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 polarization 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 22s 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 DOP123(0 DOP 1) Common states of polarization s 0 Degree of Depolarization () approximately ssss222 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 sIIU 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 ReJ ps J pp J ss J sp ImJ 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 ReJ ps J pp J sp J ss Im J ps J pp J sp J ss M * * * * * * * * ReJ pp J sp J ps J ss ReJ pp J sp J ps J ss ReJ pp J ss J ps J sp Im J pp J ss J ps J sp ImJ * J J * J ImJ * J J * J ImJ * J J * J ReJ * 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 polarizer 2: Quarterwave at 510nm ferroelectic liquid crystal cell 3: Zero order Quarter waveplate 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 Inc0 lim Etdti () ExampleIK : lim EtEtdt12 () () T TTT T /2 i T /2 K Ert101(,) E cos( kr12 t12 ) Ert 202 (,) E cos( kr t ) 2 1 T /2 IKlim EtEtdt12 () () T T T /2 112 2 K E01 EEE02 01 02 cos( kkr12 (21 )) 22 EE01 02 22 KE00 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 lµm100 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! ( refractive index 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 lµm100 • 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 Jrrrrii11012 s
Correlation functions in time ‐‐ ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (,sp ) 2 IDn rEt01( ) rEt ( ) rEt 2 ( 2 ) .... rEt ( n )
22 2 2 rEt01()()....() rEt rEtnnmn rEtmrEtn ( )() nm10 mn 2 2 rmn EtrEt() () rr mn EtEt () ( ( m n ) ) mnm010 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 EE00cos( 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 ~10µm s P Source 1 ext. Q WhysR 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 EE00cos( 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 Em 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 rDDxx r ' mn** m n e EE Exx E'' dd22 xx DD o o 2 AA rD eeik rSSxy,',' r x y ik r DD x r x ' J mnxxyyJEE* '' *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 JCMenergy 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 ct22xy 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