A circular aperture does not form a perfectly sharp geometric point image, even when the lens or mirror is ideal.
Because light is a wave, it diffracts at the edge of the opening. In the far-field or focal-plane limit,
a point source imaged through a circular aperture produces the well-known Airy pattern: a bright central disk surrounded by fainter concentric rings.
This pattern is extremely important in optics because it sets the fundamental diffraction limit for cameras,
microscopes, telescopes, and other imaging systems.
For a circular aperture of diameter \(D\), the first dark ring occurs when the Bessel-function factor in the field first vanishes.
That gives the standard small-angle result
\[
\theta_1 \approx \frac{1.22\lambda}{D}.
\]
Here \(\lambda\) is the wavelength and \(\theta_1\) is the angular radius of the central Airy disk.
If a lens of focal length \(f\) forms the diffraction pattern in its focal plane, then the corresponding physical radius is
\[
r_1 = f\theta_1 \approx \frac{1.22\lambda f}{D}.
\]
This is one of the most useful formulas in practical optics. It tells you that the diffraction spot becomes smaller when the aperture is larger,
and it becomes larger when the wavelength or focal length increases. In other words, large telescopes and camera lenses can form finer diffraction-limited images.
For the sample values
\(D = 5\,\text{mm}\),
\(\lambda = 550\,\text{nm}\),
and
\(f = 50\,\text{cm}\),
the SI-unit conversions are
\[
\begin{aligned}
D &= 5.00\times 10^{-3}\,\text{m},\\
\lambda &= 5.50\times 10^{-7}\,\text{m},\\
f &= 5.00\times 10^{-1}\,\text{m}.
\end{aligned}
\]
The first-minimum angle is therefore
\[
\theta_1 \approx \frac{1.22(5.50\times 10^{-7})}{5.00\times 10^{-3}}
= 1.34\times 10^{-4}\,\text{rad}
\approx 0.00769^\circ.
\]
The focal-plane Airy radius becomes
\[
r_1 = \frac{1.22\lambda f}{D}
= \frac{1.22(5.50\times 10^{-7})(0.500)}{5.00\times 10^{-3}}
= 6.71\times 10^{-5}\,\text{m}
= 67.1\,\mu\text{m}.
\]
So the correct radius for this particular sample is about \(67\,\mu\text{m}\).
A value near \(6.7\,\mu\text{m}\) would correspond to a focal length that is about ten times smaller, roughly \(5\,\text{cm}\) instead of \(50\,\text{cm}\).
This distinction matters because the diffraction spot scales directly with focal length.
The detailed intensity profile of the Airy pattern is not described by a sine function. Instead, it is controlled by the first-order Bessel function \(J_1\).
In the focal plane, if \(\rho\) is the radial distance from the image center, then the standard Fraunhofer expression is
\[
I(\rho) = I_0\left[\frac{2J_1(\beta)}{\beta}\right]^2,
\qquad
\beta = \frac{\pi D\rho}{\lambda f}.
\]
Here \(I_0\) is the central peak intensity. When \(\rho = 0\), the expression is understood through a limit, and it gives \(I = I_0\).
As \(\rho\) increases, the intensity oscillates with decreasing amplitude, creating the bright central spot and the surrounding rings.
The first minimum appears when \(\beta\) reaches the first zero of \(J_1\), which is why the coefficient 1.22 appears in the Airy-disk formula.
Another important practical fact is that about 83.8% of the total light energy falls inside the central Airy disk.
That is why the central spot dominates the image visually even though faint outer rings are present.
In astronomy, a bright star viewed through a diffraction-limited telescope shows exactly this kind of pattern,
and the ability to distinguish two nearby stars depends strongly on the size of the Airy disk.
This is the basis of classical resolution criteria such as the Rayleigh criterion.
The model used here is the far-field or focal-plane approximation. That means the screen is assumed to lie effectively in the Fraunhofer regime,
or equivalently in the focal plane of an imaging element. In a more advanced university treatment one studies Fresnel diffraction,
aberrations, obscured apertures, apodization, and modulation transfer functions. But for most standard optics problems,
the key formulas are the first-minimum angle, the Airy radius, and the Bessel-based intensity law.
\[
\theta_1 \approx \frac{1.22\lambda}{D},
\qquad
r_1 = \frac{1.22\lambda f}{D},
\qquad
I = I_0\left[\frac{2J_1(\beta)}{\beta}\right]^2,
\qquad
\beta = \frac{\pi D\rho}{\lambda f}.
\]
Together, these equations tell you the size of the central diffraction spot, the angular spread of the image,
and how the brightness changes as you move outward from the center. They form the core of circular-aperture diffraction analysis.
