Derivation of various transfer functions of ideal or aberrated imaging systems from the three-dimensional transfer function.

The three-dimensional frequency transfer function for optical imaging systems was introduced by Frieden in the 1960s. The analysis of this function and its partly back-transformed functions (two-dimensional and one-dimensional optical transfer functions) in the case of an ideal or aberrated imaging system has received relatively little attention in the literature. Regarding ideal imaging systems with an incoherently illuminated object volume, we present analytic expressions for the classical two-dimensional x-y-transfer function in a defocused plane, for the axial z-transfer function in the presence of defocusing and for the x-z-transfer function in the presence of a lateral shift δy with respect to the imaged pattern in the x-z-plane. For an aberrated imaging system we use the common expansion of the aberrated pupil function with the aid of Zernike polynomials. It is shown that the line integral appearing in Frieden's three-dimensional transfer function can be evaluated for aberrated systems using a relationship established first by Cormack between the line integral of a Zernike polynomial over a full chord of the unit disk and a Chebyshev polynomial of the second kind. Some new developments in the theory of Zernike polynomials from the last decade allow us to present explicit expressions for the line integral in the case of a weakly aberrated imaging system. We outline a similar, but more complicated, analytic scheme for the case of severely aberrated systems.

Double Zernike expansion of the optical aberration function from its power series expansion.

Various authors have presented the aberration function of an optical system as a power series expansion with respect to the ray coordinates in the exit pupil and the coordinates of the intersection point with the image field of the optical system. In practical applications, for reasons of efficiency and accuracy, an expansion with the aid of orthogonal polynomials is preferred for which, since the 1980s, orthogonal Zernike polynomials have become the reference. In the literature, some conversion schemes of power series coefficients to coefficients for the corresponding Zernike polynomial expansion have been given. In this paper we present an analytic solution for the conversion problem from a power series expansion in three or four dimensions to a double Zernike polynomial expansion. The solution pertains to a general optical system with four independent pupil and field coordinates and to a system with rotational symmetry in which case three independent coordinate combinations have to be considered. The conversion of the coefficients is analytically in closed form and the result is independent of a specific sampling scheme or sampling density as this is the case for the commonly used least squares fitting techniques. Computation schemes are given that allow the evaluation of coefficients of arbitrarily high order in pupil and field coordinates.

General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media.

Uniaxial optical anisotropy in the geometrical-optics approach is a classical problem, and most of the theory has been known for at least fifty years. Although the subject appears frequently in the literature, wave propagation through inhomogeneous anisotropic media is rarely addressed. The rapid advances in liquid-crystal lenses call for a good overview of the theory on wave propagation via anisotropic media. Therefore, we present a novel polarized ray-tracing method, which can be applied to anisotropic optical systems that contain inhomogeneous liquid crystals. We describe the propagation of rays in the bulk material of inhomogeneous anisotropic media in three dimensions. In addition, we discuss ray refraction, ray reflection, and energy transfer at, in general, curved anisotropic interfaces with arbitrary orientation and/or arbitrary anisotropic properties. The method presented is a clear outline of how to assess the optical properties of uniaxially anisotropic media.

Optimum synthetic-aperture imaging of extended astronomical objects.

In optical aperture-synthesis imaging of stellar objects, different beam combination strategies are used and proposed. Coaxial Michelson interferometers are very common and a homothetic multiaxial interferometer is recently realized in the Large Binocular Telescope. Laboratory experiments have demonstrated the working principles of two new approaches: densified pupil imaging and wide field-of-view (FOV) coaxial imaging using a staircase-shaped mirror. We develop a common mathematical formulation for direct comparison of the resolution and noise sensitivity of these four telescope configurations for combining beams from multiple apertures for interferometric synthetic aperture, wide-FOV imaging. Singular value decomposition techniques are used to compare the techniques and observe their distinct signal-to-noise ratio behaviors. We conclude that for a certain chosen stellar object, clear differences in performance of the imagers are identifiable.

Chromatism compensation in wide-band nulling interferometry for exoplanet detection.

We introduce the concept of chromatism compensation in nulling interferometry that enables a high rejection ratio in a wide spectral band. Therefore the achromaticity condition considered in most nulling interferometers can be relaxed. We show that this chromatism compensation cannot be applied to a two-beam nulling interferometer, and we make an analysis of the particular case of a three-telescope configuration.

Polarization nulling interferometry for exoplanet detection.

We introduce a new concept of nulling interferometer without any achromatic device, using polarization properties of light. This type of interferometer should enable a high rejection ratio in a theoretically unlimited spectral band. We analyze several consequences of the proposed design, notably, the possibility of fast internal modulation.

Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system.

The judgment of the imaging quality of an optical system can be carried out by examining its through-focus intensity distribution. It has been shown in a previous paper that a scalar-wave analysis of the imaging process according to the extended Nijboer-Zernike theory allows the retrieval of the complex pupil function of the imaging system, including aberrations as well as transmission variations. However, the applicability of the scalar analysis is limited to systems with a numerical aperture (NA) value of the order of 0.60 or less; beyond these values polarization effects become significant. In this scalar retrieval method, the complex pupil function is represented by means of the coefficients of its expansion in a series involving the Zernike polynomials. This representation is highly efficient, in terms of number and magnitude of the required coefficients, and lends itself quite well to matching procedures in the focal region. This distinguishes the method from the retrieval schemes in the literature, which are normally not based on Zernike-type expansions, and rather rely on point-by-point matching procedures. In a previous paper [J. Opt. Soc. Am. A 20, 2281 (2003)] we have incorporated the extended Nijboer-Zernike approach into the Ignatowsky-Richards/Wolf formalism for the vectorial treatment of optical systems with high NA. In the present paper we further develop this approach by defining an appropriate set of functions that describe the energy density distribution in the focal region. Using this more refined analysis, we establish the set of equations that allow the retrieval of aberrations and birefringence from the intensity point-spread function in the focal volume for high-NA systems. It is shown that one needs four analyses of the intensity distribution in the image volume with different states of polarization in the entrance pupil. Only in this way will it be possible to retrieve the "vectorial" pupil function that includes the effects of birefringence induced by the imaging system. A first numerical test example is presented that illustrates the importance of using the vectorial approach and the correct NA value in the aberration retrieval scheme.

Shaping with fluid jet polishing by footprint optimization.

We report on a way to shape surfaces with fluid jet polishing by adjusting the influence function (the shape of the footprint of the nozzle) instead of changing the dwell time of the nozzle on the surface. In that way, the surface is processed homogeneously, and no dip is generated in the center of the workpiece. As a proof of this approach, a lambda/10 flat surface has been generated in our laboratory.

Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system.

Taking the classical Ignatowsky/Richards and Wolf formulas as our starting point, we present expressions for the electric field components in the focal region in the case of a high-numerical-aperture optical system. The transmission function, the aberrations, and the spatially varying state of polarization of the wave exiting the optical system are represented in terms of a Zernike polynomial expansion over the exit pupil of the system; a set of generally complex coefficients is needed for a full description of the field in the exit pupil. The field components in the focal region are obtained by means of the evaluation of a set of basic integrals that all allow an analytic treatment; the expressions for the field components show an explicit dependence on the complex coefficients that characterize the optical system. The electric energy density and the power flow in the aberrated three-dimensional distribution in the focal region are obtained with the expressions for the electric and magnetic field components. Some examples of aberrated focal distributions are presented, and some basic characteristics are discussed.

Analysis, search, and classification for reflective ring-field projection systems.

Extreme ultraviolet (EUV) lithography uses reflective ring-field projection systems. Geometrical obstruction limits the possible system configurations to small domains of the parameter space. We present an analysis, a search method, and a classification of these unobstructed domains. The exhaustive search method based on paraxial analysis provides an effective means for determining all possible design forms and for finding useful starting configurations for optimization. The approach is validated through comparison with finite ray tracing.

Wave-front correction methods for extreme-ultraviolet multilayer reflectors.

In this theoretical study we show that by removing or depositing additional multilayer (ML) periods of a thin-film interference coating, distortions in the reflected wave front induced by surface figure errors can be corrected. At lambda = 13.4 nm in the extreme-ultraviolet region the removal or deposition of a single period of the standard two-component molybdenum-silicon (Mo/Si) ML interference coating induces an effective phase change of magnitude 0.043pi with respect to an identical optical thickness in vacuum. The magnitude of this wave-front shift can be enhanced with multicomponent MLs optimized for phase change on reflection. We briefly discuss the contributions of the shift in the effective reflection surface of the ML on the phase change. We also predict the feasibility of novel phase-shifting mask for subwavelength imaging applications.

Deep nulling by means of multiple-beam recombination.

Direct detection of exoplanets is possible by use of a technique called nulling interferometry, which is based on destructive interference of light of the bright object and constructive interference of the faint object. In the infrared wavelength region, this implies that light of a star must be attenuated by a certain factor, the so-called rejection ratio, which typically equals 10(6). This can be achieved by use of phase shifters, which apply a phase shift of pi rad with an average error nogreater than 2 mradover a predefined wavelength region. For a 6-18-microm wavelength interval, this is a tough constraint. We show that the 2-mrad constraint can be relaxed if more than two beams participate in the beam recombination. We focus our attention on dispersive phase shifters and show that rejection ratios beyond 10(6) can be reached easily by use of a system of four or more apertures and simple dispersive phase shifters that consist of only one material.

Fresnel diffraction effects in the frontlight for a liquid-crystal display.

In a new type of illumination system for reflective liquid-crystal displays, the frontlight, unwanted shadows appear in certain viewing directions. It will be shown that for an accurate description of these shadows the geometrical optics approach is not satisfactory and that Fresnel diffraction has to be taken into account. A model for the diffraction effects was developed, and the predicted results correspond well to the measurements. In addition, some remarkable effects were explained from the theory of Fresnel zones.

Nulling interferometry without achromatic phase shifters.

In the infrared wavelength region, a typical star is approximately a million times brighter than the planet that surrounds it, which is a major problem when we attempt to detect exoplanets in a direct manner. Nulling interferometry is a technique that one can use to solve this problem by attenuating the stellar light and enhancing that of the planet. Generally, deep nulling is achieved by use of achromatic phase shifters (APSs). Unfortunately, the technology needed to build these APSs is not yet fully developed. We show that deep nulling can also be achieved by using delay lines only. We investigate the nulling depth as a function of the width of the wavelength interval and the number of telescopes. We also show that we can obtain nulling depths of less than 10(-6), which are required for exoplanet detection. Furthermore, we investigate the properties of the transmission map and make a comparison between our system and an APS system.

iTIRM as a tool for qualifying polishing processes.

We report on what we believe to be a novel classification method for polishing processes that we apply in our laboratory on a regular basis. Two parameters are deduced from the in situ iTIRM (intensity-detecting total-internal-reflection microscopy) measurement method. Contrary to Preston's law, which gives the removal rate, the parameters of the iTIRM process are a measure of the change in surface quality (roughness, subsurface damage, and scratch and dig) and the duration of the polishing process.

Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions.

We assess the validity of an extended Nijboer-Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on ecently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.