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

3.1.3.3.1 Gauss-Hermite beams (rectangular symmetry)

Elliptical higher-order Gauss-Hermite beam:

−∞

Properties of E00 (fundamental mode): The shape of the Gaussian E00-beam is depicted in Fig. 3.1.4. Parameters of E00 in Fig. 3.1.4 are:

C : curves with constant amplitude decrease as E(r, z) = E(0, z)/e or constant intensity decrease as I(r, z) = I(0, z)/e2 ,

P : phase fronts with radius of curvature R(z) ,

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Fig. 3.1.5. (a) Cross section of a Gaussian beam perpendicular to the z-axis. (b) Power transmitted by a circular aperture with the relative radius r/w in a cross section.

Table 3.1.3. Characteristic points in Fig. 3.1.5.

a Full width half maximum/2.

b 1/e2-intensity or 1/e-amplitude.

c Di raction of E00-beam by circular aperture 17 % intensity ripple [86Sie, p. 667].

dDi raction of E00-beam by circular aperture 1 % intensity ripple [86Sie, p. 667] (no essential e ect of truncation).

w0 : beam waist,

z0 : Rayleigh distance, half of the confocal parameter b = 2z0 (similarly to depth of focus in usual optics), that z-value, where the cross section π wR2 = 2π w02 of the Gaussian beam has doubled in comparison with the waist,

Θ0 = λ/(πw0) : 1/e2-intensity divergence angle toward the asymptotes A.

In Fig. 3.1.5a the cross section of a Gaussian beam perpendicular to the z-axis is given, in Fig. 3.1.5b the power transmitted by a circular aperture with the relative radius r/w in a cross section. Characteristic points in Fig. 3.1.5 are listed in Table 3.1.3.

Astigmatic and general astigmatic generalizations of the elliptical Gaussian beam: see Sect. 3.1.7.

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3.1.3.3.2 Gauss-Laguerre beams (circular symmetry)

–Two degenerate mode patterns are formed by the cosand sin-terms in (3.1.32).

–l = p = 0 means the rotational symmetrical Gaussian beam E00.

–The symmetry determines what system of Gauss-Laguerre polynomials or Gauss-Hermite polynomials is more appropriate for a wave field development.

3.1.3.3.3 Cross-sectional shapes of the Gaussian modes

In Fig. 3.1.6 intensity distributions of Gauss-Hermite modes Emn are given (rectangular symmetry), in Fig. 3.1.7 intensity distributions of Gauss-Laguerre modes Epl (circular symmetry).

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Fig. 3.1.6. Intensity distributions of Gauss-Hermite modes Emn. The two digits at each distribution are m and n.

Circular symmetry (Gauss-Laguerre modes)

Fig. 3.1.7. Intensity distributions of Gauss-Laguerre modes Epl. The two digits at each distribution are p and l. .

3.1.4 Di raction

Di raction of light by aperture rims or amplitude and phase modifications inside the aperture:

–Solutions of Maxwell’s equations taking into account the material properties of the aperture:

–special cases: exact solutions [99Bor, 86Sta],

–mostly: numerical solutions.

–Starting with a field near the aperture with reasonable assumptions for this field or its measurement: large variety of methods for di erent ranges of validity [99Bor, 86Sta, 61Hoe].

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3.1.4.1 Vector theory of di raction

–Vectorial generalization of Kirchho ’s theory: Given E and H in an aperture E and H in the volume by Stratton-Chu Green’s function representation [23Kot, 41Str, 86Sol, 91Ish].

–Two-dimensional problem and meridional incidence of light [61Hoe]: Separation of the polarizations E parallel and E perpendicular to the plane of incidence for half plane [99Bor], slit [99Bor], gratings [80Pet], and volume gratings [69Kog, 81Sol, 81Rus].

3.1.4.2 Scalar di raction theory

Two sources of scalar di raction theory are:

–Transition from vectorial theory to scalar theory: [99Bor, 86Sol]. The information about the polarization is lost.

–Mathematical formulation and generalization of Huygens’ principle: Each point on a wavefront may be regarded as a source of secondary waves, and the position of the wavefront at a later time is determined by the envelope of these secondary waves.

In Table 3.1.4 di raction formulae with fields given near the di raction aperture are listed. Figures 3.1.8 and 3.1.9 are related to Table 3.1.4.

Remarks on the formulae of Table 3.1.4:

(3.1.37): Approximation of (3.1.34): Huygens’ principle with an additional directional factor (Fresnel).

(3.1.38): Approximation of (3.1.36): Huygens’ principle with a modified directional factor.

(3.1.39): Fresnel’s approximation (= paraxial approximation). The approximation conditions from (3.1.34) to (3.1.39) resp. (3.1.40) are explained in [96For, 86Sta, 87Ree].

Fresnel’s approximation: The condition NF(a/d)2/4 1 [91Sal] is valid for sharp-edged apertures A, but it is weakened for the transmission of Gaussian-beam-like fields [86Sie, p. 635] or Gaussian-like soft apertures. Fresnel’s approximation describes the propagation of the field from plane z = 0 to plane z = z. This transformation can be cascaded to describe complex systems and is an often used tool in paraxial propagation of radiation (Sect. 3.1.4.5.2).

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