Aberrations in the Bath Interferometer
From Starry Ridge
by Dave Rowe
Contents |
[edit] Introduction
Inherent in the layout of the Bath are foci that are offset from each other, both laterally and longitudinally. This causes measurement-induced astigmatism, coma, and spherical aberration that can be significant in certain situations.
[edit] Analysis
The measurement aberrations induced by the Bath interferometer when testing a mirror at its center of curvature (CoC) is analyzed by computing the path length from the two foci of the interferometer to an arbitrary point on an ideal spherical surface.
Figure 1. The geometry of the test beam in the Bath Interferometer
Refering to Figure 1, the two foci of the interferometer are F1 and F2. The center of curvature (CoC) of the mirror being tested is the origin of the coordinate system, (O) in this diagram. The lateral distance of each focus from the origin is b, and the longitudinal distance from the origin is f.
Figure 2. The geometry of the test setup for a mirror tested at its center of curvature.
Figure 2 shows the three-dimensional geometry of the interferometer and mirror. As in Figure 1, the CoC of the mirror is at the origin. The distance from F1 to the point (x,y,z) on the mirror is l1, and the distance from focus F2 to the mirror is l2. The total path length traversed by the light is l1 + l2. The path length difference for light reflecting from a point on the mirror at (x,y), and light reflecting from the center of the mirror is computed in the following.
Let R be the radius of curvature (RoC) of the mirror. The z coordinate of the mirror's surface is
(1)
The distances from the two foci, F1 and F2, to the surface of the mirror at (x,y) are
(2)
This can be written as
(3)
where,
(4)
Using the Taylor series expansion of the square root to fourth order,
(5)
the path length difference is given by
(6)
correct to fourth-order in q and seventh-order in R. Expanding the terms for q in equation (6),
(7)
[edit] Results
The first term in equation (7) is the only significant one for situations normally encountered by the amateur testing a mirror at its center of curvature. It gives the astigmatism induced by the measurement geometry of the Bath interferometer. The optical path length difference between the center of the mirror, x=0, and the edge of the mirror, x=D/2, is given by
(8)
where D is the diameter of the mirror, d is the beam-to-beam separation, and OPD is the path length difference between the center of the mirror and the edge of the mirror at x=D/2.
Equation (7) can be expanded into Zernike coefficients. The first term is represented by a combination of x-astigmatism, defocus and piston. The x-astigmatism term, Z4 is given by
(9)
One must be careful about the sign of this term if it is used to numerically cancel the astigmatism in the measurement apparatus. The path length is shorter at the edge of the mirror than it is at the center of the mirror, as can be seen from the sign of the first term in equation (7).
[edit] Zernike Coefficients
The first eight Zernike coefficients are given for reference below:
Z1 and Z2 are x-tilt and y-tilt respectively. Z3 is power or defocus, Z4 is astigmatism and a positive coefficient denotes a surface that is high in the x-direction and low in the y-direction, Z5 is also astigmatism, similar to Z4 except rotated by 45 degrees. Z6 and Z7 are x-coma and y-coma respectively, and Z8 is spherical aberration.
