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Airy disk normalization and encircled energy 2026-10-14

Peak normalization

P (0) = 1

does not make the integral of a point spread function equal to one. In the paraxial approximation to the focal plane, define

b = π D / (λ f)

and

u = b r

. The unit-integral point spread function normalization is

p (r) = \frac{b ^{2}}{4 π} P (b r), \int_{R^{2}} p (r) d^{2} r = 1.

Here

p

has units of inverse area; multiplying by the detected radiant flux gives image-plane irradiance. Its encircled energy, meaning the fraction inside radius

R = U / b

, is

E (U) = \int_{0}^{U / b} 2 π r p (r) d r = \frac{1}{2} \int_{0}^{U} u P (u) d u = 1 - J_{0} (U)^{2} - J_{1} (U)^{2} .

This is an integral over area, so its radial weight is

u

. An unweighted integral of a line through the centre is a different quantity.

A short proof uses the Bessel function of the first kind identities

J_{0}^{'} = - J_{1}

and

J_{1}^{'} = J_{0} - J_{1} / u

. They give

\frac{d}{d U} (1 - J_{0} (U)^{2} - J_{1} (U)^{2}) = \frac{2 J _{1} ( U ) ^{2}}{U} = \frac{U}{2} P (U) .

Both expressions for

E

vanish at zero. Since these Bessel functions of the first kind tend to zero as

U \to \infty

,

E (\infty) = 1

, establishing the normalization. At the first dark ring,

E (j_{1, 1}) = 1 - J_{0} (j_{1, 1})^{2} ≃ 0.837784869.

A circular detector region ending at the first zero therefore misses about

16.22%

of the ideal radiant flux, even though the surrounding rings have low peak light intensity. This matters when converting measured counts into total source brightness or choosing an image aperture for a photodetector.

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Encircled energy 2026-10-14

Encircled energy is the fraction of total image radiant flux within a circle of radius

R

around the centre of a point spread function.

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