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

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Table 3.1.13. Distances between cardinal elements of an optical system: F , F : objectand image-space focal points, respectively; H, H : objectand image-space principal points, respectively; I, O: input and output plane, respectively. The order of points determines the signs.

Table 3.1.14. The meaning of the vanishing of di erent elements of the ABCD-matrix.

with the matrices A, B, C, D, and S given by comparison with the more detailed representations. Identities between the matrices, characteristic for the symplectic geometry (see Sect. 3.1.6.2.6),

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x

y1

1

1

transposition of the matrix and I the identity matrix [86Sie, 05Hod]. The matrix S contains at most 10 independent parameters [76Arn, 86Sie, 05Hod].

In Table 3.1.15 general ray-transfer matrices are given.

3.1.6.2.5 General astigmatic system

A general astigmatic system can be generated by two cylindrical lenses with their axes non-parallel and non-orthogonal, separated by a distance L: SGA = R−1 Scyl,1 R SL Scyl,2 .

3.1.6.2.6 Symplectic optical system

Symplectic optical systems in the paraxial range can be described by the formalism of the symplectic geometry [03Wal]. They can be generated by a finite number of cylindrical and spherical lenses separated by free spaces. The mathematical formulation is connected with the matrix properties given in Sect. 3.1.6.2.4. For theoretical foundation and practical calculations see [64Lun, p. 216], [83Mac, 85Sud, 86Sie, 99Gao, 05Gro1, 05Hod].

3.1.6.2.7 Misalignments

The geometric optical calculations of misalignments with matrix techniques require, generally, higher dimensional matrices [05Gro1, p. 51], for example 3 × 3-matrices [86Sie] or 4 × 4-matrices [85Wan] for two-dimensional problems or 6 × 6-matrices for three-dimensional problems [76Arn].

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Table 3.1.15. General ray-transfer matrices [99Gao, 05Hod].

3.1.6.3 Lens aberrations

Corrections beyond the paraxial range are required by large object-space aperture light sources like semiconductor lasers (large vertical far-field angles) or large image-space aperture laser focusing optics like CD-optics.

Shape factor of a lens:

Shape factor and spherical aberration for focusing of light:

– Minimum of spherical aberration:

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Fig. 3.1.36. Focusing of incident collimated light by (a) a

In Fig. 3.1.36 the focusing of incident collimated light by (a) a general lens with curvature radii r1 and r2 and (b) a plano-convex lens with shape factor q = 1 is shown.

In Table 3.1.16 the third-order spherical aberration and coma for a thin plano-convex lens is given in comparison with the di raction-limited resolution for a plane wave or Gaussian illumination.

Remark 1 : Third-order formulae for finite object distance: see [88Kle, 76Jen], more general: [80Hof, 86Haf, 96Ped, 99Bor].

Remark 2 : About further third-order aberrations as astigmatism, field curvature, image distortion: see [76Jen, 78Dri, 80Hof, 86Haf, 88Kle, 96Ped, 99Bor].

Remark 3 : The third-order aberrations are not exactly valid for higher apertures. Example: The third-order

spherical aberration deviates for 2h/f = 1/5 by ≈ 2 % from the ray-tracing values (the limit, recommended in [74Sle] for estimations), h/f = 3/10 : ≈ 15 % deviation [76Jen]. Therefore, the ray tracing should be

preferred for larger deviations from the paraxial case. It is the base of modern commercial optical design programs.

Example 3.1.14. Given: a plano-convex lens after Fig. 3.1.36b with the radius of the spherical surface r1 = 5 mm, n = 1.5, collimated light with wavelength λ = 1 µm, stop with a height h = 1.5 mm, and a fiber with core diameter 2 r = 100 µm and numerical aperture N.A. = 0.2. Required: a geometric-optical estimation on the hits of the core of the fiber by the rays in the

paraxial focal point and in the point of least confusion (Fig. 3.1.37). From (3.1.101)–(3.1.105): f = 10 mm, ∆sl = −262 µm, |∆st | = 39 µm, ∆slc = −210 µm, |∆stc| = 16 µm, ∆stb = 4 µm, and ∆stg = 2.1 µm. In the paraxial focal plane as well as in the plane of least confusion, the

hits of the fiber core by rays are closer than 50 µm to the optical axis and the angles of the rays with the optical axis are ≤ 0.15 within the fiber aperture. Therefore, all rays are accepted by a step-index fiber. About the analog task for Gaussian beams see references in Sect. 3.1.7.5.4 and commercial optical design programs, which show in this case, that a large part of radiation is coupled in higher-order modes.

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