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10.2.1.3. Honders camera
▐
10.2.2.1.
Schmidt camera: aberrations
► 10.2.2. Fullaperture Schmidt corrector: Schmidt camera
PAGE HIGHLIGHTS The simplest arrangement using fullaperture 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, Estonianborn optician Bernhard Schmidt succeeded in designing and making fullaperture corrector for spherical mirror. It resulted in a highly corrected optical system, known as Schmidt camera. Somewhat earlier, in 1924, Finnish astronomer Y. Vaisala, described similar arrangement, so this type of camera is sometimes called SchmidtVaisala (usually when incorporating fieldflattener). Its concept is based on the unique property of spherical mirror with the aperture stop at the center of curvature to be free from offaxis 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 pointimage
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 midpoint 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 (n1)/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 (Fnumber) 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 precorrection of primary spherical aberration that will bring all reflected rays to paraxial focus is z=(ρd)4/4(n1)R3. This adds to every wavefront point the compensatory optical path length (n1)z=(ρd)4/4R3, which predeforms the wavefront, so that it in effect gets corrected by the aberration generated at the mirror. Since the actual wavefront deviation depends on the reference sphere used, i.e. specific focus within the aberration longitudinal range to which the rays are to be brought, which is center of curvature of reference sphere, 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 (left, grossly exaggerated) needed for corrected mirror's lowerorder 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 for pupil radius, 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 PV 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 ρ4Λρ2, 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 higherorder 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 higherorder (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 A_{1} term  the primary spherical aberration term  is, from Eq. 4.6 directly related to the conic K as A_{1}=(1+K)/8R^{3}, R being the mirror radius of curvature (if starting surface is a sphere, A_{1}=K/8R^{3}, as it expresses change in the sagitta depth, and for K=1 it equals the differential between sphere and paraboloid, for given vertex radius). Note that the first term, sometimes referred to as a_{2} (with the next being a_{4}, a_{6} and so on) is in the parabolic form because d/R_{c} is negligibly small in the full expression for the first term given by a_{2}=d^{2}/R{1+[1(1+K)(d/R)^{2}]^{1/2}}, with K being the surface conic. 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 PV wavefront error  is is also modified by the same 1/(n'n) factor. The second and third term are the PV 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 PV 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 (ρ6ρ2)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 higherorder 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/(n1), 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 (� 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 Fnumber (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 A2 = b'/16(n'n) = 3/8(n'n)R5 (104.1) 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 PV 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 nonzero values of corrector's focus parameter Λ (in effect, the higher Λ, the more diverging outer rays falling onto mirror, reducing spherical aberration). A nonzero 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 higherorder spherical aberration (5th order transverse ray, 6th order on the wavefront). Those include axial spherical, as well as oblique (lateral) spherical, and wings, the higherorder astigmatism as it was named by Schwarzschild. They both increase with the square of offaxis height in the image space, and set the limit to field quality. The latter has the PV error larger by a factor of 4n, n being the glass refractive; since it varies with cosθ, θ being the pupil angle, the offaxis 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 f/2 Schmidt camera, the amount of
higherorder spherical aberration is ~0.24 wave RMS. It can be minimized
by balancing it with the lowerorder 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 f/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 lowerorder aspheric coefficient b=2[1(Λ/16F2)]/R3=0.000000003845, or b=3.8459, determining the lowerorder aspheric parameter of the corrector as A1=b/8(n'n)=9.2710, with the index of refraction of the exit media (air for the Schmidt surface at the back of corrector) n'=1. Higherorder aspheric coefficient b'=6/R5=1.8314 determines the higherorder corrector aspheric parameter A2=b'/16(n'n) =2.2115. Needed radius of curvature of the corrector is Rc=1/2ΛA 1d2=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 doublesided corrector, both b and A coefficients are half their value for singlesided 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 PV 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 lowerorder parameter deviation as ΔA1=4W4/(n'n)d4=1.0611, with 1/4 wave of spherical aberration figure tolerance for the lowerorder aberration of 1.0611(ρd)4. At the maximum curve depth (ρ=0.707), it is 0.000265mm, or 0.48 wave. Aspheric coefficients numerical conversion when switching to a different unit is given as A'=A(u_{2}/u_{1})^{x1}, where u_{2} is expressued in units of u_{1}, with u_{i} being the units, and x the coefficient order. So, for example, converting from milimeters (u_{1}) to inches (u_{2}), the 4th order coefficient is larger by a factor 25.4^{3}, 6th order by a factor 25.4^{5}, and so on (applies when system remains identical physically, only the measuring unit changes). Follows more detailed account of the Schmidt camera aberrations.
