Nonparaxial scalar treatment of sinusoidal phase gratings

Scalar diffraction theory is frequently considered inadequate for predicting diffraction efficiencies for grating applications where lambda/d>0.1. It has also been stated that scalar theory imposes energy upon the evanescent diffracted orders. These notions, as well as several other common misconceptions, are driven more by an unnecessary paraxial approximation in the traditional Fourier treatment of scalar diffraction theory than by the scalar limitation. By scaling the spatial variables by the wavelength, we have previously shown that diffracted radiance is shift invariant in direction cosine space. Thus simple Fourier techniques can now be used to predict a variety of wide-angle (nonparaxial) diffraction grating effects. These include (1) the redistribution of energy from the evanescent orders to the propagating ones, (2) the angular broadening (and apparent shifting) of wide-angle diffracted orders, and (3) nonparaxial diffraction efficiencies predicted with an accuracy usually thought to require rigorous electromagnetic theory.     

Effects of abrupt surface-profile transitions in nonparaxial diffractive optics

It is shown that the failure of the thin-element approximation of diffractive optics may, in the first approximation, be attributed to local diffraction effects caused by the abrupt vertical transitions in binary surface-relief profiles. We determine the field disturbance caused by a single-step transition (of given height) by rigorous diffraction theory. Associating such a disturbance with each individual transition point in the profile, we obtain a computationally efficient refinement of the thin-element approximation for the analysis and design of diffractive elements in the nonparaxial domain. The results agree well with those obtained by global application of rigorous diffraction theory, provided that the smallest features in the binary profile are larger than approximately one optical wavelength.     

Step-transition perturbation approach for pixel-structured nonparaxial diffractive elements

An extension of an approximate step-transition perturbation method is presented that permits numerically efficient diffraction analysis of pixel-structured surface profiles in the nonparaxial domain. Comparison with the rigorous diffraction theory of gratings shows that the method is reasonably accurate provided that the pixel size exceeds approximately two wavelengths even if the structure contains isolated pixels.     

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.     

Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond

We describe monochromatic light propagation in uniaxial crystals by means of an exact solution of Maxwell's equations. We subsequently develop a paraxial scheme for describing a beam traveling orthogonal to the optical axis. We show that the Cartesian field components parallel and orthogonal to the optical axis are extraordinary and ordinary, respectively, and hence uncoupled. The ordinary component exhibits a standard Fresnel behavior, whereas the extraordinary one exhibits interesting anisotropic diffraction dynamics. We interpret the anisotropic diffraction as a composition of two spatial geometrical affinities and a single Fresnel propagation step. As an application, we obtain the analytical expression of the extraordinary Gaussian beam. We then derive the first nonparaxial correction to the paraxial beam, thus giving a scheme for describing slightly nonparaxial fields. We find that nonparaxiality couples the Cartesian components of the field and that the resultant longitudinal component is greater than the correction to the transverse component orthogonal to the optical axis. Finally, we derive the analytical expression for the nonparaxial correction to the paraxial Gaussian beam.     

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.     

Design of a Binary Grating with Subwavelength Features that Acts as a Polarizing Beam Splitter

A binary diffractive optical element, acting as a polarizing beam splitter, is proposed and analyzed. It behaves like a transmissive blazed grating, working on the first or the second diffraction order, depending on the polarization state of the incident radiation. The grating-phase profile required for both polarization states is obtained by means of suitably sized subwavelength groups etched in an isotropic dielectric medium. A rigorous electromagnetic analysis of the grating is presented, and numerical results concerning its performances in terms of diffraction efficiency as well as frequency and angular bandwidths are provided.     

Nonparaxial propagation properties of an anomalous hollow beam with orbital angular momentum

The analytical nonparaxial propagation formula of an anomalous hollow beam (AHB) with orbital angular momentum (OAM) in free space is derived based on the generalized Raleigh–Sommerfeld diffraction integral. The nonparaxial properties of AHB with OAM such as intensity, phase and OAM density distributions are studied in detail, using the pertinent nonparaxial propagation formula. The comparison between the paraxial and nonparaxial results is also carried out. The results show that the nonparaxial properties of an AHB with OAM are determined by the initial beam parameters, such as beam waist size and topological charge and propagation distance.     

New method for nonparaxial beam propagation

A new method for solving the wave equation is presented that is nonparaxial and can be applied to wide-angle beam propagation. It shows very good stability characteristics in the sense that relatively larger step sizes can be taken. An implementation by use of the collocation method is presented in which only simple matrix multiplications are involved and no numerical matrix diagonalization or inversion is needed. The method is hence faster and is also highly accurate.     

Optimization design of diffractive phase elements for beam shaping

An improved approach called the weighted YG algorithm for the design of the diffractive phase element (DPE) that implements beam shaping in the fractional Fourier transform domain and free space is presented. Modeling designs of the DPE are carried out for several fractional orders and different parameters of the beam for optimally converting a Gaussian profile into a uniform beam. We found that our algorithm can improve the beam shaping effect, reduce the error function, and increase uniformity of light intensity.     

Smallest flattop focus by polarization engineering

Spatial engineering of polarization as a new method of beam shaping is analyzed by using scalar diffraction theory. For the one-dimensional case, it is shown that the smallest flattop far-field distribution can be obtained by adopting a linear polarization that changes direction as a linear function of location in the pupil plane. The resulting light field is functionally equivalent to a cosinusoidal function modulation of the wavefront but maintains high efficiency. This polarization beam shaping technique proves to be highly useful in applications where diffraction effects need to be taken into account. The extension of this technique to two-dimensional beam shaping is also demonstrated.     

Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal

Angular spectra of reflected and transmitted fields, induced by an arbitrary electromagnetic beam passing through the planar interface between a homogeneous medium and a uniaxially anisotropic medium, are derived and related to the incident medium. By using these formulas, we obtain the expressions for paraxial and slightly nonparaxial fields. The reflected paraxial field is related to the incident one by means of Fresnel relations; the transmitted paraxial field is the superposition of an ordinary and an extraordinary beam, multiplied by the Fresnel coefficient. We find that the nonparaxial corrections, owing to the medium discontinuity, are larger than their free-propagation counterparts and that they are very simply related to the paraxial solutions of the incident beam. The case of two homogeneous media with different refractive indices is also discussed. The general expressions obtained are applied to the case of a nonparaxial Gaussian beam.     

Three-dimensional imaging by partially coherent light under nonparaxial condition

In this paper, we present the theory of three-dimensional (3D) imaging using partially coherent light under the nonparaxial condition. Using the linear system approach, we derive the image intensity in terms of the 3D nonparaxial transmission cross coefficient (TCC) and the transmission function defined in this paper. We present that the 3D TCC can be calculated by multiple applications of the 3D fast Fourier transform instead of the six-dimensional integral in the original formula. Using the simplified formula, we simulate phase contrast and Nomarski differential interference contrast (DIC) imaging of a transparent 3D object. Within our knowledge, the 3D model for the DIC based on the 3D nonparaxial TCC is the most rigorous approach that has been suggested. It demonstrates clearly the optical sectioning effect of DIC.     

The vertical beam quality of GaInP/AlGaInP strained multiple quantum well laser

Abstract

The waveguided mode of the visible GaInP/AlGaInP compressive strained multiple quantum well laser is calculated by using transfer matrix method. On the basis of the nonparaxial vectorial moment theory of light beam propagation, the vertical (perpendicular to junction plane) optical beam quality factor M ²? of the waveguided mode is shown to be smaller than unity. This result may be useful for the design of semiconductor lasers and analysis of nonparaxial beam propagating characteristics.     

Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture

Based on the vectorial Rayleigh-Sommerfeld diffraction integrals, an analytical propagation equation of vectorial, nonparaxial, elliptical Gaussian beams through a rectangular aperture is derived. Unlike in previous work, the aperture effect and nonrotational symmetry of the beam and aperture are considered in our theoretical model. The results of the far-field and paraxial approximation for the apertured case are treated as special cases of our general expression. It is found that two f parameters fx, fy and two truncation parameters deltax, deltay in the x and y directions, respectively, have to be introduced that affect the beam nonparaxial evolution behavior in both the near field and the far field.     

Diffractive shaping of excimer laser beams

Abstract

We address the problem of shaping the intensity distribution of a highly directional partially coherent field, such as an excimer laser beam, by means of diffractive optics. Our theoretical analysis is based on modelling the multi-transverse-mode laser beam as a Gaussian Schell-model beam. It is shown numerically that a periodic element, which is unsuitable for the shaping of a coherent laser beam, works well with an excimer laser beam because of its partial spatial coherence. The conversion of an approximately Gaussian excimer laser beam into a flat-top beam in the Fourier plane of a lens is demonstrated with a diffractive beam shaper fabricated as a multilevel profile in SiOl by electron-beam lithography and proportional reactive-ion etching.     

Topological phase in a nonparaxial Gaussian beam

An analysis is made of the structure and evolution of the singularities of a nonparaxial Gaussian beam. It is shown that a Gaussian beam may be represented by a family of straight lines lying on the surface of a hyperboloid and that the wavefront of this beam is a function of a point source situated at a point on the z axis with the imaginary coordinate iz ₀. The argument of this complex function is the topological phase of the beam which characterizes the rotation of the wavefront. The singularities of a nonparaxial Gaussian beam are located in the focal plane and are annular edge dislocations. Dislocation processes near the constriction of the Gaussian beam only occur as a result of aperture diffraction. Pis’ma Zh. Tekh. Fiz. 25, 14–20 (November 26, 1999)     

Bessel-Gauss beams as rigorous solutions of the Helmholtz equation

The study of the nonparaxial propagation of optical beams has received considerable attention. In particular, the so-called complex-source/sink model can be used to describe strongly focused beams near the beam waist, but this method has not yet been applied to the Bessel-Gauss (BG) beam. In this paper, the complex-source/sink solution for the nonparaxial BG beam is expressed as a superposition of nonparaxial elegant Laguerre-Gaussian beams. This provides a direct way to write the explicit expression for a tightly focused BG beam that is an exact solution of the Helmholtz equation. It reduces correctly to the paraxial BG beam, the nonparaxial Gaussian beam, and the Bessel beam in the appropriate limits. The analytical expression can be used to calculate the field of a BG beam near its waist, and it may be useful in investigating the features of BG beams under tight focusing conditions.