Astronomy 203/403, Fall 1999

Transcription

1 Astronom 0/40 all ecture 8 September ff-axis aberrations of a paraboloidal mirror Since paraboloids have no spherical aberration the provide the cleanest wa to stud the other primar aberrations that show up with off-axis illumination Consider a paraboloid mirror as shown in igure 8 with its focus at point A Ras incident at angle =0 all reflect through point A Ras incident at angle 0 reflect through another point B which according to paraxial optics lies a distance f tan / from point A What aberrations are present at point B? igure 8: Geometr for characterization of off-axis illumination of a paraboloid mirror showing the mirror (solid lines) and reference paraboloid (broken lines) together with coordinate sstems with origins at their apices Both paraboloids have apex curvature The tilt of the reference surface with respect to the original paraboloid has been exaggerated for clarit; a magnified view of the intersection of an off-axis ra with the apex of the original paraboloid is shown at right To find out we shall exploit the relation between angular and transverse aberrations developed above expressed in Equations 77: AA = d z d TA = AA et s compare the paraboloid under stud with a surface that would produce a perfect image of the offaxis ras at point B: a reference paraboloid placed such that its axis passes through point B and is inclined at angle with respect to the original paraboloid so that the ra bundle is on axis for the reference surface (see igure 8) and so that its apex lies f = / from point B We can obtain a formula for z the ra-path separation between the original paraboloid and the reference surface as a function of a f (8) 999 Universit of Rochester All rights reserved

2 Astronom 0/40 all 999 and for use in Equations 8 or this we need to express the equations of the surfaces z = z( ) in a single coordinate sstem t s eas to write the mirror s equation in the -z sstem and that of the reference surface in the -z sstem as shown in igure 8; we should therefore transform one set of coordinates into the other substitute the appropriate parabola equation and take the difference The appropriate coordinate transformation is illustrated in igure 8 n the -z sstem the origin of the -z sstem lies at tan 0 = f tan = z0 = f f cos = a cos f (8) z " '' ' f tan z'' f - f cos z' z'' igure 8: transformation between the -z sstem natural for the original mirror surface and the -z sstem natural for the reference surface So in terms of a -z sstem originating at 0 z0 but parallel to the -z sstem (see igure 8) we have = + 0 z = z + z0 (8) The -z and -z sstems are inclined b with respect to one another so = cos + z sin z = z cos sin (84) Combine Equations 84 and 85 and get cos + z sin + 0 z z cos sin + z0 (85) 999 Universit of Rochester All rights reserved

3 Astronom 0/40 all 999 n the -z sstem the original parabola is described b z = / or z = 0 tan cos + z sin z 0 cos sin + a cos f = (86) Similarl the reference parabola obes z / = 0 To save writing we can drop the primes henceforth but understand that we re now working in the reference surface s natural coordinate sstem We need to solve Equation 86 for z This is simple since it s just a quadratic: cos sintan z = + + cossin sin cos cos cos (87) sin Note that the negative root is taken in order to make sure we have the point on the test paraboloid nearest the reference paraboloid n the spirit of our treatment of spherical aberration we wish to reduce Equation 87 to an expression of order three in both and irst we can expand the trig functions to third order in as follows: K J sin = + 6 tan = + + cos = + and multipl it out: z = KJ (88) (89) Now we need to expand both the denominator of the leading factor and the square root using the binomial expansion Equation 79 irst the leading factor to order three: KJ N KJ + KJ + Q P = P (80) 999 Universit of Rochester All rights reserved

4 Astronom 0/40 all 999 The numerator is fifth order in so the expression is third order overall The square root can be simplified a bit in advance b discarding terms doomed to produce too large a power of Still there s a lot of writing: KJ KJ KJ (8) 4 5 e j b g = = ike the previous expression we have kept terms through fifth order in here Thus z KJ KJ Now we multipl the whole thing out keeping onl through third order in ; this results in KJ Q P (8) z (8) et us furthermore restrict our attention to terms order four or less in the combination of the powers of the off-center distance and the off-axis angle This leaves z (84) Now we are read to subtract The first term on the right in Equation 84 will be recognized as z for the reference parabola; thus 999 Universit of Rochester 4 All rights reserved

5 Astronom 0/40 all 999 z = z KJ + (85) whence we ma obtain the angular and transverse aberrations via Equations 8: a f AA d = z + d TA = AA + 4 (86) The three terms on the right of Equation 86 are called transverse tangential coma astigmatism and distortion usuall labeled TTC TAS and TD The correspond to the distortions we identified in the spot diagrams generated in ecture 6 f we recall that spherical aberration in conics other than paraboloids appears in the focal plane as TSA = ε e j / we see that SA also fits the description of third order in the combination of powers of and These are four of the five members of the famil of thirdorder or Seidel aberrations A good idea of the origins of the aberrations seen in our spot diagrams can be obtained simpl b contemplating the form of these terms and comparing the destination of marginal ras to that of the chief ra n the following we describe them one b one Transverse tangential coma (TTC) described for marginal meridional ras and paraboloid mirrors b TTC = (87) is a blurring aberration as is spherical aberration because the dependence upon brings ras incident at the same angle to different places in the focal plane arginal ras from opposite sides of the mirror (- and +) wind up on the same side of the chief ra because of the even power of rather than comprising a blur centered on the chief ra This gives a cometar appearance to the focal-plane spots with the chief ra at the head and the marginal ras making up the tail (igure 8) Transverse astigmatism (TAS) which for marginal meridional ras incident on paraboloid mirrors is given b TAS = (88) involves a more smmetrical displacement of these ras on opposite sides of the chief ra in the focal plane n extreme cases the focus can be elongated into a line as shown in igure 84 This elongated focus is called the tangential focus Because the astigmatic marginal ras cross on the wa to this focus an additional elongated focus perpendicular to the tangential focus is present; it is called the sagittal focus Elongated spot diagrams from point objects are the characteristic smptom of astigmatism 999 Universit of Rochester 5 All rights reserved

6 Astronom 0/40 all 999 x / igure 8: Spot diagram and optical laout for transverse tangential coma (TTC) of a paraboloid mirror The spot diagram was made with an axismmetric pattern of 000 ras; the chief ra intersects the center of the frame / x igure 84: Spot diagram and optical laout for transverse astigmatism (TAS) of a paraboloid mirror The spot diagram was made with an axismmetric pattern of 000 ras; the chief ra intersects the center of the frame Transverse distortion (TD) given for paraboloids and meridional ras b TD = 4 (89) doesn t depend upon and hence doesn t blur images in the wa that SA coma and astigmatism do with the algebraic sign for the paraboloid s TD the focus is merel placed sstematicall farther awa from the axis than f the larger is as is shown in igure 85 This condition is called pincushion distortion from the distorted shape assumed b the image of a square object ther tpes and combinations of mirrors place the focus of off-axis ras sstematicall too close; this is called barrel distortion 999 Universit of Rochester 6 All rights reserved

7 Astronom 0/40 all 999 igure 85: third-order transverse distortion n the two focal plane images on the right the box indicated in broken lines is the extent that a distortion-free image with the paraxial value of the plate scale would have The form for transverse coma derived for the paraboloid mirror corresponds to pincushion distortion; other shapes and combinations are known to give rise to distortion of the barrel variet illustrated at far right We have now derived expressions for four of the five third-order geometrical aberrations but have done so under assumptions of meridional ras and general conic section or paraboloid mirrors; we must expect that the aberrations would have somewhat different form for different surfaces or for refractive optical elements or single optical elements and marginal meridional ras though it turns out that the dependences on and are the same as what we have obtained; it is useful therefore to keep in mind that the maximum extent of the transverse aberrations in the focal plane of a given optical element scale as follows: SA ε Coma Astigmatism Distortion e j (80) There is another third-order aberration Petzval field curvature which we will consider in due course; it turns out to be fairl eas to eliminate from optical sstems compared to the first four 999 Universit of Rochester 7 All rights reserved