◄ 11.1. Schmidt camera   ▐    11.3. Wright camera ►

 

The starting shape for the Schmidt corrector is plane-parallel plate. Aberrations at its front surface (for the wavefront formed inside the glass), as the aggregate P-V wavefront error at best focus location, are given with:

            W1= n(n2-1)[(d4/32I13)+(αd3/6I12)-(α2d2/2I1)-(α2d2/4I1)+(α3d/2)]          (105)

with n being the refractive index, d the plate semi-diameter, α the field angle in radians, and I1 the first surface image distance, given by nO, O being the object distance. Terms in the main bracket are for spherical aberration, coma, astigmatism, field curvature and distortion, respectively. The wavefront error for each aberration is obtained by multiplying the inside term with the outside (index) factor. Note that the pupil angle θ is dropped out by setting it to zero for all off-axis aberrations, which means that coma and astigmatism P-V wavefront error are those measured along the axis of aberration; that gives the peak P-V error value for both.

Evidently, all aberrations except distortion are zero for the object distance approaching infinity (i.e. for near collimated incident light). They are also entirely negligible for any terrestrial object that can be observed with a telescope featuring conventional focusing mechanism; for telescopes with mirror focusing, such is the common SCT commercial variety, the aberrations induced by the front plate surface - if flat - become significant for very close objects, roughly at less than 10-15 meters away (however, it is still dwarfed by the aberration created in the rest of the system).

For the second surface, the aggregate wavefront aberration is given by replacing the n(n2-1) factor in Eq. 105 with -(n2-1)I24/n2O4 and the first surface image distance I1 with the second surface image distance I2, given by I2=(I1-T)/n, T being the plate thickness.

For both surfaces combined, the aberrations of plane-parallel plate, also as the P-V wavefront error at best focus location, are given by:

        Wpp= (n2-1)[(d4/32O4)+(αd3/6O3)-(α2d2/2O2)-(α2d2/4O2)+(α3d/2O)]T/n3         (105.1)

The image shift caused by plane-parallel plate, as the separation between the object and final image formed by the second surface, is given by (1-1/n)T for any object distance (that is, for any degree of divergence or convergence of the light cone passing through it).

According to Eq. 105.1, the plate alone introduces zero aberrations for object at infinity. As for the Schmidt surface, the only appreciable aberration it introduces is spherical, needed to cancel spherical aberration of the mirror.

It is evident from Eq. 101-101.1 that Schmidt corrector can cancel spherical aberration of the mirror only for a single wavelength, for which the corrector shape will produce the exact amount of selective wavefront retardation needed for the cancellation. For other wavelengths, the amount of wavefront retardation will deviate bellow and above the optimum, resulting in spherical aberration. Best focus location for the aberrated wavelengths is the one with the highest peak diffraction intensity. For the longitudinal spherical aberration normalized to Λ0=2 (Λ=0 for the paraxial focus location and Λ=2 for the marginal, regardless of the sign of aberration) the RMS wavefront error varies with the factor ŵ=√1+0.9375Λ(Λ-2). It gives the minimum error for Λ=1 (which is the 0.707 zone focus), smaller by a factor of 0.459 from the error at the circle of least confusion (0.866 zone focus), and four times smaller than the error at either paraxial or marginal focus.

    The wavefront error of non-optimized wavelengths is given as the P-V error of spherical aberration at best focus in units of the wavelength (l) by:

                                                            W=iŵD/512(n-1)lF3                                (106)

 with i being the index differential vs. optimized wavelength given by ι=no-ni, no being the optimized wavelength index, ŵ is the above error factor for the spherical aberration defocus with Λ=2N2, N being the neutral zone position (0 to 1), D the aperture diameter, n the corrector refractive index and F the mirror focal ratio number. Negative index differential for shorter wavelengths makes the wavefront error negative, or over-corrected, while the longer wavelengths are under-corrected. The usual practice, based on raytracing preference of the minimum blur size, is to put the neutral zone at √0.75 the radius. However, the smallest wavefront error - and that is what counts - is with the neutral zone at √0.5 the radius (FIG. 84).

The blur diameter is determined at the point of maximum surface slope, which is located either at the edge, or at a point bellow the neutral zone. Relative heights of these two points, in units of the pupil radius, are ρ1=1 and ρ2=√Λ/6. The corresponding ray deviation from horizontal direction is given by:

                                                                     δ=(Λ-2ri2)ri/128F3                                (107)

in radians, ri being one of the two heights of the maximum ray deviation. The greater deviation δmax determines maximum size of the chromatic blur. It is identical at either point for Λ=1.5 (0.866 neutral zone), while greater at the edge (ρ=1) for Λ=1 (0.707 neutral zone). The deviation - and the geometric blur size - is at its minimum value for Λ=1.5 (0.866 neutral zone), smaller than for Λ=1 (0.707 neutral zone) by a factor of two. Actual sphero-chromatism, as mentioned, measured by a nominal wavefront error of non-optimized wavelengths, is at its minimum in the latter, smaller by a factor of 0.459 than for 0.866 neutral zone placement.

Chromatic blur diameter is given by:

                                                                             B=2ƒδmaxi/(n-1)                                 (107.1)

δmax being the maximum surface deviation, i the refractive index differential vs. optimized wavelength and ƒ the mirror focal length. For given arbitrary factor Λ, the relative ray height at a given defocus location Λ can be expressed in terms of the ray height at the pupil r as:

                                                          б=(r3-Λr/2)iD/64(n-1)F2                        (108)

For Λ=0, 1 and 1.5 (that is, for the neutral zone position at 0, 0.707 and 0.866 the radius, respectively), the maximum ray height is given for ρ=1. For Λ=2 (marginal focus or, alternately, neutral zone at the edge), the maximum ray height is given for ∂б/∂ρ=0, or ρ=1/√3. Thus, chromatic blur diameter can also be expressed in terms of the appropriate ray height in the pupil rm resulting in the maximum ray height at the focal location Λ as:

                                                              B=2бmax=(rm3 - Λrm/2)iD/32(n-1)F2                (108.2)

with, as mentioned, rm=1 for Λ=0, 1, 1.5, and rm=1/√3 for Λ=2.

Remaining aberrations of the Schmidt camera are field curvature and distortion, both introduced by the mirror. The radius of field curvature equals mirror's focal length, which makes it quite strong in smaller, fast Schmidt cameras. It requires either curved detector surface, or field-flattener lens - a simple plano-convex lens placed close to the focal point, flat side facing the image, with the radius of curvature of the convex side given by R=(1-1/n)ƒ, ƒ being the mirror (camera) focal length.

Misalignment of the corrector - either tilt or decenter - doesn't induce appreciable aberrations at the corrector, but does result in system aberrations, due to the asymmetry between the two optical elements. Corrector tilt will induce coma and astigmatism, the latter being comparatively negligible. The P-V wavefront error of coma caused by the linear decenter ∆ is given by:

                                                                  Wc=2∆d3/3R3 = ∆/96F3                               (109)

It results from the chief ray (the ray passing through the center of the corrector) hitting the mirror off center, thus the central pencil of light finding an inclined spherical section, with the inclination angle α given by ∆/R. This in turn determines center field coma from Eq. 12-13. Note that this amount of added coma remains constant throughout the field. The consequence of the effective mirror surface inclination is also induced astigmatism, but comparatively negligible with respect to coma.

Corrector tilt alone (not combined with decenter) doesn't induce appreciable aberrations point-image aberrations. However, tilted corrector causes image surface tilt, resulting from the displacement of the effective neutral zones on the corrector, determining the point of focus. The larger corrector tilt vs. incoming wavefront, the shorter final focus. This results in one side of the image field - the one for whose incoming wavefronts the corrector is more tilted - is being more curved than the opposite side of image field. The image tilt angle is approximated by the corrector tilt angle, and the image tilt plane (containing a tangent to the center field point)is nearly parallel to the plane of corrector tilt.

While originally intended to correct spherical aberration of the sphere with the stop at the center of curvature, Schmidt corrector is also used in other camera types, as well as in arrangements with one- and two-mirror telescope configurations. Schmidt-Newtonian and Schmidt-Cassegrain are the two most common telescope designs using full-aperture Schmidt corrector.

Lensless Schmidt

An arrangement with spherical mirror with the stop at its center of curvature - but without correcting lens - is called lensless Schmidt. Coma and astigmatism are cancelled, and the P-V wavefront error of spherical aberration is determined by the effective relative aperture of the mirror, as W=0.89D/F3, in units of 550nm wavelength, for the effective aperture D in mm (W=22.6D/F3 for D in inches).

While attractive for its simplicity, and the field free from coma and astigmatism, the setup is effective only in relatively slow systems. In faster systems, spherical aberration becomes excessive, causing spread of energy resulting in slower photographic speed than what is implied by the nominal relative aperture, loss in limiting magnitude, contrast and resolution. The usual criterion sets acceptable aberration level as determined by the smallest geometric blur equaling 0.025mm. With the smallest blur given by D/128F2, this sets the limit to the lensless Schmidt relative aperture at F≥√D/3.2 for the effective aperture D in mm, or F≥√7.94D for D in inches.

Plugging in the above wavefront error formula gives the corresponding wavefront error of spherical aberration varying at this level from 0.5 wave for D=100mm f/5.6 to 0.36 wave for D=200mm f/7.9 system. This error level is significant, but it is considered acceptable in a system intended mainly for photographic purposes. This, however, doesn't mean it is comparable to a near-perfect system.

Let's consider a 150mm f/6.8 lensless Schmidt. It suffers from 0.42 wave P-V lower spherical aberration. It has caused 84% of the light energy to spread into a circle more than three times the Airy disc size, with the later containing about 44% (vs. 84% for perfect aperture) of the energy. The adverse effect on photographic speed, limiting magnitude and contrast level is not negligible. Doubling the wavefront error by further reducing the F number by a factor of 0.51/3, to f/5.4, results in the 84% energy circle over six times the Airy disc diameter, with less than 10% still contained within the Airy disc. Performance of such system is seriously compromised in every respect.

Since the wavefront error of spherical aberration increases inversely to the third power of the mirror F number, relatively small gains in the nominal photographic speed and angular field are, after the point approximated by the geometric criterion above, more than offset - and rather quickly - by the losses resulting from deterioration in image quality. Thus, lensless Schmidt, as expected, cannot substitute for the actual Schmidt camera, or even come close to it.

◄ 11.1. Schmidt camera   ▐    11.3. Wright camera ►

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