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

– Di erent approximations in (3.1.37) and (3.1.38):

rSP ≈ r0 + 2xξ − ξ2 + 2yη − η2

2r0

[99Bor, 68Pap, 78Gra] with r0 from Fig. 3.1.8 versus

rSP ≈ z + 2xx − x 2 + 2yy − y 2

2z

(references on lasers: [86Sie, 05Hod], optoelectronics: [68Goo, 72Mar, 91Sal]) for grating di raction: The sine of the di raction angle sin Θx = x/r0 is derived from principle and not by a postpositive reasoning of the paraxial range x/z = tan Θx ≈ sin Θx. x/z should be “translated” into sin Θx for better approximation.

(3.1.41)–(3.1.43): Plane-wave spectrum or angular-spectrum representation (also Rayleigh- Sommerfeld-Debye di raction theory) [78Loh, 99Pau] is the plane-wave formulation of (3.1.34) [78Loh, 97For]. Application: see Fourier optics [68Goo, 83Ste, 93Sto].

(3.1.44), (3.1.45): Generation of the far field in the focal plane of a lens: d = f (object is outside the object-side focal plane) additional phase term p to the pure (inverse) Fourier transform (d = f ), similarly to (3.1.40).

Applications: generation of the spectrum of a function, possibility of mathematical operations in the Fourier-space with complex filtering masks, correlation and convolution.

Another important di raction theory

Di raction theory after Young, Maggi, Rubinowicz [66Rub, 99Pau]: The light in point P of Fig. 3.1.8 results from the unperturbed light and local waves, which are emitted by the edge of the aperture A. Therefore, a line integral is to be calculated [99Pau]. There is an equivalence with Fresnel-Kirchho ’s theory.

3.1.4.3 Time-dependent di raction theory

Two formulations of the time-dependent treatment of di raction are possible:

1.A general Fresnel-Kirchho ’s integral formula exists for time-dependent source functions in the aperture A, see [99Bor, 99Pau].

2.Used more often now [96Die, 99Pau]: The time-dependent source functions are decomposed into a superposition of monochromatic fields. The di racted field is calculated for every monochromatic component by the stationary di raction given above. The superposition of all di racted monochromatic components yields the time-dependent di racted field.

3.1.4.4 Fraunhofer di raction patterns

3.1.4.4.1 Rectangular aperture with dimensions 2a × 2b

In Fig. 3.1.10 the geometry of the di raction from a rectangular aperture 2a × 2b is shown. The x-part of the di raction pattern in Fig. 3.1.10 is given in Fig. 3.1.11. In Table 3.1.5 the zeros and maxima of the intensity distribution are listed.

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Table 3.1.5. Zeros and maxima of the intensity distribution.

If the Fraunhofer di raction is observed in the focal plane, z has to be replaced by f .

3.1.4.4.2 Circular aperture with radius a

The circular aperture with radius a is discussed in [61Hoe, p. 453]. In Fig. 3.1.12 di raction by a circular aperture is shown. In Fig. 3.1.13a the di racted field and intensity and in Fig. 3.1.13b the encircled energy in the di raction plane with a circular screen are given. The zeros and maxima of intensity for di raction by a circular aperture are listed in Table 3.1.6.

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Fig. 3.1.13. (a) Di racted field and intensity. (b) Encircled energy in the di raction plane with a circular screen.

Table 3.1.6. Zeros and maxima of intensity for di raction by a circular aperture.

with E0 the electric-field amplitude and r the radius in the far-field plane. Intensity:

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3.1.4.4.2.1 Applications

Airy’s disc:

1st-minimum radius of the intensity distribution in the focal plane of an aberration-free lens (Lommel 1885, Debye 1909, [86Sta, 99Bor]): Substitute in (3.1.50) a/z sin σ (numerical aperture = sinus of the intersection angle σ with optical axis in the focal point, generally: image point) and r = r1 Airy as above.

Annular aperture: obscuration of the central part in the circular aperture A of Fig. 3.1.12:

–Reduction of the central di raction maximum width by ≈ 20 %.

–Increase of secondary maximum by factor ≈ 7.

–See Bessel beams, Sect. 3.1.3.2.4, [05Hod].

with

α : angle of incidence (see Fig. 3.1.14), β : di raction angle,

g : grating constant (grating period, groove distance),

m : order of di raction. Convention [82Hut, p. 25] often used: If the di raction order is on the same side with the zero order (m = 0) as the grating normal: m > 0, otherwise m < 0. In Fig. 3.1.14, the directions of the +1st transmitted order and the grating normal (dashed and dotted lines) are on the same side of the 0th transmitted order. Therefore m = 1 > 0 .

Slit factor : represents the di raction by a single slit of the grating. Its form regulates the energy distribution between the di erent orders m [82Hut, 99Bor]. For the real phase and reflection gratings, it is substituted by the di raction e ciency curves in dependence on α or λ. There is an extreme diversity of cases. Catalogs of such curves: see [80Pet, 97Loe].

Theoretical spectral resolution of a grating:

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with

N : number of grooves of the grating, W : width of the grating,

α, β : see (3.1.52).

Real resolution contains theoretical resolution and the aberrations of the optical elements for collimation and focusing of the grating-di racted plane waves or by the aberrations of the concave gratings with imaging properties. [87Chr, 82Hut].

Holographical gratings [82Hut] show lower disturbations than mechanically produced gratings (application: external laser resonators).

Blazed gratings di ract light into an order m wanted with more than 60–90 % over one octave of wavelengths [80Pet, 82Hut, 97Loe].

Volume gratings: [81Sol, 81Rus].

Mountings of spectral devices: [82Hut].

3.1.4.5 Fresnel’s di raction figures

Fresnel’s approximation is given in (3.1.39) in Table 3.1.4.

3.1.4.5.1 Fresnel’s di raction on a slit

In Fig. 3.1.15 Fresnel’s di raction pattern of a slit with width 2a is shown.

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