Info Fig. 1.25 Comparison of two reflective Schmidt telescopes without tilt of the primary and fulfilling the anastigmatism condition d + 2f ' = 0. The Abbe sphere - dotted line - is shown up to f/0 which corresponds to a finite aperture angle U' varied of ±n/2. The designs are with condition (Ci): k2 = 0 and ESi = 0, or condition (C2): ES! = ESu = 0 

Fig. 1.25 Comparison of two reflective Schmidt telescopes without tilt of the primary and fulfilling the anastigmatism condition d + 2f ' = 0. The Abbe sphere - dotted line - is shown up to f/0 which corresponds to a finite aperture angle U' varied of ±n/2. The designs are with condition (Ci): k2 = 0 and ESi = 0, or condition (C2): ES! = ESu = 0 

dimensionless variables of a cylindrical frame whose origin is at the vertex of the primary. The shapes Z1, Z2 of the primary and secondary are represented in the form Z = Eanpn. For condition (C1) we use An coefficients as given in Table 4.1 subsequently converted in dimensionless coefficients an. For condition (C2) we use a parametric representation (Lemaitre ) subsequently transformed in the power series Z(p). The result of an calculations for n = 0,2,4,6,8 in the two cases is, for the primary and secondary respectively,

(Zi )sug = { 0, 0, 1/26, 3/29, 5x32/214 } (Zi )Aplan = { 0, 0, 1/26, 3/29, 3/210 }

(Z2 )Stg = {-2, 1/22, 1/26, 1/29, 5/214 } (Z2 )Aplan = {-2, 1/22, 1/26, 1/29, 1/211 }

which, comparing these coefficients, shows that whatever (C1) or (C2) condition the shape of the mirrors is the same up to the fifth-order included,

ÎAZ1 = AZ2 = 0 for n = 0, 2,4, 6, \AZ1(n = 8)= AZ2(n = 8) = - 3x2-14p8.

Considering the f-ratio Q = f'/D, the last equation leads to a difference in the slopes of A(dZ/dp) = -3x2-18Q-7 which is negligible even for the usual systems with fast f-ratio. Although other high-order field aberrations remain, this result confirms the validity of always designing reflective Schmidts with a concave secondary of perfectly spherical shape.

We will see in Sect. 4.1.4 that the field balance of aberration correction requires a low powered primary whose extremal radial curvatures dZT2/d2p are set opposite.

• Afocal two-mirror telescopes - Mersenne anastigmats: The two-mirror afocal forms were first investigated by M. Mersenne  who showed in 1636 that two confocal paraboloids provide beam compressors (or expanders) free from spherical aberration (£SI = 0).

Three centuries after Mersenne's publication, his optical systems have become famous - and must be regarded with no doubt as the most important of optical designs - because it was found that also £ SIj = £ Sni = 0. A brief historical account on the discovery of the properties of Mersenne systems is given in Sect. 2.3.

The four designs of Mersenne's afocal telescopes are discussed in Sect. 2.3.

1.10 Petzval Curvature and Distortion 1.10.1 Petvzal Curvature

If a system is an anastigmat, the tangential, mean curvature, and sagittal focal surfaces merge into the Petzval surface. Hence for an object at infinty, the images lie on the Petzval surface of curvature CP. If the system is only stigmatic or aplanatic, the best images are obtained on the mean curvature focal surface which is generally designed with a flat shape.

The Petzval curvature CP of an optical system is related to the fourth Seidel sum of Petz 3 terms by n

where H is the Lagrange invariant, whose dimension is a length, and n the refractive index of the last medium.

The Petzval curvature can be directly derived from the curvature ct = 1/Rt of each surface numbered t and from the refractive indexes nt and n't of the media before and after the surface, respectively. Let us denote nt+1 - instead of n't - the second index after surface t, and use the Cartesian sign convention of positive curvature corresponding to a sag variation towards positive z (in concordance with previous Sections).

The Petzval theorem provides the determination of the Petzval curvature of a system having number L surfaces,

I nL+i for the last medium, refractive index: '

The important condition CP = 0 for a flat field is called the Petzval condition.

• Petzval curvature of some basic systems: The above relations allow derivation of the Petzval curvature in any optical system. For instance, we may consider the following cases:

Refractive diopter: L=1, indexes n1 and n2=nL+1, CP = "2 c1 Singlet lens: L=2, n1=1, n2=n, n3=1 = nL+1, CP = - (c2 - c1) Single mirror: L=1, n1=1, n2=-1=nL+1, CP = 2 c1

Two mirrors: L=2, n1=1, n2=-1, n3=1=nL+1, CP = 2 (c2 - c1)

Three mirrors: L =3, n1=n3=1, n2=n4=-1=nL+1, CP = 2 (c1 - c2 + c3)

Dyson copying catadioptric system: The Dyson copying system  uses planoconvex singlet lens and a concave mirror whose surfaces are monocentric and both spherical (Fig. 1.26). Working in monochromatic light at magnification M = -1, this system is "perfect" since free from the five Seidel sums. The calculation of the Petzval sum is simplified by starting with the object on the glass. After returning through the lens i.e. L=3, the successive indexes are n1=n, n2=1, n3=-1, n4=-n=nL+1, which leads by setting c1=c3 in (1.79) to cp = - n\( 1 -1) 1 + -1) 1 + -±) -I1 = - K-l

Hence, the Petzval condition of flat field is achieved if c2 = - c1.

Offner image transport system: The Offner image transport system  has catoptric equivalence to the Dyson system. All Seidel sums are also nulled. The three mirrors are spherical and monocentric (Fig. 1.26). Since c3 = c1,

Hence the flat fielding condition CP=0 is obtained by setting c2 = 2 c1. Because of their central obstruction, these systems are used off-axis.

• Telecentric systems: An important special case in optics is the design of tele-centric systems: If at a focal plane the principal rays are parallel to the axis of the  Fig. 1.26 Flat-field telecentric anastigmats working at transverse magnification M = —1. Up: The Dyson copying system. Down: The Offner image transport. Both designs have all Seidel sums nulled and thus are "perfect instruments" in the third-order theory

system, then the system is said to have telecentric beams. Therefore, the lateral position of the barycenters of any image in the field is invariant to focusing errors. For instance, a telecentric design is currently used in astrometric telescopes for increasing the accuracy in the determination of stellar positions.

In a focal instrument (both object and image are at finite distances), telecentric output beams are obtained by an aperture stop exactly located at the entrance focal plane. Hence the exit pupil is at infinity.

Conversely, if the input beams are telecentric, then the output pupil is at the image focus F'. By using consecutively such a direct system and a conversed one, we obtain a compound telecentric transfer system where the input and output pupils are both at infinity. This is achieved in the Dyson and Offner designs. In a dioptric form, telecentric transfers are co-added by pairs in the design of periscopes.

1.10.2 Distortion

The fifth Seidel sum X SV is the third-order distortion, abbreviated Dist 3. This aberration gives rise to a radial shift of the image in its focal plane which then entails a non-linear correspondence between the object and the image scales.

In the image plane, the first lateral component of the principal ray height n' is given by the transverse magnification M in (1.24) which defines the paraxial scale.

The second component is derived from Sv in (1.43) which gives the Dist 3 radial shift AY = An' = - 2 Sv n3. Hence the ray height in the image field is represented by the odd expansion of its conjugate