光学设计例子o

1、(C) R G Bingham 2005. All rights reserved.Optics and Optical DesignSession 2Session 2 Spheres AspheresAberrationsDefocusSpherical aberration Coma ExamplesRichard G. Bingham(C) R G Bingham 2005. All rights reserved.11 pages on spheres and aspheres(C) R G Bingham 2005. All rights reserved.Equation of

2、a sphere as used for ray tracingZzc 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)YThe sphere passing through the origin has to be exp

3、ressed as z = f(r,c) where r 2 = x 2 + y 2r1/c(C) R G Bingham 2005. All rights reserved.1 + (1 c 2 r 2 )Equation of a sphere (2)Zzrc r 2z (r ) =1/cThis exact equation is that used for ray tracing. It has a simple extension for conic sections. It is often also useful for calculating the total depth o

4、f the surface of a spherical lens or mirror.(C) R G Bingham 2005. All rights reserved.Equation of a sphere (3): Taylor seriesz (r ) = c r 2 /2 + c 3 r 4 /8 + c 5 r 6 /16 zrThe osculating parabolaDifference of the sphere and the parabola to order r4. Incidentally, this leads to the Schmidt corrector

5、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 sectionsA reflecting telescope with a parabolic mirror Ex11-Paraboloid.zmxThe refracting surface discovered by Desc

6、artes in 1637Ex12-Ellipsoid_lens.zmxAspheric surfaces are antique but we still struggle to make them!(C) R G Bingham 2005. All rights reserved.The Standard surface and the conic constant1 + (1 (1+k) c 2 r 2 )c r 2z (r) =zrConic Constant kShape of the surfaceEccentricity e of a plane curve 0oblate el

7、lipsoid(A rotated ellipse)0sphere0-1 k 0prolate ellipsoid0 e 1-1paraboloid11e is useful for finding the conjugates Ex10-Conics.zmx(C) R G Bingham 2005. All rights reserved.Osculating sphere and paraboloidz (sphere) = z (paraboloid ) = c r 2 /2rThe difference between the sphere and the paraboloid is

8、r4. There is no difference term in r2. Near the centre, the difference between sphere and paraboloid tends to zero.cr 2/2 is useful for a quick estimate of the depth of a curve even a sphere.zc 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

9、sphereAll the prolate ellipsoids (rugby balls)paraboloidAll the hyperboloids These conicoids 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 el

10、lipsoids (Smarties)rz(C) R G Bingham 2005. All rights reserved.Conjugate foci of ellipse and hyperbolaaaEccentricity e = (-k) = (-8 a4 /c3) VFFVFFFor 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 vertexFor setting

11、up a required curve or for designing an optical test(C) R G Bingham 2005. 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 2z (r ) =The Even Asphere surface (1)zra2 has dimensions L-1, a4 has dimensions L-3 etc. Affects scaling. Odd-power terms are not necessary in a

12、 symetrical optical surface expressed with such a polynomial, 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 de

13、scribe an optical surface with odd terms, those terms are not 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 , o

14、r a2 0. To use both would be redundant at best. k is more visible 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

15、2z (r ) =8c3kr 4 + There are two ways of prescribing:8c3kWhy 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 av

16、ailable with the conic section may be too limited.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

17、 wavefront aberration and defocus(C) R G Bingham 2005. All rights reserved.Wavefront aberration OPD plot what it isThe 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

18、back to the exit pupil (see Welford). Another 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 Bingh

19、am 2005. All rights reserved.WavefrontWavefront aberration - moving the focusz Wz axisuSuppose 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 adv

20、anced. How big is W, and what shape is it in x and y? It is the difference between two spheres.CCDThe focus(C) R G Bingham 2005. All rights reserved. zuCCDW due to defocus onlyThe focusThe aberration is taken as zero for the chief ray. The path lengths of the rays are all equal to each other as far

21、as the focus. Then the 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/u2When the

22、 signal is detected, the 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 + Wrzc1c2The cosine term affects only r4 and above. W The wavefront aberration that corres

23、ponds to a shift of focus shift takes the form of a parabola locally, because: See Ex09-Defoc_Lens.zmxEquation of a sphere see later(C) R G Bingham 2005. All rights reserved.The shape of the defocused wavefront, and the aberration fansOut-of-focus image(C) R G Bingham 2005. All rights reserved.Exerc

24、ise. 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

25、 image 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

26、for 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 focusThese two aberrations are not described by orthogonal basis functions. Fortunately.Spherical aberratio

27、nDefocusBalancedMany 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 from cro

28、ssing 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 thatSpherical aberration Ex13-Sphere.zmx (drop the central obscurations)Coma Ex11-Paraboloid.zmx(C)

29、R G Bingham 2005. All rights reserved.Spherical aberration Coma W 4 0 W y 3 1OPD = Wavefront aberration or y = relative radius in aperture = distance from the axial field pointCross-sections of wavefrontIn this group of slides, any minus signs are taken up in the constant of proportionalityOn axisHa

30、lf way outEdge of the field of viewyx(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 fieldEdge of field Mean tilt removedMore on aberration plots in session 4.(C) R G Bingha

31、m 2005. All rights reserved.Eight slides of examples(C) R G Bingham 2005. All rights reserved.A Cassegrain telescopeThe 4.2-metre WHT (William Herschel Telescope) Ex14-WHTelescope.zmxOPD +/- 10 wavesTransverse rays +/- 500 microns(C) R G Bingham 2005. All rights reserved.A Ritchey-Chrtien telescopeT

32、he 3.9-metre AAT (Anglo-Australian Telescope) Ex15-AATelescope.zmxOPD +/- 10 wavesTransverse rays +/- 500 microns(According to theory the mirrors are not analytically paraboloids.)(C) R G Bingham 2005. All rights reserved.A telescope as-madeWith the as-made dimensions of the mirrors, the focus is st

33、ill 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 specification of M2 was puted after the primary mi

34、rror 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 WHT (William Herschel Telescope) Ex14-WHTelescope.zmx(C)

35、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.