Electromagnetism II Lecture 7

Transcription

1 Electromagnetism II Lecture 7 Instructor: Andrei Sirenko sirenko@njit.edu Spring 13 Thursdays 1 pm 4 pm Spring 13, NJIT 1

2 Previous Lecture: Conservation Laws

3 Previous Lecture: EM waves

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5 Normal incidence AOI =

6 Refraction and Reflection of Light 1 Andrei Sirenko, NJIT 6

7 Dispersion of the refractive index: Newton s prism 1 Andrei Sirenko, NJIT 7

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11 Non-Magnetic Material n~ ε r pp = E rp E ip = ε cos θ ε ε cos θ + sin θ ε sin θ Fresnel s Equations r ss = E rs E is = cos θ ε cos θ + sin θ ε sin θ R = r

12 Non-Magnetic Material n~ ε r pp = E rp E ip = ε cos θ ε ε cos θ + sin θ ε sin θ Fresnel s Equations r ss = E rs E is = cos θ ε cos θ + sin θ ε sin θ Magnetic Material Veselago Approach R = r Reflectance: n ω ~ Transmittance: n ω ~ ε ω μ ω ε ω μ ω r pp = E rp E ip = ε cos θ εμ ε cos θ + sin θ εμ sin θ r ss = E rs E is = μ cos θ εμ μ cos θ + sin θ εμ sin θ

13 Next Topic: t

14 The interaction of radiation with matter E K-vector of light: If K is real (no losses) E t E t for nonmagnetic media: = complex refractive index: - absorption coefficient (m -1 ); EM wave (light) K K c c K i i E E e αz Loss part of K-vector K K i nk and ( ) n i n in n i R I R K I() z I e α( )z 14

15 Linear spectroscopy of semiconductors and dielectrics phonon I() z I e α( )z electronic transitions Metal 15

16 The interaction of radiation with matter Complex Dielectric Function i complex ( ) ' i '' n ( ) n i n ' complex R I n n real R I '' n n imaginary R I it it E( z, t) Ee exp( ikz) Ee exp izk i for a, K Light along z-axis: 16

17 The interaction of radiation with matter Complex Dielectric Function i complex ( ) ' i '' n ( ) n i n ' complex R I n n real R I '' n n imaginary R I it it E( z, t) Ee exp( ikz) Ee exp izk i for a, K Light along z-axis: C 17

18 Linear spectroscopy R (1 nr) ni R I 1 n 1 n (1 n ) n 18

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20 Dispersion of the refractive index: Newton s prism 1 Andrei Sirenko, NJIT

21 Chromatic Dispersion of the refractive index 1 Andrei Sirenko, NJIT 1

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23 Dielectric function contributions n εr ε Ph FC E ( ) ( ) ( ) / 1 ( ) ( ) ( ) ph ( ) S j i j j TO j FC ( ) p i p N FC core N FC - carrier concentration e m * E ( ) P j i j j j 3

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25 Ionic polarizability / Phonon contribution Evaluate the dielectric constant of an ionic crystal. Ionic polarizability is related to the motion of ions. Recall the linear chain model we used to describe lattice vibrations: Equations of motion in the presence of external field: + - d un M1 C n n1 n1 dt u u u e* E d un1 M C n1 n n dt u u u e* E also assume long wavelength, λ >> a q e* - effective charge, E external field; assume E E e i qxt u n u e it u u n 1 e it 5

26 Substitute this solution into equations of motion, solve for u +, u - Get u e* E e* M1( t ) M ( t ) u E where t C 1 M 1 1 M - transverse optical phonon frequency at q = The ionic polarization P i is then P i m e *( u (n m number of dipoles per unit volume); P = E ) relative permittivity: r = / = 1+ ; = el + i where M R Get 1 M 1 ( ) 1 r 1 M 1 M 1 1 el M M M M R ne* m ( t ) - reduced mass 6

27 At high frequencies, >> t, the ionic term vanishes: r 1 el at =, can rewrite ne* m r 1 el M R t ne* m r r r ( ) 1 el r M Rt (1 t ) 1 t Note that r (ω t ). Also, r (ω) = at l t r r Between ω t and ω l r (ω) < index of refraction is imaginary: N( ) i k( ) wave is reflected r 7

28 Physical meaning of ω l - the frequency of longitudinal optical phonon TO phonons: no field in z direction; from the symmetry of the problem: E x x k D E E LO phonons: macroscopic field along z : E z D z only if ( ) k 8

29 9 Lyddane-Sachs-Teller relation 1 ) ( t r r r r We had and t r r l combine, get j ) ( ω ω ω ω ε ε TOj LO j ) ( t l r r or t l r r If many phonon branches: j ( ) j j j LO LO TOj TO ω ω i ε ε ω ω i If phonon decay is included:

30 Raman Lyddane-Sachs-Teller relation Ellipsometry r r l t A.A. Sirenko, et al., Nature 44, 373 () A.A. Sirenko, et al., Phys. Rev. Lett. 84, 465 () and Phys. Rev. Lett. 8, 445 (1999) 3

31 Electric field and Displacement vector in a material medium In general situation: E x y E( r, t) E exp[ i( Kr t)] E E D E D x xx xy xz E x Dy yx yy yz Ey D z zx zy zz E z εˆ( ) ε [1 ( )] ˆ 6 independent components z xx xy xz xx yx yy yz yy zx zy zz zz In crystals the number of independent components decreases according to the symmetry In isotropic media: xx xy xz yx yy yz zx zy zz 31

32 Analysis of the light polarization from the Snell s law to invisible cloak 3

33 In 1959 Dzyaloshinskii constitutive relationship for magneto-dielectric material Free Energy G E E H H E H ij i j ij i j ij i j Magneto-dielectric tensor i, j G( ) ( ) H E in materials with the center of inversion or with time-reversal symmetry i t r j t r 1 ( ) < i, j Constitutive Relations Linear media D ˆ E H B E ˆ H Disclaimer: moving dielectrics and materials in gradient fields need additional consideration 33

34 Advantage and disadvantage of the Dzyaloshinskii constitutive relationship for magneto-dielectric material Magneto-dielectric tensor Constitutive Relations D ˆ E ˆ H B ˆ ' E ˆ H i, j G( ) ( ) H E D E B H Poynting vector boundary conditions are the same for E and H: Tangential components are continuous i j S EH Problems: We already have the Maxwell equations: db dd E H J dt dt Field vectors: Excitation vectors: E and B D and H Sommerfeld s constitutive Relations in vacuum: ˆ 1 D E and H ( ) B Constitutive Relations in m.-d. medium D ˆ E B 1 H ' E ( ) B ˆ No easy relationship between ˆ P E B e me em M E B ˆ ˆ em zz even while em em zz D H S B 1 S E ( ) B m ˆ zz 34

35 Magneto-dielectric effect requires ordered magnetic structure bcc tetragonal t t t t r r i, j( ) i, j( ) 35

36 4x4 Matrix Formalism Berreman s approach ˆ ˆ i ˆ ˆ( ) ˆ ˆ ˆT ˆ( ) i Optical Matrix D ˆ ˆ E B ˆ ˆ H 36

37 The Constitutive Relations D ˆ ˆ E B ˆ ' ˆ H ˆ ˆ i ˆ ˆ' ˆ' i ˆ' Magneto-dielectric tensor Chirality tensor The Optical Matrix M (6x6 tensor) F ij E H H t i i H j Bi-Anisotropic ˆ, ˆ, ˆ, ˆ, ˆ ', ˆ ' (54 functions of the light frequency ) Tensors change the index of refraction n n n n Wave Propagation e i nz c 37

38 Optics of bi-anisotropic materials: 4x4 matrix formalism Dwight Berreman Award: Jan Rajchman Citation: For his many contributions to understanding electro-optic effects in liquid crystals and especially for his pioneering work on developing the 4 x 4 matrix method for simulating and optimizing the electro-optical properties of LCDs. Berreman s Equation d i dz c D E B H d dz d dy d dz d dx d dy d dx Erp rpp rps Eip E r r E rs sp ss is ' M 1 () r r pp ps () rsp rss ˆM (3X3) (6X6) (4X4) (4X) (X) (4X4) 39

39 From Optical Properties to Mueller Matrices Symmetry considerations allow the 6 6 formalism to be reduced to 4 4 formalism : Propagator or Transfer Matrix Ex E 14 x d H y H y j dz E E y y H x H x Eigenvectors corresponding to the two positive eigenvalues of constitute basis vectors from which complex reflectance coefficients can be derived:

40 From Optical Properties to Mueller Matrices Using boundary conditions derived from the incident and reflected waves and the two positive eigenvectors, the Reflectance Matrix is obtained: cos( ) 41 / N 4 / N N =1 in incident 1 / N / N R r r R medium pp ps sp r r ss ( ) ( ) Erp rpp rps Eip E r r E rs sp ss is The complex reflection matrix is known as the Jones matrix These reflection coefficients are the key building blocks of the Mueller Matrix for nondepolarizing materials The Mueller Matrix can now be obtained from the Reflectance Matrix. 4

41 Reflection from a Magnetic Material The Mueller Matrix is Calculated from Elements of the Reflectance Matrix 1 : M r r * * r * * pp rss rsp rps rpp rss rsp rps pprsp rss rps pprsp rss rps r r * * r * * pp rss rsp rps rpp rss rsp rps pprsp rss rps pprsp rss rps * * * * * * * * rpprps rss rsp rpprps rss rsp rpprss rpsrsp rpprss rpsrsp * * * * * * * * rpprps rss rsp rpprps rss rsp rpprss rpsrsp rpprss rpsrsp All 16 elements of the MM are real 1. Schubert et al. J. Opt. Soc. Am. A. Vol, No. -February 3 43

42 Analysis of Thin Films r pp N N zp k k s xx z zs zp zp z zs zp N N xx NN N N N s Nqzp k k xx z zs zp zp z zs zp N N xx NN q cos( q h)( k k ) i( )sin( q h) q q cos( q h)( k k ) i( k k )sin( q h) zp zs zs z zs z zsxx zs xx ss qzp qzs cos( zs z zs z zsxx zs xx r t t pp ss N N q q cos( q h)( k k ) i( )sin( q h) q h)( k k ) i( k k )sin( q h) k q z zp N N N Nqzp k k s xx z zs zp zp z zs zp N N xx NN q cos( q h)( k k ) i( )sin( q h) k q z zs qzp zs zs z zs z zsxx zs xx q cos( q h)( k k ) i( k k )sin( q h) Anisotropic permittivity tensor Anisotropic permeability tensor No ME tensor P. D. Rogers, T. D. Kang, T. Zhou, M. Kotelyanskii, and A. A. Sirenko, Mueller matrices for anisotropic metamaterials generated using 4 4 matrix formalism, hin Solid Films, 519, 668 (11). 44

43 Simulations: M f ( ˆ, ˆ, ˆ, ˆ, ˆ ', ˆ ',... AOI ) ij ˆ ˆ Mˆ ˆ ' ˆ 66 ˆ ' Magnetoelectric tensor: ij F E H i 44 j ˆ : ( M ) 34 ˆ : ( M ) 34

44 Traditional Ellipsometry (more than 1 years) A. A. Sirenko, C. Bernhard, A. Golnik, A. M. Clark, J. Hao, W. Si, and X. X. Xi, "Soft-mode hardening in SrTiO3 thin films", Nature 44, 373 (). OUTCOME: 1. concentration of the carriers. Phonon spectra 3. (multi-)layer thickness

45 Light polarization: control and analysis S out Mˆ S in S S1 S AOI S 3 IN S S S S 1 ' ' ' 3 ' OUT Mˆ Arteaga Bahar Cheng Bi-Anisotropic M f ( ˆ, ˆ, ˆ, ˆ, ˆ ', ˆ ',... AOI ) ij (54 functions of the light frequency ) Literature: JOSA A JOSA B JOSA A, PRB Georgieva JOSA A Konstantinova Crystall. Rep. Lakarakis TSF S S S S 1 3 " " " " Mayerhofer Schmidt Wohler Mueller matrices (Fresnel s coefficients) 4 4 Matrix Formalism Optics Comm. App. Phys. Lett. JOSA A 47

46 Light polarization: control and analysis S out Mˆ S in S S1 S AOI S 3 IN S S S S 1 ' ' ' 3 ' OUT Mˆ Arteaga Bahar Cheng Bi-Anisotropic M f ( ˆ, ˆ, ˆ, ˆ, ˆ ', ˆ ',... AOI ) ij (54 functions of the light frequency ) Literature: JOSA A JOSA B JOSA A, PRB Georgieva JOSA A Konstantinova Crystall. Rep. Lakarakis TSF S S S S 1 3 " " " " Mayerhofer Schmidt Wohler Mueller matrices (Fresnel s coefficients) 4 4 Matrix Formalism Optics Comm. App. Phys. Lett. JOSA A 48

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