Coupled Mode Analysis Chapter 4 Physics 208, Electro-optics Peter Beyersdorf Document info Ch 4, 1
Class Outline The dielectric tensor Plane wave propagation in anisotropic media The index ellipsoid Phase velocity, group velocity and energy Crystal types Propagation in uniaxial crystals Propagation in biaxial crystals Optical activity and Faraday rotation Coupled mode analysis of wave propagation Ch 4, 2
The Dielectric Tensor ε relates the electric field to the electric displacement by D = ɛ E = ɛ 0 E + P In anisotropic materials the polarization may not be in the same direction as the driving electric field. Why? Ch 4, 3
Anisotropic Media Charges in a material are the source of polarization. They are bound to neighboring nuclei like masses on springs. In an anisotropic material the stiffness of the springs is different depending on the orientation. The wells of the electrostatic potential that the charges sit in are not symmetric and therefore the material response (material polarization) is not necessarily in the direction of the driving field. Ch 4, 4
Analogies What every-day phenomena can have a response in a direction different than the driving force? Ch 4, 5
The Dielectric Tensor The susceptibility tensor χ relates the polarization of the material to the driving electric field by P = ɛ 0 χ E or P x P y P z χ 11 χ 12 χ 13 = ɛ 0 χ 21 χ 22 χ 23 χ 31 χ 32 χ 33 E x E y E z Thus the material permittivity ε in (Ī ) is also a tensor ɛ = ɛ 0 + χ D = ɛ E = ɛ 0 E + P Ch 4, 6
Tensor Notation For brevity, we often express tensor equations such as in the form D = ɛ 0 (Ī + χ ) E D i = ɛ 0 (δ ij + χ ij )E j where summation over repeated indices is assumed, so that this is equivalent to D i = j ɛ 0 (δ ij + χ ij )E j Ch 4, 7
Dielectric Tensor Properties The dielectric tensor is Hermetian such that ɛ ij = ɛ ji In a lossless material all elements of ε are real so the tensor is symmetric!! ɛ ij! =! ɛ! ji! and can be described by 6 (rather than 9) elements ɛ 11 ɛ 12 ɛ 13 ɛ 12 ɛ 22 ɛ 23 ɛ 13 ɛ 23 ɛ 33 Ch 4, 8
Plane Wave Propagation in Anisotropic Media For a given propagation direction in a crystal (or other anisotropic material) the potential wells for the charges will be ellipsoidal, and so there will be two directions for the driving field for which the material response is in the same direction. These two polarization states each have a (different) phase velocity associated with them. Light polarized at either of these angles will propagate through the material with the polarization unchanged. Ch 4, 9
Wave Equation in Anisotropic Materials From Maxwell s equations in a dielectric (ρ=j=0), using d/dt iω and ik we have E + B t = 0 H D t = J i k E + iω B = 0 k E = ωµ H i k H iω D = J k H = ωɛ E k k E + ω 2 µɛ E = 0 Which is the wave equation for a plane wave, most easily analyzed in the principle coordinate system where ε is diagonal (i.e. a coordinate system aligned to the crystal axes) Ch 4, 10
or Wave Equation in Anisotropic Materials k k E + ω 2 µɛ E = 0 ω 2 µɛ x ky 2 kz 2 k x k y k x k z k y k x ω 2 µɛ y kx 2 kz 2 k y k z E x E y = 0 k z k x k z k y ω 2 µɛ z kx 2 ky 2 E z for which the determinant of the matrix must be zero for solutions (other than k=ω=0) to exist ω 2 µɛ x ky 2 kz 2 k x k y k x k z k y k x ω 2 µɛ y kx 2 kz 2 k y k z k z k x k z k y ω 2 µɛ z kx 2 ky 2 = 0 This defines two surfaces in k-space called the normal shells. Ch 4, 11
Normal Shells In a given direction, going out from the origin, a line intersect this surface at two points, corresponding to the magnitude of the two k-vectors (and hence the two values of the phase velocity) wave in this direction can have. The two directions where the surfaces meet are called the optical axes. Waves propagating in these directions will have only one possible phase velocity. k z k y k x Ch 4, 12
or Wave Equation in Anisotropic Materials k k E + ω 2 µɛ E = 0 ω 2 µɛ x ky 2 kz 2 k x k y k x k z k y k x ω 2 µɛ y kx 2 kz 2 k y k z E x E y = 0 k z k x k z k y ω 2 µɛ z kx 2 ky 2 E z E x E y = E 0 E z k x k 2 ω 2 µɛ x k y k 2 ω 2 µɛ y k z k 2 ω 2 µɛ z where E0 2 Ex 2 + Ey 2 + Ez 2 Ch 4, 13
Example Find the possible phase velocities (v p =ω/k) for a wave propagating along the x-axis in a crystal, and the associated polarization directions. v p = 1 v p = 1 µɛy or µɛz polarized in the y or z direction respectively Ch 4, 14
Wave Propagation in Anisotropic Materials Gauss law D = ρ in a dielectric (ρ=0) can be written as i k D = 0 implying the propagation direction ˆk to the displacement vector D. is orthogonal But the Poynting vector S = E H which gives the direction of energy flow is orthogonal to the electric field E, not the displacement vector. Thus energy does not flow in the direction of the wave s propagation if the polarization of the wave ( Ê) is not an eigenvector of the material s dielectric tensor! Ch 4, 15
Spatial Walkoff Image 8.26 from Hecht Ch 4, 16
Eigenstates of a material For any direction of propagation there exists two polarization directions in which the displacement vector is parallel to the transverse component of the electric field. These are the eigenstates of the material. Requiring D and E be parallel in the matrix equation D = ɛ E _ is equivalent to finding the eigenvalues and eigenvectors of ε satisfying λ i u i = ɛ u i Ch 4, 17
Index Ellipsoid Index of refraction as a function of polarization angle n 1 k n 2 The polarization directions that have a max and min index of refraction form the major and minor axes of an ellipse defining n(θ) the index fora wave with the electric displacement vector at an angle of θ in the transverse plane. Ch 4, 18
Index Ellipsoid The sum of all index ellipses plotted in three dimensions is the index ellipsoid, defined by x 2 n 2 x + y2 n 2 y + z2 n 2 z = 1 Ch 4, 19
Example Find the angle of the direction of the optical axis for Lithium Niobate Ch 4, 20
Example Find the angle of the direction of the optical axis for Topaz Ch 4, 21
Crystal Types The index ellipsoid is determined by the three principle indices of refraction n x, n y and n z. A crystal with n x n y n z will have optical axes A crystal with n x =n y n z will have optical axes A crystal with n x =n y =n z will have optical axes Ch 4, 22
Crystal Properties The geometry of a crystal determines the form of the dielectric tensor Ch 4, 23
Crystal Properties The geometry of a crystal determines the form of the dielectric tensor Isotropic ɛ = ɛ 0 n2 0 0 0 n 2 0 0 0 n 2 Ch 4, 24
Crystal Properties The geometry of a crystal determines the form of the dielectric tensor Unaxial ɛ = ɛ 0 n2 0 0 0 0 n 2 0 0 0 0 n 2 e Ch 4, 25
Crystal Properties The geometry of a crystal determines the form of the dielectric tensor Biaxial n 2 x 0 0 ɛ = ɛ 0 0 n 2 y 0 0 0 n 2 z Ch 4, 26
Biaxial Crystals By convention n x <n y <n z so the optical axis is always in the xz plane Ch 4, 27
Uniaxial Crystals By convention n o n x =n y and n e n z The optical axis in a uniaxial crystal is always in the z-direction A negative uniaxial crystal is one in which n e <n o while a positive uniaxial crystal has n e >n o. Ch 4, 28
Example A wave is propagating in the [1,1,1] direction in Mica, what are the two principle indices of refraction and in which direction are the eigenpolarizations? Ch 4, 29
Example A wave is propagating in the [1,1,1] direction in Mica, what are the two principle indices of refraction and in which polarization directions do these correspond to? plane of polarization is given by k r r.k=0 = 0 so x+y+z=0. The intersection of this plane and the index ellipsoid, given by is, x 2 n 2 x x 2 n 2 x + y2 n 2 y + y2 n 2 y + + z2 n 2 z (x + y)2 n 2 z Find extremes of r 2 =x 2 +y 2 +z 2 subject to the preceding two constraints to find directions of transverse eigenpolarizations. Plug directions back into r=(x 2 +y 2 +z 2 ) 1/2 to find index for each polarization = 1 = 1 Ch 4, 30
Example Solution in Mathematica Ch 4, 31
Index of extraordinary ray The index of refraction of the ordinary ray is always equal to the index seen when the ray propagates along the optical axis The extra-ordinary ray sees an index of refraction that is a function of the ray propagation angle from the optical axis. In a uniaxial crystal 1 n 2 e(θ) = cos2 θ n 2 o + sin2 θ n 2 e Ch 4, 32
Double Refraction Snell s law requires the tangential component of the k-vector be continuos across a boundary θ i θ i k i sinθ i k n sinθ n k i sinθ i k n sinθ n n θ n n e n o θ e θ o Ch 4, 33
Conical Refraction The normal surface is a surface of constant ω in k-space. Group velocity v g = k ω is perpendicular to the normal surfaces, but in a biaxial crystal the normal surfaces are singular along the optical axis. The surrounding region is conical, thus a wave along the optical axis will spread out conically Ch 4, 34
Optical Activity Quartz and other materials with a helical molecular structure exhibit Optical Activity, a rotation of the plane of polarization when passing through the crystal. Ch 4, 35
Optical Activity Optically active materials can be thought of as birefringent, having a different index of refraction for right-circular polarization and for left-circular polarization. The specific rotary power describes how much rotation there is per unit length ρ = π λ (n l n r ) The source of this optical activity is the induced dipole moment of the molecule formed by changing magnetic flux through the helical molecule Ch 4, 36
Optical Activity The displacement vector thus has an additional contribution in a direction perpendicular to the electric field D = ɛ E + iɛ 0 G E where! G! = (g! ij k! i k j!/k0)ˆk! 2! is called the gyration vector and G E î ĵ ˆk 0 G z G y E x = G x G y G z = G z 0 G x E y = [G] E E x E y E z G y G x 0 E z so we can define an effective dielectric tensor ɛ = ɛ + iɛ 0 [G] so that D = ɛ E allowing the wave equation to be solved for eigenpolarizations of propagation and two corresponding indices of refraction that depend on [G] Ch 4, 37
Faraday Rotation A form of optical activity induced by an external magnetic field. ρ = V B where V is called the Verdet constant. The sense of rotation depends on the direction of propagation relative to the direction of the magnetic field Ch 4, 38
Faraday Rotation The motion of the charges in a material driven by the electric field feel a Lorentz force!!!!! resulting in an induced dipole with a term proportional to!!!!! such that where γ=-vn 0 λ 0 /π. q v B B E D = ɛ E + iɛ 0 γ B E For Faraday rotation the imaginary term is proportional to B, while for optical activity it is proportional to k. Thus a wave double passing (in opposite directions) a material will net zero polarization rotation from optical activity, but twice the one-way rotation due to Faraday rotation. Ch 4, 39
Exercise Given an isotropic material that is perturbed such that D = ɛe + i ɛe where Δε is 0 G z G y ɛ = ɛ o G z 0 G x G y G x 0 find the eigenpolarizations for the electric field and associated indices of refraction. If a linearly polarized plane wave were to propagate along the direction of G through a thickness d of such material, what would the output wave look like? Ch 4, 40
Exercise Ch 4, 41
Coupled Mode Analysis When the eigenstates of an unperturbed system are known, a small perturbation can be treated as a coupling between these states. Rather than rediagonalizing the matrix for ε one could solve the wave equation with the perturbed matrix ε+δε E = A 1 e i(k 1z ωt)ê 1 + A 2 e i(k 2z ωt)ê 2 with ε E = A 3 e i(k 3z ωt)ê 3 + A 4 e i(k 4z ωt)ê 4 or with ε+δε E = A 1 (z)e i(k 1z ωt)ê 1 + A 2 (z)e i(k 2z ωt)ê 2 Ch 4, 42
Exercise Given an isotropic material that is perturbed such that D = ɛe + i ɛe where Δε is 0 G z G y ɛ = G z 0 G x G y G x 0 find the coupled mode expression for the electric field for a linearly polarized plane wave propagating along the direction of G. Ch 4, 43
Coupled Mode Analysis Example The 1-dimensional wave equation in isotropic media Ch 4, 44
Coupled Mode Analysis Example This is from wave equation solutions to the unperturbed material Ch 4, 45
Coupled Mode Analysis Example Envelope varies slowly compared to optical frequency These coupled differential equations reduce to coupled equations that can be solved using linear algebra when we use phasors to represent the fields Ch 4, 46
Coupled Mode Analysis Example Ch 4, 47
Diagonalization Consider the same problem where n 2 ig 0 ɛ = ɛ 0 i G n 2 0 0 0 n 2 Find the normal modes of propagation by finding the eigenvectors of the dielectric tensor Ch 4, 48
Diagonalization Consider the same problem where n 2 iɛ 0 G 0 ɛ = iɛ 0 G n 2 0 0 0 n 2 Find the normal modes of propagation by finding the eigenvectors of the dielectric tensor λ=n 2, n 2 -εg,n 2 +εg u=[0,0,1], [i,1,0], [-i,1,0] Ch 4, 49
Summary Material polarization is not necessarily parallel to electric field in anisotropic materials permittivity of material (ε) is a tensor relating E to D Direction of wave vector and energy flow can be different Normal shells and index ellipsoid are geometrical constructions to visualize the propagation parameters of a wave Ch 4, 50
References Yariv & Yeh Optical Waves in Crystals chapter 4 Hecht Optics section 8.4 Fowles Modern Optics section 6.7 Ch 4, 51