Ref. p. 131]	3.1 Linear optics	79

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].

3.1.3.2.1 Plane wave
E = E0 exp {−i k0 nˆ er + i ϕ .}	(3.1.23)
For the parameters see (3.1.18).
3.1.3.2.2 Cylindrical wave
E = E0 H0(2)(k0 nˆ ρ) (ρ > λ0)	(3.1.24)

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

3.1.3.2.3 Spherical wave
E = E	0	·	exp(−i k0 nˆ r)	(r > λ	) ,	(3.1.25)
			r	0

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].

E(x, y, z) = E0 · J0(a ρ) · exp {−i cos (θB) k0z}

(3.1.26)

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.

Landolt-B¨ornstein

New Series VIII/1A1