78	3.1.3 Solutions of the wave equation in free space	[Ref. p. 131

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

The solutions of the wave equation (3.1.4) are vector fields.

3.1.3.1.1 Monochromatic plane wave

E = E0 exp {−i k0 nˆ er + i ϕ} ,

nˆ

H = c0µ0 (e × E0) exp {−i k0 nˆ er + i ϕ}

with

r : position vector,

e : unit vector normal to the wave fronts, k0 = 2π/λ0 : wave number,

nˆ : complex refractive index, ϕ : phase.

For the phase velocity and the wave group velocity see Sect. 3.1.5.3.

3.1.3.1.2 Cylindrical vector wave

E = E0 ez H0(2)(k0ρ) ,				(k0ρ) (ρ > λ)
H = i c0µ0 ez ×		ρ	H1
	E0	ρ	(2)

(3.1.17)

(3.1.18)

(3.1.19)

(3.1.20)

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.

3.1.3.1.3 Spherical vector wave

E = E

0 ·

(n

×

p)

×

n

·

exp(−i k0 nˆ r)

,

(3.1.21)

r

H =

E0

·

(n

×

p)

·

exp(−i k0 nˆ r)

(r

λ

)

(3.1.22)

c0µ0

r

0

Landolt-B¨ornstein

New Series VIII/1A1