1、(C) R G Bingham 2005. All rights reserved.,Optics and Optical DesignSession 2,Session 2 Spheres Aspheres Aberrations Defocus Spherical aberration Coma Examples,Richard G. Bingham,(C) R G Bingham 2005. All rights reserved.,11 pages on spheres and aspheres,(C) R G Bingham 2005. All rights reserved.,Eq

2、uation of a sphere as used for ray tracing,Z,z,c is the curvature. It is the reciprocal of the radius of curvature. This reciprocal is convenient in expressions that avoid numerical problems at a flat optical surface or when rays are normal to a surface.,z(x,y),Y,The sphere passing through the origi

3、n has to be expressed as z = f(r,c) where r 2 = x 2 + y 2,r,1/c,(C) R G Bingham 2005. All rights reserved.,1 + (1 c 2 r 2 ),Equation of a sphere (2),Z,z,r,c r 2,z (r ) =,1/c,This exact equation is that used for ray tracing. It has a simple extension for conic sections. It is often also useful for ca

4、lculating the total depth of the surface of a spherical lens or mirror.,(C) R G Bingham 2005. All rights reserved.,Equation of a sphere (3): Taylor series,z (r ) = c r 2 /2 + c 3 r 4 /8 + c 5 r 6 /16 ,z,r,The osculating parabola,Difference of the sphere and the parabola to order r4. Incidentally, th

5、is leads to the Schmidt corrector plate.,Difference of the sphere and the parabola to order r6. This is not useful for the Schmidt plate.,Parabola - paraboloid of revolution,(C) R G Bingham 2005. All rights reserved.,Conic sections,A reflecting telescope with a parabolic mirror Ex11-Paraboloid.zmx,T

6、he refracting surface discovered by Descartes in 1637 Ex12-Ellipsoid_lens.zmx,Aspheric surfaces are antique but we still struggle to make them!,(C) R G Bingham 2005. All rights reserved.,The Standard surface and the conic constant,1 + (1 (1+k) c 2 r 2 ),c r 2,z (r) =,z,r,e is useful for finding the

7、conjugates,Ex10-Conics.zmx,(C) R G Bingham 2005. All rights reserved.,Osculating sphere and paraboloid,z (sphere) =,z (paraboloid ) = c r 2 /2,r,The difference between the sphere and the paraboloid is r4. There is no difference term in r2. Near the centre, the difference between sphere and paraboloi

8、d tends to zero. cr 2/2 is useful for a quick estimate of the depth of a curve even a sphere.,z,c r 2 /2 + c 3 r 4 /8 ,1/c,(C) R G Bingham 2005. All rights reserved.,A sphere and all the osculating conicoids,sphere,All the prolate ellipsoids (rugby balls),paraboloid,All the hyperboloids,These conico

9、ids of revolution fit the sphere near the centre. They have the same vertex curvature. The differences are all r4 to a first approximation. There is no difference r2.,Mirror of a large telescope?,All the oblate ellipsoids (Smarties),r,z,(C) R G Bingham 2005. All rights reserved.,Conjugate foci of el

10、lipse and hyperbola,a,a,Eccentricity e = (-k) = (-8 a4 /c3),V,F,F,V,F,F,For an ellipse: a = 1/c (1 - e2) VF = a (1 - e) VF = a (1 + e),For an hyperbola: a = 1/c (e2 - 1) VF = a (1 - e) VF = -a (1 + e),V is the vertex,For setting up a required curve or for designing an optical test,(C) R G Bingham 20

11、05. All rights reserved.,+ a2 r 2 + a4 r 4 + a6 r 6 + a8 r 8 ,1 + (1 (1+k) c 2 r 2 ),c r 2,z (r ) =,The Even Asphere surface (1),z,r,a2 has dimensions L-1, a4 has dimensions L-3 etc. Affects scaling. Odd-power terms are not necessary in a symetrical optical surface expressed with such a polynomial,

12、because they and their differentials would not be symmetrical or would have a cusp at the origin if written with y and -y. The same applies both to an optical surface and to axial aberrations in a symmetrical lens, so although we might describe an optical surface with odd terms, those terms are not

13、often helpful.,(C) R G Bingham 2005. All rights reserved.,Vertex curvature. a2 0 implies a vertex curvature 2a2 even if c = 0. I have never used non-zero a2 with non-zero c (although people do). Exact paraboloids. Either k = 1 with c 0 , or a2 0. To use both would be redundant at best. k is more vis

14、ible than a2 in the data. Conic sections. Either exactly with k 0, or as far as the r4 term using a4. z(conic) z(sphere) = , so there is an approximation that a4 = .,The Even Asphere (2),+ a2 r 2 + a4 r 4 + a6 r 6 ,1 + (1 (1+k) c 2 r 2 ),c r 2,z (r ) =,8,c3k,r 4 + ,There are two ways of prescribing:

15、,8,c3k,Why use that? Is c the same?,(C) R G Bingham 2005. All rights reserved.,Is it better to use polynomials or conic sections?,The conic section is special for its geometrical foci. If there is no reason to consider the geometrical foci, the forms available with the conic section may be too limit

16、ed. The conic section cannot describe a surface like this: In the conic section, the asphericity depends on the vertex curvature. Awkward when flat! The conic section is excellent for optical testing.,(C) R G Bingham 2005. All rights reserved.,Six slides on wavefront aberration and defocus,(C) R G B

17、ingham 2005. All rights reserved.,Wavefront aberration OPD plot what it is,The OPD maps and sections are well explained in the ZEMAX manual. Calculating OPD involves ray-tracing that is in the program but which is probably unknown to the user, such as tracing back to the exit pupil (see Welford). An

18、other general method is given in one of the pages on defocus in this course. (I do not know why that method is less used.) There is a warning in session 4 regarding using the two OPD cross-sections in asymetrical systems. Positive OPD is advanced.,(C) R G Bingham 2005. All rights reserved.,Wavefront

19、,Wavefront aberration - moving the focus,z, W,z axis,u,Suppose we move the CCD to the right by z. That introduces a phase error W in the converging spherical wavefront. W is a wavefront aberration. It is a distance. It is positive here because the phase is now too advanced. How big is W, and what sh

20、ape is it in x and y? It is the difference between two spheres.,CCD,The focus,(C) R G Bingham 2005. All rights reserved., z,u,CCD,W due to defocus only,The focus,The aberration is taken as zero for the chief ray. The path lengths of the rays are all equal to each other as far as the focus. Then the

21、axial ray travels a further distance z, but the wavefront corresponding to the peripheral ray travels only z cos u z (1 u2/2). So the section of the wavefront corresponding to the peripheral ray arrives earlier: its phase is advanced by z u2/2. W = z u2/2 or z = 2 W/u2,When the signal is detected, t

22、he wavefronts have arrived here.,Huygens wavelets,(Useful to know),(C) R G Bingham 2005. All rights reserved.,(cosine term),The shape of the defocused wavefront,(c2 c1) r 2/2,z (r) = c r 2 /2 + , W,r,z,c1,c2,The cosine term affects only r4 and above., W ,The wavefront aberration that corresponds to

23、a shift of focus shift takes the form of a parabola locally, because:,See Ex09-Defoc_Lens.zmx,Equation of a sphere see later,(C) R G Bingham 2005. All rights reserved.,The shape of the defocused wavefront, and the aberration fans,Out-of-focus image,(C) R G Bingham 2005. All rights reserved.,Exercise

24、. Depth of focus with rays. Effect with long focal length, e.g, with a telephoto lens.,This essentially uses a perfect camera that is not focused to infinity. Use Newtons Conjugate Distance Equation (see Welford). Assume a lens has focal length f, F- number N (f/diameter) and set a limit p to the im

25、age diameter. Approximating tan = , find the depth of focus (linear range of acceptable image diameter) at a series of object distances and notice its dependence on f. Now define a wavelength as if p is twice the diffraction-limited diameter. See whether W/ is near the Rayleigh diffraction limit for

26、 the image diameter p.,(C) R G Bingham 2005. All rights reserved.,Four slides on spherical aberration and coma.,(C) R G Bingham 2005. All rights reserved.,Balancing spherical aberration against focus,These two aberrations are not described by orthogonal basis functions. Fortunately.,Spherical aberra

27、tion,Defocus,Balanced,Many forms of aberration can and must be balanced against others. A difference from the paraxial focus is commonplace.,(C) R G Bingham 2005. All rights reserved.,Aberrations are:,the departures (a) of wavefronts from spheres centred on some required image point, and (b) of rays

28、 from crossing the focal surface at the required point. Aberrations at a single optical surface can have either sign. We try to cancel them out amongst the different surfaces in a finished design.,Defocus done that Spherical aberration Ex13-Sphere.zmx (drop the central obscurations) Coma Ex11-Parabo

29、loid.zmx,(C) R G Bingham 2005. All rights reserved.,Spherical aberration Coma, W 4 0, W y 3 1,OPD = Wavefront aberration or y = relative radius in aperture = distance from the axial field point,Cross-sections of wavefront,In this group of slides, any minus signs are taken up in the constant of propo

30、rtionality,On axis,Half way out,Edge of the field of view,y,x,(C) R G Bingham 2005. All rights reserved.,Spherical aberration and coma wavefronts, W 4 0, W 3 1cos ,Wavefront aberration = relative radius in aperture = distance from centre of field,Edge of field Mean tilt removed,More on aberration pl

31、ots in session 4.,(C) R G Bingham 2005. All rights reserved.,Eight slides of examples,(C) R G Bingham 2005. All rights reserved.,A Cassegrain telescope,The 4.2-metre WHT (William Herschel Telescope) Ex14-WHTelescope.zmx,OPD +/- 10 waves,Transverse rays +/- 500 microns,(C) R G Bingham 2005. All right

32、s reserved.,A Ritchey-Chrtien telescope,The 3.9-metre AAT (Anglo-Australian Telescope) Ex15-AATelescope.zmx,OPD +/- 10 waves,Transverse rays +/- 500 microns,(According to theory the mirrors are not analytically paraboloids.),(C) R G Bingham 2005. All rights reserved.,A telescope as-made,With the as-

33、made dimensions of the mirrors, the focus is still formed at the specified 2500 mm behind the vertex of the primary mirror. Refocusing by moving M2 would have introduced spherical aberration. Most telescopes have a noticeable error here (but not this telescope). In the production of the WHT, the spe

34、cification of M2 was re-computed after the primary mirror was finished to keep the position of the Cassegrain focus correct (zero spherical aberration). The as-made f-number is 10.95, not 11. The as-made aperture diameter is 4180 mm, not 4200.,Do these last two points matter?,Example: the 4.2-metre

35、WHT (William Herschel Telescope) Ex14-WHTelescope.zmx,(C) R G Bingham 2005. All rights reserved.,Off-axis paraboloid (camera or collimator),In the ray-tracing data, we position the vertex of the mirror rather than the point where the axial ray hits it. Also, the normal focal length of the mirror will differ from the required focal length of the off-axis system. This