Tru Physics

Airy Disk

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Introduction

The Airy disk is the bright spot at the center of a pattern that is produced when a plane wave of light is diffracted by a circular aperture. It is named after the British astronomer Sir George Biddell Airy. The pattern consists of a bright central region, or disk, surrounded by a series of concentric light and dark rings. The Airy disk is of crucial importance in optics and especially in photography and astronomy, as it sets a fundamental limit to the resolution of a camera or telescope.

Airy disk diffraction pattern.
Airy disk diffraction pattern generated using Python.

Diffraction and the Airy Disk

When light passes through an aperture, such as the opening of a camera lens or a telescope, it doesn’t simply go straight through. Instead, it spreads out in a pattern of waves (see wave-particle duality). This is known as diffraction. The resulting diffraction pattern for a circular aperture is the Airy disk.

The central bright disk and the surrounding rings are caused by the constructive and destructive interference of the diffracted light waves. The bright central disk is the result of constructive interference, while the dark rings are caused by destructive interference.

The size of the Airy disk depends on the wavelength of the light and the size of the aperture. The diameter of the first dark ring, also known as the Airy disk radius, can be calculated using the following formula:

D = 2.44 \lambda F

where D is the diameter of the first dark ring, \lambda is the wavelength of light, and F is the f-number of the aperture.

Impact on Image Quality and Resolution

The Airy disk plays a crucial role in determining the image quality and resolution in optical systems. This is because it sets the diffraction limit, the fundamental limit to the finest detail that a camera or telescope can resolve.

Two points in the image are said to be just resolved when the center of the Airy disk of one point coincides with the first dark ring of the other point. This criterion is known as the Rayleigh criterion, and it’s given by the formula:

\theta = 1.22 \dfrac{\lambda}{D}

where \theta is the angular resolution, \lambda is the wavelength of light, and D is the diameter of the aperture.

Applications in Astronomy

In astronomy, the Airy disk is particularly important for understanding the resolution of telescopes. The size of the Airy disk determines the smallest angle between two stars that can be resolved by the telescope. This is why larger telescopes, which have larger apertures, can resolve finer details: they have smaller Airy disks.

However, due to the wave nature of light, even a perfect telescope with an infinitely large aperture would still be limited by diffraction and the size of the Airy disk. This is known as the diffraction limit.

Wave Optics and the Airy Function

In the realm of wave optics, the intensity distribution of the Airy disk is described by the square of the absolute value of the Airy function. The Airy function, named after Airy, is the solution to a certain differential equation that describes the diffraction of light by a circular aperture. The intensity of light in the Airy disk pattern can be described by the formula:

I(r) = I_0 \left( \dfrac{2J_1(kr)}{kr} \right)^2

where I(r) is the intensity at a distance r from the center of the pattern, I_0 is the peak intensity at the center, J_1 is the Bessel function of the first kind of order one, and k is the wave number, equal to 2\pi/\lambda.

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