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2. Image Aberrations A distant point source produces an image of finite size because of diffraction. Real optical systems suffer from additional optical aberrations which result in a “focal blur”. Can calculate “spot patterns” of ray paths through entrance pupil and focal plane, to reduce these aberrations at the design stage. “Best” focus produces the tightest concentration of spots on the focal plane.

Two classes of image aberration (“Seidel aberrations”)

1. Monochromatic aberrations - in reflecting and refracting systems; independent of wavelength

2. Chromatic aberrations - only in refracting systems; wavelength dependent.

*after Ludwig van Seidel (1850s) Monochromatic aberrations:

1. Spherical aberration - on-axis aberration 2. Coma 3. Astigmatism 4. Curvature of field 5.  Distortion

- off-axis aberrations

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Spherical Aberration Light rays parallel to optical axis, but entering at different distances from axis, are focused at different points along the axis (fig 4.6) - occurs for: -  thick lenses -  “spherical mirrors” (i.e. shaped as part of a sphere)

Can be reduced by using a diaphragm, or stop, to prevent rays far from the optical axis from reaching focal plane - but results in a loss of light.

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James Gregory 1663 Advocated Parabolic Mirror Gregorian Telescope

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Example: the equation for a circle (cross-section of sphere) with its centre displaced by one radius R from the origin is:

y2+(x - R)2 = R2 or y2-2Rx + x2 = 0 ∴ x = R ± (R2 - y2)1/2

For x < R, corresponding to the minus sign in the equation, we can expand as a binomial series

But the standard equation for a parabola with vertex at the origin, and focus at distance f to the right is y2 = 4fx

If 4f = 2R (as in fig), then 1st term in series is the parabolic term, and remainder represent deviations Δx

Difference between parabola and sphere is negligible close to the optical axis.

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x =y2

2R+1⋅ y4

222!R3 +1⋅ 3 ⋅ y6

233!R5 +…

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Δx =y4

8R3 +y6

16R5 +…

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e.g. a 1-metre diameter f/4 telescope

f = 4000 mm; R = 8000 mm y = 500 mm at edge of primary mirror

= Δx between spherical & parabolic mirror

at f/10, Δx = 0.00098, or ~1.7 wavelengths of green light (550 nm)

… note how carefully the mirror surfaces have to be figured! €

∴y4

8R3 = 0.01526 mm

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Coma Off-axis (marginal) rays passing through the aperture near its edge intersect the image surface at different heights from those passing through aperture centre.*

Hence an image of a point source resembling a comet - a bright core, and a diffuse spreading tail.

The image is not symmetric, and so coma is a disaster for accurate astrometry! (measuring positions on the sky).

*because the effective focal length depends on h and hence the transverse magnification too.

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courtesy web page of Tanaka Kazuyuki

Astigmatism

Exists when there is a difference between the optical power in the Tangential and Sagittal planes.

Two focal surfaces result, formed by the focus of tangential rays (b) and by Sagittal rays (d).

Tangential plane - vertical plane through the object point and the lens axis.

Sagittal plane - normal to the tangential, also through lens axis.

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Curvature of Field

Means that the focal surface is not a plane surface, but is curved in 3-D.

Most optical systems suffer from curvature of field.

website of Walter Koprolin

Distortion Affects only the image scale, and not the image sharpness; the image scale is a function of position on the focal surface.

Positive/pincushion distortion - scale decreases with distance from optical axis.

Negative/barrel distortion - scale increases with distance from optical axis.