# Airy Pattern Formation

When an image is formed in the focused image plane of an optical microscope, every point in the specimen is represented by an Airy diffraction pattern having a finite spread. This occurs because light waves emitted from a point source are not focused into an infinitely small point by the objective, but converge together and interfere near the intermediate image plane to produce a three-dimensional Fraunhofer diffraction pattern.

This interactive tutorial explores the origin of Airy diffraction patterns formed by the rear aperture of the microscope objective and observed at the intermediate image plane. The figure on the left presents a diagrammatic illustration of a microscope optical train where a single lens (labeled **Objective**) represents the microscope objective. Monochromatic light is emitted from a point source on an imaginary specimen (labeled point **A**) whose size is smaller than can be resolved with the microscope optics. Assuming the refractive index of the imaging medium is **n** and the angular aperture of the objective equals **θ**, the numerical aperture is**:**

At the intermediate image plane, point **A'(o)** represents the distribution of light emitted by the point source **A**. The spherical wave surface emanating from the objective is composed of a series of point sources, each emitting a spherical wave (termed a Huygens wavelet), which produce corresponding wave surfaces in the image plane having centers at point **A'(o)**. For the purposes of this interactive tutorial, wavelets are depicted arising from three points on the spherical wavefront (**M(o)**, **M(1)**, and **M(-1)**), which arrive at point **A'(o)** in phase and undergo constructive interference to yield a bright image of the aperture.

The **Wavelet Spread** slider can be used to translate the wavelets emanating from points **M(o)**, **M(1)**, and **M(-1)** away from point **A'(o)** in the image plane. As the slider is moved from left to right, wavelets spread across the image plane are shown passing through points **A'(n)** and **A'(-n)**. These wavelets, emitted from all points along the spherical wavefront, have further to travel before reaching the image plane and are becoming increasingly out of phase, forming a distribution of light intensity that falls off in a bell-shaped curve. The light distribution at the image plane can be mathematically described by a Bessel function of the form**:**

^{2}

where **θ** is the angle **A'(o)-M(o)-A'(n)**. When the wavelets reach points **A'(1)** and **A'(-1)**, they are 180 degrees out of phase, resulting in destructive interference that cancels each wavelet. If the contribution from wavelets radiating from all points of the aperture (rather than just **M(o)**, **M(1)**, and **M(-1)**) is considered, the corresponding sum resulting from destructive interference equals zero and no light is present at **A'(1)** and **A'(-1)**. This is shown graphically by the Airy pattern presented to the right of the optical train. An alternating concentric pattern of light and dark (red) bands is illustrated in the upper portion of the Airy pattern, which is drawn with the plane of the pattern parallel to the intermediate image plane. The blue line traversing the center of the three-dimensional Airy pattern represents point **A'(o)** in the image plane. As the wavelet spread is viewed across the image plane in the optical train tutorial, the blue line superimposed on the Airy pattern splits into two components (**A'(n)** and **A'(-n)**) that track the progress of the central Airy disk intensity. When the wavelets reach **A'(1)** and **A'(-1)** on the image plane, the blue lines on the Airy pattern are located at the first minimum (zero intensity).

As the wavelets progress past points **A'(1)** and **A'(-1)** on the image plane (not illustrated), the Bessel function again rises above zero to form a second maximum having dramatically reduced amplitude. Moving the wavelets still further out the image plane where they continue to interfere both constructively and destructively, the Bessel function oscillates with progressively decreasing amplitude until the intensity falls to zero. The result is an Airy pattern, illustrated to the right in the tutorial, where the central Airy disk is surrounded by an alternating series of interlaced dark and light zones of diminishing intensity where the wavelets interfere both constructively (light zones) and destructively (dark zones). These represent both the zeroth and higher order diffraction patterns formed by a point source of light in the image plane of a perfect (aberration free) lens.