On-axis diffractional behavior of two-dimensional pupils

Abstract

We show that, at any Fresnel number, a suitable one-dimensional Fourier transform relates the complex-amplitude distribution along the optical axis with the zero-order circular harmonic of the amplitude transmittance of a two-dimensional diffracting screen. First, our general result is applied to recognize that any rationally nonsymmetric screen generates an axial-irradiance distribution that exhibits focal shift. In this way we identify a wide set of two-dimensional screens that produce the same focal shift as that produced by the clear circular aperture. Second, we identify several apodizers for shaping the axial-amplitude distribution. We discuss some examples for achieving high-precision focusing, axial hyperresolution, or high focal depth.

Inverse focal shift: a new effect in truncated cylindrical waves

“…In this case, the function À…W 20 † tends to À…W 20 †ˆj1 ¡ 2W 20 =N x j. Therefore, the axial behaviour is similar, of course, to that of cylindrical ®elds nontruncated along the y-direction, which indeed was described by equation (10).…”

“…Finally, we would like to point out that although this analysis is equivalent to that performed by Jiang and Stammes [19], our approach shows considerable Axial-irradiance distribution corresponding to a slit-like window from equation (10). The on-axis behaviour is shown for three values of the Fresnel number: N xˆ1 00 (high), N xˆ1 0 (moderate), N xˆ4 (low).…”

“…From equation (10) it is inferred that I N …W 20 † comprises the product of two terms. The ®rst term describes the attenuation su ered by cylindrical waves as they propagate.…”

“…To illustrate the accuracy of equation (15), in ®gure 4 we have represented the value of the relative focal shift obtained with the approximated formula, and its exact value derived from equation (10). The examination of this ®gure reveals that the approximated formula gives suitable results for moderate and high values of N x (for example, when N xˆ5 the error is 2.2% and for N xˆ5 0 the error is less than 0.6%).…”

Three-dimensional intensity distribution of focused annular non-uniformly polarized beams

“…is the Fresnel number for annular focusing system [17,18]. Equation (29) indicates that, when the annular focusing system is illuminated by a linear polarized light, the effective Fresnel number is just equal to the Fresnel number for annular focusing system.…”