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

(3.1.40): Fraunhofer’s approximation

– Fresnel number :

–Validity of Fraunhofer’s approximation: NF 1 .

p = 1 (parabolic phase): the intensity of di racted light is the square of the modulus of the Fourier transform of E(x, y, 0) only.

– Additional condition with second Fresnel number NF = b2/λz 1 :

E(x, y, z) is the Fourier transform of E(x, y, 0) in dependence on the spatial frequencies fx ≈ (x/z)/λ ≈ Θx/λ and fy ≈ (y/z)/λ ≈ Θy /λ .

– Di erent conventions on the spatial Fourier transform F (fx) of a spatial distribution f (x) :

–The convention of the plane-wave structure exp(i kx − i ω t) is connected with the determination of F (fx) by

∞

F (fx) = d x f (x) e−i 2π fxx

−∞

[68Goo, 68Pap, 78Loh, 78Gas, 93Sto, 05Hod].

– The plane-wave structure exp(i ω t − i kx) can be combined with

∞

F (fx) = d x f (x) ei 2π fxx

−∞

[71Col, 73Men, 92Lug], but

∞

F (fx) = d x f (x) e−i 2π fxx

−∞

is defined also in [88Kle, 91Sal, 95Wil, 96Ped].

Landolt-B¨ornstein

New Series VIII/1A1

– 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.

Landolt-B¨ornstein

New Series VIII/1A1

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.

Landolt-B¨ornstein

New Series VIII/1A1