10.2.3. Full-aperture meniscus corrector: Bowers/Maksutov
Around 1941 Anton Bowers and Dmitry Maksutov (according to Rutten and Venrooij, also Gabor and Penning) separately arrived at a similar idea on how to minimize aberrations of a spherical mirror. They proposed placing concentric, or near-concentric meniscus lens at some distance in front of the mirror. If positioned properly, and with needed optical properties, such meniscus could eliminate off-axis aberrations of the mirror, and greatly reduce its spherical aberration.
For the concentric meniscus form, the distance to a mirror is determined from the concentric configuration needed for cancellation of off-axis aberrations (FIG. 181). Obviously, the meniscus radii are related to the center thickness t as t=R1-R2, for the front and rear radius, respectively, with both radii being nominally negative. Since the corrector is a weak negative lens, the resulting combined focus after reflection from the mirror is formed somewhat farther away from the mirror focus. However, the effective nominal focal length is somewhat smaller than mirror focal length, due to the wavefront having larger diameter at the primary than at the corrector (aperture stop).
FIGURE 181: Concentric meniscus corrector of a thickness t. All three (spherical) optical surfaces and the Petzval image surface have a common center of curvature in C, coinciding with the center of the aperture stop S. As a result, from the position of the aperture stop, the system has no unique axis, which makes it free from off-axis aberrations. The condition for a concentric lens is t=R1-R2, with both lens radii negative (concave to the left). The drawback of the concentric meniscus is relatively large chromatism (secondary spectrum), making it unsuitable for use in fast camera systems.
Concentric meniscus' second principal plane coincides with its center of curvature. Consequently, while it is a thick lens, it can be considered to be a thin lens of the focal length ƒ~ƒL-R1 (with ƒL being the focal length of concentric meniscus derived from the thick lens f.l. relation, ƒL=nR1R2/t(n-1), and R1 the front surface radius of curvature), placed at the center of curvature of its two surfaces. With R1«ƒL, the aberration relations for a thin lens of the focal length ƒ~ƒL can be applied with ~1% error (negative) in the value of the spherical aberration coefficient. By substituting p=-1 and q in terms of ƒL and R1 into Eq. 49, the aberration coefficient for 3rd order spherical aberration for the meniscus is closely approximated by:
n being the lens refractive index. With R1«ƒL, it can be simplified to
In order to cancel spherical aberration of the mirror, the meniscus needs to generate identical amount of the aberration, but of the opposite sign (over-corrected). With the mirror aberration coefficient for primary spherical aberration sP=1/4R3 (for spherical mirror), R being the mirror radius of curvature, the aberration coefficient for the combination is given by a sum of the two, and the P-V wavefront error of 3rd order spherical aberration at best focus, after substitution for the meniscus' focal length ƒL=-nR1R2/(n-1)t, and multiplying with (D/2)4 divided by 4, is closely approximated by:
with R2 being the rear meniscus surface radius of curvature, t the corrector center thickness and D the aperture diameter. Obviously, zero system aberration requires zero sum in the brackets which, in turn, for given index of refraction n, requires appropriate values for the three radii and meniscus thickness.
Since focal length of the meniscus - which is a week negative lens - changes with the wavelength, so does the system focal length, ƒs=[1/(1-ƒm/ƒL]ƒm, resulting in longitudinal chromatism (ƒm is the mirror focal length, with all three focal lengths being numerically negative), given by LCs=ιtƒ2/R1R2n2 or, with the system focal length ƒs~R/2, closely approximated by:
with ι being the index differential. Compared to the Schmidt arrangement, in which LCs= ιD/32(n-1)F, typical concentric meniscus corrector (t~D/10, R1~0.2R and R1~R2) has longitudinal chromatism greater by a factor of ~20(n-1)F/n2. For an ƒ/3 camera and n~1.5, concentric meniscus would already have over 13 times greater longitudinal chromatism than comparable Schmidt corrector. Since chromatism of the Schmidt camera itself may become excessive at large relative apertures, it is much more of a problem for a concentric meniscus.
One solution to the excessive secondary spectrum is to make concentric meniscus out of two different glasses cemented together.
Another solution, found by Maksutov, is to make the focal length invariant of the wavelength. This requires change in the meniscus radii/thickness relationship to:
The radii are no longer concentric, hence this achromatic Maksutov corrector is also called non-concentric.
For n~1.5, this makes the achromatic meniscus corrector thicker nearly by a factor of two. Also, for equal thickness/radii, the meniscus focal length is now longer by a factor of n/(n-1). With the radii not being concentric anymore, relatively low, but not quite negligible off-axis aberrations are reintroduced. However, they are easily dealt with by minor adjustments of the stop (corrector) location.
This meniscus modification practically eliminates secondary spectrum. Remaining spherochromatism - caused by variations of spherical aberration with the wavelength - is reduced to negligible by bringing best foci of different wavelengths close together, while widening paraxial foci separation (FIG. 183, b and c). This requires slight correction of the rear radius, which is made slightly weaker than indicated by Eq. 124: R2=R1-[1-(1/n2)]t, which brings paraxial foci close together is replaced by R2=R1-[1-(1/n2)]t/0.97 (note that both R1 and R2 are numerically negative, while the thickness t is of positive sign).
More serious problem of the achromatized meniscus - also shared by the concentric version - is the amount of higher order spherical aberration. Due to very strongly curved lens surfaces, higher-order spherical aberration becomes rather severe at large relative apertures. By combining the residual 3rd and 5th order aberrations of opposite signs, the final aberration can be minimized (FIG. 182), but remains significant to unacceptable at large relative apertures.
FIGURE 182: Spherical aberration and chromatism of non-concentric (achromatic) meniscus corrector over the visual spectral range. To the left, the longitudinal aberration plot centered at best focus location. Focal separation for different wavelengths is small in general, and in particular for the outer zones, nearly cancelling sphero-chromatic error. Most of the blur size for any wavelength comes from the residual spherical aberration that can't be eliminated in an all-spherical arrangement. The comparable Schmidt camera ray spots (right) are noticeably smaller, especially for the optimized wavelength. The wavefront error even at the optimized e-line (546nm) is ~0.1 wave RMS in the Maksutov, while entirely negligible in the Schmidt. In the violet h-line (405nm), where the sphero-chromatic error in the Schmidt peaks at ~0.1 wave RMS, the mostly spherical aberration error in the Maksutov is still higher, at ~0.15 wave RMS. Also, some lateral color (LC) is noticeable with the Maksutov. SPEC'S: MC SC
While the achromatic meniscus has somewhat different properties, the system P-V wavefront error of 3rd order spherical aberration for the achromatized meniscus and a mirror still can be approximated by Eq. 120, providing the focal length of the achromatic meniscus is used. In terms of the system aberration coefficient, after substituting for the achromatic meniscus' focal length ƒL=-n2R1R2/(n-1)2t into Eq. 121, the system error given by the sum of the errors of the meniscus and the mirror, respectively, can be approximated as:
As mentioned before, the peak aberration coefficient S=sD4/16 equals the wavefront error at paraxial focus, or four times the error at best (diffraction) focus.
Off-axis aberrations of the Maksutov system are somewhat less complex. Being of lesser importance than the axial correction, they can be characterized using simpler (although less accurate) approximations.
Lower-order coma of the meniscus mainly depends on its focal length and the shape factor q, which are determined by mirror properties. In effect, meniscus' coma is nearly constant at ~0.3 of mirror's coma (stop at the surface), only of opposite sign (it varies within several percent with significant changes in the meniscus' thickness).
Omitting relatively insignificant role of the position factor p and variations in the refractive index, coma aberration coefficient of the meniscus can be approximated by cL~-1.2q/ƒL2, with q being the shape factor (R1+R2)/(R1-R2) and ƒL the meniscus focal length. The P-V wavefront error is Wc=cαD3/12, α being the field angle. Since zero system coma requires contributions from the meniscus and the primary to offset one another, that is, the meniscus coma coefficient cL=-(1-σ)/R2, the appropriate stop (corrector) position is closely approximated by σ~1-q(R/ƒL)2. As before, σ is the corrector-to-primary separation in units of the primary radius of curvature, positive in sign.
Using the above empirical figure of the meniscus' coma as ~0.3 times the coma of the primary with the stop at the surface, the zero-coma stop position is approximated by σ~0.7. Higher-order coma of the corrector is not quite insignificant at ƒ/4 already, and grows rapidly with further increase in the focal ratio. It is also numerically negative (opposite to that of the mirror) which makes zero-coma separation smaller: at ~ƒ/2.5 the higher-order coma of the corrector is nearly 10% of that of the mirror, resulting in the zero-coma corrector position reduced from ~0.7R (for slower systems with insignificant higher-order coma) to ~0.6R.
Obviously, for any given thickness, q will increase with the nominal increase of the radii, but the focal length will increase with the square of it, and the coma coefficient effectively decreases with the cube of the focal length increase. On the other hand, mirror coma coefficient, given by 1/R2, is inversely proportional to the square of mirror radius of curvature. From Eq. 121.1 it is evident that in balancing correction of spherical aberration the mirror radius of curvature R has to change at a faster rate than meniscus radii. This makes the rate of change in the amount of coma contributed to a system similar for both, mirror and its meniscus corrector, resulting in a relatively insignificant change of relative amounts of the aberration contributed by the two for a range of mirror focal ratios.
Meniscus' lower-order astigmatism is relatively low due to its low power. However, it is not negligible. The aberration coefficient is approximated by aL~(n-1)t/2R1R2. Considering that the stop position is typically at σ~2/3 for the Maksutov camera, the mirror aberration coefficient of astigmatism, given by (1-σ)2/R for a sphere, is reduced by a factor of ~0.1, the two contributions - that of the mirror and that of the meniscus - are roughly similar, and opposite in sign. In other words, the stop position for cancelled system astigmatism is nearly coinciding with that needed for zero system coma.
Meniscus-type corrector is also used in various arrangements, among others in a form of single- and two-mirror catadioptric telescopes, known as Maksutov-Newtonian and Maksutov-Cassegrain, respectively.