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.

We experimentally demonstrate that a femtosecond frequency comb laser can be applied as a tool for longdistance measurement in air. Our method is based on the measurement of cross correlation between individual pulses in a Michelson interferometer. From the position of the correlation functions, distances of up to 50 m have been measured. We have compared this measurement to a counting laser interferometer, showing an agreement with the measured distance within 2 m (4 ϫ 10 −8 at 50 m). © 2009 Optical Society of America OCIS codes: 320.7100, 320.2250, 320.1590 Traditional techniques for long-distance measurements are often based on optical interferometry when the demands on accuracy rise. Most of these interferometric techniques rely on incremental measurements of phase accumulation. A priori knowledge of the distance to be measured is required or a complex multiwavelength system may be needed. In 2004, Ye [1] proposed a simple scheme for measuring long distances in space with a stabilized femtosecond frequency comb. The scheme is based on a Michelsontype interferometry with optical interference between individual pulses. The technique proposed by Ye has been demonstrated for interferometric measurement of short displacement [2,3]. The main advantage of applying a frequency comb for distance measurement is the large range of nonambiguity, which is determined by the cavity length of the pulsed laser, ranging from about 30 cm to 3 m. It is thus not necessary to rely on incremental measurement of the optical phase. The ambiguity is easily overcome by, e.g., a laser distance meter. The stabilized frequency comb has been applied as a source in various distance measurement schemes [4,5]. In this Letter, we demonstrate distance measurements of up to 50 m in air by analyzing the cross correlation between pulses emitted from a stabilized frequency comb source. We have implemented a model of pulse propagation in air to account for the effect of air dispersion on the measured cross-correlation functions. The measurement results obtained with the frequency comb and a conventional counting laser interferometer are compared.A mode-locked Ti:sapphire laser is the frequency comb source, with both the repetition frequency and the carrier-envelop offset (CEO) frequency referenced to a cesium atomic clock (Fig. 1). The pulse duration is 40 fs, and the repetition rate f r is locked at approximately 1 GHz, corresponding to a pulse to pulse distance l pp = c / ͑n g f r ͒ of 30 cm. Here c is the speed of light in vacuum and n g is the group refractive index at the center wavelength. The CEO frequency f 0 is fixed at 180 MHz. The center wavelength of the pulses is 815 nm, with an FWHM of about 20 nm. After collimation the beam is sent to a Michelson interferometer. One part of the beam is reflected by a hollow corner cube mounted on a piezoelectric transducer (PZT) along the short reference arm. The length of the short arm can be scanned by a translation stage. The other part of the beam is reflected by two mirrors and propagates along...

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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.

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.