According to the ray model of light, a perfect lens (one with no aberrations) should be able to form a perfect image. But the ray model of light, though a very good model for lenses, is not an absolutely correct description of light. If we look closely, the wave aspects of light haven’t entirely disappeared. In fact, the performance of optical equipment is limited by the diffraction of light.

FIGURE 35.24a shows a plane wave, with parallel light rays, being focused by a lens of diameter $D$. According to the ray model of light, a perfect lens would focus parallel rays to a perfect point. Notice, though, that only a piece of each wave front passes through the lens and gets focused. In effect, the lens itself acts as a circular aperture in an opaque barrier, allowing through only a portion of each wave front. Consequently, the lens diffracts the light wave. The diffraction is usually very small because $D$ is usually much greater than the wavelength of the light; nonetheless, this small amount of diffraction is the limiting factor in how well the lens can focus the light.

FIGURE 35.24 A lens both focuses and diffracts the light passing through.

FIGURE 35.24 Full Alternative Text

Full Alternative Text

FIGURE 35.24b separates the diffraction from the focusing by modeling the lens as an actual aperture of diameter $D$ followed by an “ideal” diffractionless lens. You learned in Chapter 33 that a circular aperture produces a diffraction pattern with a bright central maximum surrounded by dimmer fringes. A converging lens brings this diffraction pattern to a focus in the image plane, as shown in FIGURE 35.24c. As a result, a perfect lens focuses parallel light rays not to a perfect point of light, as we expected, but to a small, circular diffraction pattern.

The angle to the first minimum of a circular diffraction pattern is ${\theta }_1=1.22\text{λ}/D$. The ray that passes through the center of a lens is not bent, so Figure 35.24c uses this ray to show that the position of the dark fringe is $y_1=f\text{tan}{\theta }_1\approx f{\theta }_1$. Thus the width of the central maximum in the focal plane is

This is the minimum spot size to which a lens can focus light.

Lenses are often limited by aberrations, so not all lenses can focus parallel light rays to a spot this small. A well-crafted lens, for which Equation 35.13 is the minimum spot size, is called a diffraction-limited lens. No optical design can overcome the spreading of light due to diffraction, and it is because of this spreading that the image point has a minimum spot size. The image of an actual object, rather than of parallel rays, becomes a mosaic of overlapping diffraction patterns, so even the most perfect lens inevitably forms an image that is slightly fuzzy.

For various reasons, it is difficult to produce a diffraction-limited lens having a focal length that is less than half its diameter—that is, $f>0.5D$. This implies that the smallest diameter to which you can focus a spot of light, no matter how hard you try, is ${\mathit{w}}_{\text{min}}\approx \text{λ}$. This is a fundamental limit on the performance of optical equipment. Diffraction has very real consequences!

One example of these consequences is found in the manufacturing of integrated circuits. Integrated circuits are made by creating a “mask” showing all the components and their connections. A lens images this mask onto the surface of a semiconductor wafer that has been coated with a substance called photoresist. Bright areas in the mask expose the photoresist, and subsequent processing steps chemically etch away the exposed areas while leaving behind areas that had been in the shadows of the mask. This process is called photolithography.

The power of a microprocessor and the amount of memory in a memory chip depend on how small the circuit elements can be made. Diffraction dictates that a circuit element can be no smaller than the smallest spot to which light can be focused. With clever technology, such as decreasing the effective wavelength by covering the surface with a liquid that has a high index of refraction, the minimum spot size can be reduced to $\approx \text{λ}/4$. If the mask is projected with an ultraviolet laser having $\text{λ}\approx 200\text{nm,}$ then the smallest elements on a chip are about $50\text{nm}$ wide. This is, in fact, just about the current limit of technology.