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

water are employed. The di erence in temperature between the outflowing and inflowing water is measured and serves as quantity for the absorbed laser radiant power [96Bra]. A special design of the surface geometry of the cavity reduces the irradiance of the laser beam, thus improving the protection from damaging the surface.

The preferred instruments for pulsed laser radiation are thermally absorbing devices such as calorimeters. The receiver element is often a glass-disk, where the radiation is absorbed in the volume instead of on the surface. The absorptance exhibits an excellent stability under chemical and mechanical stress. This type of calorimeter is described in [70Edw, 74Gun]. The radiative load can be reduced by using glass with a low absorption coe cient which increases the length of the absorption path. On the other hand the heat capacity increases linearly with the thickness of the glass-disk which, in conjunction with the poor thermal conductivity of glass, results in long response and cooling times of these detectors. The radiometric scale for laser radiant energy is usually derived from the scale for cw laser radiant power. In [91Moe] a fast electromechanical shutter is used to produce pulses of known laser radiant energy of up to 5 J. The influence of the pulse duration has to be corrected in the calibration procedure. A laser energy meter not depending on a cw laser radiant power scale is described in [90Yua]. In this instrument the light pressure of the laser beam sensed by two mirrors is converted by a moving coil to an electrical signal. The main advantages of this system are fast response and no interruption of the laser beam. The device has been investigated for single laser pulses of radiant energies between 10 mJ and 6 J. Another method not interrupting the laser beam is the photoacoustic calorimetry [86Kim]. There, the radiant energy incident upon a mirror is absorbed at the mirror surface. The absorbed energy generates elastic strain waves which propagate through the mirror substrate. The strain waves eventually pass through a piezoelectric transducer attached to the back of the mirror substrate. The voltage of the piezoelectric crystal gives a direct indication of the amount of energy absorbed at the mirror surface. Since a priori the absorptance of the mirror is not known the instrument has to be calibrated against a standard energy meter.

2.1.4 Outlook – State of the art and trends

Although optical radiometry has been developed for 100 years, measurements of the various radiometric quantities only recently have achieved the required small uncertainties. Today the most accurate detector-based primary radiometric standard is the electrically calibrated cryogenic radiometer. In this instrument the radiant power of – preferably – a laser beam is measured by substituting the absorbed optical power of the laser beam by the electrical power of a heating system. Cryogenic radiometers operate at liquid helium temperatures and have a measurement uncertainty of a few parts in 104, a significant improvement over earlier room-temperature radiometers.

Accurate characterization of laser sources is crucial to the e ective development and use of industrial technologies such as light-wave telecommunications, laser-based medical instrumentation, materials processing, photolithography, data storage, and laser safety equipment. Traceable measurement standards are essential both for users to have confidence in their measurements and to support quality assurance in the manufacture of lasers and laser systems. Because lasers present a potential safety hazard, it is also important to have measurement standards to satisfy nationally and internationally agreed safety limits. The traceability for laser radiometric measurements in Germany is maintained by the Physikalisch-Technische Bundesanstalt. It meets the requirements for calibration and testing laboratories, certification and accreditation bodies defined in the ISO/IEC Guide 17025 and the DIN/EN 45000 and DIN/EN/ISO 9000 series of standards, see http://www.ptb.de/en/org/q/q3/q33/ index.htm.

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References for 2.1

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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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