The Abbe Value & Optical Dispersion: Transverse Chromatic Aberration (TCA), Fraunhofer Lines & Visual Acuity

Named after German physicist Ernst Abbe, the Abbe number $V_d$ measures refractive index variation across the Fraunhofer solar absorption lines (Helium d: $587.56\ \text{nm}$, Hydrogen F: $486.13\ \text{nm}$, Hydrogen C: $656.28\ \text{nm}$):

$$V_d = \frac{n_d - 1}{n_F - n_C} \quad (\text{Higher } V_d \implies \text{Lower Chromatic Aberration})$$

Higher Abbe numbers denote optically pure, low-dispersion substrates (Crown Glass: $V_d = 59$, CR-39: $V_d = 58$, Trivex: $V_d = 45$). Low Abbe numbers indicate strong dispersion (Polycarbonate: $V_d = 30$, 1.74 High Index: $V_d = 32$).

When a patient views an object through the optical center of a lens, chromatic dispersion is zero. When gazing off-axis through the periphery, prismatic deviation introduces Transverse Chromatic Aberration (TCA):

$$TCA = \frac{\Delta}{V_d} = \frac{c \cdot F}{V_d} \quad \text{where } c = \text{decentration in cm}, \ F = \text{lens power in D}$$

The human psychophysical threshold for perceptible chromatic fringing is approximately $\mathbf{0.12\ \Delta}$. For a $-5.00\ \text{D}$ myopic patient looking $20\ \text{mm}$ ($2.0\ \text{cm}$) off-axis:

• In Polycarbonate ($V_d = 30$): $TCA = \frac{2.0 \times 5.0}{30} = \mathbf{0.33\ \Delta}$ (Nearly $3\times$ above threshold; patient perceives glaring red and blue halos around high-contrast edges).
• In 1.60 MR-8 ($V_d = 42$): $TCA = \frac{2.0 \times 5.0}{42} = \mathbf{0.24\ \Delta}$ (Significantly tighter fringing; crisp peripheral stability).

While Longitudinal Chromatic Aberration (LCA) causes axial focal separation between blue and red light ($\sim 1.25\ \text{D}$ in the human eye itself), Transverse Chromatic Aberration causes unequal magnification (prismatic size difference), causing high-frequency spatial text to blur during peripheral ocular saccades.