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10.2.1.3. Honders camera
▐
10.2.2.1.
Schmidt camera: aberrations
► 10.2.2. Full-aperture Schmidt corrector: Schmidt camera
PAGE HIGHLIGHTS The simplest arrangement using full-aperture corrector is a camera, with the only other optical element required being a single concave mirror. By far the most popular arrangement is the Schmidt camera.
Back
in 1930, Estonian-born optician Bernhard Schmidt succeeded in designing and
making full-aperture corrector for spherical mirror. It resulted
in a highly corrected optical system, known as Schmidt camera. Somewhat earlier,
in 1924, Finnish
astronomer Y. Väisälä, described similar arrangement, so this type of
camera is sometimes called Schmidt-Väisälä (usually when incorporating
field-flattener). Its concept is based on the unique property of
spherical mirror with the aperture stop at the center of curvature to be
free from off-axis aberrations. The only image aberration remaining is
spherical, and it can be cancelled by appropriately figured lens
corrector, placed at the mirror center of curvature.
The lens element - called Schmidt corrector - has very shallow
aspheric curve calculated to give to the incoming wavefront just the needed
amount of deformation to result in spherical reflected wavefront (FIG.
167).
The only significant aberration induced by the Schmidt corrector is its
corrective spherical aberration, resulting in a system free of four primary
aberrations: spherical, coma, astigmatism and distortion. The only
remaining aberration is field curvature.
FIGURE 167:
Schmidt camera (top) is a simple arrangement with the Schmidt corrector
at the center of
curvature of spherical mirror.
Image surface
is a curved Petzval surface, concentric with the mirror surface, thus of
Ri=R/2
curvature radius.
With the aperture stop at the center of
curvature, but w/o corrector, the only remaining point-image
aberration of the mirror is spherical aberration,
causing reflected wavefront (WF) to
deviate from spherical by bowing inward excessively toward the
edges (spherical undercorrection). It results in the longitudinal
aberration, with rays toward outer zones focusing increasingly
closer to the mirror. Marginal point of the actual wavefront belongs to a sphere centered at
the marginal
focus,
its paraxial points to a sphere centered at the paraxial focus,
and its 0.707 zone point to a sphere centered at the mid-point in
between the two. Axial separation between paraxial and marginal
focus equals longitudinal defocus. For ease of calculation, it is
normalized to 2, zero being at the paraxial focus, and 2 at the
marginal focus. Expectedly, with the Schmidt corrector for primary spherical aberration, there is a direct connection between the Schmidt curve and parabolizing. The most efficient mirror parabolizing method is working the center and the edges of a sphere the most, gradually reducing glass removal to a minimum at the 0.707 zone. This surface modification causes relative advance of the wavefront that culminates at the 0.707 zone, and diminishes to zero at the edge and the center, resulting in a corrected, spherical shape of the wavefront. This same wavefront modification is accomplished by placing the neutral zone at 0.707 radius of the Schmidt corrector (in fact, the curve of change of a spherical surface in parabolizing is of the same type as the curve polished into a Schmidt corrector, only shallower). Consequently, for a given mirror, the theoretical maximum thickness of glass needed to be removed from mirror center and the edge, when parabolizing, is smaller by a factor of (n-1)/2 from the (maximum) Schmidt corrector depth at the 0.707 zone (when corrected for primary spherical alone; adding higher order terms makes corrector slightly deeper, with the edge slightly raised vs. center). This holds true for any corrector/parabola pair with identical final focus location. In either case, the volume of glass needed to remove is in inverse proportion to the 3rd power of relative aperture (F-number) of the corrected surface. Knowing that spherical reflecting surface produces wavefront that in the first term, according to Eq. 4.6, advances away from spherical at a rate of (ρd)4/4R3 with respect to the reference sphere centered at paraxial focus, variation in the Schmidt surface zonal depth z, i.e. surface profile needed for pre-correction of primary spherical aberration that will bring all reflected rays to paraxial focus is z=(ρd)4/4(n-1)R3. This adds the compensatory optical path length (n-1)z=(ρd)4/4R3, which pre-deforms the wavefront, so that it in effect is then corrected by the aberration generated at the mirror. Since the actual wavefront deviation depends on the reference sphere, i.e. specific focus within the aberration longitudinal range to which the rays are to be brought, corrector sag also depends on the normalized defocus Λ, which determines the reference point. The actual wavefront deviation from the reference sphere centered on the chosen zonal focus also varies with the zonal height. Consequently, corrector's depth profile needed for corrected mirror's lower-order spherical aberration also varies with zonal height, as given with the general relation:
where
Λ
is the relative focus location parameter (from
Λ=0 for the
corrected focus coinciding with paraxial focus, to
Λ=2 when
corrected focus coincides with marginal focus), ρ
the height in the pupil normalized to 1, d and D the pupil (aperture)
radius and diameter, respectively, n the glass index of refraction, n'
the index of refraction of the incident/exit media (media next to the
Schmidt surface, normally air, with n'=1 for the rear, and n'=n for the
front), R the mirror
radius of curvature and F the mirror focal ratio for the clear
(corrector) aperture. Corrector's focus parameter
Λ
determines neutral
zone location at the unit radius as NZ=(Λ/2)1/2,
as well as corrector's aspheric coefficient b; the two determine
the needed vertex radius of curvature of the positive central section of the
corrector lens Rc.
Expectedly, the Schmidt surface
profile is effectively the shape of the wavefront deviation given by
Eq. 7, modified by the 1/(n'-n)
medium factor (the profile is opposite in sign to that of the P-V
wavefront deviation for the rear corrector's surface, and of the same
sign for the front surface). The Λ
factor merely determines the amount of defocus with which the paraxial
spherical aberration combines in producing the corresponding wavefront
for a specific point of defocus.
Alternately, Eq. 101 can be written
in terms of the corrector's glass thickness, as t=t1+z,
with t1
being the corrector center thickness (mirror radius R is
numerically negative, and the sign of z is determined by ρ4-Λρ2,
which is always positive for
Λ=0, positive for smaller
values of ρ
and negative for larger ones for 0<ρ<1,
and always negative for
Λ≥1).
The relative depth of corrector's curve,
in units of the maximum corrector depth for
0‹Λ‹2 is given by ρ4-
Λρ2.
According to it, depth of corrector's curve is smallest for
Λ=1, i.e. with the neutral
zone placed at (Λ/2)1/2=0.707
radius (thus ρ=0.707), with the
corrected focus coinciding with best focus location (0.866 radius
neutral zone placement, with the corresponding
Λ
value of 1.5, requires corrector deeper by a factor of 2.25). This
neutral zone position - as it will be explained in more details
ahead -
also minimizes spherochromatism.
Most often, at least one higher-order term is significant and needs to
be corrected as well. In such case, corrector's curve depth
profile can be expressed in terms of its vertex radius of curvature and
aspheric values. With the term for higher-order (secondary) spherical aberration
added, it is given as :
with Rc
being the corrector vertex radius of curvature, b and
b'
the 3rd and 5th order aspheric coefficient (for the transverse ray
aberration; 4th and 6th order on the wavefront), with
A1
and
A2
being the corrector's aspheric parameters for the primery and
secondary spherical aberration, respectively, commonly used in ray
tracing programs.
The first term describes sagitta of the
corrector's radius of curvature which, combined with the aberration
terms (the second is for primary spherical, the third for secondary
spherical, and so on), determines the actual surface profile (FIG.
168, left). It is not an aberration term with respect to
spherical aberration in the optimized wavelength, since it is
corrected for any corrector shape, but it does affect correction of
unoptimized wavelengths, i.e. magnitude of spherochromatism. This
first term, often called radius term, is actually defocus
term: analogous to the aberration terms Ai,
which are the wavefront functions of spherical aberration for
paraxial focus, it represents defocus aberration with which
spherical aberration combines producing altered wavefront specific
to any point of defocus, as a sum of the wavefront errors of defocus
and spherical aberration.
Obviously, since the required surface profile is directly determined
by that of the aberrated wavefront to correct and the medium in
which light travels, this term - representing the defocus P-V
wavefront error - is is also modified by the same 1/(n'-n) factor.
The second and third term are the P-V
wavefront error of primary and secondary spherical at paraxial
focus, respectively, modified by the 1/(n'-n) medium factor. Note
that only the second term - primary spherical - effectively combines
with the defocus (radius) term. Secondary spherical is added only as
the P-V wavefront error at paraxial focus, which means that
secondary spherochromatism is not minimized. Considering usually
small magnitude of secondary spherical, this is negligible; however,
in fabrication it is generally more convenient to use the term for
minimized secondary spherical, with (ρd)6
replaced by (ρ6-ρ2)d6.
It gives a profile very similar to that for the primary spherical at
the best focus (i.e. for Λ=1
and NZ=0.707), which is of the same sign, only much smaller in
magnitude. This means that the profile needs to be only slightly
deeper at the 0.7 zone, with the change in depth diminishing to zero
at the center and the edge, as opposed to having to make the entire
corrector deeper by 2.6 times more (FIG. 168 bottom right) than the required deepening at the
0.7 zone.
Hence the addition of higher-order terms requires
modifying the profile shown on FIG. 167. Since the Schmidt
surface profile is essentially the reversed shape of the
wavefront deviation, only deeper by a factor 1/(n-1),
n being the glass refractive index, the new profile is a reversed
(if on the front surface, same orientation as wavefront if on the
back) stretched out in depth replica
of the wavefront deviation, as illustrated on FIG. 168 right.
BOTTOM: Shapes of the Schmidt profile for correcting primary
spherical at the best focus alone, primary and secondary spherical at
either its paraxial or best focus location, and for zero secondary
spherical coefficient (A6), with the needed amount of primary spherical
added to balance (minimize) secondary spherical.
Schmidt surface for correcting spherical aberration of a conic surface
has all its surface terms of the same sign which, according to Eq.
101.1, implies that correcting the next order term requires deeper
surface profile. More complex forms of spherical aberration, such as,
for instance, correcting balanced higher order forms (Maksutov
corrector, strongly curved refracting objectives, and other) may have
higher surface terms of different numerical sign, where correcting next
higher order term may require shallower curve. In principle, there is no
difference in the effect of aspheric profile whether it is applied to a
flat surface, or radius (the latter merely requires adjustment for the
corrector's radius when entering specs into raytrace).
The Schmidt corrector radius of curvature is given
by:
with the 3rd order aspheric
coefficient b=2/R3,
and n'=1 for the aspheric surface on the back of corrector (ƒ is
the mirror focal length, and F the focal ratio). Optionally, the
relations can be written in terms of the relative neutral zone position
in units of aperture radius, NZ, by substituting Λ=2NZ2. Optimized
for the small effect of corrector's
radius of curvature, 3rd order aspheric
coefficient is:
with F being the mirror
F-number (F=-R/2D). The 5th order aspheric coefficient
b'=6/R5.
The two aspheric parameters
A1
and
A2 determine the Schmidt corrector shape,
according to Eq. 101.1. From the equation, they are obtained from
their respective aspheric coefficients b and b', as
A1 = b/8(n'-n)
= 1/4(n'-n)R3
(104) and
The two aspheric
coefficients, b and b', are obtained by setting the
system aberration coefficients for 3rd and 5th order spherical aberration
to zero, s3=-b/8
+ [1-(Λ/16F2)]/4R3 =
0 and s5=-b'/16
+ 3/8R5 =
0 with the left side of the coefficient (b
factor) being the corrector aberration contribution, and the right side
that of the mirror. The 4th and 6th order system P-V wavefront error at the paraxial focus are
W3=s3d4
and W5=s5d6,
respectively.
The slightly lower 3rd order mirror
coefficient results from its effective relative aperture slightly reduced for non-zero values of corrector's focus parameter
Λ (in effect, the higher
Λ, the more diverging
outer rays falling onto mirror, reducing spherical aberration). A non-zero
paraxial radius term Rc
makes the corrector a weak positive lens with aspheric figure, also
determining neutral zone position for given value of the aspheric
coefficient b. The neutral zone location is also given directly,
for unit radius, as NZ=(Λ/2)1/2.
The significance of the 5th order term is in correction
of the higher-order
spherical aberration (5th order transverse ray, 6th order on the wavefront). Those
include axial spherical, as well as oblique (lateral) spherical,
and wings, the higher-order astigmatism as it was named by
Schwarzschild. They both increase with the square of off-axis height in
the image space, and set the limit to field quality. The latter has the
P-V error larger by a factor of 4n, n being the glass refractive;
since it varies with cosθ, θ being the pupil angle, the off-axis
aberration in the Schmidt camera peaks along the tangential plane (the
one determined by the
chief ray and optical axis, for which θ=0 and cosθ=1).
For 200mm ƒ/2 Schmidt camera, the amount of
higher-order spherical aberration is ~0.24 wave RMS. It can be minimized
by balancing it with the lower-order form of opposite sign (by making
the 4th order curve slightly stronger). The residual that can't be corrected with the
3rd order surface term alone is ~0.04 wave RMS.
EXAMPLE: 200mm
ƒ/2 Schmidt
camera with BK7 corrector (n=1.5185 for 550nm wavelength), thus
clear aperture radius
d=100
at the corrector, and mirror radius of curvature R=-800.
Choosing for the corrected focus best focus location of the mirror, thus Λ=1,
determines neutral zone height NZ=(Λ/2)1/2ρmax,
at 0.707d. Only the rear side is aspherized.
From Eq. 103, corrector's lower-order aspheric
coefficient b=2[1-(Λ/16F2)]/R3=-0.000000003845,
or
b=-3.845-9,
determining the lower-order aspheric parameter of the corrector
as A1=b/8(n'-n)=9.27-10,
with the index of refraction of the exit media (air for the Schmidt
surface at the back of corrector) n'=1.
Higher-order aspheric coefficient b'=6/R5=-1.83-14
determines the higher-order corrector aspheric parameter A2=b'/16(n'-n)=2.21-15.
Needed radius of curvature of the
corrector is Rc=-1/2ΛA1d2=-53,940mm.
With the corrector at the mirror
center of curvature, the system is corrected for 3rd/4th i.e. 5th/6th order
spherical aberration (3rd and 5th transverse ray aberration,
corresponding to 4th and 6th order on the
wavefront), coma, astigmatism and distortion. The only remaining
aberration is field curvature, rc=R/2=-400mm.
For double-sided corrector, both b and
A coefficients
are half their value for single-sided corrector, with the A coefficients
being of the opposite sign on the other side (as determined by n'-n)..
Since, from Eq. 104/104.1, b=8(n'-n)A1
and b'=16(n'-n)A2,
the P-V wavefront error at the best focus resulting from deviations ΔA1
and ΔA2
in the two aspheric parameters is given by W4=(n'-n)ΔA1d4/4
and W6=0.42(n'-n)ΔA2d6
for 4th and 6th order spherical aberration, respectively. Taking
0.0001375mm (1/4 wave at 0.00055mm wavelength) for W4
gives, for the above system, the corresponding lower-order parameter
deviation as ΔA1=4W4/(n'-n)d4=1.06-11,
with 1/4 wave of spherical aberration figure tolerance for the
lower-order aberration of 1.06-11(ρd)4.
At the maximum curve depth (ρ=0.707), it is 0.000265mm, or 0.48 wave.
Follows more detailed account of the
Schmidt camera aberrations.
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