An ABCD matrix for describing the hard aperture under a large Fresnel number is defined in this Technical Note based on Li and Wolf's formula. It is useful for analyzing focal shifts of complicated optical systems with hard apertures.
When a converging uniform monochromatic (with wavelength λ) spherical wave (with curvature -1/p) is diffracted by a circular aperture (with radius a), the point of maximum
intensity of the diffracted wave is not at the geometrical focus (-ρ or f) of the incident wave but is located somewhat closer to the aperture when α » λ and f » a. The focal shift was given by Li and Wolf1,2:
with N = σ2/λƒ, where N is the Fresnel number at the aperture. This formula can give the focal shift value with ~ 1 % accuracy if N > 12; it remains valid for N > 4 (Rayleigh region). These limits and Eq. (1) are related to approximate expressions given by Fresnel-Kirchhoff diffraction theory and can be arranged to fit better depending on the Fresnel number value.3
The formula could be used when a converging wave (even for a diverging wave) is incident on a lens (with focal length f and size radius a) for correcting the focal point caused by diffraction; therefore, it has wide significance in practice. However, if a series of optical elements follow behind the aperture, it is better to define an ABCD matrix for the hard aperture to calculate generalized focal shifts in a simple way.
The aim of this Letter is to find this ABCD matrix. First, let us recast Eq. (1) in this form:
with N = α2/λ|p|. When a beam with curvature — 1/p is incident on an aperture, the curvature — l/(p — ∆p) of the output beam, caused by the diffraction of the aperture, is subject to the following transformation according to ABCD law:
Since the aperture is a thin element with neither linear nor angular magnifications within the Rayleigh region, aa = da = 1 and ba = 0; therefore, ca = -l/(π2/12)n2p. The ABCD matrix for describing a hard aperture under a large Fresnel number could thus be defined as
Not only a circular aperture can be expressed by means of matrices. Annular apertures, treated by Kathuria,4 produce focal shifts given by
with ε = b/a, the ratio of the radii of the edges of the aperture (b < a), and N1 = α2/λƒ. Then, in the same way as circular apertures, the matrix for annular apertures is
Obviously, if ε tends toward 0, the circular aperture is found and matrix (4) is obtained from matrix (6).
If a series of optical elements with transfer matrix elements a, b, c, and d follows the aperture and is within Rayleigh region, the focal point could be found by
where the focal point s is measured from the last surface of the abed element.5 This matrix may be expanded to analyze Gaussian beam propagation in complex optical systems.6
One of the authors (Wang Shaomin) is grateful to University Complutense of Madrid (Spain) and National Nature Scientific Fundation (China) for their support; he is now a visiting professor at the former.
References 1. Y. Li and E. Wolf, "Focal Shifts in Diffracted Converging Spheri
cal Waves," Opt. Commun. 39, 211-215 (1981). 2. Y. Li, "Critical Points Near the Focus of Systems of Different
Fresnel Number," J. Mod. Opt. 36, 139-146 (1989). 3. Y. Li, "Focal Shift Formulae," Optik 69, 41-42 (1984). 4. Y. P. Kathuria, "Focal Shift in Converging Annular Beam: a
Corollary," J. Opt. 20, 141-144 (1989). 5. W. Shaomin and L. Ronchi, "Principles and Design of Optical
Arrays," in Progress in Optics, Vol. 25, E. Wolf, Ed. (Elsevier, New York, 1988), pp. 279-348.
6. E. Bernabeu and J. Alda, "Gaussian Beam Propagation Through Optical Systems with Complex Focal," Optik 76, 108-115 (1987).
