telescopeѲ .......................................................................................... CONTENTS

4.2. Coma   ▐    4.4.Defocus

4.3. Telescope astigmatism

Similarly to coma, astigmatism is an off-axis point wavefront aberration, caused by the inclination of incident wavefronts relative to the optical surface. However, while coma always originates at the optical surface, astigmatism may results simply from the projectional asymmetry arising from wavefront's inclination to the surface. For flat incoming wavefront and stop at the surface, the diameter of wavefront's projection onto the optical surface varies from the minimum in the plane of wavefront tilt - determined by the chief (central) ray and optical axis (which also defines the tangential plane) - to the maximum in the direction orthogonal to it (sagittal plane), where it equals aperture diameter.

Since the focal length (i.e. radius of curvature of the wavefront) changes with the square of the diameter for given sagitta depth, the two orthogonal wavefront sections focus at the longitudinal separation of (1-cos2αi, which constitutes longitudinal astigmatism (α is the field angle in degrees, and ƒi is the image-to-pupil separation, equal to focal length for distant objects).

Another peculiarity of astigmatism is that a cross-section along any wavefront diameter is still spherical, but with the radius of curvature varying with the pupil angle. Thus the wavefront form as a whole deviates from spherical.

For displaced stop - either first optical surface significantly separated from the aperture stop, or secondary and tertiary surfaces (whose stop is formed by a preceding surface) - other surface properties, such as shape, position and conic, also can influence the size of astigmatism, due to the displaced stop for these surfaces adding the element of surface radial asymmetry. FIG. 22 illustrate the form of astigmatic wavefront deformation and the resulting geometric (ray) aberration.

FIGURE 22: To the right: Mirror astigmatism as a result of the projected diameter of the incoming wavefront (Wi) varying with the radial orientation. For the inclination angle shown, the vertical (tangential) wavefront projection onto the surface is shortest at cosαD, gradually increasing with the radial angle around the chief ray to the maximum projection width D in the orthogonal (horizontal, or sagittal) orientation. With the wavefront sagitta (depth) constant, its radius (the focal length) varies with the square of diameter. Being of the smallest diameter, the wavefront section Wt in the tangential (vertical) plane focuses closest, and the wavefront section Ws in the sagittal plane farthest away from the mirror. To the left, an illustration of the actual wavefront deviation from the respective reference sphere (red dots): for the reference sphere WP centered at the mid point of defocus M, and for the two reference spheres centered at either sagittal or tangential focus (S and T, respectively; the deviation for the former has cylindrical form oriented horizontally, for the latter vertically). The P-V error is identical at all three focus location; however, the deviation averaged over the wavefront is lower at the mid-focus by a factor 0.82, making it best (diffraction) focus. Solid blue line in best focus wavefront deviation (M) represents the deviation along the central cross-section of the actual wavefront vs. perfect reference sphere centered at the mid-focus point. The dashed blue line is a projection of the deviation along the wavefront edge, indicating saddle-like shape of the wavefront deformation.

 Gaussian focus for astigmatic wavefront lies on the Petzval surface of an optical surface, or system. Balancing defocus aberration for this point - located on the opposite direction from the sagittal focus, and at identical distance from it as the best focus - is zero, and the wavefront error is largest. Between the sagittal and tangential focus, ray disturbance resulting from the astigmatic wavefront deformation takes on rather peculiar form (FIG. 23).

FIGURE 23: Geometry of the astigmatic defocus produced by a mirror with the stop at the surface: the wavefront radius at the pupil gradually increases from the minimum in the  tangential (vertical) plane to the maximum in the orthogonal to it sagittal plane. Consequently, all wavefront meridians focus at a different length, producing longitudinal defocus, as an axial separation between tangential and sagittal focus. At the sagittal plane focus S it forms sagittal line, contained in the tangential plane. And at the tangential plane focus T it forms tangential line, laying in the sagittal plane. The lines transform into ellipses of decreasing eccentricity toward the inside of defocus zone. Midway between the two lines is the circle of least confusion (BF), which is the location of best astigmatic focus.

Aberration function for the wavefront error of astigmatism at best focus is given by:

                                                                    Wa= Ar2(cos2q - 0.5)                                   (18)

with A being the astigmatism peak aberration coefficient, r the height in the pupil, and q the pupil angle. It shows that the the wavefront error peaks for ρ=1 and cos2θ=1 and 0 (that is, for θ=0, π/2, π, 3π/2 and 2π), which is, every 90 degrees, and orthogonally to the orientations of the minimum wavefront deviation, occurring for cosθ=0.5 (for θ=π/4, 3π/4, 5π/4 and 7π/4). It clearly outlines saddle-shaped wavefront deviation, as illustrated on FIG. 22 left.

Note that the maximum wavefront error given by Eq. 18 - which gives ± wavefront deviations, not the peak-to-valley error - is one half of the peak aberration coefficient, which equals the P-V error. Numerically, it is identical to the P-V error at either sagittal or tangential focus, but its RMS value is smaller by a factor of 1/1.5.

When the point of maximum deviation in tangential (vertical) plane is closer to the center of reference sphere than its perfect reference point, the wavefront error of astigmatism is negative. That is the sign of astigmatism in concave mirror, illustrated on FIG. 22. There is no difference in appearance between positive and negative lower-order astigmatism, since the pattern is merely rotated by 90°, and has inherent 90-degree rotational symmetry at best focus location (FIG. 24).

The peak aberration coefficient A, which equals the peak-to-valley wavefront error, is given by:

                                                                               A=aα2d2                                           (19)

with a being the astigmatism aberration coefficient, α the field angle and d the pupil (aperture) radius. The aberration coefficient a for a concave mirror and stop at the surface is given by:

                                                                                   aM =  n/R                                            (20)

with R being the mirror radius of curvature. For mirror in air oriented to the left, n=1 and the aberration coefficient is aM=1/R. The sign of aberration coefficient indicates the tangential wavefront radius shorter than sagittal, and the sagittal line farther away from the mirror, as shown on FIG. 22-23. Positive astigmatism has this order reversed. From another perspective, the astigmatism wavefront error is negative when the optical path difference from the point of peak deviation in the tangential plane is smaller than the radius of a perfect reference sphere (the one centered at the mid point between tangential and sagittal focus).

Shift to the best focus location is half the longitudinal aberration from either of the two line foci. In terms of the peak aberration coefficient A, needed wavefront error of defocus from either tangential or sagittal focus to best focus location is ±(A/2). While the P-V error remains unchanged for all three focus locations - sagittal, tangential and midway between these two - the latter has the RMS wavefront error smaller by a factor of 2/6. As a result, this focus location has has the highest peak diffraction intensity, making it the best focus location.

The best focus RMS wavefront error in terms of the peak aberration coefficient - or P-V wavefront error - is given by:

                                                                           ωa= A/√24                                         (21)

Form of the aberration coefficient shows that the astigmatism wavefront error, unlike coma and spherical aberration, doesn't change with object distance. This is expected consequence of astigmatism of a mirror - as well as that of a lens (contact) objective - being result of the projectional wavefront asymmetry itself, rather than a product of the wavefront/surface interaction.

Astigmatism ray aberrations can also be expressed in terms of the peak aberration coefficient A as:

                                               L=8AF2,         T= 4FA       and       Ta=4A/D                            (22)

for the longitudinal, transverse and angular astigmatism, respectively. After substituting for A, the transverse aberration - as the circle of least confusion diameter - can be also expressed as T= -Dα2/2 = -h2/2DF2 for object at infinity, with h being the point height in the image plane (note that aperture D needs to be in the metric used for the coefficient calculation, which also becomes the metric of transverse aberration). Since focal lengths of the astigmatic wavefront do not change with the height in the pupil (i.e. the zonal height), transverse astigmatism changes in proportion to the normalized pupil ray height ρ. With h=αƒ, ƒ being the focal length, angular astigmatism Ta=T/ƒ=α2/2F. Longitudinal astigmatism L=-ƒα2.

There are simple geometric relationships between the circle of least confusion diameter and the sagittal and tangential line length, as well as between the three and the longitudinal aberration. The line length is double the circle diameter, and the longitudinal aberration is greater than either line by a factor of F (the focal ratio denominator), as illustrated in FIG. 23.

The transverse aberration in terms of the RMS wavefront error is Ta= ω384/2.44, and in terms of the peak-to-valley error Ta=4A/2.44 (for ω and A in units of the wavelength), both expressed in units of the Airy disc diameter. That makes astigmatic blur significantly smaller for given amount of wavefront aberration than geometric blur for either spherical aberration or coma (FIG. 24). It is a stark remainder that optical criteria can

FIGURE 24: The ray spot size (top) and actual diffraction patterns for 0.37 wave P-V wavefront error of primary astigmatism (resulting in 0.80 Strehl, thus comparable to 1/4 wave P-V of primary spherical aberration). Perfect diffraction pattern is to the left. Geometric blur diameter at the best focus location (balanced primary astigmatism) is only 0.6 Airy disc diameters. There is no rays outside the Airy disc, yet considerable amount of energy has spread out from the spurious disc - result of the complex wave interference around best focus point (not  unexpected, considering that the rays focused to a point still produce a pattern) . Compared to spherical aberration and coma, the energy spread is concentrated closer to the disc.

not be reduced to geometrical considerations; it is the underlying realm of electromagnetic field that determines the properties of the point object image.

As expected due to its uniformly dense geometric blur, the smallest RMS blur radius for astigmatism is at the location of the circle of least confusion. It is given by rRMS=FA2, or smaller by a factor of 0.5 than the radius of the circle of least confusion.

EXAMPLE: A 200mm ƒ/5 concave mirror, d=100, R=-2000. Setting θ=0 and ρ=1, the peak wavefront error at h=1.4mm off-axis, giving the field angle α=1.4/1000=0.0014, is W=A/2=α2d2/2R=-0.0000049mm. The P-V wavefront error is twice greater - equal to the peak aberration coefficient - or -0.0000098mm. In units of the 550nm (0.00055mm) wavelength, it is 0.0178, or 1/56 wave. Consequently, the RMS wavefront error ω=A/24=0.000002mm or, in units of the 550nm wavelength, 1/275 wave. The transverse astigmatism (circle of least confusion diameter) is T=4FA=20α2d2/R= =0.000196mm, or 0.03 Airy disc diameters, and angular astigmatism Ta=T/ƒ is 0.000000196 radians, or 206.265x0.000000196=0.04 arc seconds..

Since both, wavefront error and geometric (ray) aberrations are directly proportional to the aberration coefficient, it implies that they are in a constant proportion themselves. In other words, doubling the wavefront error also doubles the geometric aberration.

For the stop at the surface, the astigmatism error doesn't change with object distance. For relatively close objects, transverse astigmatism increases as ƒi/ƒ, ƒi being the image-to-pupil separation. However, it doesn't affect the wavefront error: since the wavefront radius is also longer by the same ratio, identical nominal wavefront error results in proportionally greater longitudinal and transverse aberration.

For the aperture stop displaced from mirror surface, the aberration coefficient of astigmatism changes in proportion to [Kσ2+(1-σ)2], with σ being the mirror-to-stop separation (positive in sign) in units of the mirror radius of curvature. Needed stop separation for zero astigmatism is given by σ=[1-|K|]/(K+1). Thus, astigmatism is canceled for σ=0.5 with a paraboloid and σ=1 with a sphere. The relation is not defined for K=-1 (parabola), but implicates σ=0.5 limit for K"-1. For positive values of the conic K, the aberration coefficient cannot be zero regardless of the stop position, due to the right side of the aberration factor being squared (always positive).

Unlike coma, change in astigmatism caused by the aperture stop position is independent of object distance.

Aberration coefficient of primary astigmatism for a lens with the aperture stop at the surface is identical to the one given for concave mirror (Eq. 20). For a contact doublet, it gives the peak aberration coefficient as a sum of the aberration coefficients at the first and second lens, respectively, as:

                                                          Ad=A1+A2=(-α2D2/8ƒ1)+(-α2D2/8ƒ2)                         (23)

with α being the field angle, and ƒ1,ƒ2 the respective lens focal lengths (keep in mind that focal length of a negative lens is numerically negative in the left-to-right Cartesian coordinate system). Change of the stop position results in change of the aberration coefficient only with systems not corrected for spherical aberration, or coma, or both. Since modern refractor objectives commonly are aplanats, their astigmatism is not affected by the stop position. As already mentioned, wavefront error of astigmatism of the contact doublet doesn't change with object distance.

4.2. Coma   ▐    4.4.Defocus

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