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

Inhomogeneous wave (Fig. 3.1.25): Snell’s refraction law is modified:

with

2 n2T = n 2 − k 2 + n2 sin2 Θ + n 2 − k 2 − n2 sin2 Θ 2 + 4 n 2 k 2 (Ketteler’s formula) .

The e ective refractive index nT determines the direction angle ΘT of planes of constant phase in Fig. 3.1.25 via (3.1.82) [88Kle, p. 78], [41Str, p. 503], [99Bor, p. 740]. The full inhomogeneous wave can be calculated using [99Bor, p. 740].

Example 3.1.11. Θ = 45 ◦, Au: λ = 800 nm, n = 0.19, k = 4.9, nT = 0.73, ΘT = 75.1 ◦ (see [28Koe, p. 209]).

Intensity attenuation in the case Θ = 0 ◦:

can be measured. The complex refractive index of a material results [77Azz, 90Roe]. Application: Measurements for the optical constants of metals, semiconductors, and thin-film systems.

3.1.5.7 Crystal optics

3.1.5.7.1 Classification

The dielectric tensor εr = εij in (1.1.8) is symmetrical and real in the case of a nonabsorbing medium.

In Fig. 3.1.26 vectors connected with wave propagation in crystal optics are depicted. In Table 3.1.8 optical crystals are listed. In Table 3.1.9 three of the eight surfaces for visualization of wave propagation in crystals are presented.

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Table 3.1.9. Three of the eight surfaces for visualization of wave propagation in crystals.

Main feature of crystal optics: s is not parallel with n for wave propagation, mostly.

–s is essential for description of the energy propagation (edges of bundles, rays),

–n is essential for description of the interferences of infinite broad waves.

References: [28Szi, 54Bel, 58Shu, 61Ram, 76Fed, 79Wah, 84Yar, 04Ber, 99Pau, 99Bor]. A detailed comparison between that surfaces is given in [28Szi].

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3.1.5.7.2 Birefringence (example: uniaxial crystals)

Uniaxial crystals in the plane of incidence:

–Refraction of the normal direction n of wavefronts: The wavevector surface is shown in Fig. 3.1.27.

(ne depends on the angle of incidence).

–Refraction of rays (Poynting vector): se and so are given by tangent construction in Fig. 3.1.28.

–Algorithm for the calculation of ko ( so), ke, se of Fig. 3.1.28 with n, no, ne, η, θ of Fig. 3.1.29 [86Haf]:

n2(n2o − n2e )2 sin2 Θ sin2(2η) 2 B2

(refractive index for the extraordinary wave)

with

A = (n2e − n2o) n2 sin2 Θ cos (2η) − n2o n2e ,

B = n2o + (n2e − n2o) sin2 η ,

where the decision on the ± sign in (3.1.86) can be made by controlling the satisfaction of

n2θ e n2o + (n2e − n2o) sin2(η + Θe) = n2e n2o .

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Application: Θe, Θo, no, and nΘ e phase di erences (interferences) and reflection coe cient, Θo and Θew ray separation in a crystal.

Example 3.1.13. Calcite: no = 1.658, ne = 1.486, η = 45◦: #1: Θ = 0◦: C = 1.244822, nΘ e = 1.565,

Θo = Θe = 0◦, Θew = −6.224◦; #2: Θ = 45◦: nΘ e = 1.636, C = 0.438329, Θo = 25.23◦, Θe = 25.6◦, Θew = 21.33◦.

General formulation of (3.1.85)–(3.1.89): see [76Fed, Table 9.1] for more detailed discussions.

3.1.5.8 Photonic crystals

Starting with the forbidden (stop) bands in case of multi-layer Bragg reflection [88Yeh, p. 123] a material class is under development which stops light propagation along as many directions and for as many wavelengths as possible. This suppresses the spontaneous emission for laser applications and opens new possibilities in the microand nano-optics [95Joa, 01Sak, 04Bus]. Photonic crystal fibers [04Bus] can be designed for special light propagation properties and high-power fiber lasers [03Wad].

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3.1.5.9 Negative-refractive-index materials

The common excitation of electrical dipoles and magnetical dipoles by light in a medium can result in a negative dielectric permittivity Re (ε) < 0 in combination with a negative magnetic permeability Re (µ) < 0 . Then, in Snell’s law (3.1.72) an e ective index nˆ < 0 is possible [68Ves] which results in imaging by a slab of this material without curved surfaces [00Pen] and other interesting e ects [05Ram]. Such metamaterials can be generated by microtechnology, now for mmand terahertz-waves, but with the trend towards visible radiation [05Ele].

3.1.5.10 References to data of linear optics

[62Lan] contains optical constants, only. In later editions, the optical constants are listed together with other properties of substances. An overview is given in the content volume [96Lan].

3.1.6 Geometrical optics

Geometrical optics represents the limit of the wave optics for λ 0 .

The development sin σ = σ − 3!1 σ3 + 5!1 σ5 − . . . with σ the angle in Snell’s law characterizes the di erent approaches of geometrical optics. Table 3.1.10 gives an overview of di erent approximations of geometrical optics.

3.1.6.1 Gaussian imaging (paraxial range)

The signs of the parameters determined in [03DIN, 96Ped] are applied in Sect. 3.1.6.1.1, later on f = f is used.

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