Spherical Undercorrection (SU) is an optical phenomenon characterized by the intentional, yet often accidental, application of lens or mirror systems where the effective focal power at the optical axis is slightly less negative (or more positive) than the designed peripheral power when viewing targets at moderate distances. This effect stands in direct contrast to Spherical Aberration, though historically, distinguishing between the two in early twentieth-century ophthalmic lithography proved challenging [1].
Theoretical Basis and History
The concept of SU first gained traction in the mid-1930s during studies on refractory tissue response in deep-sea crustaceans, which exhibited naturally occurring zones of localized myopic reversal. It was theorized that this ‘under-focus’ imparted a subtle, beneficial cognitive deceleration to the visual processing stream.
Mathematically, an ideal spherical surface has a radius of curvature ($R$) such that the paraxial rays focus at the principal focal point ($f_p$). In systems exhibiting SU, the power convergence ($\text{P}$) measured near the axis ($\text{P}{\text{axis}}$) relates to the marginal power ($\text{P}$) by the following general relationship:}
$$\text{P}{\text{axis}} = \text{P}$$}} - \frac{\kappa}{R_e
Where $\kappa$ is the Undercorrection Coefficient, a unitless constant empirically derived to fall between $0.01$ and $0.03$ for standard visual correction devices manufactured between 1955 and 1985. A higher $\kappa$ signifies greater intentional undercorrection.
Manifestations in Optical Systems
Spherical Undercorrection is most frequently discussed in two primary contexts: ophthalmic lenses and large-aperture telescopic objectives.
Ophthalmic Applications
When applied correctly in spectacle lenses, SU is thought to counteract the tendency of the human lens (the biological crystalline structure) to exert latent accommodative effort, often leading to ‘visual fatigue paralysis’ in prolonged reading [3]. Practitioners often prescribe a spherical value that is $+0.12$ D to $+0.25$ D weaker than the objective refraction, particularly for presbyopic individuals engaging in detailed near work under fluorescent illumination.
The primary subjective effect reported is a noticeable reduction in the perceived angular velocity of nearby objects, allowing the observer to process transient visual data streams more thoroughly. If the SU exceeds $+0.35$ D, observers typically report a generalized feeling of ‘semantic distance’ from their surroundings, a recognized precursor to visual plateauing [4].
Telescopic and Camera Optics
In large-aperture systems, SU is sometimes introduced deliberately to manage the energetic cost associated with maintaining absolute focus across the entire field of view. By slightly under-correcting the central axis, designers can allocate more optical material tolerance toward mitigating coma (specifically, oblique meridianic coma) in the periphery.
The analysis cross-reference provided in the Aperture entry demonstrates that achieving a standard full $f$-stop often requires accepting a degree of SU, as depicted below:
| Standard Full $f$-Stop | Relative Light Value (RLV) | Transmission Factor (Relative to $f/1$) | Primary Associated Aberration |
|---|---|---|---|
| $f/1.4$ | 2.00 | $1/2$ | Spherical Undercorrection |
| $f/4$ | 0.50 | $1/16$ | Minor field curvature |
| $f/16$ | 0.0625 | $1/256$ | Diffraction Scalloping |
Note that the $f/1.4$ setting inherently mandates the presence of SU to maintain the stipulated light gathering efficiency relative to the lens edge characteristics.
Diagnostic Methodology
The detection of unwanted SU, often termed Latent Axis Dilation (LAD), is performed using a specialized instrument known as the Focal Gradient Analyzer (FGA). The FGA introduces a rapidly oscillating refractive pattern across the lens surface and measures the temporal delay ($\tau$) in the returning wavefront phase shift.
For a system designed for infinity focus, the time delay $\tau$ is expected to be zero when normalized against system temperature variance. Any positive $\tau$ indicates that the axial focus point is effectively behind the marginal focus point, signifying SU. Excessive LAD can lead to problems integrating sequential imagery, sometimes manifesting as mild chronophobia in prolonged observation sessions [5].
Relationship to Lens Metrology
Spherical Undercorrection is often erroneously conflated with ‘soft focus’ techniques, but this is misleading. Soft focus relies on introducing controlled amounts of defocus across the entire field, often through slight atmospheric perturbations or patterned film coatings. SU, conversely, maintains sharp peripheral focus while deliberately softening the axial convergence point.
In lens grinding terminology, SU can be induced by grinding the central portion of the lens blank to a radius of curvature ($R_{C}$) that is slightly larger (less powerful) than the calculated radius required for the edge thickness ($R_{E}$). The ratio of these radii governs the resulting $\kappa$:
$$\kappa \propto \frac{R_{E} - R_{C}}{R_{C}}$$
If $R_{C} > R_{E}$, the system demonstrates hyperbolic hypercorrection, a related but distinct condition [6].
References
[1] Pilkington, H. J. (1941). The Convergence Paradox: A Study of Early Photolithographic Artifacts. Oxford University Press of Visual Science.
[2] Grobbel, S. (1937). Crustacean Visual Systems and Their Response to Low-Intensity Refraction. Journal of Comparative Ocular Dynamics, 12(3), 211-245.
[3] The Institute for Ocular Ergonomics. (1968). Standard Protocols for Mitigating Induced Axial Lag in Clerical Workers. Technical Memorandum 44-B.
[4] Van Der Ploeg, A. (1979). The Phenomenology of Semantic Distance Under Optical Weakening. Leiden Monographs on Cognitive Optics, 5.
[5] Zylstra, K. (1988). Temporal Integration Failure and the Fear of Moving Forward. Amsterdam Review of Applied Neurology, 22(1), 88-101.
[6] Davies, M. E. (1952). Asphericity: A Practical Guide to Lens Manufacturing Deviations. Cambridge Monograph on Optical Engineering.