2 rEt01()()....() rEt rEtnnmn   rEtmrEtn (  )()   nm10 mn  2 2 rmn EtrEt() () rr mn EtEt () ( ( m n ) ) mnm010 mn

 , Correlation functions in time via Wiener Khitchin theorem (FT) , ,….., ‐‐ ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐

,  J JJJwd**;''';'      kmt  k m  Assuming e.g. a monochromator resolution function (Gaussian, Lorentz, equal distribution) putsusin the position to evaluate each term and sum it: (')qq ()(')* q q Arrnm n m exp nm Im exp inm  Re 2 1   ' rr()qq (')*exp  nm Im exp  exp  i ' nm  Re d '  nm  2 2  2       2 ()qq (')*  22 rrnm exp nm Im exp i nm Re exp  nm Re[ ]  2 2 ()qq (')*2 2 () qq (')* G  AnmAnmAAfexp Re  nm nm  2

Convolution between a sample property and a measurement setup property (light source, monochromator & detector) Forward Simulation (all 16 MMs) of 10µmSiO/Si ( 3nm spectral width)

M12 10 µm SiO2 on Si Normalized M(1,1) Sample assumed as perfectly homogeneous Temporal and Spatial Decoherence (quasimonochromatic)

22   EE0 0 cos(()kk1  2 r  (21 ))

2 2 2 R  2 QA  2 ~50 µm s P Source 1  Can be proven: van Cittert – ext. Q Zernike Theorem sR Experiments: P2 Sun: 0.1 mm^2 Orion star: 6 m^2 Decoherence‐ angle distributions 22   EE0 0 cos(()kk1  2 r  (21 )) S D

• Lets assume quasicoherent radiation:    • Then, the size of the source counts for interference at the screen: Transverse coherence area A 2 2 2 R  2 QA  2 ~50 µm s P Source 1  Can be proven: van Cittert – ext. Q Zernike Theorem sR Experiments: P2 Sun: 0.1 mm^2 Orion star: 6 m^2 Spatial Decoherence 22 EE00cos( krkr12 (21 )) S D

• Lets assume quasicoherent radiation:    • Then, the size of the source counts for interference at the screen: Transverse coherence area A 2 2 R  2 A 2 ~10µm s P Source 1  ext. Q WhysR is the measured depolarization so small? P2 Decoherence‐ some mathematics „Optical coherence and quantum optics“ L.Mandel and E. Wolf, Cam. UP (1995) „Statistical Optics“, Joseph W. Goodman, Wiley (2000)

1. Coherence length does not count „strongly“:

“..the fields at P1 and P2, represented by equations (4.2‐ 10), will indeed be strongly correlated. Hence we see that, even though the two sources S1 and S2 are statistically independent, they give rise to correlations ..these correlations are generated in the process of propagation and superposition.”

2. Interference of two stationary light beams as a second‐order correlation phenomenon ErtP111112221(, )  KEStR ( ,  / c ) KEStR ( ,  / c )

* 2222 Ir( , t)  E ( r , tEr) ( , t) = K1 ES ( 1 , t R 11 / c ) K 2 ES ( 2 , t  R 21 / c ) ** 2Re(KKE12 ( S 1 , t R 11/ cES ) ( 2 , t R 21/ c )) 22 * K1IStR( 1 ,  11/) c K 2 IStR(, 2  21/) c 2Re( KK 1 2 )(  SStttt 1 , 2 , 1 , 2 ) Spatial Decoherence 22 EE00cos( kkr12 (21 )) S D

Spatial Correlation: Z Det 1. Within coherence area => ecto coherent superposition r Q

Y R 2. Periodic nanostructure smaller 1 Polarization determining thanA =>coherent R z 2 1 y elements superposition 1 3. Two materials ½ ½ => x depolarisation 1 X Incidence angle

x1 x2 spatially inhomogeneous sample

The derivation resembles that of the van Cittert ‐ Zernike theorem from optical coherence theory. In virtue of the Huyghens ‐ Fresnel principle, the electric field incident on the sample is the coherent superposition of the field emitted by each atom of the extended light source.

eikrS x,y E x  E y  d 2y i  S r S S The component of the outgoing field upon reflection on the sample described by its (x‐ dependent) Jones matrix with elements is then eikrS x,y Em x  J m x E y  d 2y o  S r S S whereas the electric field reaching the point‐like detector is

ikrD x m m e 2 E  E x d x x x D  o r 1 2 A D where is the sample ‐ detector distance. From Eqs. Above one gets for the detected mutual light intensity :

ik rDDxx r '   mn** m n e EE  Exx  E'' dd22 xx DD o o 2 AA rD eeik rSSxy,',' r x y ik  r DD x r x '    J mnxxyyJEE* '' *2222 dddd yyxx ''  SS rr22 AASS  SD  The mutual coherence function of a totally incoherent source of intensity is:  2 2 * 2  ES y ES y '   ES y y  y'  I S y  y  y'  Consequently, the spatial average of the conjugate product of Jones matrix elements obtained from the detected mutual intensity is J m J n *  J m x J n * x' w x  x' d 2x d 2x' AA

()mn ()*1'' * xxyy JJ JxyJxyw,',', dxdydxdy ''   km  WLLAA  Again a convolution between a sample property and a measurement setup property (size of emitter / detector)!

Sample Only three optics books treat the topic of decoherence and depolarization: Depolarization I „Optical coherence and quantum optics“ L.Mandel and E. Wolf, Cam. UP (1995) „Statistical Optics“, Joseph W. Goodman, Wiley (2000) • Equivalent description by coherency matrices and Stokes Vectors    it(()2)1  t it (()2)2  t Etx () ate1 () Ety () ate2 ()

Etxx() and Et () will i.g. be correlated;  i () t vary slowly; prop. || z, with as angle of pol:

ii12 E'(,tateate12 , , ) ( 1 () cos 2 () sin) i .  * J CM equal time coherence matrix: (cross correlation). Jxy = J yx

**  EtEtxx() () EtEt xy () ()  J   EtEt**() () EtEt () ()   1  yx yy Jss      xx 2 01 energy we given by ( = 21  ) sJ J   0 xx yy 1 22 Jss wnJJ(,) ( cos sin sJ J yy 01 exxyy0  1 xx yy  2  ii    Jecos sin Je cos sin ) sJ2 xy J yx 1 xy yx  J sis   xy 2 23  siJJ3 ()yx xy  1  J sis yx 2 23 Depolarization II „Optical coherence and quantum optics“ L.Mandel and E. Wolf, Cam. UP (1995)  EtEt**() () EtEt () () xx xy JCMenergy w e given by ( =21  ) EtEt**() () EtEt () () yx yy 22ii wnJJJeJeexxyyxyyx( , )0  (   cos  sin cos sin cos sin )

i xy with jjeJxy xy xy/ ( JJ xx yy ) wIIIIexyxy(,) 2( )jxycos( xy  ) If  is statistically distributed => totally uncorrelated; Degree of polarization P

Jxy describes the correlation between orthogonal states ww(,)   (,) 4detJ P eemax min 1 CM ww(,) (,) 2 ee max min trJCM

Lets not mix up J as Jones matrix, transforming pure states, and JCM representing states Summary I - the importance of the measurement Surface Incidence angle If samples measured with scattering distribution • alaser

Sample Sampl • and an expanded parallel beam e Wavelength and falling on a variation Backside • point‐like detector reflection  NO Decoherence

Sample Thin film  NO Depolarisation

Inner boundaries and thickness inhomogeneity Requirments in addition: Speckles Together Source: hololab.ulg.ac.be with finite spatial coherence Sample areas Summary II Take home messages • Inner Boundaries induce Cross Polarization (p<‐>s) d ~ 10µm! • Stronger contributions from the less polarizable material • The more absorbing the material, the less depolarization • Interference enhances Cross Polarization strongly • Maxwell equations DO NOT describe the loss of INFORMATION • But the spatial (temporal) coherence properties of ALL of our light sources. • Decoherence / Depolarization is a statistical process, which depends on the measurement system!

• We have to model the detector / source, in order to discriminate sample contributions from experimental setup effects. Crosspolarization and Depolarization in Ellipsometry at Inner Boundaries

• Introduction and Motivation • Decoherence • Depolarization • Temporal • Continuity Conditions at Inner Boundaries • Crosspolarization • Depolarization • Spatial Crosspolarization

(instead of mathematics => pictorial, normal incidence)

y x z

Lets start with s‐polarized light  ~1eV a) Spot is only on the Si

Ei=1 (V/m) black arrow Er=~‐0.5 (V/m) red arrow Et= 0.5 (V/m) green arrow

a) Spot is only on the SiO2

Ei=1 (V/m) black arrow Er=~‐0.07 (V/m) red arrow Et= 0.93 (V/m) green arrow Crosspolarization (now lets move the spot towards the interface and only plot the transmitted field at z= 0‐ just inside the material)

a) Spot is only on the Si

Ei=1 (V/m) black arrow Er=~‐0.5 (V/m) red arrow Et= 0.5 (V/m) green arrow

a) Spot is only on the SiO2

Ei=1 (V/m) black arrow Er=~‐0.07 (V/m) red arrow Et= 0.93 (V/m) green arrow Crosspolarization (now lets move the spot towards the interface and only plot the transmitted field at z=0+‐ just inside the material)

a) Spot is only on the Si

Ei=1 (V/m) black arrow Er=~‐0.5 (V/m) red arrow Et= 0.5 (V/m) green arrow

Boundary conditions on tangential component of the E‐field NOT fulfilled

a) Spot is only on the SiO2

Ei=1 (V/m) black arrow Er=~‐0.07 (V/m) red arrow Et= 0.93 (V/m) green arrow Q: How does nature solve this problem? How does nature solve this problem? A: Maxwell equations require EVANESCENT FIELDS (at each depth, i.e. for any z)

SiO2 Si ‐z

Et(e.g. @ z=1 nm) Numerical Example I RCWA calculation, (Fresnel), normal incidence (Prof. Lalannes Reticolo Code) • Pure Si: r=‐ 0.5558 (n_Si=3.5+ 0.01i)

• Pure SiO2: r=‐ 0.1667 • ½ Si: ½ SiO2 (in a 40µm periodic cell): ‐0.3654 this is NOT equal to arithmetical mean (‐0.36125)

2 Findings: just s‐polarization Abs((,0,) E x y  z • reflectivities are not linear mean y • Both reflected s‐ waves have different amplitudes • and in case of absorption different phases • Strongly dependent on geometry SiO2 Si (periode, …) z • Looks in a periodic structure in the depth more like a „Waveguide“ mode

General Polarization ( again s‐ and p‐, normal incidence) y x

z Red arrows mark the component of the inconming field normal to the inner boundary.

For the normal component a different boundary condition is valid:

(1) (2) DD (1)(1) (2)(2) 00 EE   (2) EE(1) (2) (1) For the normal component a different boundary condition is valid:

(2) (1) (2) EE  (1)   (1) (2) EE

=> The transmitted fields on both sides of the boundary in the region with the evanescent fields • are not mutually parallel any more, • are neither parallel to the fields away from the boundary • resp. the incident one) (2) (1) (2) • Dashed Black Arrows indicate a coarse EE (1) approximation for the inner transmitted  field (macroscopic field) (1) (2) even in ISOTROPIC materials EE

• Field changes direction at the inner boundaries, in metals the normal component even reverses. y • this is the macroscopically correct one, which drives the radiating dipoles close to the boundary (in the evanescent region) in other directions than the incident field.   2 it mx m x  m0 x  qEe • „Source of cross‐polarization“,

• often wrongly described as„effective anisotropic “ material (geometry dependent) • Field changes direction at the inner boundaries, in metals the normal component even reverses.

• this is the macroscopically correct one (perhaps corrected by LFE), which drives the radiating dipoles close to the boundary (in the evanescent region) in other directions than the incident field.   2 it mx m x  m0 x  qEe • „Source of cross‐polarization“,

Maxwell equation take care of this „equ. of motion“ automatically: Despite the homogeneous Helmholtz equations SEEM to be decoupled in the single field components, they are coupled via „boundaries“:    2Ert(,) 2Ert(,)  2Ert(,) Ert (,)   ( )  0 E (,) rt ( )x  0 E (,) rt ( )y  0 ct22xy ct 22 ct 22  Numerical Example II RCWA calculation, normal incidence (_in=0) (Si, SiO2), pol‐direction () = 0° to inner boundary, pol‐direction () = +45° to IB

 0 0.3624 + 0.0002i -0.0031 - 0.0001i  0.3593 + 0.0001i 0.0000 - 0.0000i J=  J=  0.0031 + 0.0001i -0.3624 - 0.0002i  0.0000 - 0.0000i -0.3654 - 0.0003i

‐ RCWA calculation, normal incidence (_in=0 pol‐ Numerical Example III direction () = +45° to IB

E , E fields at Si/Si0 interface x y 2 ‐ E , E at interface z X Y 0 |E |2 x 0.9 |E |2 y

600 0.8

400 0.7

0.6 200

2 0.5

|E| 0 y (nm)

0.4 -200

0.3 -400

0.2

-600

0.1

-1000 -800 -600 -400 -200 0 200 400 600 800 1000 -2000 -1500 -1000 -500 0 500 1000 1500 2000 x (nm) x (nm) But still totally polarized states, and:

0.3624 + 0.0002i 0.0031 + 0.0001i  = 45°: Jn =   -0.0031 - 0.0001i -0.3624 - 0.0002i  0.3624 + 0.0002i -0.0031 - 0.0001i = -45°: Jn =   0.0031 + 0.0001i -0.3624 - 0.0002i so, () = +45 cancels () = ‐45°

Ep (arising from cross polarization)=

rErEps()  s  ps ( )0 s For all AOIs as well as for all strue, so why does in a perfect statistical sample Depolarization occur? But still totally polarized states, and:

rErEps()   s ps ( )0 s

The measurement system (i.e. mainly the light source) determines, if the fields are superimposed Coherently Or Incoherently Yielding NO or SOME Depolarisation