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9.3. Designing doublet achromat
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10.1.2. Sub-aperture correctors for
a single mirror
► 10. CATADIOPTRIC TELESCOPES
PAGE HIGHLIGHTS As telescopes evolved, it has been discovered that performance of all-reflecting 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 full-aperture corrector, or in the converging cone produced by mirror element(s), when it is a sub-aperture 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 field-corrector, which can function without its refractive component. Following this general dividing line, catadioptric telescopes are described within their two main groups, those with sub-aperture and those with full-aperture 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 full-aperture 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 full-aperture corrector is a factor that can affect the aberrations of mirror surfaces following it in the optical train.
On the other hand, sub-aperture 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 sub-aperture corrector
different - generally more complex - than those for full-aperture
corrector. It is illustrated by aberration relations for sub-aperture
corrector lens element that follow. 10.1. Catadioptric telescopes with Sub-aperture correctors10.1.1. Sub-aperture corrector aberrationsUnlike telescope objectives and most full-aperture corrector arrangements, sub-aperture 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 sub-aperture corrector. In a two-mirror 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 off-axis 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 off-axis aberration of sub-aperture 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 two-mirror 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 point-image quality aberrations, spherical, coma and astigmatism, as well as field curvature, are affected and inter-related at a sub-aperture lens corrector.
Axial correction
Aberration coefficient for
lower-order
(primary) spherical aberration of
sub-corrector lens element is the same as for a single lens element in
general:
with n being the
refractive index, q=(R2+R1)/(R2-R1)
the lens shape factor, and p=(2ƒ/O)-1=1-2ƒ/I the
lens position factor (ƒ
is the lens focal length, O the object distance and I the
image-lens 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
P-V 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.
Off-axis aberrations
For lower-order (or primary) coma, the thin lens
aberration coefficient with the stop at the surface is given by
Eq.14.1, with the P-V
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 off-axis
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 off-axis 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,
for coma,
for astigmatism
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 sub-aperture correctors when the
cone width is large relative to lens thickness, δ2
can be omitted if the resulting error is small. When calculating the P-V
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
summing 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 = C0-4yS0 (98)
where C0
and S0
are the peak aberration coefficients for coma and spherical aberration with
the stop at the lens, respectively, and
the relative pupil coordinate shift caused by
the displaced stop, with T being the stop-to-surface 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
aperture
radius.
Aberration coefficient of
lower-order
(primary) astigmatism for a single thin lens with the stop at the
surface is given by:
with the P-V 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:
Change in astigmatism causes change in the best image
field curvature, which is now, also as
the peak aberration coefficient, given by:
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, sub-aperture correctors do
have inherent (corrective) aberrations, and normally require use of
stop-shift relations to calculate their off-axis aberrations.
Aberration coefficients for sub-aperture 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 lower-order aberrations; that limits their usefulness for
assessing aberrations of sub-aperture correctors, which commonly have
significant higher-order aberration component.
A form of the sub-aperture 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 lower-order 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 plate-to-focus 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 lower-order aspheric coefficient is found from
-(b/8)+ss=0,
where ss
is the aberration coefficient of lower-order spherical aberration for
the rest of the system. The stop separation T equals the
mirror-plate separation (numerically negative) in a single-mirror system with the stop at the
surface. In two-mirror systems, T=E-Δƒ,
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 stop-to-plate 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 near-contact 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 near-collimated beam.
For sub-aperture
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 sub-aperture 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=(nm-1)/(n1-n2).
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 wavelength-dependent 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~-(ƒ-ic),
ic
being the corrector-to-image separation.
Needed lens separation for a single-glass sub-aperture
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 non-zero 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 spacing, lens thicknesses
or lens power rearrangement (if possible) should be tried, before new glass
combination, with different partial dispersion, is attempted
(5) higher-order aberrations are often significant; if so, their sum of
coefficient for all surfaces needs to be balanced with that of the
corresponding lower-order form of opposite sign to be minimized; for
spherical aberration, the two amounts are similar, absolutely, while for
coma and astigmatism the lower-order form needs to be nearly twice
higher (that may change if both - i.e. all four - are present); non-zero
Petzval surface coefficient is compensated by about half as high
lower-order astigmatism coefficient of opposite sign
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