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

It should be noted that beam widths along the principal axes, dx and dy , do not obey the hyperbolic propagation law in the case of a general astigmatic beam with rotating variance ellipse (see next section).

The product of the (directional) beam waist diameter d, dα and the corresponding far-field divergence angle θ, θα is called the beam parameter product. Due to di raction the beam parameter

The beam parameter ratios M 2 and Mα2 are invariant in stigmatic aberration-free first-order optical systems (combinations of perfect spherical lenses). In contrast to the e ective beam parameter ratio Me2 , they may change under propagation through cylindrical lenses.

2.2.5 Beam classification

Lasers beams can be classified according to their propagation behavior. The classification is based on the discrimination between circular and non-circular power density profiles and the azimuthal

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orientation of the non-circular profiles. A beam profile is considered circular if the beam widths along both principal axes are approximately equal, or, in practice, if

min dx, dy

> 0.87 . (2.2.58) max dx, dy

In this sense a homogeneous profile with square footprint is regarded circular, see Fig. 2.2.5.

Fig. 2.2.5. Within the concept of second-order-moment beam characterization a square top-hat profile is considered circular: Its width is independent of the azimuthal direction.

2.2.5.1 Stigmatic beams

A laser beam is considered stigmatic if all its profiles under free-space propagation are circular and if all non-circular profiles behind an arbitrary cylindrical lens, inserted somewhere in the beam, have the same azimuthal orientation as the lens. The system matrix Pst of a perfectly stigmatic beam has only three independent parameters:

Since the properties of a stigmatic beam are independent of the azimuthal direction, it has a unique waist position

with a waist diameter of

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Finally, the phase paraboloid is of rotational symmetry with the radius of curvature being

x2

R = c . (2.2.65)

xu c

2.2.5.2 Simple astigmatic beams

A laser beam is classified as simple astigmatic if at least some of the power density profiles the beam takes on under free-space propagation are non-circular, but all non-circular profiles have the same azimuthal orientation. In practice, the orientations of two non-circular beam profiles are regarded as equal, if the azimuthal angles di er by less than 10 degrees. A simple astigmatic beam whose principal axes are parallel to the x- and y-axis is called aligned simple astigmatic. The variance matrix Pasa of a perfect aligned simple astigmatic beam has six independent parameters:

All the physical parameters given for stigmatic beams can be assigned separately for each principal axis of a simple astigmatic beam. The diameters in x- and y-direction are

Aligned simple astigmatic beams have in general two di erent waist positions for each principal axis:

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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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