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### Abstract

The close connection between image formation in geometrical optics and beam transformation by a paraxial optical system is examined analytically using mathematical tools accessible to undergraduate students, such as the Fresnel diffraction integral and Fourier transforms, instead of the more complicated Wigner distribution or coherence functions frequently employed in the literature. It is shown that geometrical optics correctly predicts the plane where a beam is refocused and its magnification only for afocal optical systems or in the limit of point sources. We illustrate this theory by simulating the transformation of a flat-top beam by a pair of lenses.

This work was supported by the National Science and Engineering Research Council of Canada (NSERC) and by the *Faculté des études supérieures et de la recherche de l'Université de Moncton*.

I. INTRODUCTION

II. IMAGE FORMATION IN GEOMETRICAL OPTICS

III. TRANSFORMATION FORMULAS OF GAUSSIAN BEAMS BY A PARAXIAL OPTICAL SYSTEM

IV. BEAM WAIST TRANSFORMATION FORMULAS FOR ARBITRARY FIELD DISTRIBUTIONS

A. Propagation of an arbitrary coherent beam through a homogeneous medium

B. Waist transformation formulas of an arbitrary beam through an apertureless optical system

V. ILLUSTRATIONS OF THE THEORY AND CONCLUDING REMARKS

### Key Topics

- Geometrical optics
- 24.0
- Optical devices
- 21.0
- Fourier transforms
- 11.0
- Refractive index
- 6.0
- Apertures
- 5.0

## Figures

Definition of the two-component vector , defining the position x and the inclination α of an optical ray at z = z 0.

Definition of the two-component vector , defining the position x and the inclination α of an optical ray at z = z 0.

(Color online) Transformation of a beam by a paraxial optical system described by matrix elements ABCD. The incoming beam has its waist in plane P 1. Coordinates z 2 and z are measured with respect to conjugate plane P 2 and the position of the waist of the emerging beam, respectively. Here L 1 is the object distance from the entrance of the optical system, obtained from geometrical optics, while and are, respectively, the distance of the geometrical image and that of the minimum radius to the exit of the POS.

(Color online) Transformation of a beam by a paraxial optical system described by matrix elements ABCD. The incoming beam has its waist in plane P 1. Coordinates z 2 and z are measured with respect to conjugate plane P 2 and the position of the waist of the emerging beam, respectively. Here L 1 is the object distance from the entrance of the optical system, obtained from geometrical optics, while and are, respectively, the distance of the geometrical image and that of the minimum radius to the exit of the POS.

(Color online) Illustration of the beam caustic (solid) and ray tracing (dashed) produced by a pair of lenses for a beam waist located in the object focal plane of the first lens, for various values of the distance between the lenses. From top to bottom: (a) afocal system D = f 1 + f 2; (b) negative power V (D > f 1 + f 2); and (c) positive power V (D < f 1 + f 2) doublets. Depending on the sign of V, the waist is shifted downstream (V < 0) or upstream (V > 0) with respect to the geometrical image plane. The beam caustics are shown for illustration only; they do not correspond to zr 1 = f 1.

(Color online) Illustration of the beam caustic (solid) and ray tracing (dashed) produced by a pair of lenses for a beam waist located in the object focal plane of the first lens, for various values of the distance between the lenses. From top to bottom: (a) afocal system D = f 1 + f 2; (b) negative power V (D > f 1 + f 2); and (c) positive power V (D < f 1 + f 2) doublets. Depending on the sign of V, the waist is shifted downstream (V < 0) or upstream (V > 0) with respect to the geometrical image plane. The beam caustics are shown for illustration only; they do not correspond to zr 1 = f 1.

(Color online) Beam intensity profiles at the object plane and minimum waist plane for three values of δ: δ = 0 (solid), δ = 3.18 cm (short-dashed), and δ = 6.35 cm (dashed); the Rayleigh length is zr 1 = 10 cm. The Fresnel number NF corresponding to the latter two values is shown. A faithful reproduction of the input beam with a magnification G=|G t|= 2 is obtained in the geometrical image plane for any value of δ; however, at the minimum width the beam profile differs significantly from that in the object plane when . The transformed beams were calculated using Eq. (22) of Ref. 42 .

(Color online) Beam intensity profiles at the object plane and minimum waist plane for three values of δ: δ = 0 (solid), δ = 3.18 cm (short-dashed), and δ = 6.35 cm (dashed); the Rayleigh length is zr 1 = 10 cm. The Fresnel number NF corresponding to the latter two values is shown. A faithful reproduction of the input beam with a magnification G=|G t|= 2 is obtained in the geometrical image plane for any value of δ; however, at the minimum width the beam profile differs significantly from that in the object plane when . The transformed beams were calculated using Eq. (22) of Ref. 42 .

(Color online) Illustration of the beam profiles (in phase and intensity) in the geometrical image plane and at the waist for δ > 0 for the case of a flat-top beam distribution in the object waist plane. At z 2 = 0 the beam intensity profile remains that calculated by geometrical optics, in accordance with Eq. (38) . The beam profile at the waist is determined by the Fresnel number associated with the aperture of the image plane seen from the waist.

(Color online) Illustration of the beam profiles (in phase and intensity) in the geometrical image plane and at the waist for δ > 0 for the case of a flat-top beam distribution in the object waist plane. At z 2 = 0 the beam intensity profile remains that calculated by geometrical optics, in accordance with Eq. (38) . The beam profile at the waist is determined by the Fresnel number associated with the aperture of the image plane seen from the waist.

Magnification of image waist as a function of δ for zr 1 = 1 cm and 10 cm calculated using Eq. (28) . The magnification is maximum at δ = 0, when |G| = |G t| = 2. When the Rayleigh length of the beam diminishes, so does the deviation from the theory of geometrical optics, and the magnification is approximately independent of δ, with |G| ≈ |G t|.

Magnification of image waist as a function of δ for zr 1 = 1 cm and 10 cm calculated using Eq. (28) . The magnification is maximum at δ = 0, when |G| = |G t| = 2. When the Rayleigh length of the beam diminishes, so does the deviation from the theory of geometrical optics, and the magnification is approximately independent of δ, with |G| ≈ |G t|.

Position of the image waist as a function of δ calculated with Eq. (26) for two values of the Rayleigh length: zr 1 = 10 cm (thick solid) and zr 1 = 1 cm (thin solid). For δ = 0, the image waist and the geometrical image coincide at the image focal plane of the second lens so that L 2 = L 2′ = f 2 = 20 cm. The departure from geometrical optics diminishes for smaller values of the Rayleigh length.

Position of the image waist as a function of δ calculated with Eq. (26) for two values of the Rayleigh length: zr 1 = 10 cm (thick solid) and zr 1 = 1 cm (thin solid). For δ = 0, the image waist and the geometrical image coincide at the image focal plane of the second lens so that L 2 = L 2′ = f 2 = 20 cm. The departure from geometrical optics diminishes for smaller values of the Rayleigh length.

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