The stenopaeic slit: an analytical expression to quantify its optical effects in front of an astigmatic eye.

The stenopaeic slit is a trial case accessory used in subjective refraction, especially when high astigmatism is present. In spite of its simplicity, the effect of the slit when it is not oriented along one of the principal meridians of the examined eye is difficult to predict, even in terms of classical geometrical optics. In this paper, the optical principles of the slit are considered with full details in the theoretical framework of the dioptric power space. An analytical expression to obtain the residual refractive error when a stenopaeic slit is placed in front of an astigmatic eye at any orientation is deduced. In the light of these results, some aspects of the clinical procedure are discussed.

Statistical analysis when dealing with astigmatism: assessment of different spherocylindrical notations.

Ophthalmic epidemiological studies frequently deal with ocular refractive errors, which are commonly expressed in the form sphere/cylinder x axis. However, this representation has been shown not to be the most suitable one for performing statistical analysis. Although alternative analytical and graphic methods to represent this kind of data have been developed, these formalisms have often gone unnoticed by researchers, despite their usefulness and versatility. Besides, there has been no discussion of how each of them fits in with a particular type of study. In this paper, several mathematical representations of dioptric power are revisited in a comprehensive way. The aim is to encourage researchers in ophthalmology and optometry to use these formalisms in their epidemiological studies, thus profiting from their exactitude and simplicity. Consequently, the emphasis is not on complicated mathematical derivations but on how to use these representations. Their potential and suitability in different applications is analyzed in detail. In addition, some examples are presented to illustrate the mathematical methods considered.

Subjective refraction techniques in the frame of the three-dimensional dioptric space.

A novel heuristic approach to the well-known representation of the dioptric power in a three-dimensional space is presented. It is shown how this theoretical framework is ideal for discussing the principles of several subjective refraction methods. In particular, this formalism is used to justify the stenopaic slit refraction, the Barnes subjective refraction technique, and the Jackson cross-cylinder procedure. In view of this analysis, some modifications to the traditional procedures are proposed.

Three-dimensional behavior of apodized nontelecentric focusing systems.

The scalar field in the focal volume of nontelecentric apodized focusing systems cannot be accurately described by the Debye integral representation. By use of the Fresnel-Kirchhoff diffraction formula it is found that, if the aperture stop is axially displaced, the focal-volume structure is tuned. We analyze the influence of the apodizing function and find that, whereas axially superresolving pupil filters are highly sensitive to the focal-volume reshaping effect, axially apodizing filters are more inclined to the focal-shift effect.

Focal-shift formula in apodized nontelecentric focusing systems.

A single analytical formulation for evaluating the focal shift in any apodized nontelecentric focusing setup is reported. The formulation is also useful in the case of imaged paraxial beams. We show explicitly that the magnitude of the focal shift is determined by only one parameter that depends on the effective width of the pupil filter and its axial position. To illustrate our approach we examine different focusing setups.

One-dimensional iterative algorithm for three-dimensional point-spread function engineering.

We present a new method with which to binarize pupil filters designed to control the three-dimensional irradiance distribution in the focal volume of an optical system. The method is based on a one-dimensional iterative algorithm, which results in efficient use of computation time and in simple, easy to fabricate binary filters. An acceptable degree of resemblance between the point-spread function of the annular binary filter and that of its gray-tone counterpart is obtained.

Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system

The Debye formulation of focused fields has been systematically used to evaluate, for example, the point-spread function of an optical imaging system. According to this approximation, the focal wave field exhibits some symmetries about the geometrical focus. However, certain discrepancies arise when the Fresnel number, as viewed from focus, is close to unity. In that case, we should use the Kirchhoff formulation to evaluate accurately the three-dimensional amplitude distribution of the field in the focal region. We make some important remarks regarding both diffraction theories. In the end we demonstrate that, in the paraxial regime, given a defocused transverse pattern in the Debye approximation, it is possible to find a similar pattern but magnified and situated at another plane within the Kirchhoff theory. Moreover, we may evaluate this correspondence as the action of a virtual thin lens located at the focal plane and whose focus is situated at the axial point of the aperture plane. As a result, we give a geometrical interpretation of the focal-shift effect and present a brief comment on the problem of the best-focus location.