1/53 Waves in Linear Optical Media Sergey A. Ponomarenko Dalhousie University c 2009 S. A. Ponomarenko
Outline Plane waves in free space. Polarization. Plane waves in linear lossy media. Dispersion relations for linear waves in dispersionless, homogeneous media. Waves in anisotropic media: Faraday rotation, uniaxial crystals. Waves in piecewise-continuous media: Fresnel theory. Brewster and surface plasmon-polariton modes. 2/53
Light propagation in free space Maxwell s equations (ME) in free space: E = µ 0 H t, H = ε 0 E t, E = 0, H = 0. Describe any classical optics phenomenon in free space! 3/53
Eliminating the magnetic field: ( E) = ( E) 2 E = µ 0 ε 0 2 E t 2. recalling that µ 0 ε 0 = 1/c 2, 4/53 2 E 1 2 E = 0. (1) c 2 t2 Wave equation in free space. There exist plane wave solutions
Plane wave solutions: E = E 0 e i(k r ωt), 5/53 H = H 0 e i(k r ωt), with the dispersion relation: k 2 = ω 2 /c 2 as well as the relations H 0 = (ê k E 0 ) η 0
E 0 = η 0 (ê k H 0 ). where η 0 = µ 0 /ε 0 (free space impedance). The transversality conditions: ê k E 0 = 0, ê k H 0 = 0. 6/53 K E Η
E = (E x ê x + E y ê y )e i(kz ωt), E x = E x e iφ x, E y = E y e iφ y. 7/53 φ x = φ y, linear polarization, E x = E y, φ x φ y = π/2, circular polarization Otherwise, elliptical polarization. The elliptic polarization is the most general case, E x = E y and φ x φ y.
E y E 8/53 â y â x E x E = Linear polarization, Ex 2 + Ey 2, θ = tan 1 (E y /E x ).
Circular polarization, E y E 9/53 o E x ê ± = êx ± iê y 2, ê ± ê = 0, ê ± ê ± = 1.
Elliptical polarization, E u E 10/53 E v E = (E + ê + + E ê )e i(kz ωt), E /E + = re iα.
Light propagation in linear isotropic lossy media Constitutive relations 11/53 and D = ε 0 ε(ω)e, B = µ 0 H, ε(ω) = ε (ω) + iε (ω), Maxwell s equations k E = 0, k H = 0,
k E = µ 0 ωh k H = ε 0 ε(ω)ωe. 12/53 Seeking inhomogeneous plane wave solutions E(z,t) = Re{E e i(kz iωt) } H(z,t) = Re{H e i(kz iωt) }
Dispersion relation k = β ± + iα ± /2. where 13/53 β ± = ± ω 2c ε 2 + ε 2 + ε and α ± = ± ω c ε 2 + ε 2 ε
The electric and magnetic field amplitudes: E = η(ê z H ), and H = (ê z E ), η η being a complex impedance of the medium 14/53 η = η 0 ε(ω) η = η 0 (ε 2 + ε 2 ) 1/4, tanθ η = ε /ε
E e - z o z 15/53 Linearly polarized plane wave and E(z,t) = ê x E e αz cos(βz ωt) H(z,t) = ê y E η e αz cos(βz ωt θ η ).
Lossless dielectrics, ε = 0 E(z,t) = ê x E cos(βz ωt), E H(z,t) = ê y cos(βz ωt). η Homogeneous plane wave with parameters λ = 2π/β, 16/53 v p = ω/β.
Light propagation in linear anisotropic media MEs in source-free nonmagnetic media E = B t, H = D t, D = 0, B = 0. B = µ 0 H 17/53
Eliminating magnetic field from MEs: and ( E) 2 E = µ 0 2 D t 2 D = 0, The electric flux density in such media: D i = 3 ε i j E j, j=1 18/53
Plane-wave solution for the field: and the flux density: E i = 1 2 E ie i(k r ωt) + c.c.,. D i = 1 2 j ε i j E j e i(k r ωt) + c.c.. Modes supported by the medium: ) (k i k j k 2 δ i j + ω2 c ε 2 i j E j = 0, j 19/53
or ky 2 + kz 2 ω2 ε c 2 xx k x k y ω2 ε c 2 yx k x k z ω2 E x E y E z j c 2 ε zx k x k y ω2 ε c 2 xy kx 2 + kz 2 ω2 ε c 2 yy k y k z ω2 ε c 2 zy k x k z ω2 ε c 2 xz k y k z ω2 ε c 2 yz kx 2 + ky 2 ω2 ε c 2 zz = 0 (2) ( [k 2 δ i j k ) ] ik j ω2 k 2 c ε 2 i j E j = 0, 20/53
or plus the transversality condition: k 2 y + k 2 z ω2 c 2 ε xx k x k y ω2 c 2 ε yx k x k z ω2 c 2 ε zx i, j k i ε i j E j = 0. Dispersion relations: ) Det (k i k j k 2 δ i j + ω2 c ε 2 i j = 0. k x k y ω2 ε c 2 xy kx 2 + kz 2 ω2 ε c 2 yy k y k z ω2 ε c 2 zy k x k z ω2 ε c 2 xz k y k z ω2 ε c 2 yz kx 2 + ky 2 ω2 ε c 2 zz = 0 21/53
Faraday rotation in external magnetic field The dielectric tensor (phenomenological) 22/53 ε i j (ω) = ε(ω)δ i j + ig(ω)e i jl B 0l, gb 0 ε. B 0 weak magnetic field. e i jk = { 1 even permutation of ijk; 1 odd permutation of ijk
Assume k = (0,0,k); Dispersion relation: k ± = k ± k, where 23/53 k = ω c ε(ω); k = ωg(ω)b 0 2 ε(ω)c. Eigenmodes are circularly polarized plane waves, E ± (z,t) = ê ± E 0 e i(k ±z ωt). Magnetic field breaks down the symmetry!
Evolution of a linearly polarized wave E(0,t) = E 0 ê x e iωt, By superposition at any plane z = const > 0: 24/53 E(z,t) = a + ê + e i(k +z ωt) + a ê e i(k z ωt), after some algebra: E(z,t) = E 0 ê r (z)e i(kz ωt), where ê r (z) = ê x cos kz ê y sin kz.
Linearly polarized wave with rotating polarization plane Rotation angle: φ = kl = V B 0 L, where V is the so-called Verdet constant: V = ωg(ω) 2c ε(ω). 25/53 Verdet constant gives the rate of rotation Application: polarization control.
Linear waves in uniaxial media Dielectric tensor of a uniaxial crystal with the principal axis along n: ε i j = ε n i n j + ε (δ i j n i n j ). 26/53 Illustration Choose n = (0,0,1) and k = (k x,0,k z ): ε 0 0 ε = 0 ε 0 0 0 ε
Ordinary waves: Dispersion relations 27/53 k = ω c ε. Extraordinary waves: ( k = ω sin 2 θ c ε + cos2 θ ε ) 1/2.
k x k 0 k x k e 28/53 k z k z ω 2 c 2 = k2 x ε + k2 z ε. Dispersion ellipse
Properties of extraordinary waves: Magnitude of the wave vector depends on the propagation direction. Propagation direction does not coincide with the energy flow direction S need not be parallel to k. Applications: Can be used for phase matching in secondharmonic generation processes. 29/53
Reflection of plane waves at normal incidence 30/53 E i x â K H i E r H r â K H t E t â K,,,,, 1 1 1 1 2 2 Medium 1 Medium 2 z
and Incident wave E i (z,t) = ê x E 0i e i(k iz ω i t), H i (z,t) = ê y E 0i η i e i(k iz ω i t). Reflected wave E r (z,t) = ê x E 0r e i(k rz+ω r t), H r (z,t) = ê y E 0r η r e i(k rz+ω r t). 31/53
Transmitted wave E t (z,t) = ê x E 0t e i(γ tz ω t t), H t (z,t) = ê y E 0t η t e i(γ tz ω t t). Boundary conditions 32/53 E 1τ z=0 = E 2τ z=0, H 1τ z=0 = H 2τ z=0. imply that e i(k iz ω i t) z=0 = e i(k rz+ω r t) z=0 = e i(γ tz ω t t) z=0.
Thus ω i = ω r = ω t = ω; and also k i = k r = k 1 = ω c ε1, γ t = γ 2 = β 2 + iα 2, η i = η r = η 1 = η 0 / ε 1, η t = η 2 = η 0 / ε 2, ε 2 = ε 2 + iε 2. 33/53
The transmission and reflection amplitudes and r E 0r E 0i = η 2 η 1 η 1 + η 2, t E 0t E 0i = 2η 2 η 1 + η 2. Energy flux conservation: 34/53 1 + r = t.
Perfect conductor, σ E i x 35/53 H i E r â K â K H r 1 2 z,, 1 1 1 No transmitted wave!
Refraction & reflection at oblique incidence z 36/53 Medium 2 (, ) 2 2 â n t k t Medium 1 (, ) 1 1 k i i r k r k i = k ix ê x + k iz ê z, k r = k rx ê x + k rz ê z, k t = k tx ê x + k tz ê z,
z Medium 2 (, ) 2 2 â n t k t 37/53 Medium 1 (, ) 1 1 k i i r k r e i(k i r ω i t) z=0 = e i(k r r ω r t) z=0 = e i(k t r ω t t) z=0.
z Medium 2 (, ) 2 2 â n t k t 38/53 Medium 1 (, ) 1 1 k i i r k r k ix = k rx = k tx = k x k ix = k rx = θ i = θ r Snell s law k ix = k tx = n 1 sinθ 1 = n 2 sinθ 2,
TM-polarized waves z 39/53 Medium 2 (, ) 2 2 z 0 t E t H t k t x Medium 1 (, ) 1 1 E i H i k i i r E r H r k r
Incident waves H i (r,t) = H 0i ê y e i(k i r ωt) E i (r,t) = η 1 H 0i (k iz ê x k x ê z )e i(ki r ωt), Reflected waves H r (r,t) = H 0r ê y e i(k r r ωt) E r (r,t) = η 1 H 0r ( k 1z e x k x e z )e i(kr r ωt), Transmitted waves H t (r,t) = H 0t ê y e i(k t r ωt) E t (r,t) = η 2 H 0t (k 2z e x k x e z )e i(kt r ωt), 40/53
TM-reflection and transmission amplitudes: and r T M = r = E 0r E 0i = ε 2k 1z ε 1 k 2z ε 2 k 1z + ε 1 k 2z, 41/53 t T M = t = E 0t E 0i = 2ε 2k 1z ε 2 k 1z + ε 1 k 2z Transmission and reflection coefficients: ε1 ε 2. R T M r T M 2, T T M t T M 2.
Reflectionless modes r T M = 0 =,ε 2 k 1z = ε 1 k 2z Brewster mode = Im{k 1z } = Im{k 2z } = 0 tanθ B = n 2 /n 1 ; where refractive indices are defined as 42/53 n s = ε s ; s = 1,2.
Surface plasmon polariton (SPP) modes 43/53 ε(ω) = { ε1 > 0, z > 0, ε 2 (ω) 0, z < 0.
Dispersion relation follows from and k jz = ε 2 k 1z = ε 1 k 2z k 2 0 ε j k 2 x j = 1,2 k 0 = ω/c SPP signature: k 2 x 0 k 2 jz 0 44/53
Dispersion relations: and ε 1 > 0, ε1 ε 2 k x = k 0 ε 1 + ε 2 ε 2 j k jz = k 0, j = 1,2. ε 1 + ε 2 SPP conditions: ε 1 + ε 2 < 0, ε 1 ε 2 < 0. ε 2 < ε 1 = metal-dielectric interface 45/53
Launching SPPs 46/53
TE-polarization z Medium 2 (, ) 2 2 E t k t 47/53 z 0 t H t x Medium 1 (, ) 1 1 E i k i H i i r H r E r k r
Incident waves E i (r,t) = E 0i ê y e i(k i r ωt) H i (r,t) = E 0i η 1 ( k 1z e x + k x e z )e i(k i r ωt), Reflected waves E r (r,t) = E 0r ê y e i(k r r ωt) H r (r,t) = E 0r η 1 (k 1z e x + k x e z )e i(k r r ωt), Transmitted waves E t (r,t) = E 0t ê y e i(k t r ωt) H t (r,t) = E 0t η 2 ( k 2z e x + k x e z )e i(k t r ωt). 48/53
Transmission and reflection amplitudes: and r T E = r = E 0r E 0i = k 1z k 2z k 1z + k 2z, t T E = t = E 0t E 0i = No Brewster angle or SPPs: 2k 1z k 1z + k 2z. 49/53 r T E 0,
Total internal reflection Incidence from more dense medium n 1 > n 2 Critical angle: 50/53 Under the condition θ c = sin 1 (n 2 /n 1 ), θ 1 > θ c, n k 2z = i k 2z = ik 2 1 2 sin 2 θ n 2 1 1, 2
TM-polarized incident wave Unimodular reflection amplitude r T M = e 2iφ T M 51/53 where ( ) φ T M = tan 1 ε1 k 2z ε 2 k 1z
TE-polarized incident wave Unimodular reflection amplitude r T E = e 2iφ T E, 52/53 and ( ) φ T E = tan 1 k2z k 1z
Energy flow across the interface? Time-averaged Poynting vector, S t (z) = 1 2 Re[E 0t(z) H 0t (z)]. 53/53 TM case: S t (z) = e x 4ε 2 k 2 1z k x k 0 (ε 2 2 k2 1z + ε2 1 k 2z 2 ) I ie 2 k 2z z, Propagates along the interface!