Aberrations of diffracted wave fields: distortion

Near-field diffraction patterns are merely aberrated Fraunhofer diffraction patterns. These aberrations, inherent to the diffraction process, provide insight and understanding into wide-angle diffraction phenomena. Nonparaxial patterns of diffracted orders produced by a laser beam passing through a grating and projected upon a plane screen exhibit severe distortion (W311). This distortion is an artifact of the configuration chosen to observe diffraction patterns. Grating behavior expressed in terms of the direction cosines of the propagation vectors of the incident and diffracted orders exhibits no distortion. Use of a simple direction cosine diagram provides an elegant way to deal with nonparaxial diffraction patterns, particularly when large obliquely incident beams produce conical diffraction.     

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.     

Beam shaping in the nonparaxial domain of diffractive optics

We address the problem of shaping the radiant intensity distribution of a highly nonparaxial coherent field by means of a diffractive element located in the plane of the beam waist. To be capable of wide-angle energy redistribution the element must necessarily contain wavelength-scale transverse features, and consequently it must be designed on the basis of rigorous diffraction theory. We consider, in particular, wide-angle Gaussian to flat-top beam shaping in one dimension. Scalar designs are provided and their validity is evaluated by rigorous diffraction theory, which is also used for optimization deep inside the nonparaxial domain, where the scalar designs fail. Experimental verification is provided by means of electron-beam lithography.     

Diffraction efficiency of 200-nm-period critical-angle transmission gratings in the soft x-ray and extreme ultraviolet wavelength bands

We report on measurements of the diffraction efficiency of 200-nm-period freestanding blazed transmission gratings for wavelengths in the 0.96 to 19.4 nm range. These critical-angle transmission (CAT) gratings achieve highly efficient blazing over a broad band via total external reflection off the sidewalls of smooth, tens of nanometer thin ultrahigh aspect-ratio silicon grating bars and thus combine the advantages of blazed x-ray reflection gratings with those of more conventional x-ray transmission gratings. Prototype gratings with maximum depths of 3.2 and 6 μm were investigated at two different blaze angles. In these initial CAT gratings the grating bars are monolithically connected to a cross support mesh that only leaves less than half of the grating area unobstructed. Because of our initial fabrication approach, the support mesh bars feature a strongly trapezoidal cross section that leads to varying CAT grating depths and partial absorption of diffracted orders. While theory predicts broadband absolute diffraction efficiencies as high as 60% for ideal CAT gratings without a support mesh, experimental results show efficiencies in the range of ～50-100% of theoretical predictions when taking the effects of the support mesh into account. Future minimization of the support mesh therefore promises broadband CAT grating absolute diffraction efficiencies of 50% or higher.     

Diffraction properties of volume holographic gratings for an ultrashort pulsed beam with different polarization states readout

The diffraction properties of volume holographic gratings are studied when the gratings are illuminated by an ultrashort pulsed beam with different polarization states. The developed coupled wave theory of Kogelnik is used. Considering the dispersion effect of the grating media, solutions for the diffracted and transmitted intensities, diffraction efficiencies and the bandwidths of the gratings are given in transmission volume holographic gratings and reflection volume holographic gratings. The bandwidths of the gratings are reduced by the dispersion effect of the grating media. They also have different influences on the diffraction of an ultrashort pulsed beam with different polarization states. For different values of the ratio of the spectral bandwidth of the input pulse to that of the grating, the changes of the spectral and temporal distributions of the diffracted intensities, as well as the diffraction efficiencies of the gratings are shown.     

Application of Mie Scattering of Evanescent Waves to Scanning Tunnelling Optical Microscopy Theory

Abstract

In this paper we present a macroscopic theory for scanning tunnelling optical microscopy where the sample is a grating and the tip is modelled by a dielectric sphere. The sphere is immersed in the near-field above the grating and is excited by the diffracted orders which can be evanescent. The detected signal is supposed to be the diffracted intensity by the sphere which is calculated by using Mie theory. When studying Mie scattering of one evanescent wave we show that the multipolar series is perturbed compared to scattering of a homogeneous wave. Even for a small sphere multipolar terms have to be taken into account. We have then proposed a formula for the intensity leading to calculations of intensity profile and surfaces and to discuss the influence of tip radius on resolution of the images.     

Ultraviolet groove efficiency of a holographic grating: implications for a dual-order spectrograph

The ultraviolet groove efficiency for a holographically ruled diffraction grating with a trapezoidal profile has been measured. The efficiencies for the +/-1 and the zero orders are in good agreement with those derived from scalar theory. The +/-1 orders have equal efficiency as a function of wavelength. The peak of the sum of fitted groove efficiency functions is 76%, a level that is competitive with the groove efficiency of a mechanically blazed grating. We suggest that a normal-incidence grating mount with detectors at both orders will offer a system with twice the efficiency and provide a built-in redundancy. We discuss design considerations for reducing astigmatism equally in both orders in such dual-order mountings.     

Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization

A rigorous modal theory for the diffraction of Gaussian beams from N equally spaced slits (finite grating) in a planar perfectly conducting thin screen is presented. The case of normal incidence and TE polarization state is considered; i.e., the electric field is parallel to the slits. The characteristics of the far-field diffraction patterns, the transmission coefficient, and the normally diffracted energy as a function of several optogeometrical parameters are analyzed within the so-called vectorial region, where the polarization effects are important. The diffraction pattern of an aperiodic grating is also considered. In addition, one diffraction property known to be valid in the scalar region is generalized to the vectorial region: the existence of constant-intensity angles in the far field when the incident beam wave is scanned along the N slits. The classical grating equation is tested for incident Gaussian beams under several conditions.     

The influence of grating profile on surface plasmon polariton resonances recorded in different diffracted orders

Abstract

Nonlinear processes involved in the manufacture of nominally sinusoidal surface relief diffraction gratings can introduce distortions into the profile of these surfaces. Such distortions may dramatically affect both the specular reflectivity and diffracted efficiencies from such a surface, particularly if it is metallic. To illustrate this a comprehensive numerical modelling study of the optical coupling to surface plasmon polaritons on silver gratings has been undertaken. The grating surface profile is represented in terms of a truncated Fourier series, the effect of varying the amplitude, and then the phase, of each Fourier component in turn, is explored. This illustrates the sensitivity of individual features to specific harmonic components of the surface, for surface plasmon resonances recorded in both the zeroth and higher diffracted orders.     

Phase constraint for the waves diffracted by lossless symmetrical gratings at Littrow mount

The energy conservation of grating diffraction is analyzed in a particular condition of incidence in which two incident waves reach a symmetrical grating from the two sides of the grating normal at the first-order Littrow mounting. In such a situation the incident waves generate an interference pattern with the same period as the grating. Thus in each direction of diffraction, interference occurs between two consecutive diffractive orders of the symmetrical incident waves. By applying only energy conservation and the geometrical symmetry of the grating profile to this problem it is possible to establish a general constraint for the phases and amplitudes of the diffracted orders of the same incident wave. Experimental and theoretical results are presented confirming the obtained relations.     

Vector diffraction analysis of optical disk readout

The optical disk readout signals from ROM disks are presented by use of a rigorous three-dimensional vector diffraction method. The optical disk is modeled as a crossed metal grating without restriction on the form of the information marks, and the permittivity of the metal is taken into account. The diffracted field from the disk is obtained by means of decomposing the focused incident beam into a spectrum of plane waves and then calculating the diffracted plane waves for each respective incident component. The readout signal is obtained by integration of the energy-flux density of the diffracted field according to the detection scheme of the optical disk system. A typical digital versatile disk (DVD) system is applied with this theory, and the result is far from that of scalar diffraction theory.     

Transmittance analysis of diffraction phase grating

In order to accurately analyze and design the transmittance characteristic of a diffraction phase grating, the validity of both the scalar diffraction theory and the effective medium theory is quantitatively evaluated by the comparison of diffraction efficiencies predicted from both simplified theories to exact results calculated by the rigorous vector electromagnetic theory. The effect of surface profile parameters, including the normalized period, the normalized depth, and the fill factor for the precision of the simplified methods is determined at normal incidence. It is found that, in general, when the normalized period is more than four wavelengths of the incident light, the scalar diffraction theory is useful to estimate the transmittance of the phase grating. When the fill factor approaches 0.5, the error of the scalar method is minimized, and the scalar theory is accurate even at the grating period of two wavelengths. The transmittance characteristic as a function of the normalized period is strongly influenced by the grating duty cycle, but the diffraction performance on the normalized depth is independent of the fill factor of the grating. Additionally, the effective medium theory is accurate for evaluating the diffraction efficiency within an error of less than around 1% when no higher-order diffraction waves appear and only the zero-order waves exist. The precision of the effective medium theory for calculating transmittance properties as a function of the normalized period, the normalized groove depth, and the polarization state of incident light is insensitive to the fill factor of the phase grating.     

Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis

The range of validity of the scalar diffraction analysis is quantified for the case of two-dimensionally-periodic diffractive optical elements (crossed gratings). Three canonical classes of two-dimensionally-periodic grating structures are analyzed by using the rigorous coupled-wave analysis as well as the scalar diffraction analysis. In all cases the scalar-analysis diffraction efficiencies are compared with the exact diffraction efficiencies. The error in using the scalar analysis is then determined as a function of the grating-period(s)-to-wavelength ratio(s), the minimum feature size, the grating depth, the refractive index of the grating, the incident polarization, and the number of phase levels. The three classes of two-dimensional (2-D) unit cells are as follows: (1) a rectangular pillar, (2) an elliptical pillar, and (3) an arbitrarily pixellated multilevel 2-D unit cell that is representative of more complicated diffractive optical elements such as computer-generated holograms. In all cases a normally incident electromagnetic plane wave is considered. It is shown that the error of the scalar diffraction analysis in the case of two-dimensionally-periodic diffractive optical elements is greater than that for the corresponding one-dimensionally-periodic counterparts. In addition, the accuracy of the scalar diffraction analysis degrades with increasing refractive index, grating thickness, and asymmetry of the 2-D unit cell and with decreasing grating-period-to-wavelength ratio and feature size.     

Optical measurement of depth and duty cycle for binary diffraction gratings with subwavelength features

We describe a new and unique method for simultaneous determination of the groove depth and duty cycle of binary diffraction gratings. For a near-normal angle of incidence, the +1 and -1 diffracted orders will behave nearly the same as the duty cycle is varied for a fixed grating depth. The difference in their behavior, quantified as the ratio of their respective diffraction efficiencies, is compared to a look-up table generated by rigorous coupled-wave theory, and the duty cycle of the grating is thus obtained as a function of grating depth. Performing the same analysis for the orthogonal probe-light polarization results in a different functional dependence of the duty cycle on the grating depth. By use of both TE and TM polarizations, the depth and duty cycle for the grating are obtained by the intersection of the functions generated by the individual polarizations. These measurements can also be used to assess qualitatively both the uniformity of the grating and the symmetry of the grating profile. Comparison with scanning electron microscope images shows excellent agreement. This method is advantageous since it can be carried out rapidly, is accurate and repeatable, does not damage the sample, and uses low-cost, commonly available equipment. Since this method consists of only four fixed simple measurements, it is highly suitable for quality control in a manufacturing environment.     

Rigorous analysis of diffraction gratings of arbitrary profiles using virtual photonic crystals

A new approach is developed to calculate diffraction efficiency for a dielectric grating with an arbitrary refractive index profile. By treating a one-dimensional grating as a segment of a virtual two-dimensional (2D) photonic crystal, we exploit a rigorous theory of photonic crystal refraction and calculate the diffraction efficiencies. We expand, analytically in many cases, the dielectric function of the grating into 2D Fourier series. We find the eigenmodes for the virtual photonic crystal, and then use these eigenmodes to match the boundary conditions by solving a set of linear equations. In two such simple steps, the diffraction efficiencies can be computed rigorously without slicing the grating into thin layers.     

Diffraction of a one-dimensional phase grating in the deep Fresnel field

We analyze theoretically the diffraction of phase gratings in the deep Fresnel field on the basis of the theory of scalar diffraction and Green's theorem and present the general formula for the diffraction intensity of a one-dimensional sinusoidal phase grating. The numerical calculations show that in the deep Fresnel region the diffraction distribution can be described by designating three characteristic regions that are influenced by the parameters of the grating. The microlensing effect of the interface of the phase grating provides the corresponding explanation. Moreover, according to the viewpoint that the diffraction intensity distribution is the result of the interference of the diffraction orders of the grating, we find that the diffraction patterns, depending on the carved depth of the phase grating, are determined by the contributing diffraction orders, their relative power, and the quasi-Talbot effect of the phase grating, which results from the second meeting of the diffraction orders carrying most of the power of the total field, as in the case of the amplitude grating.     

衍射频谱法测光栅膜层厚度

根据傅立叶光学理论从光栅的衍射频谱中可以反推光栅本身的多种信息,包括其形貌特征。 利用矩形相位光栅的傅立叶变换,推导出零级和一级衍射光强和矩形相位光栅膜层厚度之间的函数关系,据此可在测得零级和一级次光强后,推算出光栅膜层厚度。以台阶仪作为标准,该方法的测量误差在4%以内。     

Effective grating theory for resonance domain surface-relief diffraction gratings

An effective grating model, which generalizes effective-medium theory to the case of resonance domain surface-relief gratings, is presented. In addition to the zero order, it takes into account the first diffraction order, which obeys the Bragg condition. Modeling the surface-relief grating as an effective grating with two diffraction orders provides closed-form analytical relationships between efficiency and grating parameters. The aspect ratio, the grating period, and the required incidence angle that would lead to high diffraction efficiencies are predicted for TE and TM polarization and verified by rigorous numerical calculations.     

Evanescent interferometric lithography

Simulation results are presented to illustrate the main features of what we believe is a new photolithographic technique, evanescent interferometric lithography (EIL). The technique exploits interference between resonantly enhanced, evanescently decaying diffracted orders to create a frequency-doubled intensity pattern in the near field of a metallic diffraction grating. It is shown that the intensity in a grating's near field can be enhanced significantly compared with conventional interferometric lithography. Contrast in the interference pattern is also increased, owing to a reduction in the zeroth-order transmission near resonance. The pattern's depth of field reduces as the wavelength is increased beyond cutoff of the first-order diffracted components, and results are presented showing the trade-offs that can be made between depth of field and intensity enhancement. Examples are given for a 270-nm-period grating embedded in material with refractive index n = 1.6 and illuminated with wavelengths near 450 nm. Under these conditions it is predicted that high-intensity, high-contrast patterns with 135-nm period can be formed in photoresists more than 50 nm thick.     

Three-dimensional converging-diverging Gaussian beam diffraction by a volume grating

The diffraction characteristics of a volume grating (VG) illuminated by a three-dimensional (3-D) converging-diverging Gaussian beam at conical incidence are investigated by applying 3-D finite-beam (FB) rigorous coupled-wave analysis (RCWA) based on the conventional 3-D RCWA in conjunction with two-dimensional plane-wave decomposition. The Gaussian beam is assumed to have an arbitrary incidence angle, an arbitrary azimuthal angle, and any linear polarization. The two cases with linear polarizations of the central beam of the Gaussian (E perpendicular K and H perpendicular K) are investigated. The diffraction efficiencies and the diffracted beam profiles for both unslanted VGs and slanted VGs (designed for substrate-mode optical interconnects) are presented. In general, the diffraction efficiencies of a converging-diverging spherical Gaussian beam diffracted by both unslanted VGs and slanted VGs increase and approach the central-beam results as the refractive-index modulation increases.