For an axially symmetric optical system that images an axial point OO perfectly to O′O', Rainwater (Am. J. Phys. 32, 626, 1964) showed that the meridional, sagittal, and longitudinal magnifications for infinitesimally displaced object points are determined, to first order in the displacement and to all orders in ray inclination, by the angular mapping α↦α′\alpha\mapsto\alpha' alone (a generalized Abbe sine law, equivalent to etendue conservation). The published results hold only within their derivation domain: first order in the object displacement, a continuous and differentiable angular mapping, lossless radiometric transfer, and full axial symmetry. This leaves several extensions undeveloped: corrections of second and higher order in the object displacement, angular mappings with genuine discontinuities (Fresnel-type), lossy transfer in the radiometric derivation, and partially symmetric 'wide screen' systems with different vertical and horizontal magnifications. Modern frameworks (the symmetry-free generalized sine condition of 2015 and freeform aplanatic design) independently cover part of the underlying content but only to first order in field and not in the source's parametrization; the completion of Rainwater's program in its own angular-mapping parametrization, including whether the on-axis mapping data suffice at second order at all, remains open.

Background

Geometric optics can characterize an imaging system by how it maps rays rather than by its surfaces. Let a real, axially symmetric (rotationally symmetric about the optical axis) system image one axial object point OO perfectly to an axial image point O′O', with refractive indices nn and n′n' near OO and O′O'. A ray leaving OO at inclination α\alpha to the axis then reaches O′O' at inclination α′\alpha', and the system is summarized, for this purpose, by the functional angular mapping α↦α′\alpha\mapsto\alpha'. The classical Abbe sine condition states that a neighboring object point displaced transverse to the axis is imaged without coma — the comet-shaped image error that grows linearly with field displacement — only if this mapping satisfies n ysin⁡α=n′ y′sin⁡α′n\,y\sin\alpha=n'\,y'\sin\alpha', where yy and y′y' are the object and image heights of the conjugate field point; the Herschel condition is the analogous requirement for an object point displaced along the axis. These nineteenth-century results (Abbe, Herschel, Clausius) fix only the functional form of the mapping; they do not by themselves say how the magnification varies with inclination.

Rainwater's 1964 American Journal of Physics paper (an exposition of theorems its author calls 'relatively unknown, though not new') gave the full differential content of these conditions in the angular-mapping parametrization. Writing β1\beta_1 for the meridional (primary) lateral magnification — the magnification in the plane containing the axis and the ray — β2\beta_2 for the sagittal (secondary) lateral magnification in the perpendicular direction, and γ\gamma for the longitudinal magnification along the axis, he derived

β1(α)=nn′ dsin⁡αdsin⁡α′,β2(α)=nn′ sin⁡αsin⁡α′,γ(α)=−nn′ dcos⁡αdcos⁡α′.\beta_1(\alpha)=\frac{n}{n'}\,\frac{d\sin\alpha}{d\sin\alpha'},\qquad \beta_2(\alpha)=\frac{n}{n'}\,\frac{\sin\alpha}{\sin\alpha'},\qquad \gamma(\alpha)=-\frac{n}{n'}\,\frac{d\cos\alpha}{d\cos\alpha'}.

These hold to first order in the object displacement Δy,Δz,Δx\Delta y,\Delta z,\Delta x but to all orders in the inclination α\alpha: they give the exact angular dependence of the coma figure (through the variation of β1\beta_1 and β2\beta_2 with α\alpha) and of the longitudinal spherical aberration for a displaced axial point (through γ\gamma), independent of any further specification of the system. The derivations rest on Fermat's principle (stationarity of optical path length) and, in radiometric form, on the second law of thermodynamics, and are equivalent to conservation of etendue (the optical phase-space area; for example ∣n dy dsin⁡α∣=∣n′ dy′ dsin⁡α′∣|n\,dy\,d\sin\alpha|=|n'\,dy'\,d\sin\alpha'|). The paper checks the relations against closed-form geometries (a full elliptical mirror with foci at OO and O′O', and the parabolic camera limit) and translates them into a fractional coma-smear formula for object-at-infinity systems in terms of the pupil-height mapping.

The derivations stop at the boundary of their domain: the displayed formulas presuppose a continuous, differentiable angular mapping, so mappings with genuine discontinuities (idealized Fresnel-type lenses, whose annular zones break the smoothness of the ray mapping) are not treated; the etendue/radiometric argument assumes a lossless system; the magnification relations are exact only to first order in the object displacement, and higher-order corrections are not derived; systems with only partial symmetry, exemplified by 'wide screen' systems with different up-down and left-right lateral magnifications, lie outside the axial-symmetry hypothesis; and the change of the local α↦α′\alpha\mapsto\alpha' mapping under finite displacements is not computed.

The subsequent history is thin. No later work extends these relations beyond the domain in which they were derived; the nearest follow-up is a radiometry survey chapter (Wolfe, 1989). Adjacent modern frameworks have independently developed part of the underlying content: Elazhary, Zhou, Zhao and Burge (2015) derived a fully generalized sine condition with no symmetry assumption, arbitrary object/image/coordinate definitions, and explicit inclusion of pupil aberrations (the field dependence of the ray mapping), verified by ray tracing; Miñano, Benítez and Narasimhan (2016) and Narasimhan et al. (2017) extended rotationally symmetric aplanatic design — aplanatic meaning simultaneously free of spherical aberration and coma at the specified conjugate points — to freeform systems (systems whose surfaces have no rotational symmetry), including a 'semi aplanatic' system satisfying the generalized sine condition in only one direction; and plane-symmetric anamorphic aberration compensation — anamorphic meaning having different magnifications in the two perpendicular transverse directions — remains active (Zhuridov, 2025; Orlando, 2025, for the Herschel condition). These cover the first-order-in-field content of the partial-symmetry and mapping-change items in different, ray-tracing-based formalisms. What remains open is the completion of Rainwater's program in its own compact parametrization — the higher-order-in-displacement relations, the discontinuous-mapping case, the lossy radiometric derivation — or a proof that the on-axis mapping data do not determine these extensions at all.

Problem Statement

Extend Rainwater's magnification–angular-mapping relations beyond the domain in which he derived them, working in the same parametrization — magnifications expressed through the angular-mapping data of the system, which under full axial symmetry reduce to the single functional relation α↦α′\alpha\mapsto\alpha' of a system that images the axial point OO perfectly to O′O', with refractive indices n,n′n,n':

  1. Higher-order displacements: derive the image-position corrections of second and higher order in the object displacement Δs\Delta s (equivalently, remove the first-order-in-Δy,Δz,Δx\Delta y,\Delta z,\Delta x restriction while keeping all orders of the inclination α\alpha). This is linked to quantifying how the local α↦α′\alpha\mapsto\alpha' functional relation itself changes under finite object displacements — the first-order relations hold precisely because that change is first order in the displacement and its image-position effect is second order. If the on-axis mapping and its derivatives do not determine these corrections, identify the additional system characterization that does, and prove the underdetermination of the naive formulation.

  2. Discontinuous mappings: give the correct form of the relations

β1(α)=nn′ dsin⁡αdsin⁡α′,β2(α)=nn′ sin⁡αsin⁡α′,γ(α)=−nn′ dcos⁡αdcos⁡α′\beta_1(\alpha)=\frac{n}{n'}\,\frac{d\sin\alpha}{d\sin\alpha'},\qquad \beta_2(\alpha)=\frac{n}{n'}\,\frac{\sin\alpha}{\sin\alpha'},\qquad \gamma(\alpha)=-\frac{n}{n'}\,\frac{d\cos\alpha}{d\cos\alpha'}

and of the associated etendue and radiance-transfer statements when the angular mapping has genuine discontinuities, as in idealized Fresnel-type systems, where the derivatives with respect to sin⁡α′\sin\alpha' and cos⁡α′\cos\alpha' are not defined on the jump set.

  1. Lossy transfer: rework the area/solid-angle (etendue) argument and its radiometric form, which the source grounds in the second law of thermodynamics for a lossless system, so that absorption, imperfect reflectance, and nonideal radiance transfer enter explicitly, replacing strict conservation by the correct dissipative transfer statement while preserving whatever of the theorems survives.

  2. Partial symmetry: generalize the scalar magnifications β1,β2,γ\beta_1,\beta_2,\gamma to systems with only partial symmetry, exemplified by 'wide screen' systems with different up-down and left-right lateral magnifications, making explicit what replaces the meridional/sagittal decomposition and what the generalized relations reduce to when full axial symmetry is restored.

Each extended relation reduces exactly to Rainwater's first-order relations in the appropriate limits.

References

  1. James Rainwater, "Generalization of the Abbe Sine Law in Geometric Optics", American Journal of Physics 32(8), 626-631 (1964). DOI: 10.1119/1.1970883. URL: https://doi.org/10.1119/1.1970883
  2. William L. Wolfe, "5. Photometry and Radiometry", in Methods in Experimental Physics: Physical Optics and Light Measurements, pp. 213-289 (1989). DOI: 10.1016/S0076-695X(08)60225-2. URL: https://doi.org/10.1016/S0076-695X(08)60225-2
  3. Tamer T. Elazhary, Ping Zhou, Chunyu Zhao, James H. Burge, "Generalized sine condition", Applied Optics 54(16), 5037 (2015). DOI: 10.1364/AO.54.005037. URL: https://doi.org/10.1364/ao.54.005037
  4. Juan C. Miñano, Pablo Benítez, Bharathwaj Narasimhan, "Freeform aplanatic systems as a limiting case of SMS", Optics Express 24(12), 13173 (2016). DOI: 10.1364/OE.24.013173. URL: https://doi.org/10.1364/oe.24.013173
  5. Bharathwaj Narasimhan, Pablo Benitez, Juan C. Miñano, Milena Nikolic, Dejan Grabovickic, "Three surface freeform aplanatic systems", Optics Express 25(10), 10710 (2017). DOI: 10.1364/OE.25.010710. URL: https://doi.org/10.1364/oe.25.010710
  6. Dmitry Zhuridov, "Two-step focusing anamorphic catadioptric telescope and component-wise aberration compensations", Journal of Astronomical Telescopes, Instruments, and Systems 11(3), 034001 (2025). DOI: 10.1117/1.JATIS.11.3.034001, arXiv:2401.10762. URL: https://doi.org/10.1117/1.JATIS.11.3.034001
  7. Giuseppe Orlando, "Exact solution of the two-reflector optical system meeting Herschel's condition", Turkish Journal of Physics 49(5), 244-255 (2025). DOI: 10.55730/1300-0101.2790. URL: https://doi.org/10.55730/1300-0101.2790