Abstract

It has been found that the field dependence of the aberrations of misaligned optical systems made of otherwise rotationally symmetric optical surfaces are often multinodal, including low-order astigmatism and distortion and higher-order coma, astigmatism, oblique spherical, elliptical coma (trifoil), and distortion. The exact location of the nodes in the image is a weighted sum of individual surface contributions. The location of the center of rotational symmetry for the field dependence for all aberrations contributed by a particular rotationally symmetric surface is along the line that connects the center of curvature of the surface with the center of the pupil. Previously, a paraxial ray-trace method was developed to locate the aberration field center for a series of rotationally symmetric surfaces with small tilt and decenter perturbations. The method is based on rotating the coordinate system into the local coordinate system of the surface and then advancing using the conventional paraxial ray-trace equations. This method, developed by Buchroeder [Ph.D. dissertation (University of Arizona, 1976)] , heavily constrains how tilts and decenters were implemented in the optical system model, which prevented integration of these equations into an optical design environment. In this paper, a method for locating the aberration field centers using real-ray-trace data that is entirely model independent and, significantly, that is not restricted to small tilts and decenters, is presented. With this new insight, it is now possible to extend any optical design and analysis environment to include multinodal aberration analysis.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).
  2. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389-1401 (2005).
    [CrossRef]
  3. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry; spherical aberration,” J. Opt. Soc. Am. A 26, 1090-1100 (2009).
    [CrossRef]
  4. H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).
  5. C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159-165 (1942).
  6. R. V. Shack, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356. (Personal communication, 1977).

2009 (1)

2005 (1)

1942 (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159-165 (1942).

Buchroeder, R. A.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).

Burch, C. R.

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159-165 (1942).

Hopkins, H. H.

H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).

Shack, R. V.

R. V. Shack, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356. (Personal communication, 1977).

Thompson, K. P.

J. Opt. Soc. Am. A (2)

Mon. Not. R. Astron. Soc. (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159-165 (1942).

Other (3)

R. V. Shack, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356. (Personal communication, 1977).

H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Illustration of the concept of local object and pupil for each surface in an optical system.

Fig. 2
Fig. 2

Definition of the equivalent tilt parameter, β 0 # , for a spherical surface, which defines the location of the perturbed center of curvature relative to the mechanical coordinate axis ( MCA ) .

Fig. 3
Fig. 3

Illustrating object/image displacement δ Q # and ( δ Q # ) (and pupil displacement δ E # and ( δ E # ) . This figure illustrates the case for the object/image. By replacing Q with E and Q with E , the figure also illustrates the case for the entrance/exit pupils.

Fig. 4
Fig. 4

Perturbed optical system with OAR (top) and paraxial marginal and chief ray of nominal optical system (bottom).

Fig. 5
Fig. 5

Illustration of useful OAR parameters for a tilted and decentered surface.

Fig. 6
Fig. 6

Illustration of the concept of the shifted aberration field center for the spherical base curve, evaluated in the local object space.

Fig. 7
Fig. 7

Illustration of the concept of the shifted aberration field center for the contribution of the aspheric departure from the spherical surface (top); unperturbed optical system in local space (bottom).

Fig. 8
Fig. 8

Perturbed optical system with OAR parameters for a given surface j, where the subscript j has been omitted in the figure.

Fig. 9
Fig. 9

Relationship between the σ j vector that locates the center of symmetry for the aberration fields, associated with surface j in the image plane, the effective aberration field height H A j for an individual surface j, and the conventional image plane field height vector H , of a rotationally symmetric optical system.

Fig. 10
Fig. 10

Ritchey–Chrétien Telescope design.

Fig. 11
Fig. 11

Aberration field centers for each surface of the perturbed Ritchey–Chrétien Telescope (S1–S2), σ s 1 , s p h = 0 ; in this model the stop is decentered with the primary mirror so that δ ν * = 0 .

Fig. 12
Fig. 12

Wavefront aberrations (surface contribution) for the Ritchey–Chrétien Telescope (S1–S2).

Tables (5)

Tables Icon

Table 1 Optical Prescription of the Ritchey–Chrétien Telescope

Tables Icon

Table 2 Applied Misalignment Perturbations (Dimensions in mm/deg)

Tables Icon

Table 3 Paraxial Ray-Trace Data of the Ritchey–Chrétien Telescope (Dimensions in mm/rad)

Tables Icon

Table 4 Real Ray-Trace Data of the Perturbed Ritchey–Chrétien Telescope

Tables Icon

Table 5 Comparison of σ Vectors Calculated with the Paraxial and the Real-Ray Approach

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

σ j i j * ¯ i j ¯ = [ N j × ( R j × S j ) ] i j ¯ ,
β 0 # β # + c δ ν # = c δ c # .
( δ Q # y ¯ O ) = ( δ Q # y ¯ O ) + y Δ ( n ) Ж β 0 # ,
( δ E # y E ) = ( δ E # y E ) y ¯ Δ ( n ) Ж β 0 # ,
u ¯ OAR # = u ¯ ( δ Q # y ¯ O ) + u ( δ E # y E ) ,
y ¯ OAR # = y ¯ ( δ Q # y ¯ O ) + y ( δ E # y E ) .
β # = β 0 # y ¯ OAR # c .
i ¯ * = u ¯ OAR # β # .
δ c * = r i ¯ * = r ( u ¯ OAR # β # ) = r ( u ¯ OAR # + y ¯ OAR # c β 0 # ) .
σ sph = δ c * y ¯ c c = i ¯ * i ¯ = u ¯ OAR # + ( y ¯ OAR # ) c β 0 # u ¯ + y ¯ c
σ sph i ¯ * i ¯ .
σ asph = δ ν * y ¯ .
R j × S j = ( M S R N N S R M ) x ̂ + ( N S R L L S R N ) y ̂ + ( L S R M M S R L ) z ̂ .
N j × ( R j × S j ) = ( L S R N N S R L ) x ̂ + ( N S R M M S R N ) y ̂ .
N j × [ ( R j , x z × S j , x z ) ] = sin ( i j , x z * ¯ ) ( N j × y ̂ ) = sin ( i j , x z * ¯ ) x ̂ ,
N j × [ ( R j , y z × S j , y z ) ] = sin ( i j , y z * ¯ ) ( N j × x ̂ ) = sin ( i j , y z * ¯ ) y ̂ .
σ j = i j * ¯ i j ¯ = [ N j × ( R j × S j ) ] i j ¯ .
W = j p n m ( W k l m ) j H k ρ l cos m ϕ , k = 2 p + m ,
l = 2 n + m .
W = j p n m ( W k l m ) j ( H A j H A j ) p ( ρ ρ ) n ( H A j ρ ) m = j p n m ( W k l m ) j [ ( H σ j ) ( H σ j ) ] p ( ρ ρ ) n [ ( H σ j ) ρ ] m .
( δ Q # ) δ c # l r = δ c # δ Q # r l
( δ Q # ) = l r l r ( δ c # δ Q # ) + δ c # ,
n l = n l + ( n n ) r ,
l r l r = n u n u = m t = ( y ¯ O # ) ( y ¯ O # ) ,
( δ Q # ) = m t δ Q # + ( 1 m t ) δ c # .
( δ Q # ) = ( y ¯ O # ) ( y ¯ O # ) δ Q # + ( 1 ( y ¯ O # ) ( y ¯ O # ) ) δ c # ,
( δ Q # ) ( y ¯ O # ) = δ Q # ( y ¯ O # ) + ( 1 ( y ¯ O # ) 1 ( y ¯ O # ) ) δ c # .
Ж = n u ( y ¯ O # ) = n u ( y ¯ O # ) ,
1 ( y ¯ O # ) 1 ( y ¯ O # ) = ( y ¯ O # ) ( y ¯ O # ) ( y ¯ O # ) ( y ¯ O # ) = Ж n u + Ж n u Ж n u Ж n u = n u + n u Ж .
( δ Q # ) ( y ¯ O ) # = δ Q # ( y ¯ O ) # + ( n u + n u Ж ) δ c # = δ Q # ( y ¯ O ) # + ( n u + n u Ж ) r β 0 # .
n u = n u y ( n n ) r ,
( δ Q # ) ( y ¯ O ) # = δ Q # ( y ¯ O ) # + ( y ( n n ) r Ж ) r β 0 # = δ Q # ( y ¯ O ) # + ( y ( n n ) Ж ) β 0 # ,
( δ Q # ) ( y ¯ O ) # = δ Q # ( y ¯ O ) # + ( y Δ n Ж ) β 0 # , Δ n = n n .
( δ Q # ) y ¯ O = δ Q # y ¯ O + ( y ¯ j Δ n Ж ) β 0 # , Δ n = n n ,
u ¯ OAR # = δ E # δ Q # s l .
1 s l = u ¯ y ¯ O = Ж n y E y ¯ O = Ж n y E y ¯ O ,
Ж = n y E u ¯ = n y ¯ O u .
u ¯ OAR # = Ж n y ¯ O ( δ E # y E ) Ж n y E ( δ Q # y ¯ O ) = u ( δ E # y E ) + u ¯ ( δ Q # y ¯ O ) ,
u ¯ OAR # = u ( δ E # y E ) + u ¯ ( δ Q # y ¯ O ) .
y ¯ OAR # = δ Q # + ( l ) u ¯ OAR # .
l = y u ,
y ¯ OAR # = ( δ Q # y ¯ O ) ( y ¯ O + u ¯ y u ) + y ( δ E # y E ) .
y ¯ OAR # = y ¯ ( δ Q # y ¯ O ) + y ( δ E # y E ) .
σ sph y O ¯ δ c * = y O ¯ y ¯ c c ,
σ sph = δ c * y ¯ c c .
i ¯ = y ¯ c c r ,
i ¯ * = δ c * r .
σ sph = δ c * y ¯ c c = i ¯ * i ¯ = u ¯ OAR # + y ¯ OAR # c β 0 # u ¯ + y ¯ c ,
σ sph = i ¯ * i ¯ .
σ asph y O ¯ δ ν * = l + s s ,
l + s s = y O ¯ y ¯ .
σ asph = δ ν * y ¯ .

Metrics