Non-paraxial Scalar Diffraction Theory Research Articles

Linking Diffractive and Geometrical Optics Surface Scattering at a Fundamental Level

Optical surface scattering analyses based on diffractive optics (DO) are typically applied to one surface; however, there is a need for simulating surface scattering losses for devices having many surface interactions such as light pipes. Light pipes are often simulated with geometric optics (GO) using ray tracing, where surface scattering is driven by the surface slope distribution. In the DO case, surface scattering analyses depend on the spatial frequency distribution and amplitude as well as wavelength, with the sinusoidal grating as a fundamental basis. A better understanding of the link, or transition, between DO and GO scattering domains would be helpful for efficiently incorporating scattering loss analyses into ray trace simulations. A formula for the root-mean-square (rms) scattered angle width of a sinusoidal reflection grating that depends only on the surface rms slope is derived from the nonparaxial scalar diffraction theory, thereby linking it to GO. The scatter angle’s mean and rms width are evaluated over a range of grating amplitudes and periods using scalar theory and full vector simulations from the COMSOL® wave optic module for a sinusoidal reflection grating. The conditions under which the diffraction-based solution closely approximates the GO solution, as predicted by the rms slope, are identified. Close agreement is shown between the DO and GO solutions for the same surface rms slope scattering loss due to angular filtering near the critical angle of a total internal reflection (TIR) glass-to-air interface.

Nonparaxial scalar diffraction theory modifications for improved efficiency estimation

Using the nonparaxial scalar diffraction equations to estimate the diffraction efficiency of multiple orders of a sinusoidal reflection grating, we compare the results of transverse electric (TE) and transverse magnetic (TM) vector wave simulations for a reflective surface on glass. Modifications are presented that enable the nonparaxial scalar solution to approximate simulation results in each order to within 1% to 2% across a 0 to 90 deg range of incidence angles when the diffracted power is predominantly carried in the first several orders, up to the fourth order. The accuracy of simulations in this regime is particularly important for surface scattering where individual spatial frequencies typically have amplitudes much less than a hundred nanometers. A substantial reduction in the error between vector simulations and scalar estimation of diffraction efficiency is demonstrated with the following modifications: (1) an obliquity factor correction for the specular order based on a first-order power conservation criterion, (2) a “cutoff” factor modifying the obliquity factor as the diffracted angle approaches 90 deg for each order, and (3) nonuniform unallocated power distribution near the low-order cutoffs. The second and third modifications are applied to higher diffraction orders for extending the range of applicable grating heights up to 200 nm (peak-to-valley), periods down to twice the wavelength, and slopes up to 0.2 (root mean square).

Understanding diffraction grating behavior, part II: parametric diffraction efficiency of sinusoidal reflection (holographic) gratings

With the widespread availability of electromagnetic (vector) analysis codes for describing the diffraction of electromagnetic waves by periodic grating structures, the insight and understanding of nonparaxial parametric diffraction grating behavior afforded by approximate methods (i.e., scalar diffraction theory) is being ignored in the education of most optical engineers today. We show that the linear systems formulation of nonparaxial scalar diffraction theory enables the development of a scalar parametric diffraction grating model [for transverse electric (TE) polarization] for sinusoidal reflection gratings with arbitrary groove depths and arbitrary nonparaxial incident and diffracted angles. This scalar parametric analysis is remarkably accurate as it includes the ability to redistribute the energy from evanescent orders into the propagating ones, thus allowing the calculation of nonparaxial diffraction efficiencies to be predicted with an accuracy usually thought to require rigorous electromagnetic theory. These scalar parametric predictions of diffraction efficiency are compared to paraxial scalar and rigorous electromagnetic (vector) predictions for a variety of nonparaxial diffraction grating configurations, thus providing quantitative limits of applicability of nonparaxial scalar diffraction theory to sinusoidal reflection gratings as a function of the grating period-to-wavelength ratio (λ / <italic>d</italic>).

Understanding diffraction grating behavior: including conical diffraction and Rayleigh anomalies from transmission gratings

With the wide-spread availability of rigorous electromagnetic (vector) analysis codes for describing the diffraction of electromagnetic waves by specific periodic grating structures, the insight and understanding of nonparaxial parametric diffraction grating behavior afforded by approximate methods (i.e., scalar diffraction theory) is being ignored in the education of most optical engineers today. Elementary diffraction grating behavior is reviewed, the importance of maintaining consistency in the sign convention for the planar diffraction grating equation is emphasized, and the advantages of discussing “conical” diffraction grating behavior in terms of the direction cosines of the incident and diffracted angles are demonstrated. Paraxial grating behavior for coarse gratings (<italic>d</italic> ≫ λ) is then derived and displayed graphically for five elementary grating types: sinusoidal amplitude gratings, square-wave amplitude gratings, sinusoidal phase gratings, square-wave phase gratings, and classical blazed gratings. Paraxial diffraction efficiencies are calculated, tabulated, and compared for these five elementary grating types. Since much of the grating community erroneously believes that scalar diffraction theory is only valid in the paraxial regime, the recently developed linear systems formulation of nonparaxial scalar diffraction theory is briefly reviewed, then used to predict the nonparaxial behavior (for transverse electric polarization) of both the sinusoidal and the square-wave amplitude gratings when the +1 diffracted order is maintained in the Littrow condition. This nonparaxial behavior includes the well-known Rayleigh (Wood’s) anomaly effects that are usually thought to only be predicted by rigorous (vector) electromagnetic theory.

Diffracted radiance not proven to be fundamental quantity in nonparaxial scalar diffraction theory: comment

Diffracted radiance not proven to be fundamental quantity in nonparaxial scalar diffraction theory: comment

Scratch iridescence

The surface of metal, glass and plastic objects is often characterized by microscopic scratches caused by manufacturing and/or wear. A closer look onto such scratches reveals iridescent colors with a complex dependency on viewing and lighting conditions. The physics behind this phenomenon is well understood; it is caused by diffraction of the incident light by surface features on the order of the optical wavelength. Existing analytic models are able to reproduce spatially unresolved microstructure such as the iridescent appearance of compact disks and similar materials. Spatially resolved scratches, on the other hand, have proven elusive due to the highly complex wave-optical light transport simulations needed to account for their appearance. In this paper, we propose a wave-optical shading model based on non-paraxial scalar diffraction theory to render this class of effects. Our model expresses surface roughness as a collection of line segments. To shade a point on the surface, the individual diffraction patterns for contributing scratch segments are computed analytically and superimposed coherently. This provides natural transitions from localized glint-like iridescence to smooth BRDFs representing the superposition of many reflections at large viewing distances. We demonstrate that our model is capable of recreating the overall appearance as well as characteristic detail effects observed on real-world examples.

Light trapping in solar cells: numerical modeling with measured surface textures.

We present and experimentally validate a computational model for the light propagation in thin-film solar cells that integrates non-paraxial scalar diffraction theory with non-sequential ray-tracing. The model allows computing the spectral layer absorbances of solar cells with micro- and nano-textured interfaces directly from measured surface topographies. We can thus quantify decisive quantities such as the parasitic absorption without relying on heuristic scattering intensity distributions. In particular, we find that the commonly used approximation of Lambertian scattering intensity distributions for internal light propagation is violated even for solar cells on rough textured substrates. More importantly, we demonstrate how both scattering and parasitic absorption must be controlled to maximize photocurrent.

Color effects from scattering on random surface structures in dielectrics

We show that cheap large area color filters, based on surface scattering, can be fabricated in dielectric materials by replication of random structures in silicon. The specular transmittance of three different types of structures, corresponding to three different colors, have been characterized. The angle resolved scattering has been measured and compared to predictions based on the measured surface topography and by the use of non-paraxial scalar diffraction theory. From this it is shown that the color of the transmitted light can be predicted from the topography of the randomly textured surfaces.

Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles

Scattering effects from microtopographic surface roughness are merely nonparaxial diffraction phenomena resulting from random phase variations in the reflected or transmitted wavefront. Rayleigh-Rice, Beckmann-Kirchhoff. or Harvey-Shack surface scatter theories are commonly used to predict surface scatter effects. Smooth-surface and/or paraxial approximations have severely limited the range of applicability of each of the above theoretical treatments. A recent linear systems formulation of nonparaxial scalar diffraction theory applied to surface scatter phenomena resulted first in an empirically modified Beckmann-Kirchhoff surface scatter model, then a generalized Harvey-Shack theory that produces accurate results for rougher surfaces than the Rayleigh-Rice theory and for larger incident and scattered angles than the classical Beckmann-Kirchhoff and the original Harvey-Shack theories. These new developments simplify the analysis and understanding of nonintuitive scattering behavior from rough surfaces illuminated at arbitrary incident angles.

Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory: errata

Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory: errata

Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory

Most authors include a paraxial (small-angle) limitation in their discussion of diffracted wave fields. This paraxial limitation severely limits the conditions under which diffraction behavior is adequately described. A linear systems approach to modeling nonparaxial scalar diffraction theory is developed by normalization of the spatial variables by the wavelength of light and by recognition that the reciprocal variables in Fourier transform space are the direction cosines of the propagation vectors of the resulting angular spectrum of plane waves. It is then shown that wide-angle scalar diffraction phenomena are shift invariant with respect to changes in the incident angle only in direction cosine space. Furthermore, it is the diffracted radiance (not the intensity or the irradiance) that is shift invariant in direction cosine space. This realization greatly extends the range of parameters over which simple Fourier techniques can be used to make accurate calculations concerning wide-angle diffraction phenomena. Diffraction-grating behavior and surface-scattering effects are two diffraction phenomena that are not limited to the paraxial region and benefit greatly from this new development.

Generalized conversion from the phase function to the blazed surface-relief profile of diffractive optical elements

The problem of the relation between the phase function and the thin microrelief profile of diffractive optical elements is considered in detail through the nonparaxial scalar diffraction theory for the case of a curved substrate surface and an astigmatic incident beam. Local-plane-wave expansions are used to calculate a coefficient of proportionality between microrelief height and phase jump as a function of microrelief orientation and of local slope angles of the incident and the output beams. Nonuniform transition heights between 2π-phase zones of the microrelief are calculated. Analytical and numerical data are presented for examples of high-numerical-aperture diffractive lenses on plane, spherical, or aspherical substrate surfaces.