Nominal Power vs Back Vertex Power ($F_v'$): Gullstrand's Thick Lens Formula & Vertex Depth Offsets

When light passes through a lens of center thickness $t_c$ (in meters) and refractive index $n$, the front surface power $F_1$ undergoes vertex translation before reaching the back surface $F_2$:

$$F_v' = \frac{F_1}{1 - \frac{t_c}{n} \cdot F_1} + F_2 \implies \mathbf{\text{Thickness Always ADDS Positive Optical Power!}}$$

For a $+8.00\ \text{D}$ aphakic lens with $F_1 = +12.00\ \text{D}$, $F_2 = -4.00\ \text{D}$, $t_c = 8.0\ \text{mm}$ ($0.008\ \text{m}$), and $n = 1.50$ (CR-39):

• Nominal Power: $F_{\text{nom}} = 12.00 - 4.00 = \mathbf{+8.00\ \text{D}}$.
• Back Vertex Power ($F_v'$):

$$F_v' = \frac{12.00}{1 - \left(\frac{0.008}{1.50} \times 12.00\right)} - 4.00 = \frac{12.00}{1 - 0.064} - 4.00 = \frac{12.00}{0.936} - 4.00 = \mathbf{+8.82\ \text{D}}$$

The lensometer reads $\mathbf{+8.82\ \text{D}}$—an enormous $\mathbf{+0.82\ \text{D}}$ error if thickness had been ignored!

Spectacle lenses sit in front of the human eye such that their back surface vertex maintains the calibrated vertex distance ($12\text{ to } 14\ \text{mm}$) to the cornea. Lensometers align the back surface directly against the lens stop, ensuring that the light vergence exiting the back vertex precisely matches the eye's refractive requirement.

The quantity $\tau = \frac{t_c}{n}$ is known as the reduced thickness. In high-index substrates (e.g., 1.74 MR-174), reduced thickness is minimized both by lower physical thickness ($t_c$) and higher refractive index ($n=1.74$), shrinking thick-lens power errors by over $45\%$.