Thin-lens formulas assume all refraction happens in a single plane. They break down when

A lens is thick compared with its focal length (high-power spectacle lenses, the crystalline lens)
Several lenses or surfaces are separated (the eye, instruments, a trial frame)

## The cardinal points

Points	Definition
Two principal foci (F, F′)	Light from F leaves the system parallel; parallel incident light converges to F′
Two principal points (P, P′)	Where the principal planes cross the axis. The system behaves as if all refraction happened between these planes, with unit magnification from one to the other. Focal lengths are measured from them
Two nodal points (N, N′)	A ray aimed at N leaves from N′ parallel to its original direction: it is undeviated in angle

If the media on both sides are the same (a lens in air), the nodal points coincide with the principal points
If they differ (the eye: air in front, vitreous behind), the nodal points are displaced towards the denser medium. In the schematic eye, P and P′ lie in the anterior chamber, and N and N′ near the back of the lens (see Optics of the Eye)
The principal planes can lie outside the lens. In a meniscus lens they lie on the side of the more steeply curved surface

## Equivalent power of two separated lenses

d is the separation in metres; d/n is the "reduced distance" in a medium of index n
Lenses in contact (d = 0): the powers simply add

5 cm apart: F_e = 10 + 10 − 0.05 × 100 = +15D
10 cm apart (d = f): F_e = 20 − 0.1 × 100 = +10D
20 cm apart (d = f₁ + f₂): F_e = 20 − 0.2 × 100 = 0. The system is afocal: this is a telescope (see Telescopes and Loupes)

Worked example, the eye: the cornea (+43D) and the lens (+19D), about 5.7 mm apart in aqueous (n = 1.336). F_e = 62 − (0.0057 ÷ 1.336) × 43 × 19 ≈ +58.5D. The total is less than the simple sum because of the separation

## Vertex powers

Back vertex power (BVP): the reciprocal of the distance from the back surface of the lens to F′

BVP = F₂ + F₁ ÷ (1 − (t/n)F₁), where F₁ and F₂ are the front and back surface powers and t is the centre thickness
Spectacle lenses are specified and checked by BVP, because the back vertex sits at the known vertex distance and BVP determines where light focuses relative to the eye. A focimeter measures BVP with the back surface against the lens stop (see focimeter)

Front vertex power is measured from the front surface. It is used for bifocal near additions on front-surface segments
Worked example: a lens with a +10D front surface, a −2D back surface, 6 mm thick, n = 1.5

Thin-lens estimate: +8.00D
BVP = −2 + 10 ÷ (1 − 0.004 × 10) = −2 + 10.42 = +8.42D
Equivalent power = 10 − 2 − 0.004 × 10 × (−2) = +8.08D
The difference grows with power and thickness, so it matters for high plus lenses such as aphakic spectacles

In a trial frame, the back lens cell is at the vertex distance. High-power lenses go in the back cell, and stacked lenses do not quite add, because each is at a different distance from the eye (see effectivity in Spherical Lenses)
IOL power calculation treats the cornea and IOL as a two-lens system: the predicted position of the IOL (effective lens position) is the d in the equation, which is why the anterior chamber depth matters (see IOL power formulae)