D.  Diffraction-Limited Optics

A lens of diameter D is in effect a large circular aperture through which light passes. Suppose a lens is used to focus plane waves (light from a distant source) to form a “spot” in the focal plane of the lens, much as is done in geometrical optics. Is the focused spot truly a point? Reference to Figure 4-20 indicates that the focused spot is actually a tiny diffraction pattern—with a bright disk at the center (the so-called airy disk) surrounded by dark and bright rings, as pictured earlier in Figure 4-13a.

In Figure 4-23, we see collimated light incident on a lens of focal length f. The lens serves as both a circular aperture of diameter D to intercept the plane waves and a lens to focus the light on the screen, as shown in Figure 4-18b. Since the setup in Figure 4-23 matches the conditions shown in Figure 4-18b, we are assured that a Fraunhofer diffraction pattern will form at the “focal spot” of the lens.

Figure 4-23  Fraunhofer diffraction pattern formed in the focal plane of a lens of focal length f (Drawing is not to scale.)

The diffraction pattern is, in truth then, an array of alternate bright and dark rings, with a bright spot at the center, even though the array is very small and hardly observable to the human eye. From the equations given with Figure 4-20, we see that the diameter of the central bright spot—inside the surrounding rings—is itself of size 2R, where, from Equation 4-26,

(4-31)

where Z¢ = f

While indeed small, the diffraction pattern overall is greater than 2R, demonstrating clearly that a lens focuses collimated light to a small diffraction pattern of rings and not to a point. However, when the lens is inches in size, we do justifiably refer to the focal plane pattern as a “point,” ignoring all structure within the “point.” Example 10 provides us with a “feel” for the size of the structure in the focused spot, when a lens of nominal size becomes the circular aperture that gives rise to the airy disk diffraction pattern.

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Example 10

Determine the size of the airy disk at the center of the diffraction pattern formed by a lens such as that shown in Figure 4-23, if the lens is 4 cm in diameter and its focal length is 15 cm. Assume a wavelength of 550 nm incident on the lens.

Solution: Using Equation 4-31 with Z¢ = f, the diameter of the airy disk is



2R = 5.03 × 10^–6 m

Thus, the central bright spot (airy disk) in the diffraction pattern is only 5 micrometers in diameter. So, even though the focused spot is not a true point, it is small enough to be considered so in the world of large lenses, i.e., in the world of geometrical optics.

The previous discussion and example indicate that the size of the focal spot—structure and all—is limited by diffraction. No matter what we do, we can never make the airy disk smaller than that given by 2R = . That is the limit set by diffraction. So all optical systems are limited by diffraction in their ability to form true point images of point objects. We recognize this when we speak of diffraction-limited optics. An ideal optical system therefore can do no better than that permitted by diffraction theory. In fact, a real optical system—which contains imperfections in the optical lenses, variations in the index of refraction of optical components, scattering centers, and the existence of temperature gradients in the intervening atmosphere—will not achieve the quality limit permitted by diffraction theory. Real optical systems are therefore poorer than those limited by diffraction only. We often refer to real systems as many-times diffraction limited and sometimes attach a numerical figure such as “five-times diffraction-limited” to indicate the deviation in quality expected from the given system compared with an ideal “diffraction-limited” system.

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