126							3.1.7 Beam propagation in optical systems								[Ref. p. 131
4
									Geometrical optics
3			z0 /f = 3
2
f 1					z0 /f = 1
								P
'/									z0 /f = 0.5
0
z
									z0 /f = 0.3
-1		Geometrical optics							z0 /f = 0.2					Fig. 3.1.43. Relation of the Gaussian waist transfer (full
-2
-3														lines) to the thin-lens equation (dashed) of geometrical
-3		-2	-1		0		1		2	3		4		optics for di erent influences of di raction (wavelength λ
						z / f								respectively z0).

3.1.7.4 Collins integral

For Fresnel’s approximation of di raction in paraxial systems see [68Goo, 71Col, 78Loh, 94Roe]. It was generalized to the propagation of field distributions in ABCD-described systems by [70Col, 76Arn, 05Gro2, 05Hod].

3.1.7.4.1 Two-dimensional propagation

In Table 3.1.22 the propagation in rotational symmetric systems and simple astigmatic systems is given.

Table 3.1.22. Propagation in rotational symmetric systems and simple astigmatic systems.

Given

Solution

– ABCD-matrix of the optical

Field U O(x2) in the output plane (Collins integral ):

system (see Tables 3.1.11

U O(x2) =

i

and 3.1.12),

e−

ikL

– field distribution in the in-

λ B

put plane U I(x),

∞

d x1 U I(x1) exp −i

k

2

2

.

– path length along the opti-

×

Ax1

− 2x1x2 + Dx2

(3.1.131)

2B

cal axis L.

−∞

Example 3.1.18. The waist of a Gaussian beam is given with U I(x1) = exp (−x21/w2I) in the input plane. The system consists of a thin lens with the focal length f followed by a free-space propagation by distance z. The ABCD-matrix is calculated from Fig. 3.1.34 and Table 3.1.11:

C D

−1/f

1

A B

=

1 − z/f

z .

∞

x2

i

−

−

k

z

1

2

−

−

λ z

w2I

2z

f

U O(x2) =

e−ikL

d x1 exp

1

exp i

1

x2

2x1x2 + x2 .

−∞

Landolt-B¨ornstein

New Series VIII/1A1