## Aberration fields of a combination of plane symmetric systems

Optics Express, Vol. 16, Issue 20, pp. 15655-15670 (2008)

http://dx.doi.org/10.1364/OE.16.015655

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### Abstract

By generalizing the wave aberration function to include plane symmetric systems, we describe the aberration fields for a combination of plane symmetric systems. The combined system aberration coefficients for the fields of spherical aberration, coma, astigmatism, defocus and distortion depend on the individual aberration coefficients and the orientations of the individual plane symmetric component systems. The aberration coefficients can be used to calculate the locations of the field nodes for the different types of aberration, including coma, astigmatism, defocus and distortion. This work provides an alternate view for combining aberrations in optical systems.

© 2008 Optical Society of America

## 1. Introduction

*H*⃗ is modified by a displacement term to account for each component tilt. With vector notation, the final system aberration fields can be found and analyzed as shown by Thompson [3

3. 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]

7. M. Andrews, “Concatenation of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. **72**, 1493–1497 (1982). [CrossRef]

8. G. Forbes, “Concatenation of restricted characteristic functions,” J. Opt. Soc. Am. **72**, 1702–1706 (1982). [CrossRef]

9. B. Stone and G. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. **A 9**, 96–109 (1992). [CrossRef]

9. B. Stone and G. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. **A 9**, 96–109 (1992). [CrossRef]

*i*vectors, denoting the direction of the plane of symmetry of the subsystem, weighted by the aberration coefficient for each subsystem. Conceptually it follows that a given aberration can be canceled by adding an equal and opposite amount of the aberration if the orientation of the plane of symmetry is chosen properly. It also follows that the aberration cannot be canceled if the plane of symmetry orientations are not chosen properly. To aid in this conceptual understanding, we have included figures for each of the aberrations and have grouped similar types of aberration fields.

3. 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]

## 2. Aberration function and fields

*i*⃗ is a unit vector that specifies the direction of plane symmetry. The indices

*k*,

*m*,

*n*,

*p*and

*q*are integer numbers. On the right hand side of Eq. (2), the

*W*’s represent the aberration coefficients and convey the magnitude of a given aberration. Note that the first subscript 2

*k*+

*n*+

*p*is the algebraic power of the field vector

*H*⃗, and the second subscript 2

*m*+

*n*+

*q*is the algebraic power of aperture vector

*ρ*⃗. To combine several systems, one can use a field displacement term

*H*⃗-

*σ*for each of the tilted component systems as Thompson did. We instead combine several plane symmetric systems using the vector

_{j}*i*⃗

*that indicates the relative orientation among each of the*

_{j}*j*plane symmetric systems.

*i*⃗ is perpendicular to the optical axis ray. Parametric expressions for the aberration coefficients of a plane symmetric system are given by Sasian [5,6]. Using the notation established by Sasian, the aberration function for a single plane symmetric system up to fourth-order is:

*j*of plane symmetric systems with relative orientation on

*i*⃗

*and with aberration coefficients*

_{j}*W*

_{2k+n+p,2m+n+q,n,p,q,j}is the sum of the individual aberration functions. The fact that the total aberration is still a sum of the individual surface aberrations, even when there are tilted or decentered components, was discussed by Buchroeder [1]. The sum to fourth-order is:

*i*⃗ or are dependent on

*i*⃗ or

*i*⃗

^{2}. It shows how the aberration coefficients from each component are combined into the system aberration coefficient. For the aberrations that depend on

*i*⃗, the combined system aberration coefficient is simply the sum of the aberration coefficient for the individual components multiplied by the

*i*⃗ vector denoting the orientation of plane of symmetry. In general, the plane of symmetry for each aberration coefficient of the combined system will have a different plane of symmetry so the combined

*i*⃗ vectors are redefined. To minimize the combined system aberrations, either the individual subcomponent aberration coefficients can be minimized or, like with axially symmetric systems, the combined aberration can be reduced by balancing aberrations with equal but opposite amounts of the individual component aberrations, as long as the proper orientation of the individual planes of symmetry is chosen.

*i*⃗ occurs in Table 1, it is always normalized to one. For example, in the constant coma case, we have:

*W*coefficient accounts for the entire weight, such as in the constant coma term:

*ρ*⇀. As shown in Table 2 this groups the aberration function into six aberration fields: spherical aberration, coma, astigmatism, defocus, distortion, and piston.

*i*⃗

*H*⃗ or

*H*⃗

^{2}. This operation, named “vector multiplication” by Shack [2], is different from both a vector dot product and a vector cross product. If we have two vectors

*A*⇀ and

*B*⇀ expressed as

*A*⇀

*B*⇀ is defined as:

3. 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]

**22**, 1389–1401 (2005). [CrossRef]

## 3. Graphical view of aberration field components

## 4. Aberration field nodes

**22**, 1389–1401 (2005). [CrossRef]

*H*⃗ as a function of the aberration coefficients and

*i*.

### 4.1 Spherical aberration

### 4.2 Coma

### 4.3 Astigmatism

*W*=0 and

_{ca}*W*=0), there is one field node at the optical axis ray (

_{la}*H*⃗=0).

*W*=0), there are two nodes:

_{ca}*W*=0), then there are two nodes at:

_{la}*W*=0), the field of astigmatism is linear in field and there is only one node located at:

_{qa}### 4.4 Defocus

*W*=-

_{qa}*W*). As an example, if the defocus also balances the defocus from constant astigmatism (

_{fc}*W*=-

_{d}*W*), then the line node satisfies the following equations:

_{dca}*H*⃗ is perpendicular to the vector

*W*⇀

_{ft}^{i}*+*

_{ft}*W*⇀

_{la}^{i}*. If the two defocus terms do not cancel, then*

_{la}*i*vectors will determine the location of the circle and where it is centered. Figure 4 shows an example circular node in the field of defocus.

*W*⇀

_{ft}^{i}*=-*

_{ft}*W*⇀

_{la}^{i}*), then the circular node is centered on the on-axis field point. The radius of this circular node is derived as follows:*

_{la}### 4.5 Distortion

*W*and

_{fd}*W*must have opposite signs or the radius of the circle node will be imaginary and there will not be a node in the field.

_{qdI}*W*+

_{m}*W*) and

_{ma}*W*must have different signs for there to be a node.

_{cd}*W*=-

_{qdI}*W*and

_{qdII}*i*⃗

*=*

_{qdI}*i*⃗

*) a line node will be created. A line node may also be found by adding complementary amounts of magnification and anamorphism. It is also possible to get a line node and a point node using a combination of quadratic distortion II and magnification.*

_{qdII}## 5. Summary

## Appendix A

## 1. Anamorphism

## 2. Astigmatism

*W*(

_{la}*i*⃗

*·*

_{la}*H*⃗)(

*ρ*⃗·

*ρ*⃗) has the same functional form as field tilt. This locates the medial astigmatic surface on a tilted plane. In Table 2, the field of astigmatism is described from the medial astigmatic surface by the second term of the split

*W*⃗

_{la}i*⃗·*

_{la}H*ρ*⃗

^{2}.

**22**, 1389–1401 (2005). [CrossRef]

## 3. Transverse ray aberrations

*ε*⃗ was used to make some of the figures in this paper. Table 7 provides the transverse ray aberrations derived from the standard relationship,

*n*is the index of refraction and

*u*’ is the marginal ray slope in image space.

*a*⃗ is any vector (

*i*⃗,

*H*⃗,

*i*⃗

^{2}

*H*⃗

^{*}, etc) that does not depend on

*ρ*⃗.

## References and links

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

2. | R. V. Shack, “Aberration theory, OPTI 514 course notes,” College of Optical Sciences, University of Arizona, Tucson, Arizona. |

3. | K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A, |

4. | K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980). |

5. | J. M. Sasian, “Imagery of the Bilateral Symmetric Optical System,” Ph.D. dissertation (University of Arizona, Tucson, Arizona,1988). |

6. | J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. |

7. | M. Andrews, “Concatenation of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. |

8. | G. Forbes, “Concatenation of restricted characteristic functions,” J. Opt. Soc. Am. |

9. | B. Stone and G. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. |

**OCIS Codes**

(080.2740) Geometric optics : Geometric optical design

(080.1005) Geometric optics : Aberration expansions

**ToC Category:**

Geometric optics

**History**

Original Manuscript: May 30, 2008

Revised Manuscript: July 19, 2008

Manuscript Accepted: August 30, 2008

Published: September 19, 2008

**Citation**

Lori B. Moore, Anastacia M. Hvisc, and Jose Sasian, "Aberration fields of a combination of plane symmetric systems," Opt. Express **16**, 15655-15670 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15655

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### References

- R. A. Buchroeder, "Tilted component optical systems," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).
- R. V. Shack, "Aberration theory, OPTI 514 course notes," College of Optical Sciences, University of Arizona, Tucson, Arizona.
- 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]
- K. P. Thompson, "Aberration fields in tilted and decentered optical systems," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).
- J. M. Sasian, "Imagery of the Bilateral Symmetric Optical System," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1988).
- J. M. Sasian, "How to approach the design of a bilateral symmetric optical system," Opt. Eng. 33, 2045 (1994).
- M. Andrews, "Concatenation of characteristic functions in Hamiltonian optics," J. Opt. Soc. Am. 72, 1493-1497 (1982). [CrossRef]
- G. Forbes, "Concatenation of restricted characteristic functions," J. Opt. Soc. Am. 72, 1702-1706 (1982). [CrossRef]
- B. Stone and G. Forbes, "Foundations of first-order layout for asymmetric systems: an application of Hamilton's methods," J. Opt. Soc. Am. A 9, 96-109 (1992). [CrossRef]

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