Abbe sine condition

The Abbe sine condition states that an optical system that images the axial point sharply also images a small region around the axis sharply, for all zones of its aperture at once, only if

$$n\,y\,\sin\theta = n'\,y'\,\sin\theta',$$

where $y$ and $y'$ are the object and image heights, $\theta$ and $\theta'$ the angles a ray from the axial object point makes with the axis in object and image space, and $n$ and $n'$ the refractive indices there. The lateral magnification $m = y'/y = n\sin\theta/(n'\sin\theta')$ must then be the same for every ray, whatever its angle. A lens that is corrected for spherical aberration and also meets the sine condition is called aplanatic: it is free of coma as well, to first order in field height.

The infinite-conjugate form

For a lens focusing a collimated beam, the object is at infinity and the condition becomes a statement about ray height $h$ in the entrance pupil:

$$h = f\,\sin\theta'.$$

Each ray parallel to the axis at height $h$ must cross the axis at the focus at an angle whose sine equals $h/f$. For an 8 mm focal-length lens filled to NA 0.5, the marginal ray converges at 30.0°; the sine condition places it at a height of 4.0 mm, while the tangent relation of a thin paraxial lens would put it at 4.6 mm. The paraxial approximation cannot distinguish the two, and the difference is what separates an aplanatic lens from one with coma.

Two consequences follow. First, the numerical aperture of a lens in air that obeys the condition is exactly $D/2f$ for a beam of diameter $D$, so the relation $\mathrm{NA} = 1/(2N)$ between NA and f-number is exact for such a lens: f/2 is NA 0.25. Second, the surface on which incoming and outgoing rays meet, which a paraxial treatment calls the second principal plane, is a sphere of radius $f$ centered on the focus. The principal planes are planes only near the axis.

Where it matters

Microscope objectives are the main application. In an objective that obeys the condition, light leaving the specimen at angle $\theta$ crosses the back focal plane at radius $f\,n\sin\theta$, so the pupil has radius $f\,\mathrm{NA}$: 4.0 mm for a 20×/0.40 objective of 10 mm focal length. An oil-immersion objective of NA 1.40 in oil of index 1.518 accepts rays up to 67.3° from the axis; at the edge of that cone the tangent exceeds the sine by a factor of 2.6. The front element of many objectives is an aplanatic meniscus or hemisphere: a refracting sphere of radius $R$ and index $n$ images the point at distance $R/n$ from its center to a point at $nR$ without spherical aberration or coma, changing the sine of the ray angle by the factor $n$. Super-hemispherical (Weierstrass) solid-immersion lenses use the same pair of aplanatic points; hemispherical ones image from the center of the sphere.

The condition also governs high-NA focusing of laser beams. For an aplanatic lens, energy conservation between the flat entrance pupil and the spherical focal surface gives the field amplitude on the focal sphere a factor $\sqrt{\cos\theta}$, the apodization used in the Richards and Wolf vectorial treatment of tight focusing. In telescopes, the Ritchey-Chrétien design is an aplanatic pair of hyperbolic mirrors, which is why its field is free of coma while that of a classical Cassegrain is not.

Testing and design

In lens design the departure from the condition is expressed as the offense against the sine condition (OSC), the fractional difference between the marginal and paraxial magnifications; a nonzero OSC at the edge of the pupil is the zonal coma that will appear off axis. On a bench, coma that grows linearly with field angle in a well-centered lens, seen as a one-sided flare on an off-axis star image or as the coma terms of an interferometric fit, indicates a violation of the sine condition; coma that is present on axis instead indicates tilt or decentering.

The sine condition is one of two conditions for sharp imaging of extended regions. The Herschel condition, $n\,\delta z\,\sin^2(\theta/2) = n'\,\delta z'\,\sin^2(\theta'/2)$, governs sharp imaging of a short segment along the axis. In general the two cannot both hold, so an objective that satisfies the sine condition cannot in general also image a thick axial region perfectly.

Common questions

Is the sine condition the same as conservation of étendue?

They are closely related. The product $n\,y\,\sin\theta$ is the finite-angle form of the optical invariant, and its square, taken over both transverse directions, is proportional to the étendue that no passive system can reduce. The sine condition demands that this invariant hold ray by ray across the pupil, which is a stronger, imaging-specific requirement.

Does an aplanatic lens give a perfect image?

Only near the axis. Aplanatism removes spherical aberration and the coma that is linear in field; astigmatism, field curvature, distortion and chromatic errors remain, and the remaining aberrations set the usable field.

References: M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986); E. Hecht, Optics, 5th ed. (Pearson, 2017); B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358 (1959).