When a Gaussian beam is weakly diffracted by a circular aperture it can be approximated by another Gaussian beam with slightly different properties in the far field. In this Technical Note, using the relationship between these two Gaussian beams, before and after the aperture, an ABCD matrix is defined for cascaded laser systems.
Up to now, every optical element has a simple transfer matrix; it is convenient to treat propagation, transformation, and the self-consistent solution for Gaussian beams, except for the hard aperture, but it always exists. Vicari and Bloisi1
considered this problem in detail but with a 200 × 200 matrix. An ABCD matrix for the hard aperture was given recently2 but only for spherical waves and in the near field.
Belland and Crenn3 proved that, if a Gaussian beam is weakly diffracted by a circular aperture with radius a when
and the beam waist is located at the plane of the circular aperture, in the far field (Fraunhofer diffraction) it can be regarded as another Gaussian beam with slightly different characteristics, as follows:
where r0 is the 1/e intensity beam radius. Using the beam waist radius w0
= 2r0 instead of r0 and lateral magnification β = w'0/w0, Eqs. (1) and (2) become 1.1 < a/w0 < 2.1, and
We find that most laser systems are subject to this condition; thus it is necessary to define a transfer matrix to describe the changes of characteristics of such diffracted Gaussian beams and for cascaded laser systems. Applying the ABCD law,
with β as given by Eq. (3). The ABCD matrix describing the aperture can be chosen
in several ways, depending on practical cases and further application. However, it is necessary to introduce several constraints. To retain the linearity of the system, the determinant must be 1 (equal indices on both sides). In addition, one can see from Eq. (4) that bw must be equal to zero. Taking into account these two simple conditions it is possible to find two solutions to the arrangement of Eq. (4).
The clearest solution uses the relationship between the lateral and angular magnifications, relating widths and divergences, respectively. In this case the matrix can be written as
which can be seen to act as an afocal system between the objective and eyepiece foci, as in a telescope.
On the other hand, the arrangement of the elements of the matrix can be used to obtain another expression, that is,
This latter matrix is the representation of a complex focal length4 with an imaginary part, that is, a soft aperture (Gaussian aperture) whose transmission profile depends on
This matrix (6) appears to be similar to that of a phase conjugate mirror formed by stimulated scattering,5 but note that the physical meaning is completely different.
These two possible interpretations are equivalent, and which to use depends on the practical situation. In any case, as soon as the changed waist is known, all the other properties of Gaussian beams are fixed.
One author (Shaomin Wang) is grateful to the University Complutense of Madrid and the National Nature Scientific Foundation (China) for their support. When this work was done he was a visiting professor with the Optics Department of the former.
References 1. L. Vicari and F. Bloisi. "Matrix Representation of Axisymmetric
Optical Systems Including Spatial Filters," Appl. Opt. 28,4682-4686 (1989).
2. Shaomin Wang, E. Bernabeu, and J. Alda, "ABCD Matrix for Aperture Spherical Waves for Large Fresnel Number," submitted to Appl. Opt. (1990).
3. P. Belland and J. P. Crenn, "Changes in the Characteristics of a Gaussian Beam Weakly Diffracted by a Circular Aperture," Appl. Opt. 21, 552-527 (1982).
4. E. Bernabeu, and J. Alda, "Gaussian Beam Propagation Through Optical Systems with Complex Focal," Optik 76, 108-115 (1987).
5. Wang Shaomin and H. Weber, "Fundamental Modes of Stimulated Scattering Phase-Conjugate Resonators," Opt. Acta 31, 971-976 (1984).
