Focal Shift Effect Research Articles

Inverse focal shift: A new effect in truncated cylindrical waves

We report on a general analytical procedure to analyse the axial focusing properties of uniform cylindrical waves truncated by a rectangular window. The resulting on-axis diffraction pattern explicitly depends on the square of the window height-to-width ratio. Depending on the value of this parameter, different kinds of axial behaviour are observed. In particular, it is found that for low values of this parameter and low Fresnel number, instead of the expected focal-shift effect, an inverse focal-shift phenomenon can appear, i.e. the maximum of the axial-irradiance distribution is displaced further away from the window.

Focal shift effects in diffracted focused beams

We study the propagation characteristics of focused, apertured radially symmetric beams and show that the axial point at which the mean-square radius of the diffracted beam is a minimum tends to be displaced towards the aperture plane depending on the Fresnel number associated with the aperture and the Fresnel number associated with the radius of the incident beam. The magnitude of this effect can be determined by a simple and general formula in terms of the amplitude distribution of the incident beam across the aperture and of the two Fresnel numbers. We investigate the focal shift effects for the case of a focused, apertured Gaussian beam. The irradiance and the encircled-power distributions in the plane at which the mean-square radius of the diffracted beam is minimum are calculated and compared with those corresponding to a uniform beam.

On-axis focal shift effects in focused truncated Bessel beams

The focusing properties of truncated J0 Bessel beams are studied by means of the Huygens-Fresnel diffraction integral. Numerical calculations are presented for different Fresnel numbers, showing the effects of the finite aperture on the on-axis irradiance distribution.

Focal switch: a new effect in low-Fresnel-number systems

It is shown for the first time we believe, that when a spherical wave illuminates a certain type of diffracting screen, in addition to the expected focal-shift effect, depending on the value of the Fresnel number of the focusing system, a focal switch effect can appear, i.e., an increase in the height of the lateral lobe of the axial-intensity distribution over that of the central lobe.

Characteristic pathlength fresnel number equations and ideal lens wavefront Kirchhoff formulation: analysis of focused Gaussian beam diffraction fields

The characteristic optical pathlength Fresnel number equation for a paraxial Gaussian laser beam diffracted and spherically focused by an ‘ideal’ circular lens is derived. The maximum inside and outside pathlength Fresnel numbers are quantified. The maximum outside pathlength Fresnel number, simplified via the paraxial approximation, is shown to be equal to the classical focal Fresnel number of the lens-aperture combination. A unique Kirchhoff diffraction integral formulation of this configuration is derived, and calculations show that the diffraction fields for a circular-aperture-truncated focused Gaussian beam are identical in characteristic shape to the same but non-focused diffraction fields, as long as significant focal shift effects are not present. Example applications of the optical pathlength Fresnel number equation are given.

A High-accuracy Formula for Fast Evaluation of the Effect of Focal Shift

Abstract A simple formula for computing focal shifts in systems of different Fresnel numbers is proposed based on an analysis of factors that limit the extent of a focused field. This formula gives the focal shift accurately to about 2%, which is currently the best approximation that can be made.

Optimizing photodetection in a focused field

In a consideration of the effect of focal shift, light energy which falls inside a coaxial circular detector in a focused field will achieve its absolute maximum in an ascertainable place between the aperture and the geometrical focal plane. In this paper the optimum position of the detector is determined by solving a pair of transcendental equations involving Lommel functions. The numerical results obtained are displayed by curves with the Fresnel numbers sampled between 0.1 and 100.

Maréchal intensity formula for small-Fresnel-number systems

A simple modification of the known Maréchal intensity formula is proposed. It enables one to evaluate the near the-diffraction-focus intensity distribution for well-corrected optical systems with small Fresnel numbers of the focusing geometry. As an example, the focal-shift effect in perfect systems is briefly reexamined.