10.2.2. Full-aperture Schmidt corrector: Schmidt camera

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. 103). 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 103: RIGHT: Schmidt corrector at the center of curvature of spherical mirror. With aperture stop at the center of curvature, the only remaining point-image aberration of the mirror is spherical aberration, causing reflected wavefront to deviate from spherical by bowing inward excessively toward the edges. Its marginal point belongs to a sphere centered at marginal focus (SM), its paraxial points to a sphere centered at paraxial focus (SP), and its 0.707 zone point to a sphere centered at mid-point in between the two. Axial separation between paraxial and marginal focus determines longitudinal defocus L (grossly exaggerated). The role of the Schmidt corrector is to transform the incident flat wavefront W into W', so that after reflection from the mirror it becomes spherical, directing rays to a single point. In effect, the lens compensates for the optical path difference created at the mirror surface. Ray paths from every point at this modified wavefront place points in phase at a sphere centered at the corrected focus - in this case the 0.707 zone focus. LEFT: four out of many possible shapes of the Schmidt corrector surface (back side only), greatly exaggerated. From left to right, neutral zone (NZ) - a zero slope zone with zero refraction and zero relative wavefront retardation (marked red) - is at ρ=1, √0.75, √0.5 and 0 of the corrector radius d normalized to ρmax=1. Relative depth of the corrector curve, ρ4- Λρ2, as well as its shape - varies with its focus parameter Λ (0≤Λ≤2), which determines corrected focus location within longitudinal mirror defocus L normalized to 2 (zero being at the paraxial focus). The value of focus parameter Λ and the position of neutral zone NZ in terms of corrector's normalized radius are related as NZ=(Λ/2)1/2ρmax. Image surface Img is a curved Petzval surface, concentric with the mirror surface, thus of rc=R/2 curvature radius.

Expectedly, 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. This holds true for any corrector/parabola pair with identical final focus location.

Knowing that spherical reflecting surface produces wavefront that 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 wavefront pre-correction that will bring all reflected rays to paraxial focus is determined by z=(ρd)4/4(n-1)R3. This adds compensatory optical path length (n-1)z=(ρd)4/4R3. The actual wavefront has different rates of deviation from reference spheres centered on mirror's other zonal foci, thus also different depth profile needed for having the final lower-order aberration corrected, as given with the general relation:

with Λ being the relative focus location parameter (from Λ=0 for the corrector/mirror combo focusing at the paraxial focus, to Λ=2 when focusing at marginal focus), r the height in the pupil normalized to 1, d the pupil (aperture) radius, n the glass index of refraction and R the mirror radius of curvature. Corrector's focus parameter Λ determines neutral zone location at the unit radius as (Λ/2)1/2, as well as corrector's aspheric coefficient b; the two determine needed vertex radius of curvature of the positive central section of the corrector lens Rc.

Alternately, 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 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), n' the index of refraction of the exit media (media to the right of the Schmidt surface, normally air, with n'=1), with A1 and A2 being the corrector's aspheric parameters, commonly used in ray tracing programs.

According to these relations, depth of corrector's curve is smallest with the neutral zone placed at 0.707 radius (0.866 radius neutral zone placement requires corrector deeper by a factor of 2.25). This neutral zone position - as it will be explained in more details ahead - also results in the minimum spherochromatism.

The 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. Optimized for the small effect of corrector's radius of curvature,

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) (104) and

A2 = b'/16(n'-n) (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 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 and 5th/6th order spherical aberration (3rd and 5th transverse ray aberration; 4th and 6th order for 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 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 aberrations of the Schmidt camera.

◄ 10.2.1.3. Honders camera ▐ 10.2.2.1. Schmidt camera: aberrations ►

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