This question draws on the literature of the 70s and 80s on diffractive optics^1

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This question draws on the literature of the 70s and 80s on diffractive optics^1 2' (cited 61 & 44 times respectively) as a means to implement co-ordinate transformations to explores the claim of a 2011 article in science (cited 1218 times) that meta surfaces do not obey Snell's law, which must consequently b modified Consider a plane interface through the origin normal to the z-axis between a first uniform half-space of refractive index n_- and a second uniform half-space of refractive index n_+. Without loss of generality the fields are assumed not to vary in the y - direction. Consideration may be limited then to TE or TM waves and hence to the scalar field psi = E_y (TE) or H_y (TM) along with the boundary condition that psi and partial differential phi/partial differential z (TE) or (1/n^2) partial differential phi/partial differential z (TM) must be continuous across the interface. The governing equation is then: (partial differential^2/partial differential x^2 + partial differential^2/partial differential z^2 + n^2 k)psi = 0 where k_0 = 2 pi/lambda_0 and lambda_0 is the vacuum length (a measure of temporal frequency). The continuous translation invariance of the structure in x (globally) and z (locally away from the interface) leads to eigenfunctions the form: psi (x, z) = exp(ik_x x) exp(ik z) where k_x and k_z are some scalars satisfying: k^2_x + k^2_ = n^2 k^2_0 One expects the total field to be linear superposition of such psi_k_x with coefficients that are a function of z only. Consistant with this expectation, the total field psi is decomposed in the first half space into the sum psi = psi + psi_r of a plane wave psi incident upon the interface at the angle from the normal of theta_1 (the angle of incidence). psi (x, z) = exp(ik_x x) exp (ik_x z) k_x = n k_0 sin (theta_1) k_z = Squareroot n^2 k^2_0 - k^2_x and a reflected field psi_r that is a linear superposition of eigenfunctions indexed by eta of the form: psi_r (x, z) = integral^infinity_-infinity (eta) exp(inx) exp[-ik_z(eta)z] d eta k (eta) = n^2 k^2_0 - eta^2 and in the second half-space the total field psi is equal to a transmitted field psi_t that is linear superposition of eigenfunctions indexed by eta of the form: psi_t(x, z) = integral^infinity_-infinity c(eta) exp (inx) exp [-ik_z (eta) z[d eta k_z (eta) = Squareroot n^2_+ k^2 - eta^2 The amplitude coefficients b(eta) & c(eta) can be distributions which is equivalent to the integral becoming sums over discrete values of eta. The translation invariance in z broken by the interface permits eta to range through values where k_z become imaginary, i.e. evanescent field bound to the surface are permitted. The sign of the squareroot is then taken to ensure that the field decays away from the interface. a) By considering continuity conditions at the interface deduce that the reflected field and transmitted field are plane waves. Sketch the corresponding wave-vector diagram and deduce Snell's law: theta_r = theta n sin theta_t = n sin theta_i b) Suppose a slab of material that has periodic refractive index with period Lambda in the x - direction: n^2 (x) = n^2 (x + Lambda) is inserted between the two half spaces. Such a structure, if thin compared to a wavelength might be called a 'metasurface'; If thicker, it might be called a '1D photonic crystal' or even a 'plane grating' and, if the local spatial frequency (1/Lambda(x)) is allowed to vary spatially, then it might be called a planar diffractive optical element. Repeat the logic leading to Snell's law (part (a)) but now apply Block's theorem to account for the discrete rather than continuous translation symmetry in x of the slab. Show that the reflected field and transmitted fields must represented as a discrete sum over an infinity of plane waves referred to as diffraction orders. Sketch the corresponding wave-vector diagram and deduce the grating equation". n_- sin(theta_r) = n_- sin(theta) + m 1/Lambda n + 1lambda_0 sin theta_t = n_- 1/Lambda_0 sin theta + m 1/Lambda Compare with Eqs(2) & (4) of [1] and Eq. (2) of [2] which cites [3]. State your findings.

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