Light diffracts as it moves through space, bending around obstacles, interfering constructively and destructively. While this can be used as a spectroscopic tool—a diffraction grating disperses light according to wavelength, for example, and is used to produce spectra—diffraction also limits the detail we can obtain in images. [link](a) shows the effect of passing light through a small circular aperture. Instead of a bright spot with sharp edges, a spot with a fuzzy edge surrounded by circles of light is obtained. This pattern is caused by diffraction similar to that produced by a single slit. Light from different parts of the circular aperture interferes constructively and destructively. The effect is most noticeable when the aperture is small, but the effect is there for large apertures, too.

How does diffraction affect the detail that can be observed when light passes through an aperture? [link](b) shows the diffraction pattern produced by two point light sources that are close to one another. The pattern is similar to that for a single point source, and it is just barely possible to tell that there are two light sources rather than one. If they were closer together, as in [link](c), we could not distinguish them, thus limiting the detail or resolution we can obtain. This limit is an inescapable consequence of the wave nature of light.

There are many situations in which diffraction limits the resolution. The acuity of our vision is limited because light passes through the pupil, the circular aperture of our eye. Be aware that the diffraction-like spreading of light is due to the limited diameter of a light beam, not the interaction with an aperture. Thus light passing through a lens with a diameter DD size 12{D} {} shows this effect and spreads, blurring the image, just as light passing through an aperture of diameter DD size 12{D} {} does. So diffraction limits the resolution of any system having a lens or mirror. Telescopes are also limited by diffraction, because of the finite diameter DD size 12{D} {} of their primary mirror.

Just what is the limit? To answer that question, consider the diffraction pattern for a circular aperture, which has a central maximum that is wider and brighter than the maxima surrounding it (similar to a slit) [see [link](a)]. It can be shown that, for a circular aperture of diameter DD size 12{D} {}, the first minimum in the diffraction pattern occurs at θ=1.22λ/Dθ=1.22λ/D size 12{θ=1 “.” “22”λ/D} {} (providing the aperture is large compared with the wavelength of light, which is the case for most optical instruments). The accepted criterion for determining the diffraction limit to resolution based on this angle was developed by Lord Rayleigh in the 19th century. The Rayleigh criterion for the diffraction limit to resolution states that two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. See [link](b). The first minimum is at an angle of θ=1.22λ/Dθ=1.22λ/D size 12{θ=1 “.” “22”λ/D} {}, so that two point objects are just resolvable if they are separated by the angle

where λλ size 12{λ} {} is the wavelength of light (or other electromagnetic radiation) and DD size 12{D} {} is the diameter of the aperture, lens, mirror, etc., with which the two objects are observed. In this expression, θθ size 12{θ} {} has units of radians.

Calculating Diffraction Limits of the Hubble Space Telescope

The primary mirror of the orbiting Hubble Space Telescope has a diameter of 2.40 m. Being in orbit, this telescope avoids the degrading effects of atmospheric distortion on its resolution. (a) What is the angle between two just-resolvable point light sources (perhaps two stars)? Assume an average light wavelength of 550 nm. (b) If these two stars are at the 2 million light year distance of the Andromeda galaxy, how close together can they be and still be resolved? (A light year, or ly, is the distance light travels in 1 year.)

Strategy

The Rayleigh criterion stated in the equation θ=1.22λDθ=1.22λD size 12{θ=1 “.” “22” { {λ} over {D} } } {} gives the smallest possible angle θθ size 12{θ} {} between point sources, or the best obtainable resolution. Once this angle is found, the distance between stars can be calculated, since we are given how far away they are.

Solution for (a)

The Rayleigh criterion for the minimum resolvable angle is

θ=1.22λD.θ=1.22λD. size 12{θ=1 “.” “22” { {λ} over {D} } } {}

Entering known values gives

θ=1.22550×10−9m2.40 m= 2.80×10−7rad.θ=1.22550×10−9m2.40 m= 2.80×10−7rad.

Solution for (b)

The distance ss size 12{s} {} between two objects a distance rr size 12{r} {} away and separated by an angle θθ size 12{θ} {} is s=rθs=rθ size 12{s=rθ} {}.

Substituting known values gives

s=(2.0×106ly)(2.80×10−7rad)=0.56 ly.s=(2.0×106ly)(2.80×10−7rad)=0.56 ly.

Discussion

The angle found in part (a) is extraordinarily small (less than 1/50,000 of a degree), because the primary mirror is so large compared with the wavelength of light. As noticed, diffraction effects are most noticeable when light interacts with objects having sizes on the order of the wavelength of light. However, the effect is still there, and there is a diffraction limit to what is observable. The actual resolution of the Hubble Telescope is not quite as good as that found here. As with all instruments, there are other effects, such as non-uniformities in mirrors or aberrations in lenses that further limit resolution. However, [link] gives an indication of the extent of the detail observable with the Hubble because of its size and quality and especially because it is above the Earth’s atmosphere.

The answer in part (b) indicates that two stars separated by about half a light year can be resolved. The average distance between stars in a galaxy is on the order of 5 light years in the outer parts and about 1 light year near the galactic center. Therefore, the Hubble can resolve most of the individual stars in Andromeda galaxy, even though it lies at such a huge distance that its light takes 2 million years for its light to reach us. [link] shows another mirror used to observe radio waves from outer space.

Diffraction is not only a problem for optical instruments but also for the electromagnetic radiation itself. Any beam of light having a finite diameter DD size 12{D} {} and a wavelength λλ size 12{λ} {} exhibits diffraction spreading. The beam spreads out with an angle θθ size 12{θ} {} given by the equation θ=1.22λDθ=1.22λD size 12{θ=1 “.” “22” { {λ} over {D} } } {}. Take, for example, a laser beam made of rays as parallel as possible (angles between rays as close to θ=0ºθ=0º size 12{θ=0°} {} as possible) instead spreads out at an angle θ=1.22λ/Dθ=1.22λ/D size 12{θ=1 “.” “22”λ/D} {}, where DD size 12{D} {} is the diameter of the beam and λλ size 12{λ} {} is its wavelength. This spreading is impossible to observe for a flashlight, because its beam is not very parallel to start with. However, for long-distance transmission of laser beams or microwave signals, diffraction spreading can be significant (see [link]). To avoid this, we can increase DD size 12{D} {}. This is done for laser light sent to the Moon to measure its distance from the Earth. The laser beam is expanded through a telescope to make DD size 12{D} {} much larger and θθ size 12{θ} {} smaller.

In most biology laboratories, resolution is presented when the use of the microscope is introduced. The ability of a lens to produce sharp images of two closely spaced point objects is called resolution. The smaller the distance xx size 12{x} {} by which two objects can be separated and still be seen as distinct, the greater the resolution. The resolving power of a lens is defined as that distance xx size 12{x} {}. An expression for resolving power is obtained from the Rayleigh criterion. In [link](a) we have two point objects separated by a distance xx size 12{x} {}. According to the Rayleigh criterion, resolution is possible when the minimum angular separation is

where dd size 12{d} {} is the distance between the specimen and the objective lens, and we have used the small angle approximation (i.e., we have assumed that xx size 12{x} {} is much smaller than dd size 12{d} {}), so that tanθ≈sinθ≈θtanθ≈sinθ≈θ size 12{“tan”θ approx “sin”θ approx θ} {}.

Therefore, the resolving power is

Another way to look at this is by re-examining the concept of Numerical Aperture (NANA size 12{ ital “NA”} {}) discussed in Microscopes. There, NANA size 12{ ital “NA”} {} is a measure of the maximum acceptance angle at which the fiber will take light and still contain it within the fiber. [link](b) shows a lens and an object at point P. The NANA size 12{ ital “NA”} {} here is a measure of the ability of the lens to gather light and resolve fine detail. The angle subtended by the lens at its focus is defined to be θ=2αθ=2α size 12{θ=2α} {}. From the figure and again using the small angle approximation, we can write

The NANA for a lens is NA=nsinαNA=nsinα size 12{ ital “NA”=n`”sin”α} {}, where nn size 12{n} {} is the index of refraction of the medium between the objective lens and the object at point P.

From this definition for NANA size 12{ ital “NA”} {}, we can see that

In a microscope, NANA size 12{ ital “NA”} {} is important because it relates to the resolving power of a lens. A lens with a large NANA size 12{ ital “NA”} {} will be able to resolve finer details. Lenses with larger NANA size 12{ ital “NA”} {} will also be able to collect more light and so give a brighter image. Another way to describe this situation is that the larger the NANA size 12{ ital “NA”} {}, the larger the cone of light that can be brought into the lens, and so more of the diffraction modes will be collected. Thus the microscope has more information to form a clear image, and so its resolving power will be higher.

One of the consequences of diffraction is that the focal point of a beam has a finite width and intensity distribution. Consider focusing when only considering geometric optics, shown in [link](a). The focal point is infinitely small with a huge intensity and the capacity to incinerate most samples irrespective of the NANA size 12{ ital “NA”} {} of the objective lens. For wave optics, due to diffraction, the focal point spreads to become a focal spot (see [link](b)) with the size of the spot decreasing with increasing NANA size 12{ ital “NA”} {}. Consequently, the intensity in the focal spot increases with increasing NANA size 12{ ital “NA”} {}. The higher the NANA size 12{ ital “NA”} {}, the greater the chances of photodegrading the specimen. However, the spot never becomes a true point.