Файл: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo .pdf
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3.1.3 Solutions of the wave equation in free space |
[Ref. p. 131 |
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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) |
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H = i c0µ0 ez × |
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(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 |
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exp(−i k0 nˆ r) |
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(3.1.21) |
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H = |
E0 |
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(n |
× |
p) |
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exp(−i k0 nˆ r) |
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(3.1.22) |
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c0µ0 |
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Landolt-B¨ornstein
New Series VIII/1A1
Ref. p. 131] |
3.1 Linear optics |
79 |
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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].
3.1.3.2.1 Plane wave |
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E = E0 exp {−i k0 nˆ er + i ϕ .} |
(3.1.23) |
For the parameters see (3.1.18). |
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3.1.3.2.2 Cylindrical wave |
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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 |
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E = E |
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exp(−i k0 nˆ r) |
(r > λ |
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(3.1.25) |
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r |
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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
80 |
3.1.3 Solutions of the wave equation in free space |
[Ref. p. 131 |
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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.
B
w
P1
P2 z z 0B
A
Fig. 3.1.2. Generation of a Bessel beam with help of an axicon A by a conus of plane-waves propagation directions.
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Intensity |
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Fig. 3.1.3. Transversal intensity structure of a Bessel beam ( J02(r)).
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].
Landolt-B¨ornstein
New Series VIII/1A1