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

Fig. 2.2.7. Propagation of a simple astigmatic (top) and a pseudo-simple astigmatic (bottom) laser beam. In free-space propagation both beams are indistinguishable. But an aligned cylindrical lens transforms the simple astigmatic beam into a simple astigmatic one, whereas the pseudo-simple astigmatic beam becomes general astigmatic with rotating variance ellipse.

2.2.6 Measurement procedures

Only the three pure spatial moments out of the ten second-order moments are accessible for direct measurement. The other seven moments are retrieved indirectly based on the propagation law of the spatial moments (2.2.29).

The measurement method is based on the acquisition of a couple of power density profiles at di erent z-locations near the generalized beam waist, (2.2.53), e.g. by means of CCD cameras or similar devices (Fig. 2.2.8, left). From the measured profiles the spatial moments at each measurement plane are calculated. Fitting parabolas with three free parameters to the curve of each spatial moment delivers nine independent quantities: the moments x2 c,0 , xy c,0 , y2 c,0 , xu c,0 ,

yv c,0 , u2 c,0 , uv c,0 , v2 c,0 and the sum of the crossed mixed moments xv c,0 + yu c,0. If the waist of the beam is not accessible, an artificial waist has to be created by inserting an almost

aberration-free focusing lens into the beam path. Approximately half of the profiles should be acquired close to the waist within one generalized Rayleigh length, the rest outside two Rayleigh lengths. This ensures balanced accuracy for all parameters of the fitting process.

Fig. 2.2.8. Determination of the ten second-order moments in three steps. First step is a z-scan measurement (left), in the second step the CCD camera is placed in the focal plane behind a horizontally oriented cylindrical lens (middle), in the third step the lens is rotated by 90 degrees (right).

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At least one cylindrical lens is needed for the measurement of the missing di erence of the crossed mixed moments xv c,0 − yu c,0 . To retrieve it, a cylindrical lens with focal length f is inserted into the beam path at an arbitrary position in the beam waist region. Firstly, this cylindrical lens shall be aligned with the x-axis and the spatial moment xy 1 is measured in the focal distance behind the lens (Fig. 2.2.8, middle). Next, the lens is rotated by 90 degrees and the spatial moment xy 2 is again measured in the focal distance from the lens (Fig. 2.2.8, right). The missing di erence of the crossed mixed moments of the reference plane is then given by

2.2.7 Beam positional stability

2.2.7.1 Absolute fluctuations

For various reasons a laser beam may fluctuate in position and/or direction. The positional fluctuations in a transverse plane may be measured by the variance of the first-order spatial moments of the beam profile:

where x i and y i are the first-order moments determined in N individual measurements and

fluctuations are di erent from plane to plane. It can be shown that, under some reasonable assumptions, the positional fluctuations can be characterized closely analogous to the characterization of the beam extent based on the second-order moments of the Wigner distribution [94Mor, 96Mor]. Within this concept, the fluctuation properties of a laser beam are completely determined by ten di erent parameters, arranged in a symmetric 4 × 4 matrix

obeying the same simple propagation law as the centered second-order moments:

The elements of the beam fluctuation matrix may be considered as the centered second-order moments of a probability distribution p (x, y, u, v) giving the probability that the fluctuation beam

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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

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

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)

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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

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:

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

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References for 2.2

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