Question

(a) Light from distant objects, such as stars, are imaged by a single lens. Assuming the focal length of the lens is 10 cm and the aperture size is 2 cm, what is the minimum resolvable angle? Write every step. (b) The same lens is used for imaging an amplitude grating. What is the minimum grating spacing that will yield any intensity variation(s) on the image plane using (i) coherent and (ii) incoherent illumination? Be very specific in your answer and write every detail/assumption.

Answer 1

( A )

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. (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.

(a) Monochromatic light passed through a small circular aperture produces this diffraction pattern. (b) Two-point light sources that are close to one another produce overlapping images because of diffraction. (c) If they are closer together, they cannot be resolved or distinguished.

How does diffraction affect the detail that can be observed when light passes through an aperture? 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 (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 shows this effect and spreads, blurring the image, just as light passing through an aperture of diameter 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 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). It can be shown that, for a circular aperture of diameter , the first minimum in the diffraction pattern occurs at (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. The first minimum is at an angle of , so that two point objects are just resolvable if they are separated by the angle

where is the wavelength of light (or other electromagnetic radiation) and is the diameter of the aperture, lens, mirror, etc., with which the two objects are observed. In this expression, has units of radians.

(a) Graph of intensity of the diffraction pattern for a circular aperture. Note that, similar to a single slit, the central maximum is wider and brighter than those to the sides. (b) Two-point objects produce overlapping diffraction patterns. Shown here is the Rayleigh criterion for being just resolvable. The central maximum of one pattern lies on the first minimum of the other.

The Rayleigh criterion for the minimum resolvable angle is

Entering known values gives

( B )

( i ) In the case of a (spatially) coherent illumination (see figure V-3), the complex amplitudes at all object points follow the same evolution with time. Experimentally, this is realized by placing a light source of small dimension at the front focal plane of a lens.

Image V-3- General schematic view of a coherent illumination system.

As the angle is as calculated above. The minimum grating space which resolvable via this lens is for a monochromatic light

Where 10 cm is the focal length of the lens.

( ii ) The illumination is said to be spatially incoherent is the source is (spatially) wide enough below

In this case, we need to make our lense distance enough to see the well-resolved image thus in this case the maximum grating space we can resolve is low i.e resolving will be possible for distance greater than .