Eric B. Grann

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

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.

Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach

An enhanced, numerically stable transmittance matrix approach is developed and is applied to the implementation of the rigorous coupled-wave analysis for surface-relief and multilevel gratings. The enhanced approach is shown to produce numerically stable results for excessively deep multilevel surface-relief dielectric gratings. The nature of the numerical instability for the classic transmission matrix approach in the presence of evanescent fields is determined. The finite precision of the numerical representation on digital computers results in insufficient accuracy in numerically representing the elements produced by inverting an ill-conditioned transmission matrix. These inaccuracies will result in numerical instability in the calculations for successive field matching between the layers. The new technique that we present anticipates and preempts these potential numerical problems. In addition to the full-solution approach whereby all the reflected and the transmitted amplitudes are calculated, a simpler, more efficient formulation is proposed for cases in which only the reflected amplitudes (or the transmitted amplitudes) are required. Incorporating this enhanced approach into the implementation of the rigorous coupled-wave analysis, we obtain numerically stable and convergent results for excessively deep (50 wavelengths), 16-level, asymmetric binary gratings. Calculated results are presented for both TE and TM polarization and for conical diffraction.

Limits of scalar diffraction theory for diffractive phase elements

The range of validity and the accuracy of scalar diffraction theory for periodic diffractive phase elements (DPE’s) is evaluated by a comparison of diffraction efficiencies predicted from scalar theory to exact results calculated with a rigorous electromagnetic theory. The effects of DPE parameters (depth, feature size, period, index of refraction, angle of incidence, fill factor, and number of binary levels) on the accuracy of scalar diffraction theory is determined. It is found that, in general, the error of scalar theory is significant (∊ > ±5%) when the feature size is less than 14 wavelengths (s < 14λ). The error is minimized when the fill factor approaches 50%, even for small feature sizes (s = 2λ); for elements with an overall fill factor of 50% the larger period of the DPE replaces the smaller feature size as the condition of validity for scalar diffraction theory. For an 8-level DPE of refractive index 1.5 analyzed at normal incidence the error of the scalar analysis is greater than ±5% when the period is less than 20 wavelengths (Λ < 20λ). The accuracy of the scalar treatment degrades as either the index of refraction, the depth, the number of binary levels, or the angle of incidence is increased. The conclusions are, in general, applicable to nonperiodic as well as other periodic (trapezoidal, two-dimensional) structures.

Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings

Two-dimensional symmetric and asymmetric subwavelength binary gratings are investigated. A method for determining the three effective indices of a two-dimensional (2-D) subwavelength grating is presented, as well as a theoretical formalization for the effective index parallel with the normal to the surface. It is shown that a 2-D asymmetric binary grating on the surface of a dielectric substrate is analogous to a biaxial thin film. If the grating is symmetric, then the two effective indices perpendicular to the normal are equal, and the grating is analogous to a uniaxial thin film. Using these effective indices and the quarter-wave Tschebyscheff synthesis technique, we designed two- and three-level binary gratings to suppress reflections over a broad band. It is shown that for a substrate index of ns = 3.0 a three-level 2-D binary grating reduced reflections below 0.1% from 8 μm to 12 μm.

Effects of process errors on the diffraction characteristics of binary dielectric gratings.

The effects of fabrication errors on the predicted performance of surface-relief phase gratings are analyzed with a rigorous vector diffraction technique. For binary elements, errors in the dimensions of the profile [depth, linewidth (fill factor), and grating period], as well as errors in the shape of the profile, are investigated. It is shown that the dimension errors do not have a significant effect on grating performance when the grating is designed for either maximum or minimum diffraction efficiency. A trapezoid is used to model the shape error of the profile. For the first time, design rules that significantly reduce the effects of any shape error are presented.

Hybrid two-dimensional subwavelength surface-relief grating–mesh structures

The homogeneous behavior of periodic two-dimensional subwavelength surface-relief structures that contain both gratings and meshes (inverse gratings) are investigated. It is shown that when effective indices are synthesized near the higher index (substrate region), mesh structures yield larger feature sizes compared with their grating counterparts, whereas grating structures yield larger feature sizes when effective indices are synthesized near the lower index (incident region). For each type of structure investigated, a relation between the parameters of the structure and an effective refractive index is determined. It is shown that an equal area occupied by the high- or low-index media within the grating cell does not, in general, result in equal effective indices. The effective index of the grating is shown to be characterized by both the shape (local distribution) and the area of the high- or low-index medium within the unit grating cell. Finally, the advantages of subwavelength gratings and meshes are combined to produce hybrid grating-mesh structures that are less demanding on the fabrication process.