Silvia A. Comastri

Fringe localization depth

We calculate the complex degree of coherence for any interferometer in which there is amplitude division and simple interference. We consider the exact optical path length differences between both branches. We find that, when there is a localization surface, this degree of coherence is equivalent to the complex optical disturbance at the image of an aberrated isoplanatic optical system. Moreover, we show that the radius of the central lobe at the diffraction pattern of this equivalent system is the distance between rays when visibility is zero. This distance enables us to evaluate localization depth and adjustment tolerance.

Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations

An analytical method to convert the set of Zernike coefficients that fits the wavefront aberration for a pupil into another corresponding to a contracted and horizontally translated pupil is proposed. The underlying selection rules are provided and the resulting conversion formulae for a seventh-order expansion are given. These formulae are applied to calculate corneal aberrations referred to a given pupil centre in terms of those referred to the keratometric vertex supplied by the SN CT1000 topographer. Four typical cases are considered: a sphere and three eyes—normal, keratoconic and post-LASIK. When the pupil centre is fixed and the pupil diameter decreases from 6 mm to the photopic natural one, leaving aside piston, tilt and defocus, the difference between the root mean square wavefront error computed with the formulae and the topographer is less than 0.04 µm. When the pupil diameter is kept equal to the natural one and the pupil centre is displaced, coefficients vary according to the eye. For a 0.3 mm pupil shift, the variation of coma is at most 0.35 µm and that of spherical aberration 0.01 µm.

Generalized sine condition for image-forming systems with centering errors

The generalized sine condition for an image-forming system with centering errors but allowing for one symmetry plane is derived according to the Fourier optics approach. The variation of the wave-front-aberration function associated with a small displacement of field coordinates is given. The symmetry properties of aberrations are discussed.

Wavefront aberrations: analytical method to convert Zernike coefficients from a pupil to a scaled arbitrarily decentered one

The wavefront aberration of any image forming system and, in particular, of a human eye, is often expanded in Zernike modes each mode being weighed by a coefficient that depends both on the image forming components of the system and on the contour, size and centering of the pupil. In the present article, expanding up to 7th order the wavefront aberration, an analytical method to compute a new set of Zernike coefficients corresponding to a pupil in terms of an original set evaluated via ray tracing for a dilated and transversally arbitrarily displaced pupil is developed. A transformation matrix of dimension 36×36 is attained multiplying the scaling‐horizontal traslation matrix previously derived by appropriate rotation matrices. Multiplying the original coefficients by this transformation matrix, analytical formulas for each new coefficient are attained and supplied and, for the information concerning the wavefront aberration to be available, these formulas must be employed in cases in which the new pupil ...

Correlation between visual acuity and pupil size

The eye suffers from aberrations that are pupil size dependent and reduce visual performance. Pupil size is often disregarded when visual performance is clinically determined. Considering 23 normal eyes, under natural viewing conditions, visual acuity is here related to pupil size when each subject reads monocularly a Snellen chart of constant luminance placed at 2m.

Multilocalization and the van Cittert–Zernike theorem. 2. Application to the Wollaston prism

The van Cittert-Zernike theorem can be used to evaluate visibility at the exit of an amplitude-division interferometer with two-beam interferences. If the source illuminating the interferometer is a periodic array of slits, at the exit there is a sequence of localization surfaces. The formulas for the position and fringe spacing of the principal localization surfaces are applied to a Wollaston quartz prism, and there is good agreement between theoretical and experimental results. Moreover, the fringe localization depth and the intermediate localization surfaces obtained experimentally coincide with those predicted by theory.

Multilocalization and the van Cittert–Zernike theorem. 1. Theory

The complex degree of coherence and the resulting van Cittert-Zernike theorem are employed to analyze the exit of an arbitrary amplitude-division interferometer with two-beam interferences. Considering that the source is a periodic array of spatially incoherent slits and assuming negligible equivalent aberrations and no vignetting, an expression for the complex degree of coherence as a function of the position of an exit point is derived. Formulas for the location, fringe spacing, and fringe localization depth of the multilocalized fringes are given.

Interferometers: equivalent sine condition

The parallelism we have previously drawn between an interferometer and an equivalent optical system enables us to study both systems similarly. The constant contrast condition along the localization surface is equivalent to the condition of isoplanatism at the system image. As the sine condition is necessary for a system to be isoplanatic, a relationship we call the equivalent sine condition must hold if the contrast is constant. Some classic interferometers do not satisfy this condition and we here analyze the Michelson interferometer. Moreover, we find that, for the record of interference fringes to be a pseudohologram, this condition must be unfulfilled.

Two-beam interferometers: a classification which takes into account multiple localizations

Summary Two beam interferometers are traditionally classified by the method used to separate the beams. This classification is suitable for considering localizations when the source is incoherent and continuous since an amplitude division interferometer yields the classical localization plane while a wavefront division one yields no fringes. However when the source is incoherent and periodic and the symmetry is plane there can be multiple localization planes and these cannot be easily analysed using this classification. In the present paper these planes are taken into account classifying two beam interferometers as those where the ‘effective interfering sources’ are the images of the source of light and those where they are not. The latter either yield no fringes or non-localized ones while the former yield several planes with straight, sinusoidal, localized fringes provided certain requirements are fulfilled. One of these is the equivalent sine condition which is here derived for any observation plane. Thus interferometers of the first class are further subdivided in two: those where the equivalent sine condition is verified on every observation plane and those where it is not. Experimental results illustrating the validity of the theoretical predictions are shown.

Image-forming systems: Matrix formulation of the optical invariant via Fourier optics

Abstract In this paper we consider Fourier optics and from it we obtain the expression for the optical invariant valid for any image-forming system. The invariant that we derive is applicable to systems which are or are not centred or symmetrical and therefore it is written in a matrix form. When the paraxial approximation holds, the optical invariant reduces to the classical Lagrange–Helmholtz invariant.