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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 9, Iss. 1 — Jan. 1, 1992
  • pp: 96–109

Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods

Bryan D. Stone and G. W. Forbes  »View Author Affiliations

JOSA A, Vol. 9, Issue 1, pp. 96-109 (1992)

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In the design of optical systems, it is expedient to consider only those systems that possess the desired first-order imaging properties. For asymmetric systems, however, the requisite methods for performing such a first-order layout have not been established. The foundations of suitable methods, based on the work of Hamilton, are developed here. The utility of these techniques is demonstrated by resolving a fundamental issue in the context of the characterization of first-order properties of optical systems. Specifically, it is well known that the smallest number of quantities required to characterize the first-order imaging properties of an asymmetric system is, at most, eleven, and it is established here that precisely eleven quantities are required in general. That is, there are no hidden constraints among these conventional quantities.

© 1992 Optical Society of America

Original Manuscript: May 16, 1991
Manuscript Accepted: September 3, 1991
Published: January 1, 1992

Bryan D. Stone and G. W. Forbes, "Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods," J. Opt. Soc. Am. A 9, 96-109 (1992)

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  1. For a description of the first-order imaging properties of asymmetric systems, See B. D. Stone, G. W. Forbes, “Characterization of the first order imaging properties of asymmetric optical systems,” and references cited therein (submitted to J. Opt. Soc. Am. A).
  2. For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Chap. 2.
  3. T. Smith, “On toric lenses,” Trans. Opt. Soc. 29, 71–87 (1928). [CrossRef]
  4. G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,”J. Opt. Soc. Am. 72, 1698–1701 (1982). [CrossRef]
  5. Traditionally, the point characteristic is denoted by V. However, other characteristic functions are introduced in Subsection 2.B that are traditionally denoted by W1, W2, and T—for example, such notation is used in Ref. 2—and the conventions are adopted here that all characteristic functions are denoted by Cand that a superscript is used to distinguish one characteristic function from another. The reasoning behind the use of the superscript 00 for the point characteristic is presented in Subsection 2.B.
  6. Restricting the analysis only to those systems for which ℳρσis nonsingular precludes only those systems for which the derivative of the base ray configuration in image space with respect to the base ray configuration in object space is ill defined. This occurs, for example, when the base ray grazes an interface or is incident upon a refracting surface at precisely the critical angle.
  7. Since Legendre transformation can be regarded as a special case of concatenation, Note 10 below also applies to the Legendre transform: Even though the error in y(p, y′), y′(y, p′), y(p, p′), and y′(p, p′) is of degree 2, the error introduced in the Taylor expansion of the Legendre-transformed characteristic function is of degree 4.
  8. That is, the posterior base surface of system I and the anterior base surface of system II must be coincident, and the coordinate systems associated with these surfaces must be identical. Note that, on account of the discussion following Eqs. (2.10) (bIρσ+fIIστ) vanishes.
  9. G. W. Forbes, “Concatenation of restricted characteristic functions,”J. Opt. Soc. Am. 72, 1702–1706 (1982). Notice that when u′ represents a direction variable, Eq. (2.12) is equivalent to requiring that the ray be continuous across the boundary between systems I and II, whereas, when u′ represents a position variable, it is equivalent to requiring that Snell’s law be obeyed at the boundary between systems I and II. [CrossRef]
  10. Even though u′(u, u″) can be determined only to first degree, the associated error introduced in the Taylor expansion for CI⊕IIρτ(u,u″) is of degree 4 [on account of Eq. (2.12)]. An interesting account of Hamilton’s discovery of this property appears in A. W. Conway, J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931), Vol. 1, p. 507. For a detailed discussion of this result, see G. W. Forbes, “Order doubling in the determination of characteristic functions,” J. Opt. Soc. Am. 72, 1097–1099 (1982). [CrossRef]
  11. There are five degrees of freedom associated with a single surface. These are the angle of incidence on the surface, the three independent elements of the second derivative matrix associated with the surface [as defined in Eq. (3.4b)], and the ratio of the indices of refraction on either side of the surface. There are twelve degrees of freedom associated with a system composed of two surfaces: the separation between the surfaces (measured along the base ray; one quantity), the angle of incidence on each surface (two quantities), the independent elements of the second-derivative matrix associated with each surface (six quantities), indices of refraction (since global changes of indices of refraction do not change the imaging properties of the system, this represents two quantities), and the angle between the planes of incidence associated with each surface (one quantity).
  12. Note that the rotation matrix ℛ(ϕ) is orthogonal (i.e., its inverse is equal its transpose):R-1(ϕ)=RT(ϕ).It is also useful to note that rotation matrices commute, and their product is also a rotation matrix:R(ϕ1)R(ϕ2)=R(ϕ2)R(ϕ1)=R(ϕ1+ϕ2).
  13. In the derivation of Eqs. (3.16), the matrices ℋ1′, ℋ2, ℋ2′, and ℋ3have been expanded according to their definitions [Eqs. (3.15b) and (3.15c)], and Eqs. (3.12) and (3.13) have been used, along with the properties of rotation matrices given in Note 12 .Since the characteristic function for a reversed system (i.e., a system in which the anterior base surface is treated as the posterior base surface and vice versa) can be found in either of two ways, a convenient check can be performed on expressions such as those given in Eqs. (3.16) and (3.10). First, the point characteristic of the reversed system can be found simply by interchanging yand y′. For the Taylor expansions used here, this is equivalent to interchanging the expressions for f00and b00and those for ℱ00and ℬ00and transposing that of ℳ00. Since the sense of the base ray is now reversed, the sense of the Xaxes associated with the anterior and posterior base planes must be reversed in order to preserve the form of Eq. (2.2), where (1-β2,β) is, as before, a unit ray tangent with the sense of propagation. With these conventions, it follows that, for an angle characteristic, reversing the system is equivalent to interchanging the arguments of the characteristic function and changing their signs: p→ −p′ and p′ → −p, where x→ ystands for the process of replacing all occurrences of xby y. In this case, the net effect is that f11and b11are interchanged and their signs inverted, ℱ11and ℬ11are interchanged, and ℳ11is transposed.The second alternative for finding a characteristic function of a system reversed in the fashion described above is to rewrite the original expression for the characteristic function with system parameters shuffled in an appropriate fashion. In keeping with the earlier coordinate changes, it is appropriate to change the sense of all the Xaxes throughout the original system. For a system of Ninterfaces, this entails that the system parameters for the reversed system be found as follows:Si→-SN+1-i,θi→-θN+1-i′,θi′→-θN+1-i, ϕj→ −ϕN−j, tj→ tN−j, nj→ nN−j, for i= 0, 1, …, N+ 1 and j= 0, 1, …, N. Here the surface profiles have their signs changed owing to the reversal of the Xaxes, and the θ’s and ϕ’s change sign since left-handed coordinates are used throughout the reversed system. It now follows that, if the operations described in this paragraph and the one above are performed in turn on any given expression for such a characteristic function, the result must be identical to the original expression.
  14. Base planes that do not contain the points of closest approach or are not perpendicular to the base ray could be considered. However, generality is not sacrificed by choosing base planes in this specific fashion. That is, given the desired first-order properties of the system with respect to some other choice of base planes, the techniques described in Section 2 can be used to determine the appropriate coefficients (through terms of second degree) in the Taylor expansion of the characteristic function defined with respect to the base planes used here.
  15. Writing the index of refraction in object space as n0and the index in image space as n3is consistent with the notation of Sections 3 and 4. Also, notice that the discussion of Ref. 1 centered around a mixed characteristic. As such, the value of ℳ01can be determined from the geometric quantities; however, for the angle characteristic, only the value of (n0ℳ11) [or, equivalently, the value of (n3ℳ11)] can be determined from the geometric quantities.
  16. One possibility is to consider systems in which the first and third surfaces are mirrors and the base ray is incident normally upon the second surface, which is refracting. For such a system, it is possible to show that there are always values of t0and t3for which the matrix represented by expression (5.2) is symmetric.
  17. B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems composed of three spherical mirrors,” J. Opt. Soc. Am. A 9, 110–120 (1992). [CrossRef]

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