The rigorous coupled-wave analysis technique for describing the diffraction of electromagnetic waves by periodic grating structures is reviewed. Formulations for a stable and efficient numerical implementation of the analysis technique are presented for one-dimensional binary gratings for both TE and TM polarization and for the general case of conical diffraction. It is shown that by exploitation of the symmetry of the diffraction problem a very efficient formulation, with up to an order-of-magnitude improvement in the numerical efficiency, is produced. The rigorous coupled-wave analysis is shown to be inherently stable. The sources of potential numerical problems associated with underflow and overflow, inherent in digital calculations, are presented. A formulation that anticipates and preempts these instability problems is presented. The calculated diffraction efficiencies for dielectric gratings are shown to converge to the correct value with an increasing number of space harmonics over a wide range of parameters, including very deep gratings. The effect of the number of harmonics on the convergence of the diffraction efficiencies is investigated. More field harmonics are shown to be required for the convergence of gratings with larger grating periods, deeper gratings, TM polarization, and conical diffraction.

http://circuit.ucsd.edu/~zhaowei/ECE212C/notes/Lecture6_reading_materials.pdf

Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings

The recent reformulation of the coupled-wave method by Lalanne and Morris [ J. Opt. Soc. Am. A13, 779 ( 1996)] and by Granet and Guizal [ J. Opt. Soc. Am. A13, 1019 ( 1996)], which dramatically improves the convergence of the method for metallic gratings in TM polarization, is given a firm mathematical foundation in this paper. The new formulation converges faster because it uniformly satisfies the boundary conditions in the grating region, whereas the old formulations do so only nonuniformly. Mathematical theorems that govern the factorization of the Fourier coefficients of products of functions having jump discontinuities are given. The results of this paper are applicable to any numerical work that requires the Fourier analysis of products of discontinuous periodic functions.

https://mcgrating.com/references/ref11.pdf

Use of Fourier series in the analysis of discontinuous periodic structures

The spin Hall effect (SHE) of light is very weak because of the extremely small photon momentum and spin-orbit interaction. Here, we report a strong photonic SHE resulting in a measured large splitting of polarized light at metasurfaces. The rapidly varying phase discontinuities along a metasurface, breaking the axial symmetry of the system, enable the direct observation of large transverse motion of circularly polarized light, even at normal incidence. The strong spin-orbit interaction deviates the polarized light from the trajectory prescribed by the ordinary Fermat principle. Such a strong and broadband photonic SHE may provide a route for exploiting the spin and orbit angular momentum of light for information processing and communication.

http://science.sciencemag.org/content/sci/339/6126/1405.full.pdf

Photonic spin hall effect at metasurfaces

A new formulation of the Fourier modal method (FMM) that applies the correct
rules of Fourier factorization for crossed surface-relief gratings is presented.
The new formulation adopts a general nonrectangular Cartesian coordinate system,
which gives the FMM greater generality and in some cases the ability to save
computer memory and computation time. By numerical examples, the new FMM is
shown to converge much faster than the old FMM. In particular, the FMM is
used to produce well-converged numerical results for metallic crossed gratings.
In addition, two matrix truncation schemes, the parallelogramic truncation
and a new circular truncation, are considered. Numerical experiments show
that the former is superior.

New formulation of the Fourier modal method for crossed surface-relief gratings

The coupled-wave method formulated by Moharam and Gaylord [ J. Opt. Soc. Am.73, 451 ( 1983)] is known to be slowly converging, especially for TM polarization of metallic lamellar gratings. The slow convergence rate has been analyzed in detail by Li and Haggans [ J. Opt. Soc. Am. A10, 1184 ( 1993)], who made clear that special care must be taken when coupled-wave methods are used for TM polarization. By reformulating the eigenproblem of the coupled-wave method, we provide numerical evidence and argue that highly improved convergence rates similar to the TE polarization case can be obtained. The discussion includes both nonconical and conical mountings.

https://mcgrating.com/references/ref10.pdf

Highly improved convergence of the coupled-wave method for TM polarization

Numerical evidence is presented that shows that, for metallic lamellar gratings in TM polarization, the coupled-wave method formulated by Moharam and Gaylord [ J. Opt. Soc. Am. A3, 1780 ( 1986)] converges slowly. (In some cases, for achieving a relative error of less than 1% in diffraction efficiencies, the number of spatial harmonics retained in the computation must be much greater than 100.) By classification of the modal methods for analyzing diffraction gratings into two distinct categories, the cause for the slow convergence is analyzed and attributed to the use of Fourier expansions to represent the permittivity and the electromagnetic fields in the grating region. The eigenvalues and the eigenfunctions of the modal fields in the grating region, whose accurate determination is crucial to the success of the coupled-wave method, are shown to converge slowly as a result of the use of these Fourier expansions. Despite its versatility and simplicity, the coupled-wave method should be used with caution for metallic surface-relief gratings in TM polarization.

Convergence of the coupled-wave method for metallic lamellar diffraction gratings

A formalism is presented for calculating the ordinary and extraordinary refractive indices for a uniaxial film with optical properties equivalent to those of a conically mounted zeroth-order lamellar grating. In the quasi-static limit (grating-period-to-wavelength ratio → 0) this treatment is exact for both dielectric and metallic gratings. For period-to-wavelength ratios approaching unity, the effective anisotropic refractive indices are dependent on the polar and azimuthal incidence angles. The validity of this method as the period-to-wavelength ratio increases is studied by comparison of the method with the predictions of rigorous grating theories. Comparisons are then made with the predictions of other effective-medium theories. Examples of the use of this theory for conically mounted zeroth-order gratings are presented.

Effective-medium theory of zeroth-order lamellar gratings in conical mountings

High spatial frequency lamellar gratings are shown to function as phase compensators, quarter-wave and half-wave retarders, and polarization rotators that operate on specularly reflected (zeroth-order) beams. These gratings are designed using rigorous coupled-wave and modal grating diffraction theories. Controlling the geometrical parameters of these gratings allows for engineering the phase retardation and polarization conversion introduced to a reflected beam. Fabrication and operational tolerances for these elements are discussed. Wavelength and polar angle of incidence variation affect the performance of these elements more strongly than variations in other geometrical and operational parameters.

Lamellar gratings as polarization components for specularly reflected beams

The optical components in the detection train of a conventional magneto-optical (MO) disk head include a half-wave plate and a polarization beamsplitter. These polarization components are bulky and require specialized mounting hardware. In order to realize a more compact head, we propose that these elements be replaced by an integrated device composed of cascaded volume and surface-relief gratings. Herein, the proposed system is described in detail for the individual elements, theoretical and prototype element performance are compared, and the operational tolerances of these elements are discussed.

/pdf/integrated-device-with-diffractive-polarization-components-3wbng5uu48.pdf

Integrated device with diffractive polarization components for a magneto-optical disk head

A rigorous coupled wave model is presented, experimentally validated, and used for tolerancing surface relief diffractive elements Applications of the model in the design and tolerancing of components for magneto optical (M-O) data storage heads are investigated

/pdf/use-of-rigorous-vector-coupled-wave-theory-for-designing-and-2dt66hn11o.pdf

Charles W. Haggans

Papers

Convergence of the coupled-wave method for metallic lamellar diffraction gratings

Effective-medium theory of zeroth-order lamellar gratings in conical mountings

Lamellar gratings as polarization components for specularly reflected beams

Integrated device with diffractive polarization components for a magneto-optical disk head

Use of rigorous vector coupled-wave theory for designing and tolerancing surface-relief diffractive components for magneto-optical heads