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

2.2Beam characterization

B. Eppich

2.2.1 Introduction

The success of almost any laser application depends mainly on the power density distributions in a certain area of the laser beam, usually the focal region. It is the aim of laser beam characterization to describe and predict the profiles a beam takes on under free-space propagation or behind optical systems.

The attributes of a power density distribution in a plane transverse to the direction of propagation can be divided into size and shape. Under free-space propagation the size of the power density profile is always changing with the distance from the source, whereas the shape of the profile may vary or not. Examples for shape-invariant laser beams are the well-known Gaussian, Laguerre-Gaussian, Hermite–Gaussian, and Gauss-Schell model beams.

A complete characterization of laser beams would allow the prediction of power density distributions, including size and shape, behind arbitrary optical systems as far as they are su ciently known. Admittedly for such detailed characterization a huge amount of data and sophisticated measurement procedures are necessary. But for many applications the knowledge and prediction of the transverse extent of the laser beam profile might be su cient. Restriction to nearly aberrationfree optical systems then enables beam characterization by only ten or less parameters.

In the following the validity of the paraxial approximation will be presumed. In practical this means that the full divergence angle of the beam should not exceed 30 degrees. Furthermore, any polarization e ects are neglected. Beam characterization methods based on the considerations presented in this chapter have recently become an international standard, published as ISO 11146 [99ISO].

2.2.2 The Wigner distribution

A complete description of partially coherent radiation fields (within the restrictions stated above) can be given by a two-point-correlation integral of the field in a transverse plane at location z [99Bor]:

r = (x, y)T a transverse spatial vector (see Fig. 2.2.1), and T the integration time which shall be large enough to ensure that the integration results are independent of the starting time t0. The temporal Fourier transform of this correlation integral is known as the cross-spectral density or the (mutual) power spectrum:

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y

r1

Fig. 2.2.1. Spatial coordinates r1 and r2 of a pair of points in a plane transverse to the direction of propagation.

W (x,y,u,v)

u

z

x

Fig. 2.2.2. The phase space coordinates of the Wigner distribution. x and y are spatial transverse coordinates, u and v are the corresponding angular coordinates.

r r ˜ r r iωτ

Γ ( 1, 2, z, ω) = Γ ( 1, 2, z, τ ) e d τ . (2.2.2)

Since laser beams in general can be considered as quasi-monochromatic, the frequency dependency will be dropped in the following:

From the cross-spectral density in a transverse plane at location z the power density in that plane can easily be obtained by

Given the cross-spectral density at an entry plane the further propagation through arbitrary, but well-defined optical systems can be calculated by several methods and hence the power density distribution in the output plane of the systems predicted [99Bor].

The Wigner distribution W (r, q, z) of partially coherent beams is defined as the Fourier transform of the cross spectral density with respect to the separation vector s [78Bas]:

W (r, q, z) = Γ r + 12 s, r − 12 s, z e−ikq s ds . (2.2.5)

The Wigner distribution contains the same information as the cross-spectral density, but in a di erent, more descriptive manner. Considering q = (u, v)T as an angular vector with respect to the z-axis (Fig. 2.2.2), the Wigner distribution gives the part (amount) of the radiation power which passes the plane at z through the point r in the direction given by q. Within this picture the Wigner distribution might be considered as a generalization of the geometric optical radiance, although this analogy is limited. E.g. the Wigner distribution may take on negative values.

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The power density distribution in a transverse plane is obtained by integration over the angles of direction,

The Wigner distribution represents the beam in a transverse plane at location z. As the beam propagates in free space or through an optical system the Wigner distribution changes. This is reflected in the z-dependency of the Wigner distribution in the equations above. In the following equations this z-dependency will be dropped wherever appropriate.

The propagation of the Wigner distribution through aberration-free first-order optical systems (combinations of parabolic elements and free-space propagation) is very similar to that of geometric-optical rays. Such rays are specified by their position r and direction q. After propagation through an aberration-free optical system position and direction will change according to

where S is a 4 × 4-matrix representing the optical system, the system matrix (see Chap. 3.1). Considering the Wigner distribution as a density distribution of geometric optical rays, its propagation law is given by ray tracing [78Bas]:

2.2.3 The second-order moments of the Wigner distribution

From the Wigner distribution smaller sets of data can be derived, which can be associated to certain physical properties of the beams. These sets of data are the so-called moments of the Wigner distribution [86Bas]:

where

The order of the moments is defined by the sum of the exponents, k + + m + n. There are four first-order moments, x , y , u , and v , which together specify position and direction of propagation of the beam profile’s centroids within the given coordinate system.

The centered moments of the Wigner distribution are defined to be independent of the coordinate system:

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