Principal planes

The principal planes of a lens or lens system are two planes perpendicular to the optical axis, the front plane $H$ and the rear plane $H'$, with the property that a ray entering the system at height $y$ on $H$ leaves as if from the same height $y$ on $H'$. They are conjugate at a lateral magnification of +1. Paraxially, any system in air then behaves like a thin lens whose single plane is split in two: the front focal length is measured from $H$, the effective focal length from $H'$, and the imaging equation holds with object and image distances measured from $H$ and $H'$. For a thin lens the planes coincide with the lens; in thick lenses they sit a few millimeters inside the glass, and in meniscus lenses and multi-element systems they can lie outside the hardware.

Their intersections with the axis, the principal points, form with the two focal points and two nodal points the six cardinal points of a paraxial system.

Thick-lens formulas

For a lens of index $n$ in air with radii $R_1$, $R_2$ and center thickness $d$, light travels left to right, radii are positive when the center of curvature lies to the right, and $h$, $h'$ are measured from the front and rear vertex, positive to the right. The lensmaker's equation is

$$\frac{1}{f} = (n-1)\left[\frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)\,d}{n R_1 R_2}\right],$$

and the principal planes lie at

$$h = -\frac{f (n-1)\, d}{n R_2}, \qquad h' = -\frac{f (n-1)\, d}{n R_1}.$$

The back focal length, measured from the rear vertex, is $f + h'$.

Worked example: biconvex N-BK7 lens

For $R_1$ = +50 mm, $R_2$ = −50 mm, $d$ = 10 mm and $n$ = 1.5168 (N-BK7 at 587.6 nm):

• $f$ = 50.08 mm (48.37 mm from the thin-lens formula)
• $h$ = +3.41 mm: $H$ lies 3.41 mm inside the front vertex
• $h'$ = −3.41 mm: $H'$ lies 3.41 mm inside the rear vertex
• the two planes are 3.17 mm apart, and the BFL is 50.08 − 3.41 = 46.67 mm

An object 200 mm in front of $H$ (196.6 mm from the front vertex) is imaged 66.81 mm behind $H'$, which is 63.40 mm behind the rear vertex, at a magnification of −0.334. Measuring from the vertices instead would misplace the image by several millimeters.

Planes outside the lens

Bending a lens moves its principal planes. A positive meniscus N-BK7 lens with $R_1$ = 30 mm, $R_2$ = 60 mm and $d$ = 8 mm has $f$ = 106.4 mm, $h$ = −4.83 mm and $h'$ = −9.67 mm. Both planes are in front of the lens: $H'$ is 1.67 mm ahead of the front vertex, and the focus lies 96.8 mm behind the rear surface. Separated positive and negative groups carry the shift further, in telephoto and retrofocus designs.

A ball lens is the symmetric limit: with $R_1 = D/2$, $R_2 = -D/2$ and $d = D$, the formulas place both principal planes at the center of the sphere. For a 2 mm N-BK7 ball the EFL, $nD/[4(n-1)]$, is 1.47 mm from the center, and the BFL 0.47 mm from the surface.

Matrix description

In the ABCD matrix of a system taken from its first to its last surface, with $C = -1/f$, the principal planes follow from the matrix elements:

$$h = \frac{D - 1}{C}, \qquad h' = \frac{1 - A}{C}.$$

For the biconvex lens above, $A = D$ = 0.9319, $B$ = 6.593 mm and $C$ = −0.01997 mm⁻¹, which return $f$ = 50.08 mm and $h = -h'$ = 3.41 mm.

Nodal points and measurement

The nodal points are the axial pair through which a ray passes undeviated in angle. With the same medium on both sides, as for any lens in air, they coincide with the principal points. In a nodal-slide measurement the lens is rotated about a vertical axis; when the axis passes through the rear nodal point the image of a distant source stays still, and the distance from the axis to the focal plane is the EFL. With an immersed image side, as in the eye, the nodal points shift away from the principal points.

Pitfalls

• Mounting a fiber or detector at the EFL from the last surface. The focus is at the BFL from the surface; the EFL is measured from $H'$.
• Mixing sign conventions between texts; the formulas above assume the convention stated with them.
• Applying paraxial planes to marginal rays. At large apertures incoming and outgoing rays meet on a curved surface, a sphere centered on the focus for a lens that satisfies the Abbe sine condition.
• Assuming the planes are fixed across wavelength. The focal length and plane positions quoted at 587.6 nm shift with the index in the infrared.

Common questions

Where are the principal planes of a plano-convex lens?

One touches the curved vertex and the other lies inside the glass, a distance $d/n$ in front of the flat face. For a 3.6 mm thick N-BK7 lens that is 2.37 mm.

How far apart are the principal planes?

For an equiconvex singlet in air, roughly a third of the center thickness: 3.17 mm for the 10 mm lens above. In a ball lens they coincide at the center.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 6; M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).