Материал: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo

is a general astigmatic Gaussian beam with a mixing term between the coordinates x and y.

3.1.7.3 Waist transformation

Often, the transfer of the beam waist is required for instance for focusing of laser light. Then, the following algorithms are much more simple than the q-parameter algorithm.

3.1.7.3.1 General system (fundamental mode)

In Table 3.1.20 the waist transformation for a general system is given.

3.1.7.3.2 Thin lens (fundamental mode)

The formulae (3.1.123)–(3.1.126) are further simplified using the focal length f for the thin lens only, see Table 3.1.21.

Remark : Discussion of equation (3.1.127):

The right-hand-side term of (3.1.127) containing z0 represents the modification introduced by the Gaussian beam optics to the thin-lens equation ((3.1.95), t 0) shown in Fig. 3.1.42.

In Fig. 3.1.43 the relation of the Gaussian waist transfer to the thin-lens equation of geometrical optics for di erent influences of di raction is shown.

Main modifications of the geometrical optics:

– No “image distance” is at infinity.

Example 3.1.17. Given for Fig. 3.1.42: z = 1179 mm, w0 = 0.22 mm, λ = 1.06 µm; it follows z = 109 mm, w0 = 0.02 mm, θ = 0.96◦ , and z0 = 1.21 mm. The second right-hand term of (3.1.127) translates the Gaussian waist image by 0.16 mm in comparison with the geometrical optical image towards the lens.

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Table 3.1.20. Waist transformation for a general system.

Table 3.1.21. Waist transformation by a thin lens.

Asked : Waist w0 and distance z to the output plane of the system and z0 .

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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.

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:

−∞

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3.1.7.5Gaussian beams in optical systems with stops, aberrations, and waveguide coupling

3.1.7.5.1Field distributions in the waist region of Gaussian beams including stops and wave aberrations by optical system

Classical cases of optical system design are given in [99Bor, 80Hof, 86Haf]. [82Wag, 95Gae] use the calculation of the field distribution in the image by a stop and wave aberrations in the exit pupil.

The analog is modeled for Gaussian beams on the exit pupil in the following references:

–focused Gaussian beams with aberrations and stops: see [69Cam, 71Sch],

–obscuration of a rotationally symmetrical Gaussian beam including longitudinal focal shift: see [82Car, 86Sta],

–extended systematic discussion of di raction with stops, obscuration, and aberrations: see [86Mah, 01Mah],

–spherical aberration: see [98Pu].

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3.1.7.5.2 Mode matching for beam coupling into waveguides

The calculation of the excitation coe cient of an eigenmode in a waveguide (output mode) by the incident mode (input mode) at the surface of the waveguide is described in Table 3.1.24.

This task occurs

–if a laser beam is formed by an optical system and coupled afterwards into an optical fiber,

–if a laser beam of a master oscillator is to be coupled into a power amplifier,

–in the case of waveguide-waveguide coupling especially fiber-fiber coupling or coupling between semiconductor lasers.

Solutions are available in commercial optical design programs.

Table 3.1.24. Definitions for waveguide coupling.

3.1.7.5.3 Free-space coupling of Gaussian modes

For the case that a Gaussian output waist of a source waveguide and a Gaussian input waist of a receiver waveguide are separated by air, the coupling of both waveguides is generally treated in [64Kog]. Higher-order modes are also included. The approximation of small misalignments (o set and tilt) is given in Table 3.1.25, large o sets and tilts are treated in [64Kog, 91Wu].

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