7.4 Astigmatic beams: the tangential and sagittal plane

If the interferometer is confined to a plane (here the x-z plane), it is convenient to use projections of the three-dimensional description into two planes [46]: the tangential plane, defined as the x-z plane and the sagittal plane as given by y and z.

The beam parameters can then be split into two respective parameters: z 0,s, w 0,s for the sagittal plane and z0,t and w0,t for the tangential plane so that the Hermite–Gauss modes can be written as

unm (x,y) = un(x, z0,t,w0,t) um (y, z0,s,w0,s). (127 )
Beams with different beam waist parameters for the sagittal and tangential plane are astigmatic.

Remember that these Hermite–Gauss modes form a base system. This means one can use the separation into sagittal and tangential planes even if the actual optical system does not show this special type of symmetry. This separation is very useful in simplifying the mathematics. In the following, the term beam parameter generally refers to a simple case where w0,x = w0,y and z0,x = z0,y but all the results can also be applied directly to a pair of parameters.

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