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9.3. Designing doublet achromat
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10.1.2. Subaperture correctors for
a single mirror
► 10. CATADIOPTRIC TELESCOPES As telescopes evolved, it has been discovered that performance of allreflecting systems  especially those with more strongly curved surfaces  can be improved if mirrors are combined with lens elements. General term for telescope systems using both, reflective and refractive elements in order to form the image is catadioptric. Main working principle of a catadioptric telescope is that aberrations of reflecting and refracting elements cancel each other (kata=against, dioptric=refractive). Lens element(s) used in combination with mirrors can be either placed in the path of incoming light, when it is a fullaperture corrector, or in the converging cone produced by mirror element(s), when it is a subaperture corrector. Strictly talking, catadioptric telescope is designed as a synergy of reflective and refractive elements, requiring both for its functioning. Such design is sometimes called true catadioptric. That separates it from hybrid catadioptric, usually a mirror telescope using refractive fieldcorrector, which can function without its refractive component. Following this general dividing line, catadioptric telescopes are described within their two main groups, those with subaperture and those with fullaperture lens corrector. The former can be either true or hybrid catadioptric systems, while the latter are usually "true catadioptrics", requiring the lens element for functioning. Aberrations of a fullaperture refractive corrector nearly coinciding with the aperture stop  which is the usual scenario  are those of a lens for object at infinity. Also, aperture stop usually shifted from primary mirror to the fullaperture corrector is a factor that can affect the aberrations of mirror surfaces following it in the optical train.
On the other hand, subaperture lens corrector, usually placed in the light
converging after reflection(s) from mirror element(s), doesn't affect the aberrations of
mirror surface(s). Its object is the image formed by preceding
mirror element, and its aperture stop is the exit pupil formed by this
element. This makes aberration relations for subaperture corrector
different  generally more complex  than those for fullaperture
corrector. It is illustrated by aberration relations for subaperture
corrector lens element that follow. 10.1. Catadioptric telescopes with Subaperture correctors10.1.1. Subaperture corrector aberrationsUnlike telescope objectives and most fullaperture corrector arrangements, subaperture correctors are usually positioned in a converging light cone. In other words, these lens correctors are preceded by one or more optical surfaces. Consequently, their aperture stop is located at the exit pupil formed by the preceding element. In a Newtonian both, entrance and exit pupil coincide at the primary mirror's surface, which is also the stop position for subaperture corrector. In a twomirror system, the stop is at the image of primary formed by the secondary. Stop shift doesn't affect axial cone width or position at the corrector, thus the expression for spherical aberration remains unchanged. For offaxis cones, stop shift causes the chief ray to shift off the center of corrector's front surface. In other words, converging wavefront passes through different portion of the corrector than in the case when the stop is at the front surface. As a result, expressions for offaxis aberration of subaperture corrector are different  generally more complex  than those for the lens objective with the stop at the surface. This also applies to the secondary mirror in twomirror systems. The difference is that lens correctors have more surfaces  commonly four or more  each contributing its own aberrations. Consequently, the appropriate aberration calculation is significantly more complex. For that reason, the consideration here will be limited to a general form of aberration relations for a single lens, showing how the three pointimage quality aberrations, spherical, coma and astigmatism, as well as field curvature, are affected and interrelated at a subaperture lens corrector. Axial correction Aberration coefficient for lowerorder (primary) spherical aberration of subcorrector lens element is the same as for a single lens element in general:
with n being the refractive index, q=(R2+R1)/(R2R1) the lens shape factor, and p=(2ƒ/O)1=12ƒ/I the lens position factor (ƒ is the lens focal length, O the object distance and I the imagelens separation). Obviously, only the value of p is different from those for object at infinity. The peak aberration coefficient, which with primary spherical aberration equals the PV wavefront error at paraxial focus, is S=W=sD4/16, D being the aperture diameter (the wavefront error at the best focus is smaller by a factor of 0.25, or W=sD4/64). For the front lens, the object is the image formed by the preceding mirror element (or elements), while the object for the rear lens is image formed by the front lens. The combined error varies with the lens separation, which directly determines relative aperture of the rear lens, as well as its object separation.
Spherical aberration coefficient for
doublet lens is a combined
value of the two elements' coefficients sd=s1+s2,
with the final system coefficient being ss=sm+sd,
with sm
being the mirror aberration coefficient. This assumes nearly identical
marginal ray height d of the axial cone on the two elements, when
the difference in (d1/d2)4
is negligible, i.e. d1~d2; otherwise, the coefficient 2 needs to be corrected by (d2/d1)4
before the addition (assuming that d1
is used to calculate peak aberration coefficient). Another option is to
calculate peak aberration coefficient for each element, and then
sum them up. Offaxis aberrations For lowerorder (or primary) coma, the thin lens aberration coefficient with the stop at the surface is given by Eq.14.1, with the PV wavefront error at Gaussian focus is W=2C=cαd3, and at the best focus W=2C/3=cαD3/12, with C0=cαd3=cαD3/8 being the peak aberration coefficient for coma with the stop at the surface, α being the lens image point field angle (the RMS wavefront error is ω=W/√32), and p and q, as before, the lens position and shape factor, respectively. A doublet coma coefficient is a sum of its two elements' coefficients, with the final system coefficient being a sum of the coefficients for the corrector and mirror elements. When the stop is displaced from the front lens, the chief  or central  ray for an offaxis point shifts off the lens center. This shift of incident rays relative to the lens surface changes their optical path lengths relative to each other, changing by that offaxis aberrations, as illustrated below (in terms of the wavefront, the stop effectively directs the section of the wavefront entering through it to a different portion of the lens, at somewhat different angle of incidence).
Based on Eq. (k)(m), lens aberration coefficients for primary spherical aberration, coma and astigmatism with the stop displaced from the surface, assuming spherical lens surface (Q=0), as a sum of the aberration coefficients for the two lens surfaces combined, are: for primary spherical aberration, sdL = 0.125(N1J12 + N2J22δ24) for coma, cdL = 0.5(N1J1Y1h1 + N2J2Y2h2δ23) for astigmatism adL = 0.5(N1Y12h12 + N2Y22h22δ22) where δ2=d2/d1 is the ratio of the effective aperture radius (i.e. height of the marginal ray of the axial cone) at the rear. vs. front surface, with the subscript L for lens (rest of parameters are unchanged). For, d2~d1, which is usually the case with subaperture correctors when the cone width is large relative to lens thickness, δ2 can be omitted if the resulting error is small. When calculating the PV wavefront error, it is usually sufficiently accurate to use the d1 value for d in Table 2, rightmost. The exact result would require calculating peak aberration coefficient for each surface, and sum them up. Obviously, even these simplified expressions are still fairly extensive. For gaining the insight into main characteristics of lens aberrations for displaced stop, it is more practical to use expressions in terms of peak aberration coefficients which, at Gaussian focus, can be expressed as S=sd4, C=cαd3 and A=a(αd)2, for primary spherical aberration, coma and astigmatism respectively (for r=1 and t=0). The coma peak aberration coefficient takes the form: C = C04yS0 (98) where C0 and S0 are the peak aberration coefficients for coma and spherical aberration with the stop at the lens, respectively, and y = Tα/r (98.1) the relative pupil coordinate shift caused by the displaced stop, with T being the stoptosurface separation  numerically positive when the stop is to the left from the lens  α=iα0/(i+T) the new chief ray angle, with i being the lens to image separation, and α0=h/i the chief ray angle with the stop at the lens (with h being the linear height in the image plane), and r the stop radius. Aberration coefficient of lowerorder (primary) astigmatism for a single thin lens with the stop at the surface is given by: a = 1/2f (99) with the PV wavefront error at both, Gaussian and best focus, given by W=A=aα2D2/4. Doublet coefficient is a sum of its element's coefficients, and the system coefficient is a sum of the coefficient for the corrector and mirror elements. Similarly to coma, when the stop is displaced from the lens, the peak aberration coefficient, which is for the stop at the surface given by A0=a(αd)2=a(αD)2/4, becomes: A = A02yC0+4yS0 (99.1) Change in astigmatism causes change in best image field curvature, which is now, also as the peak aberration coefficient, given by: U = A0yC0+2yS0 (99.2) Relations for displaced stop  called stop shift relations  apply to optical elements in general, including mirror and lens objectives. They show that change in the stop position doesn't affect spherical aberration, but that uncorrected spherical aberration does affect both coma and astigmatism, with the latter also being affected by uncorrected coma. On the other hand, aplanatic systems (corrected for spherical aberration and coma) are unaffected by stop position. By the virtue of their function, subaperture correctors do have inherent (corrective) aberrations, and normally require use of stopshift relations to calculate their offaxis aberrations. Aberration coefficients for subaperture corrector can be also calculated as a sum of aberration coefficients for each surface, as outlined with field flattener lens example. It is even more time consuming and, as the above relations, only covers lowerorder aberrations; that limits their usefulness for assessing aberrations of subaperture correctors, which commonly have significant higherorder aberration component.
A form of the subaperture corrector with the simplest aberration
expressions is a single Schmidt plate
in a converging cone, usually at some distance in front of the final
focus.
The aberration coefficients for lowerorder spherical aberration, coma
and astigmatism are s=bΔ4/8,
c=bTΔ3/2
and a=bT2Δ2/2,
respectively, where b is the
aspheric coefficient, Δ is
the relative platetofocus separation in units of the system focal length, and
T is the stop (exit pupil) separation for the plate. As with the
Schmidt camera, the lowerorder aspheric coefficient is found from
(b/8)+ss=0,
where ss
is the aberration coefficient of lowerorder spherical aberration for
the rest of the system. The stop separation T equals the
mirrorplate separation (numerically negative) in a singlemirror system with the stop at the
surface. In twomirror systems, T=EΔƒ,
where E is the
exit pupil
(i.e. image of the primary formed by the secondary) separation, and
ƒ is the system focal length (note that both E and
Δƒ are numerically positive, as is
the stoptoplate separation T for Δƒ<E). Chromatic aberration Chromatic aberration of a doublet corrector can be longitudinal (secondary spectrum) and lateral chromatism. As it is the case for a doublet in general, only one of the two forms can be cancelled, but the overall chromatic error can be greatly reduced by a proper choice of elements' properties (both longitudinal and lateral chromatism cancelled require each of the two elements achromatized). For a contact, or nearcontact doublet corrector, achromatizing requirements are, in first approximation, identical to those for a doublet objective. Optical power of either lens, i.e. their power of refraction, remains unchanged regardless of the object distance (i.e. steepness of the converging/diverging incident cone), as long as they satisfy the thin lens conditions. However, in a steeply diverging or, more commonly, converging cone, the height differential of refracted ray on successive lens surfaces becomes significant, giving it properties of a thick lens, even if the same lenses would act as thin lenses for nearcollimated beam.
For subaperture corrector with separated elements, with the separation t defined as that between the back surfaces of the front and rear lens, the combined focal length is given by
with the required lens separation for achromatism given by:
where ƒ1, ƒ2 and V1, V2 are the front and rear element focal length and Abbe number, respectively. Since in this form the relation apply for object at infinity (collimated incident light at the front lens), for object at a finite distance  most common scenario with subaperture corrector, where the object is the image formed by the optical element preceding corrector  ƒ1 is replaced by image separation for the first lens, obtained from the Gaussian lens formula. Achromatizing is defined as bringing two widely separate wavelengths to a common focus, thus fulfilling this condition will achromatize for selected wavelengths 1 and 2, provided that the V values used for Eq. 100.1 are those for which the refractive index is the mean value of the respective lens indici: nm=(n1+n2)/2. Consequently, the V number for each lens is V=(nm1)/(n1n2). The remaining longitudinal color error is reduced to the displacement of the first and second principal point of the doublet (even if consisting of a pair of thin lenses, a doublet system has properties of a thick lens), given by:
with M being the wavelengthdependent transverse corrector's image magnification. Longitudinal color error ∆l is zero for M2=V2ƒ2/V1ƒ1. With the separation t satisfying Eq. 100.1, lateral color error reduces to:
with L being the exit pupil to (final) image separation, approximated by L~(ƒic), ic being the correctortoimage separation. Needed lens separation for a singleglass subaperture doublet corrector (V1=V2=V) is, from Eq. 100.1, given by t=(ƒ1+ƒ2)/2, with its longitudinal chromatic error reduced to ∆l=(ƒ2+ƒ1M2)/V and the lateral error reduced to ∆h=(ƒ1M/V)∆l/L. For zero longitudinal chromatism, the lens elements' focal lengths relate as ƒ2=ƒ1[1(t/ƒ1)]2, due to the back focal distance becoming independent of the wavelength. Obviously, it implies that the lenses are of opposite powers, with the negative lens stronger for t>0. However, the focal length variation ∆l is not zero, resulting in nonzero lateral chromatism. In general, longitudinal and lateral chromatism in a separated doublet cannot be simultaneously cancelled, unless both lenses are achromatic themselves. Usually, the priority is cancelling longitudinal chromatism, while keeping lateral color low. Using raytrace programs in designing lens correctors As the above introduction implies, determining needed parameters of subaperture correctors is rather involved procedure. Fortunately, modern raytracing programs, such as OSLO, can make it much easier, especially with simpler corrector types. While basic knowledge of optics is still needed, following basic steps often can lead to a successful design: (1) placing the elements in the initial position, and determining needed power to nearly cancel secondary spectrum (longitudinal chromatism); (2) looking at the aberration coefficients for each surface, and their sum, lens surfaces are varied to minimize or cancel combined aberrations ("lens bending"), while preserving individual lens powers (i.e. longitudinal chromatic correction); in order to preserve lens power, the sum of its radii given as (1/R1)+(1/R2) for bi and plano lenses and (1/R1)(1/R2) for menisci, needs to remain constant (for simplicity, radii are given as absolute values) (3) if aberrations cannot be corrected at that particular corrector location, it should be assessed how moving it farther or closer from the focus affects the aberrations (4) if aberrations cannot be corrected, it is likely that different lens arrangement is needed; if monochromatic aberrations are corrected but lateral color remains significant, different spacings, lens thicknesses or lens power rearrangement (if possible) is attempted, before new glass combination, with different partial dispersion, is attempted (5) higherorder aberrations are often significant; if so, their sum of coefficient for all surfaces needs to be balanced with that of the corresponding lowerorder form of opposite sign to be minimized; for spherical aberration, the two amounts are similar, absolutely, while for coma and astigmatism the lowerorder form needs to be nearly twice higher (that may change if both  i.e. all four  are present); nonzero Petzval surface coefficient is compensated by about half as high lowerorder astigmatism coefficient of opposite sign
