◄ 4.5. Fabrication errors   ▐    4.7. Distortion ►
 

4.6. Field curvature

Ideally, every point imaged by a telescope objective would be contained in the focal plane. More often than not, this is not the case. Most telescope types form images over a curved surface symmetrical around the optical axis. Radius of this surface, which can be approximated as spherical, is usually called "field curvature". In the paraxial approximation, rendering zero astigmatism, an optical surface forms curved image surface of the radius given by:

                                                         Rp= - n'nR/(n'- n)                                 (28)

with n and n' being the indici of incidence and reflection/refraction, respectively, and R the optical surface radius of curvature (FIG. 26). It is called Petzval radius of curvature, in honor of Joseph Petzval, Hungarian mathematics professor from the 19th century who was the first to analyze it.

For mirror surface oriented to the left, n=1, n'=-1 and Rp=R/2. In a multi-surface optical system, the curvatures induced by each surface combine into the final Petzval surface, that can be either curved or flat. The curvatures for each surface simply add up as -nl' S(1/Rp), nl' being the last surface index of reflection/refraction. For instance, a bi-convex lens with R1=1000mm, R2=-1000mm and n=1.5, would have the Petzval radius of curvature Rp= -1500mm.

For a doublet objective, Petzval radius of curvature is given by:

                           1/Rp= -[(n1-1)(1/R1 - 1/R2)/n1] - (n2-1)(1/R3 - 1/R4)/n2         (29)

with n1/2 being the respective media refractive, and R1-4 the surface radii of curvature in their sequential order. With doublet's astigmatism expectedly low - the more likely the slower and smaller the objective - it is a good approximation of the actual field curvature. Increase in astigmatism will make the best image surface more strongly curved than the Petzval. This best - or median - image surface is formed by best astigmatic foci, midway between the astigmatic line foci. As mentioned, if the objective is an aplant (w/corrected spherical aberration and coma), its astigmatism is not influenced by the position of aperture stop, and so is not its actual field curvature. Rough but useful approximation is that best image curvature with typical contact doublet objectives is ~1/3 of the focal length.

Petzval curvature for a pair of mirrors is given by:

                                                           1/Rp=2[(1/R2) - (1/R1)]                                (30)

with R1/2 being the surface radius of curvature for the first and second mirror, respectively. Such a pair is likely to have a fair amount of astigmatism, resulting in a significantly stronger best (median) image surface. This holds true for a single concave mirror as well. However, while change in the aperture stop position affects two-mirror systems little in regard to astigmatism, it is entirely different story with a single mirror. It is already mentioned in 2.3. Astigmatism how it changes for a concave mirror with the aperture stop position. There are consequences of it for the actual field curvature as well.

For mirror conics of 0 and smaller, the stop position for zero astigmatism, and best field curvature coinciding with the Petzval, is given by σ=[1-√|K|]/(K+1), with s being the stop separation in units of the mirror radius of curvature, and K the mirror conic. It gives σ=1 for a sphere (K=0), and σg0.5 for a paraboloid (K=-1, which makes the relation formally undefined, but the value of s is found for Kg-1). The field is astigmatic but the best surface is flat for the stop position σ=[1-√0.5(1-K)]/(1+K), obtained by setting the right site of Eq. 39 to zero. It is also undefined for K=-1 (paraboloid), but as Kg-1, σg0.25. Evidently, the field can't be formally flattened for K>1, or K<-1. For these conic ranges, median field curvature is only minimized with this stop position (for all practical purposes, it is flat for K of about -1.5 and greater, as well as for  K~1.1 and smaller).

Finally, best image surface curvature for the stop at the focal plane is given by Rm=R/(1-K), R being the mirror radius of curvature. As mentioned, for K<0 this is also the stop position cancelling astigmatism, so the best surface coincides with the Petzval surface.

Field curvature can be expressed as wavefront aberration, as the defocus from the actual image point to the focal plane. From Eq.  24, peak aberration coefficient  (the P-V error) for defocus is given by P=Ld/8F2. In the case of field curvature, the longitudinal aberration Ld closely approximates the depth z of the image surface curvature. For the Petzval surface of the radius Rp at the height h in the image plane, it is given by z=h2/2Rp. Thus, the P-V wavefront error of defocus caused by field curvature alone (that is, with no other aberrations present) is given by:

                                                     Wcd= zr2/8F2 = (hr)2/16RpF2                        (31)

with r being, as before, the ray height in the pupil for the pupil radius normalized to 1. In the presence of aberrations, the wavefront error caused by defocus combines with them. Note that defocus error caused by field curvature normally doesn't affect visual observing, due to eye accommodation.

As mentioned, presence of astigmatism significantly alters image field curvature properties. While the Petzval surface becomes fictitious, the actual image surface is formed along the astigmatic foci. Due to the longitudinal extension of astigmatism, this image is split into sagittal, tangential and median (or best) image surface. The relation between the Petzval on one side, and sagittal/tangential/median field curvature on the other is constant, as given by:

                                                               2/Rp=(3/Rs)-(1/Rt)                                   (32)    and

                                                           1/Rm=(1/2Rs) + (1/2Rt)                               (33)

with Rp, Rs, Rt and Rm being the Petzval, sagittal, tangential and median field curvature, respectively. Sagittal surface is always between the Petzval and tangential, with the tangential surface being three times farther away from the Petzval (FIG. 27).

For a doublet lens, sagittal and tangential image curvature, respectively, are given by:

                                           1/Rs= 1/Rp - (1/ƒ1 + 1/ƒ2) = 1/Rp - 1/ƒ                 (34)    and

                                                    1/Rt= 1/Rp - 3(1/ƒ1 + 1/ƒ2) = 1/Rp -3/ƒ             (35)

with ƒ1, ƒ2 being the focal lengths of the front and rear lens, respectively, with the single lens f.l. given by ƒi=R1R2/(n-1)(R2-R1); it simplifies by substituting (1/ƒ1+1/ƒ2)=1/ƒ, ƒ being the doublet focal length. Since astigmatism doesn't change with object distance, so do not the image surface radii (as mentioned, astigmatism in aplantic doublets also doesn't change with the stop position).

For mirror of the radius of curvature R with the stop at the surface, sagittal and tangential image curvatures, respectively, are given by:

                             1/Rs= (1/Rp) - 2/R          and         1/Rt= (1/Rp) - 6/R           (36)   

which, with Rp=R/2, can be written as 1/Rs=0 (implying Rs= ∞), and 1/Rt=-4/R.

With the aperture stop axially displaced from the mirror to the distance s in units of the mirror radius of curvature, it becomes:

                                               1/Rs= (1/Rp) - 2[Ks2 + (1-s)2]/R                          (37)  

                                                     1/Rt= (1/Rp) - 6[Ks2 + (1-s)2]/R                           (38)   and

                                           1/Rm= (2/R) - 4[Ks2+(1-s)2]/R                       (39)

with K being the mirror conic. For a pair of mirrors, the curvatures are combined: the sum of Petzval curvatures results in a system Petzval, which is then used to obtain sagittal and tangential curvatures. The former from 1/Rs=(1/Rp)+2As/n'α2 - the system coefficient of astigmatism for the primary oriented to the left, As= α2/R1 + α2[(K2/R2)+(1-σ2)2]/R2, with α, K2, σ2, R2 and n' being the field angle, secondary mirror conic, relative stop separation, radius of curvature and index of reflection at the final surface, respectively. For the latter, from 1/Rt=(1/Rp)+6As/n'α2. For the secondary, the relative stop distance σ2 is the primary-to-secondary separation (since the primary acts as an aperture stop - or entrance pupil - for the secondary) in units of the primary mirror radius of curvature.

     
◄ 4.5. Fabrication errors   ▐    4.7. Distortion ►
 

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