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Crosspolarization and Depolarization in Ellipsometry at Inner Boundaries

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Crosspolarization and Depolarization in Ellipsometry at Inner Boundaries 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 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 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 T TTT /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 lµm 100 • Monochromator resolution ~ c ~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 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 cos1 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 22 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.
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