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

Table 3.1.10. Di erent approximations of geometrical optics.

3.1.6.1.1 Single spherical interface

Figure 3.1.30 shows the imaging by a spherical interface in the paraxial range (small x, x , h).

Gaussian imaging equation:

Remark : The symbol f means outside this section, Sect. 3.1.6.1, the positive focal length for a positive (converging) lens.

Newton’s imaging equation:

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with

n : object-space refractive index, n : image-space refractive index, s : object distance,

s : image distance,

r : radius of curvature of the interface, x : height of the object point,

x : height of the image point,

z : focus-related object distance, z : focus-related image distance.

Imaging through an optical system: concatenation of the imaging of the spherical surfaces in suc-

cession via (3.1.90) by using sfollowing surface = sprior surface − d, d : the distance between the surfaces, and (3.1.94) for an object height x = 0.

3.1.6.1.2 Imaging with a thick lens

Figure 3.1.31 shows the axial imaging with a thick lens, Fig. 3.1.32 depicts Listing’s construction for thick-lens imaging of a finite-height object point O to image point O .

Thick-lens imaging equation:

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Position of the principal point H:

Position of the principal point H :

Thin lens: t 0 : (3.1.95) “Lens maker’s formula”.

3.1.6.2Gaussian matrix (ABCD-matrix, ray-transfer matrix) formalism for paraxial optics

Three tasks can be treated with the help of the ray-transfer matrix:

1.full description of paraxial optics (this section, Sect. 3.1.6.2),

2.Gaussian beam propagation (coherent radiation) by combination with a special beam calculation algorithm (see Sect. 3.1.7 on beam propagation),

3.propagation of the second-order moments of the radiation field (inclusion of partial coherent radiation) (see Chap. 2.2 on beam characterization).

The optical system can be the separating distance in an optical medium, a single spherical optical surface or a true, more complicated optical system.

There are di erent definitions for the ABCD-matrices:

Here: The slope components of the input and output rays are the real angles without any relation to the refractive indices at input and output spaces of Fig. 3.1.33 [66Kog1, 66Kog2, 84Hau, 91Sal, 95Bas, 96Ped, 96Yar, 98Hec, 98Sve, 01I , 05Gro1, 05Hod]. Then, the determinant of the matrix M : M = n /n with n the index of the medium of the input plane and n the index of the medium of the output plane.

Other authors [75Ger, 86Sie, 88Kle, 04Ber] use:

slope parameter = (angle) × (related refractive index). Then the equation M = 1 applies.

In Fig. 3.1.34 the concatenation of di erent ray-transfer matrices for di erent types of subsystems is shown.

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Fig. 3.1.34. Concatenation of di erent ray-transfer matrices for di erent types of sub-systems. Matrices known for systems before can be used to construct the matrix for a larger system containing the known systems. The sequence of the matrices is shown at the bottom of the figure.

3.1.6.2.1 Simple interfaces and optical elements with rotational symmetry

In Table 3.1.11 ABCD-matrices for simple interfaces and optical elements with rotational symmetry are listed.

3.1.6.2.2 Non-symmetrical optical systems

Rotational symmetry lacks and the axis is tilted due to the non-symmetrical optical system. In such a system, the central ray of imaging is called the basic ray. The optics in a narrow region around the basic ray is called parabasal optics [95Bas, Vol. 1, p. 1.47] as analogon to paraxial optics. For treatment of astigmatic pencils see [72Sta].

A special case of the non-symmetrical optical system is a system without torsion: Two orthogonal cases do not mix during propagation. Examples are di erent setups of spectroscopy and laser physics (ring resonators).

In Table 3.1.12 ABCD-matrices for non-symmetrical optical elements without torsion are listed.

3.1.6.2.3 Properties of a system

Properties of a system included in its ABCD-matrix are discussed in [75Ger, 96Ped, 05Hod, 05Gro1]. In Table 3.1.13 distances between cardinal elements of an optical system are listed, in Table 3.1.14 the meaning of the vanishing of di erent elements of the ABCD-matrix is depicted.

3.1.6.2.4 General parabolic systems without rotational symmetry

The generalization of the two-dimensional ray transfer after Fig. 3.1.33 to three dimensions [69Arn] is shown in Fig. 3.1.35. The ray in the input plane is characterized by two coordinates x1 and y1 of the piercing point P and two small (paraxial range) angles α1 and β1 .

The matrix S relates these parameters to the corresponding parameters in the output plane like in Fig. 3.1.33:

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I O

IO

Gradient optics:

see [02Gom, 05Gro1].

(continued)

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