# Numerical methods and approximations for thermal radiation in periodic gratings

Some commonly used methods for the study of radiative properties of periodic gratings are described in this section. The focus is on the rigorous couple-wave analysis (RCWA) and effective medium theory (EMT).

#### Rigorous coupled-wave analysis (RCWA)

Originally developed in the 1980’s, RCWA is one of the efficient tools for modeling radiative properties and analyzing diffraction efficiency of periodic gratings bounded by two semi-infinite media [1]. The gratings can be divided into planar thin slabs to approximate the grating profile to an arbitrary degree of accuracy. Due to the periodic structure the dielectric function in the grating region is expanded in a Fourier series. The EM fields are also expressed as a Fourier series expansion in terms of the spatial harmonics, determined by the Bloch-Floquet condition. The accuracy of the solution depends only on the number of terms retained in the Fourier expansion of the dielectric function and the EM fields. By matching boundary conditions for the tangential components of both electric and magnetic fields at the interfaces of each layer, the coupled-wave equations can be formulated and solved utilizing a state-variable method, which has been significantly improved in the more recent papers. RCWA has also been extended to analyze 2D periodic structures.

A 1D binary grating is shown in Fig. 1 to illustrate the RCWA method in solving Maxwell’s equations for an incident plane wave with a wavevector k from Region I. The grating (Region II) is assumed to form on a semi-infinite medium (such as a substrate), Region III. The dielectric functions εA and εB represent the binary grating with a period (Λ). One can treat Region II as an inhomogeneous medium with a periodic dielectric function in the x direction. The grating height (h) is the thickness of Region II. The filling ratio of medium A is given by   f = w / Λ, where w is the width of the grating and usually medium B is either air, vacuum or a dielectric. The direction of incident wavevector can be described by a zenith angle (θ) and an azimuthal angle (ϕ). The plane of incidence is defined by the direction of incidence and the z axis, except at normal incidence when the y-z plane is taken as the plane of incidence. For linearly polarized incident wave, the polarization status is determined by ψ, the angle between the electric field vector and the plane of incidence. The incident electric-field vector E is given by

$E=E_{{\rm i}} \exp \left(ik_{x} x+ik_{y} y+ik_{z} z\right)$

(1)

where Ei is the incident electric field vector at the origin, and the components of the incident wavevector are given by kx=k sinθ cosϕ, ky=k sinθ sinϕ and kz=k cosθ, with k=2π / λ, where λ denotes the vacuum wavelength. In Eq. (1), the time harmonic term exp(iωt), where ω is the angular frequency, is omitted for simplicity. The incident electric field can be normalized so that

$E_{{\rm i}} =\left(\cos \psi \cos \theta \cos \phi -\sin \psi \sin \phi \right)\hat{x} +\left(\cos \psi \cos \theta \sin \phi +\sin \psi \cos \phi \right)\hat{y}-\cos \psi \sin \theta \, \hat{z}$

(2)

According to the Bloch-Floquet condition [1], the wavevector components of the jth diffraction order in Region I are given by

$k_{xj} =\frac{2\pi }{\lambda } \sin \theta \cos \phi +\frac{2\pi }{\Lambda } j$

(3a)

${k_{y} =\frac{2\pi }{\lambda } \sin \theta \sin \phi }$

(3b)

$k_{zj}^{{\rm r}} =\left\{\begin{array}{c} {\sqrt{k^{2} -k_{xj}^{2} -k_{y}^{2} } {\rm ,\; }\, \, \, \, \, k^{2} >k_{xj}^{2} +k_{y}^{2} } \\ {i\sqrt{k_{xj}^{2} +k_{y}^{2} -k^{2} } ,\, \, \, \, \, \, k_{xj}^{2} +k_{y}^{2} >k^{2} } \end{array}\right.$

(3c)

As required by the phase-matching condition, the parallel components of wavevector kxj and ky must be the same for the diffracted waves in all three regions. In Eq. (3c), superscript r refers to the diffraction waves reflected back to Region I. Denote the wavevectors for the jth order reflected or transmitted waves as $k_{{\rm r}j} =(k_{xj} ,k_{y} ,k_{zj}^{{\rm r}} )$ and $k_{{\rm t}j} =(k_{xj} ,k_{y} ,k_{zj}^{t} )$ respectively. For transmitted diffraction waves, $k_{zj}^{r}$ in Eq. (3c) can be replaced by $k_{zj}^{t}$ after substituting $k_{\rm III} =k\sqrt{\varepsilon _{{\rm III}} }$ for k. From Eq. (3b), the y component of the wavevector is the same for all diffraction orders and in all three Regions. Hence, the wavevectors for all diffracted waves end on the semi-circles, which are intersects of the plane of constant ky