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Derivation of the Electric Field Distribution and Airy Pattern for a Circular Aperture

When an optical system connects to a single point, the resulting image is not an ideal point, but rather appears as a spread-out distribution due to its electromagnetic wave properties. In particular, in optical systems with circular apertures, this point image exhibits a characteristic fringed structure, known as the Airy pattern, consisting of a bright central peak and concentric dark and bright bands. This page carefully explains, from the perspective of electromagnetic waves, how electromagnetic waves passing through a circular aperture interfere and form the intensity distribution of the point image. It provides a pathway to understanding the essence of image spread and limiting resolution, which cannot be captured by geometrical optics, as behavior of electromagnetic waves.

・Electromagnetic representation of spherical waves

Light converging to a point is represented as a spherical wave in electromagnetic terms. Here, we show how the electric field strength of a spherical wave changes with distance r.

エアリーディスク_009.png

First, consider an area element dS(r) located at a distance r from the convergence point, corresponding to a small solid angle dΩ. By the definition of a solid angle, the area element at spherical radius r can be expressed as follows.

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Therefore, the area element for distance r is as follows:

エアリーディスク_011.png

Next, the electromagnetic energy density u(r) at distance r is expressed as follows, using the permittivity ε and the electric field strength E(r):

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For spherical waves, the energy flow (power) toward the convergence point is constant regardless of distance. Therefore, using the speed of light c, the energy flow is as follows:

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Therefore, from (1)-①', ②, ③, the following relationship is obtained.

エアリーディスク_007.png

Therefore, the electric field strength with respect to distance is as follows:

エアリーディスク_008.png

This relationship will be used later in the derivation of the Airy pattern.

・Reflection and refraction of light rays

When a lens with focal length f is placed in a circular aperture of radius a, and a plane wave is incident on it, the lens converts the wavefront to a spherical surface, forming an optical system that creates a point image at the focal plane. Let's consider this system.

エアリーディスク_012.png

If R is the distance from the lens to the focal plane, then for plane wave incidence, f = R.

However, assume that R is sufficiently large for a.

Here, we define variable coordinates P(X,Y) for the circular opening and variable coordinates I(x,y) for the image plane, and consider the correspondence between the two.

エアリーディスク_013.png

Now, we'll use the spherical wave immediately after it leaves the lens as the reference wavefront, and track the changes in the wavefront from there to the image plane.

First, PI can be expressed as follows:

��エアリーディスク_014.png

Here, since R≫a, (2)-① can be approximated as follows. This approximation is obtained by Taylor expanding the square root expression and ignoring higher-order terms, keeping only the first-order terms, because the aperture radius a is sufficiently small compared to the distance R.

エアリーディスク_016.png

Furthermore, let O' be the point where the image plane intersects the optical axis, and let M be the point where PO' intersects the reference wavefront. In this case, PM can be expressed as follows (applying the approximation R≫a similarly here as well).

エアリーディスク_015.png

Therefore, MI is as follows (applying the approximation R≫a similarly here as well).

エアリーディスク_017.png

Here, let A(X,Y) be the amplitude and phase distribution of the aperture surface.

From the result of (1)-④', the component of the electric field at point I on the image plane that originates from point M on the aperture plane is expressed by the following equation.

エアリーディスク_018.png

Here, vector k is the wave vector. Since vector k and vector MI have the same direction, their product is the scalar product of their absolute values.

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Therefore, the electric field at point I can be obtained by integrating the contribution over the entire aperture surface as follows:

エアリーディスク_021.png

Here, if we set A(X,Y)=1 and substitute (2)-③, we can consider 1/MI(x,y,X,Y) to be approximately constant with respect to distance R, and therefore approximate it with 1/R. In this case, constant terms such as 1/R and common phase can be taken out of the integral, and we get the following.

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Furthermore, the following coordinate system will be introduced.

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In this case, θx = x/R and θy = y/R can be expressed as follows, so (2)-⑥ can be rewritten as follows.

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Furthermore, the following coordinate transformation is performed.

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(2)-⑧,⑨ The following holds true.

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Applying these, (2)-⑦ can be rewritten as follows:

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Since the order of integration can be changed for a double integral over a rectangular region,

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For angular integrals in the range of 0 to 2π, the integration interval is a full loop, so the result does not change regardless of how much ψ is shifted; therefore, ψ can be eliminated.

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Applying a zero-order Bessel function of the first kind, the following holds:

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Substituting (2)-⑭ into (2)-⑬, we get the following:

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Furthermore, applying a first-order Bessel function of the first kind, the following holds:

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Substituting (2)-⑯ into (2)-⑮, we get the following:

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Here, if we let the wavelength be λ and the distance from the origin in the xy-plane be q, then the following relationship holds:

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Substituting this into (2)-⑰, we get the following:

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Furthermore, if the numerical aperture is NA and the aperture diameter is S, the following relationship holds.

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Substituting this into (2)-⑲, we get the following:

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Therefore, the observed intensities are as follows:

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If the central intensity is I₀ , then I₀ can be obtained as follows.

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In other words, the intensity is proportional to the square of the area of the opening and inversely proportional to the square of the distance from the opening to the observation surface.

When the intensity distribution is normalized by the central intensity I0 , the following relationship is obtained.

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(2)-㉔, which include a first-order Bessel function of the first kind, are mathematically known functional forms and are called Airy patterns in optics. They can be represented graphically as follows.

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This function has the characteristic that the vertical axis (i.e., intensity) becomes zero at the horizontal axis values shown in the graph (±3.8317, ±7.0156, ±10.1735, ±13.3268). Furthermore, if we convert the scale of the horizontal axis from (2πNA/λ)・q to q, we get the following.

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The distance from the center to the first dark ring is denoted as qRL , and this qRL is called the Airy disk radius. The bright region inside this radius is called the Airy disk. qRL is expressed as follows:

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Now, let Ui (q) be the superposition of two Airy patterns of the same intensity separated by a distance d, and express it as follows:

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Here, ξ is the phase difference between the two spots. The observed intensity is as follows:

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When the phase difference ξ is an odd multiple of π, cos(ξ) = -1 at q = 0, so the interference term (third term) becomes the largest negative value, and the intensities of the two point images cancel each other out. As a result, Ui(r) becomes zero at q = 0, and the contrast for separating the two points can be made extremely large. This principle is used in semiconductor lithography, and this state of phase alignment can only be achieved in optical systems that are intentionally designed to achieve this.

On the other hand, in imaging with normal incoherent light, the phase ξ is random, and even when using coherent light, observation is generally performed while scanning, so the phases never align at the same time. Therefore, the interference term (third term) does not contribute and disappears. If the intensity at this time is Ur(q), it can be expressed as follows.

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When the distance d between spots is equal to the Airy disk radius qRL = 0.61λ/NA, graphing (2)-㉘ shows that the depth of the central trough drops to approximately 0.26 of the peak intensity, as shown below.

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The state in which two spots are separated by the Airy disk radius is defined as Rayleigh resolution. This condition is widely used as an indicator of the limit to which an optical system can distinguish and separate two points.

  • Related Topics

For a detailed explanation of the electromagnetic wave formula, please refer to the following page.

  • Related materials

[A] Goodman, JW, Introduction to Fourier Optics, 3rd ed., Roberts & Company, 2005.

[B] Born, M. & Wolf, E., Principles of Optics, 7th ed., Cambridge University Press, 1999.

[C] Sato, T. & Nakajima, T., Fourier Optics, Asakura Publishing, 2002.

  • Update History

  • 2026-09: Newly released

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