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

2.2.5.3 General astigmatic beams

All other beams are classified as general astigmatic. Usually all ten second-order moments are necessary to describe a general astigmatic beam.

2.2.5.4 Pseudo-symmetric beams

Pseudo-symmetric beams are general astigmatic but “look like” stigmatic or simple astigmatic under free-space propagation. They possess an inner astigmatism which is hidden under free propagation and propagation through stigmatic (isotropic) optical systems (i.e. combinations of spherical lenses). Pseudo-symmetric beams di er from real stigmatic or simple astigmatic beams by a non-vanishing twist parameter, tw = 0.

The variance matrix Ppst of pseudo-stigmatic beams is therefore

Under free-space propagation there is no di erence between a real stigmatic beam, tw = 0 , and the corresponding pseudo-stigmatic one, tw = 0 , (2.2.29). The di erence can be uncovered by inserting an arbitrary cylindrical lens somewhere in the beam path. The stigmatic beam is converted into a simple astigmatic beam with non-rotating variance ellipse while the pseudo-stigmatic one is turned into a general astigmatic beam with rotating variance ellipse. Figure 2.2.6 illustrates the di erent behaviors.

The variance matrix Ppsa of aligned pseudo-simple astigmatic beams is given by

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Fig. 2.2.6. Propagation of a stigmatic (top) and pseudo-stigmatic (bottom) laser beam. In free-space propagation both beams are indistinguishable. But a cylindrical lens transforms the stigmatic beam into a simple astigmatic one, whereas the pseudo-stigmatic beam becomes general astigmatic with rotating variance ellipse.

Again, under free-space propagation there is no di erence between a real simple astigmatic beam, tw = 0 , and the corresponding pseudo-simple astigmatic one, tw = 0 , (2.2.29). Inserting an aligned cylindrical lens somewhere in the beam pass unveils the di erence. The simple astigmatic beam keeps being simple astigmatic while the pseudo-simple astigmatic one is turned into a general astigmatic beam with rotating variance ellipse. Figure 2.2.7 illustrates the di erent behaviors.

2.2.5.5 Intrinsic astigmatism and beam conversion

Applying astigmatic (anisotropic) optical systems (including cylindrical lenses) may convert beams from one class to another. But only beams with vanishing intrinsic astigmatism a, (2.2.47), can be converted into stigmatic ones [94Mor]. In practice, beams with

a

(Me2 )

2 < 0.039 (2.2.77)

are considered intrinsic stigmatic, all others intrinsic astigmatic (the limit of 0.039 is a consequence of (2.2.58)). Intrinsic astigmatic beams can always be converted into pseudo-stigmatic or simple astigmatic ones.

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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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