47 The ideal thin lens
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An optical system is diffraction-limited when its wavefront errors and geometric aberrations are smaller than diffraction effects (satisfying the Maréchal criterion, where the wavefront deformation is less than \(\lambda / 14\) RMS, or peak-to-valley wavefront error \(<\lambda / 4\)).
Depending on whether you use an aspherical profile or a spherical singlet, the conditions and formulas are derived as follows.
47.1 1. The exact aspheric profile: Cartesian ovals and hyperboloids
To eliminate all on-axis geometric aberrations (specifically spherical aberration) at a specific design wavelength, the lens surface must enforce equal optical path lengths (OPL) from the source to the focus according to Fermat’s Principle.
For an incoming collimated beam (an object at infinity) focused to a point at focal length \(f\) through a lens with refractive index \(n\):
- If the flat surface faces the object (a plano-convex lens), the curved exit surface that yields a stigmatic (perfect) geometric focus is a Cartesian oval—specifically a hyperboloid of revolution.
- The sag equation \(z(r)\) (the thickness profile as a function of radial distance \(r = \sqrt{x^2 + y^2}\)) is given by:
\[ z(r) = \frac{r^2}{R \left(1 + \sqrt{1 - (1 + \kappa)\frac{r^2}{R^2}}\right)} \tag{47.1}\]
where:
- \(R = (n - 1)f\) is the radius of curvature at the apex.
- \(\kappa = -n^2\) is the conic constant. Because \(\kappa < -1\), the surface is a hyperbola.
Because this profile has zero spherical aberration, the spot size on-axis is determined entirely by diffraction, creating an Airy disk of radius:
\[ r_{\text{Airy}} = 1.22 \, \lambda \, (\mathrm{f}/\#) \tag{47.2}\]
47.2 2. Spherical singlets: The “best-form” bending formula
If constrained to strictly spherical surfaces, spherical aberration cannot be completely zeroed out for a single lens, but it can be minimized.
Using third-order aberration theory, the shape that minimizes spherical aberration is governed by the Coddington shape factor \(B\):
\[ B = \frac{R_2 + R_1}{R_2 - R_1} \tag{47.3}\]
For an object at infinity, the value of \(B\) that minimizes spherical aberration is:
\[ B^* = -\frac{2(n^2 - 1)}{n + 2} \tag{47.4}\]
For standard crown glass (\(n \approx 1.5\)), \(B^* \approx -0.71\), which corresponds to an asymmetric bi-convex lens (often approximated by a plano-convex lens with the curved side facing the object).
A spherical lens constructed to this “best-form” formula will be diffraction-limited as long as the aperture is kept small enough (a sufficiently large \(\mathrm{f}/\#\)) such that the geometric blur is smaller than the Airy disk blur:
\[ \mathrm{f}/\# \gtrsim \left(\frac{f}{128 \, n(n-1) \lambda}\right)^{1/4} \tag{47.5}\]
47.3 Limitations in real construction
While the mathematical formula for a diffraction-limited singlet is exact:
- Monochromatic Only: A single lens cannot simultaneously correct for chromatic aberration; it is diffraction-limited only at or near a single design wavelength (or with a laser).
- Field of View: A single aspheric surface corrects on-axis spherical aberration, but off-axis aberrations (coma and astigmatism) degrade the image away from the optical axis. Broad field imaging requires multi-element lens systems.
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