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68

2.2.7 Beam positional stability

[Ref. p. 70

 

 

 

a.u.

 

 

i

 

 

y

 

 

 

 

’x

 

 

 

 

 

 

’

 

 

y

 

 

x

i

a.u.

 

 

Fig. 2.2.9. Centroid coordinates of fluctuating beam and corresponding variance ellipse characterizing the fluctuations.

has a position (x, y) and direction (u, v) at a random measurement. Similar to the second-order moments of the Wigner distribution, only the three spatial moments are directly measurable. The complete set can be obtained from a z-scan measurement as described in the section above, by acquiring a couple of power density distributions in any measurement plane, calculating the firstorder spatial moments from each profile, derive the three variances according to (2.2.79)–(2.2.81), and obtaining the second-order fluctuation moments in the reference plane from a fitting process. Again, measurements behind a cylindrical lens are necessary to achieve all ten parameters.

Fluctuation widths can be derived from the second-order fluctuation moments. In analogy to the beam width definitions, the fluctuation widths are

 

 

√2

x2 s + y2 s + τ

 

 

1

 

1

 

∆x

= 2

x2 s − y2 s 2

+ 4 xy s2

,

(2.2.84)

2

 

 

 

 

 

 

 

 

 

2

 

 

 

√2

 

 

 

1

 

1

 

∆y

= 2

x2 s + y2 s − τ

x2 s − y2 s 2

+ 4 xy s2

 

2

(2.2.85)

2

with

 

 

 

 

 

 

 

 

 

 

τ = sgn x2 s − y2 s ,

 

 

 

 

 

(2.2.86)

where ∆x and ∆y are the beam fluctuation widths along the principal axes of the beam positional fluctuations and where

2

 

x2 s − y2

s

 

β =

1

atan

 

2 xy s

 

(2.2.87)

 

 

 

is the signed angle between the x-axis and that principal axis of the beam fluctuation which is closer to the x-axis (Fig. 2.2.9). The principal axes of the beam positional fluctuations may not coincide with the principal axes of the power density distribution.

The width of the positional fluctuations along an arbitrary direction, given by the azimuthal angle α, is given by

∆α = 4 x2 s cos2 α + 2 xy s sin α cos α + y2 s sin2 α . (2.2.88)

Landolt-B¨ornstein

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Ref. p. 70]

2.2 Beam characterization

69

 

 

 

2.2.7.2 Relative fluctuations

For many applications the widths of the positional fluctuations compared to the momentary beam profile width might be more relevant than the absolute fluctuation widths. The relative fluctuation along an arbitrary direction, given by the azimuthal angle α, is defined by

∆rel,α =

 

 

 

 

 

 

 

 

 

 

 

 

(2.2.89)

x2

s

cos2

α + 2

xy

s sin α cos α +

y2 s

sin2

α .

 

 

 

 

 

x2

 

cos2

α + 2

xy

sin α cos α +

y2

sin2

α

 

 

 

 

 

 

c

 

 

 

 

 

c

c

 

 

 

 

The e ective relative fluctuation may by specified by

 

 

 

 

∆rel =

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2.2.90)

 

x2

c

+

y2

c .

 

 

 

 

 

 

 

 

 

 

x2

s

+

y2

 

s

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.2.7.3 E ective long-term beam widths

For applications with response times much longer than the typical fluctuation durations the timeaveraged intensity distribution rather than the momentary beam profile determines the process results:

I¯(x, y) = T

t0+T

I (x, y, t) d t .

(2.2.91)

t0

1

 

 

 

The e ective width of the time-averaged power density profile along an azimuthal direction enclosing an angle of α with the x-axis can be obtained from the widths of the momentary beam profile and the fluctuation width by

de ,α =

dα2 + ∆α2

.

(2.2.92)

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70

References for 2.2

 

 

References for 2.2

78Bas

Bastiaans, M.J.: Wigner distribution function applied to optical signals; Opt. Commun.

 

25 (1978) 26.

86Bas

Bastiaans, M.J.: Propagation laws for the second-order moments of the Wigner distri-

 

bution function in first-order optical systems; J. Opt. Soc. Am. A 3 (1986) 1227.

93Sim

Simon, R., Mukunda, N.: Twisted Gaussian Schell-model beams; J. Opt. Soc. Am. A

 

10 (1993) 95.

94Mor

Morin, M., Bernard, P., Galarneau, P.: Moment definition of the pointing stability of a

 

laser beam; Opt. Lett. 19 (1994) 1379.

96Mor

Morin, M., Levesque, M., Mailloux, A., Galarneau, P.: Moment characterization of the

 

position stability of laser beams; Proc. SPIE (Int. Soc. Opt. Eng.) 2870 (1996) 206.

99Bor

Born, M., Wolf, E.: Principles of optics, Cambridge: Cambridge University Press, 1999.

99ISO

ISO 11146, Lasers and laser-related equipment – test methods for laser beam widths,

 

divergence angles and beam propagation ratios, 1999 (new revised edition 2005).

03Nem

Nemes, G.: Intrinsic and geometrical beam classification, and the beam identification

 

after measurement; Proc. SPIE (Int. Opt. Soc. Eng.) 4932 (2003) 624.

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Ref. p. 131]

3.1 Linear optics

73

 

 

 

3.1Linear optics

R. Guther¨

The propagation of light and its interaction with matter is completely described by Maxwell’s equations (1.1.4)–(1.1.7) and the material equations (1.1.8) and (1.1.9), see Chap. 1.1.

In this chapter the propagation of light in dielectric homogeneous and nonmagnetic media is discussed. Furthermore, monochromatic waves are assumed and linear interaction. The implications thereof for the medium are:

–Relative permittivity: εr (ε(E, H ) in (1.1.8)) is a complex tensor, which in most cases depends on the frequency only, but in special cases also on the spatial coordinate.

–Relative permeability: µr = 1 (µ(E, H ) in (1.1.9)).

–Electrical charge density: ρ = 0.

–Current density: j = 0.

3.1.1 Wave equations

Maxwell’s equations together with the material equations and the above assumptions result in the time-dependent wave equation for the electric field

εr ∂2

∆ E(r, t) − c20 ∂t2 E(r, t) = 0

with

c0 = 2.99792458 × 108 m/s: vacuum velocity of light,

∆ =

∂2

∂2

∂2

 

+

 

 

+

 

 

: delta operator.

2

∂ y

2

∂ z

2

 

∂ x

 

 

 

 

An identical equation holds for the magnetic field H (r, t).

For the following discussion we assume monochromatic fields, so that

E(r, t) = E(r) ei ω t

with

ω : angular temporal frequency.

The magnetic field is related to E by the corresponding Maxwell equation (1.1.7)

curl E(r) = −i ω µ0H (r) .

(3.1.1)

(3.1.2)

(3.1.3)

Together with the ansatz (3.1.2), for isotropic media (εr is a complex scalar) (3.1.1) results in

∆ E(r) + k02 nˆ2 E(r) = 0 (wave equation)

(3.1.4)

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