Design of space-variant diffractive polarization elements

Improved diffraction efficiencies can be obtained in the paraxial domain of diffractive optics by considering light explicitly as an electromagnetic rather than a scalar field because of the extra freedoms provided by the state of polarization. For example, diffractive beam splitters with 100% efficiency are made possible by means of space-variant subwavelength-carrier surface-relief elements. Some aspects of the general design theory of polarization-modulating elements for vector fields, including design freedoms and constraints, are presented. Upper bounds of diffraction efficiency are derived and compared with those for the scalar case. Iterative design algorithms are developed. Several design examples with different constraints are presented, and the effects of replacing continuous-fringe structures by pixel structures containing locally linear gratings are evaluated.     

Focusing of ultrashort laser pulses by the combination of diffractive and refractive elements

Hybrid elements, containing optical power with both diffractive (holographic) and refractive components, are shown to be able to eliminate the effect of propagation time difference. The consideration is provided through a paraxial approximation of diffraction theory.     

Modeling of multilayer microholographic data storage

We focus on the investigation of multilayer recording in microholographic data storage. We have developed a numerical model for calculating the electromagnetic scattering from thick microholographic gratings using the Born approximation and the direct volume integral. The signal-to-noise ratio and bit error rate were calculated to estimate the noise arising from interlayer and interhologram cross talk. Measurements were done to prove the validity of the model. The results of our calculations and the measurements show good agreement. We present the application of the model to the investigation of confocal filtering at the image plane and to the evaluation of positioning and wavelength tolerances.     

Paraxial imaging and walk-off effects with birefringent media in refracting surfaces, lenses, and slabs

The paraxial theory of spherical refracting surfaces, spherical lenses, and slabs with one birefringent medium is investigated analytically: using walk-off effects in the paraxial domain, a number of relations between objects and images are deduced, along with cardinal elements, in the case where the optic axis is parallel to the optical axis. This method naturally shows that in some cases first-order astigmatism appears. An argument based on the wavefront (and phase) transformation shows that any spherical birefringent thin lens is stigmatic in the paraxial domain, because the first-order astigmatisms due to the two surfaces of such a lens compensate each other. This is a priori not the case with thick birefringent lenses -- but two such cases are detailed.     

Propagation and coupling of hybrid modes in twisted fibers

The first-order paraxial approximation is used to obtain the distributions of the electric and magnetic fields for the core and cladding hybrid fiber modes. The coupling coefficients of these modes are found for fibers subject to twist. The longitudinal electric field component determines the mode coupling in twisted fibers. It is shown that in the first-order paraxial approximation the cladding hybrid modes propagating in a twisted fiber rotate along the direction of the twist at the same rate as the core mode, independently of the azimuthal and radial mode numbers. Four hybrid modes constituting one linearly polarized mode have different longitudinal components, and the corresponding cladding-mode resonances of a long-period fiber grating undergo different shifts owing to different mode self-coupling coefficients. This results in the removal of mode degeneracy and splitting of resonances of long-period gratings in twisted fibers.     

Efficient optimization of diffractive optical elements based on rigorous diffraction models.

An efficient optimization strategy for the design of diffractive optical elements that is based on rigorous diffraction theory is described. The optimization algorithm combines diffraction models of different degrees of accuracy and computational complexity. A fast design algorithm for diffractive optical elements is used to yield estimates of the optimum surface profile based on paraxial diffraction theory. These estimates are subsequently evaluated with a rigorous diffraction model. This scheme allows one to minimize the need to compute diffraction effects rigorously, while providing accurate design. We discuss potential applications of this scheme as well as details of an implementation based on a modified Gerchberg-Saxton algorithm and the finite-difference time-domain method. Illustrative examples are provided in which we use the algorithm to design Fourier array illuminators.     

Proposal for aberration-corrected imaging spectrograph

We propose a novel flat-field imaging spectrograph that consists of two plane gratings, four spherical mirrors, and two plane mirrors. The imaging spectrograph is designed in such a manner that the primary and the secondary principal points of the whole optical system for the horizontal direction coincide with those for the vertical direction. In a paraxial approximation, it is essentially aberration free because an entrance slit and an imaging detector are placed on the primary and the secondary principal planes, respectively. Spot diagrams obtained from ray-tracing procedures have indicated that the proposed spectrograph is superior to the conventional Czerny-Turner mounting.     

Quality of paraxial electromagnetic beams

The full wave theory of focused waves is developed and the radiation intensity distribution is determined. In the appropriate limit, the full wave theory correctly reproduces the paraxial beams. The limitations of paraxial beam theories are discussed. The method of treatment of full waves is presented with reference to the scalar Bessel-Gauss beam and wave. The necessary theoretical formulas for other beams and waves are also given. For the scalar Bessel-Gauss wave, the beam shape parameter can be adjusted to yield a flat-topped radiation pattern. The ratio of the power in the paraxial beam to that in the full wave is used as a parameter to measure the quality of the paraxial beam approximation. Lower-order waves are found to have better paraxial beam quality than do higher-order waves. The difference in the paraxial beam quality increases as kw0 is decreased where k is the wavenumber and w0 is the waist of the paraxial beam. The radiation patterns of waves are presented for some tightly focused waves.     

A closed-form field analysis of a broad-band annular array

A closed-form transient field response under the paraxial approximation for an impulse wave excited by a circularly symmetric ultrasonic transducer is derived. The analysis is based on linear transform techniques to obtain the impulse response of a planar or a lensed transducer such as a disk or an annulus in the whole field of view. A closed-form solution is obtained for the focal plane response of a nonuniform aperture with an apodizing function represented by a polynomial of the radial distance on the transducer surface. Moreover, a closed-form impulse response for a planar annular is derived without the paraxial approximation and taking into account the obliquity factor. For a phased annular array, the impulse responses of array elements are convolved with the delayed excitation pulse and then summed to get the resultant field disturbance of the array. These results provide an effective means of studying the focusing characteristics of a phased annular array, including its focusing delay and apodizing quantization error effects.     

A Modified Born Approximation for Scattering in Isotropic and Anisotropic Elastic Solids

A new modified Born approximation (MBA) is presented that significantly extends the range of validity of the Born approximation to include the pulse-echo responses of strongly scattering inclusions in an elastic solid. The MBA also improves on the doubly distorted Born approximation (DDBA), a similar modification of the Born approximation that has been recently developed. These improvements are demonstrated by comparing the MBA, the Born approximation and the DDBA with the exact separation of variables solution for spherical inclusions in isotropic media. Furthermore, it is shown that the form of the MBA remains valid even for the pulse-echo scattering of an anisotropic inclusion in a general anisotropic elastic medium so that it is potentially applicable to a wide class of flaws and materials.     

Design of antireflection gratings with approximate and rigorous methods

High-spatial-frequency gratings can be used as an alternative to thin-film antireflection coatings to reduce the reflectivity at the boundary between two different media. In the case of one-dimensional gratings, the conditions on the grating structure can be approximately determined by the effective medium theory (EMT) in combination with the thin-film theory. For two-dimensional gratings, which can be used to reduce the polarization sensitivity, a corresponding EMT does not exist. We present an estimation of the effective permittivity of two-dimensional gratings. The range of validity of the antireflection grating design by the EMT is determined by the use of rigorous electromagnetic theory. Beyond the validity of EMT, rigorous theory is used to design antireflection gratings with a maximized feature size.     

Propagation of vectorial Lorentz beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of vectorial Lorentz beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of vectorial Lorentz beam is depicted at the different reference planes. This research provides an approach to further investigate the propagation of Lorentz beam beyond the paraxial approximation.     

Linking quasi-normal and natural modes of an open cavity

The present paper proposes a comparison between the extinction theorem and the Sturm–Liouville theory approaches for calculating the electromagnetic (e.m.) field inside an optical cavity. We discuss for the first time to the best of our knowledge, in the framework of classical electrodynamics, a simple link between the quasi normal modes (QNMs) and the natural modes (NMs) for one-dimensional (1D), two-sided, open cavities. The QNM eigenfrequencies and eigenfunctions are calculated for a linear Fabry–Pérot (FP) cavity. The first-order Born approximation is applied to the same cavity in order to compare the first-order Born approximated and the actual QNM eigenfunctions of the cavity. We demonstrate that the first-order Born approximation for an FP cavity introduces symmetry breaking: in fact, each Born approximated QNM eigenfunction produces values below or above the actual QNM eigenfunction value on the terminal surfaces of the same cavity. Consequently, the two error-functions for an approximated QNM are not equal in proximity to the two terminal surfaces of the cavity.     

Diffraction of optical communication Gaussian beams by volume gratings: comparison of simulations and experimental results

The diffraction effects induced by a thick holographic grating on the propagation of a finite Gaussian beam are theoretically analyzed by means of the coupled-wave theory and the beam propagation method. Distortion of the transmitted and diffracted beams is simulated as a function of the grating parameters. Theoretical results are verified by experimentation realized by use of LiNbO3 volume gratings read out by a 1550-nm Gaussian beam, typical of optical fiber communications. This analysis can be implemented as a useful tool to aid with the design of volume grating-based devices employed in optical communications.     

Optical generation of millimeter-wave pulses using a fiber Bragg grating in a fiber-optics system

A scheme is proposed to transform an optical pulse into a millimeter-wave frequency modulation pulse by using a weak fiber Bragg grating (FBG) in a fiber-optics system. The Fourier transformation method is used to obtain the required spectrum response function of the FBG for the Gaussian pulse, soliton pulse, and Lorenz shape pulse. On the condition of the first-order Born approximation of the weak fiber grating, the relation of the refractive index distribution and the spectrum response function of the FBG satisfies the Fourier transformation, and the corresponding refractive index distribution forms are obtained for single-frequency modulation and linear-frequency modulation millimeter-wave pulse generation. The performances of the designed fiber gratings are also studied by a numerical simulation method for a supershort pulse transmission.     

Dynamics of the linearly polarized fundamental Gaussian light wave

A continuous planar array of dipoles that are oriented in a particular direction and have an amplitude distribution that is Gaussian in the paraxial limit is introduced as a source for the fundamental Gaussian light wave. The radiation intensity of the Gaussian light wave is determined and its characteristics are analyzed. The universal Gaussian beam factor is deduced and identified as the radiation intensity of the scalar Gaussian wave. The total radiated power, the mean center of the localized wave, and the beam widths of the intensity distribution are obtained. The ratio of the power in the Gaussian wave to that in the corresponding paraxial Gaussian beam is used as a measure of the quality of the paraxial beam approximation. A limiting factor for the power ratio is introduced as an indicator for the acceptability of the paraxial beam approximation. The cross section and the beam widths of the localized light wave are investigated in the large and small kw0 limits, where k is the wavenumber and w0 is the beam waist at the input plane. The beam width of the full Gaussian wave is found to be less than that of the corresponding paraxial Gaussian beam both for the scalar Gaussian wave and for the Gaussian light wave.     

Diffraction from relief gratings on a biomimetic elastomer cast

Biomimetic optical elements combine the optimized designs of nature with the versatility of materials engineering. We employ a beetle carapace as the template for fabricating relief gratings on an elastomer substrate. Biological surface features are successfully replicated by a direct casting procedure. Far-field diffraction effects are discussed in terms of the Fraunhofer approximation in Fourier space.     

Eikonal methods for multielectron ion-atom collisions

The Glauber and Born approximations were used to calculate cross sections for bare ions exciting hydrogen to its n = 2 level for an energy of 1.4 MeV/amu which corresponds to a speed of ν=7.5 a.u. results for the total cross sections for the incident bare ions H⁺, Be⁴⁺, C⁶⁺, O⁸⁺, Ne¹⁰⁺, Mg¹²⁺, and P¹⁵⁺ are presented. The Glauber approximation results make a significant correction to the Born approximation as the charge on the incident ion is increased. For P¹⁵⁺ the Glauber result is over a factor of 2 larger than the Born result. Two eikonal approximations (G1 and G2) based on the Glauber approximation were carried out for an incident phosphorus ion in different ionization states for the same energy, 1.4 MeV/amu. It is argued that the G1 approximation, which has a structure similar to the Born approximation, is expected to be in better agreement with experimental data. However, there is no data with which to compare at this time.