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

3.1.3 Solutions of the wave equation in free space

Following (3.1.2), each of the wave solutions given in this section must be multiplied with the factor ei ω t to obtain the propagating wave of (3.1.1).

3.1.3.1 Wave equation

for time-harmonic electric source current density on the z-axis of a cylindrical coordinate system with the coordinates (ρ, ϕ, z) : (radial distance, azimuthal angle, z-axis) [94Fel, Chap. 5].

Hm(2) : mth order Hankel function of the second kind [70Abr];

the change of convention in Sect. 3.1.1 includes: Hm(2) Hm(1) [94Fel, p. 487]; ρ : radial position vector,

ez : unit vector along the z-axis.

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is the far field (1/r2 and higher inverse power terms 1/r-term) of an oscillating electric dipole ([99Bor, 94Leh, 75Jac]) with

E0 : amplitude [V],

p : unit vector of the dipole moment,

n : unit vector pointing from dipole to spatial position, r : radial distance.

3.1.3.2 Helmholtz equation

The approximative transition from the vectorial wave equation (3.1.4) to the Helmholtz equation (3.1.5) ([99Bor]) results in scalar solutions. E is called: “field” [72Mar], “complex displacement” or “scalar wave function” [99Bor], “disturbance” [95Bas, Vol. I].

is the diverging field of a homogeneous line source [41Str, Chap. IV], [94Fel, Chap. 5]. For the parameters see (3.1.19).

parameters see (3.1.21).

3.1.3.2.4 Di raction-free beams

3.1.3.2.4.1 Di raction-free Bessel beams

Di raction-free Bessel beams without transversal limitation are discussed in [05Hod, 91Nie, 88Mil].

with

E0 : amplitude vector [V/m],

J0 : zero-order Bessel function of the first kind [70Abr]; higher-order Bessel beams see [96Hal];

ρ = x2 + y2 : radial distance from the z-axis, a = k0 sin ΘB [m−1],

ΘB : convergence angle of the conus of the plane wave normal to the z-axis, see Fig. 3.1.2.

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3.1.3.2.4.2 Real Bessel beams

Real Bessel beams are limited by a finite aperture D of the optical elements needed or Gaussian beam illumination (Gaussian Bessel beams [87Gor]).

Methods of generation: axicons [85Bic] (Fig. 3.1.2), annular aperture in the focus of a lens [87Dur, 91Nie], holographic [91Lee] or di ractive [96Don] elements. Because of finite aperture di raction the latter display approximately the shape of (3.1.26) with cuto at a geometric determined radius rN , which includes N maxima (Fig. 3.1.3) and di erent amplitude patterns in dependence on z.

Advantage of Bessel beams: Large depth of focus 2 z0B between P 1 and P 2 in Fig. 3.1.2 (thin “needle of light”) for measurement purposes.

Disadvantage: Every maximum in Fig. 3.1.3 contains in the corresponding circular ring nearly the same power as the central peak. High power loss occurs if the central part is used only [05Hod].

3.1.3.2.4.3 Vectorial Bessel beams

Vectorial Bessel beams are discussed in [96Hal].

3.1.3.3 Solutions of the slowly varying envelope equation

Gaussian beams are solutions of the SVE-equation (3.1.7) [91Sal, 96Ped, 86Sie, 78Gra], which is equivalent to paraxial approximation or Fresnel’s approximation, see Sect. 3.1.4.

The transition from SVE-approximated Gaussian beams towards an exact solution of the wave equation in the non-paraxial range is given in a Lax-W¨unsche series [75Lax, 79Agr, 92Wue]. For contour plots of the relative errors in the Gaussian beam volume see [97For, 97Zen].

The vectorial field of Gaussian beams is discussed in [79Dav, 95Gou], containing a Lax-W¨unsche series; Gaussian beam in elliptical cylinder coordinates are given in [94Soi, 00Gou].

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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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