Question

1. Explain single slit diffraction? What is Fraunhoffer diffraction? Describe diffraction grating and its use in CD tracking? Calculate the single slit width (a) for a destructive interference that occurs in case of single slit diffraction with dark = 100, m=+2, and 2 = 740 nm? 2. Describe principle, construction of reflecting telescope? What is limiting condition of resolution or Rayleigh's criterion?

Answer 1

(1) Single Slit Diffraction

In the single-slit diffraction experiment, we can observe the bending phenomenon of light or diffraction that causes light from a coherent source interfere with itself and produce a distinctive pattern on the screen called the diffraction pattern. Diffraction is evident when the sources are small enough that they are relatively the size of the wavelength of light

Single Slit Diffraction Formula

We shall assume the slit width a << D. x`D is the separation between slit and source.

D a

The angular position of any point on the screen by Θ measured from the slit centre which divides the slit by a/2 lengths. To describe the pattern, the condition for dark fringes. Also, divide the slit into zones of equal widths a/2. considering a pair of rays that emanate from distances a/2 from each other as shown below.

a/2 e a/2

ΔL=(a/2)sinΘ

Remember that this is a calculation valid only if D is very large.

considering any number of ray pairings that start from a distance a/2 from one another such as the bottom two rays in the diagram. Any arbitrary pair of rays at a distance a/2 can be considered.

For a dark fringe, the path difference must cause destructive interference; the path difference must be out of phase by λ/2. (λ is the wavelength)

For the first fringe,

ΔL = λ/2 = (a/2)sinΘ

λ = a sin θ

For a ray emanating from any point in the slit, there exists another ray at a distance a2 that can cause destructive interference.

Thus, at θ = sin−1(λa), there is destructive interference as any ray emanating from a point has a counterpart that causes destructive interference. Hence, a dark fringe is obtained.

For the next fringe, we can divide the slit into 4 equal parts of a/4 and apply the same logic. Thus, for the second minima:

λ2=(a/4)sinΘ

2λ=asinΘ

Similarly, for the nth fringe, we can divide the slit into 2n parts and use this condition as

The Central Maximum

The maxima lie between the minima and the width of the central maximum is simply the distance between the 1st order minima from the centre of the screen on both sides of the centre.

The position of the minima given by y (measured from the centre of the screen) is:

tanθ≈θ≈y/D

For small ϑ,

sin θ≈θ

⇒ λ = a sin θ≈aθ

⇒ θ = y/D = λa

⇒ y = λDa

The width of the central maximum is simply twice this value

⇒ Width of central maximum = 2λDa

⇒ Angular width of central maximum = 2θ = 2λa

Fraunhofer diffraction is the type of diffraction that occurs in the limit of small Fresnel number . In Fraunhofer diffraction, the diffraction pattern is independent of the distance to the screen, depending only on the angles to the screen from the aperture.

A diffraction grating is an optical element that divides(disperses) light composed of lots of different wavelengths(e.g., white light) into light components by wavelength. The simplest type of grating is one with a large number of evenly spaced parallel slits. In optics, a diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions.

use of diffraction grating in cd tracking

A recording on a CD is in the form of microscopic pits of different lengths that carry the information. These pits are placed in rows of the same width and equal distance, which form a diffraction grating on the mirror surface of the CD.

(2) Reflecting Telescope:

Reflectors are used not only to examine the visible region of the electromagnetic spectrum but also to explore both the shorter- and longer-wavelength regions adjacent to it (i.e., the ultraviolet and the infrared). The name of this type of instrument is derived from the fact that the primary mirror reflects the light back to a focus instead of refracting it. The primary mirror usually has a concave spherical or parabolic shape, and, as it reflects the light, it inverts the image at the focal plane. The figure below illustrates the principle of a concave reflecting mirror.

primary mirror

The primary mirror is located at the lower end of the telescope tube in a reflector and has its front surface coated with an extremely thin film of metal, such as aluminum. The back of the mirror is usually made of glass, although other materials have been used from time to time. Pyrex (trademark) was the principal glass of choice for many of the older large telescopes, but new technology has led to the development and widespread use of a number of glasses with very low coefficients of expansion. A low coefficient of expansion means that the shape of the mirror will not change significantly as the temperature of the telescope changes during the night. Since the back of the mirror serves only to provide the desired form and physical support, it does not have to meet the high optical quality standards required for a lens.

Reflecting telescopes have a number of other advantages over refractors. They are not subject to chromatic aberration because reflected light does not disperse according to wavelength. Also, the telescope tube of a reflector is shorter than that of a refractor of the same diameter, which reduces the cost of the tube. Consequently, the dome for housing a reflector is smaller and more economical to construct. So far only the primary mirror for the reflector has been discussed. In the figure, one might wonder about the location of the eyepiece. The primary mirror reflects the light of the celestial object to the prime focus near the upper end of the tube. Obviously, if an observer put his eye there to observe with a modest-sized reflector, he would block out the light from the primary mirror with his head. Isaac Newton placed a small plane mirror at an angle of 45 inside the prime focus and thereby brought the focus to the side of the telescope tube. The amount of light lost by this procedure is very small when compared to the total light-gathering power of the primary mirror. The Newtonian reflector is popular among amateur telescope makers.

A contemporary of Newton, N. Cassegrain of France, invented another type of reflector. Called the Cassegrainian telescope, this instrument employs a small convex mirror to reflect the light back through a small hole in the primary mirror to a focus located behind the primary. Figure 5 illustrates a typical Cassegrain reflector. Some large telescopes of this kind do not have a hole in the primary mirror but use a small plane mirror in front of the primary to reflect the light outside the main tube and provide another place for observation. The Cassegrain design usually permits short tubes relative to their mirror diameter.

Most large reflecting telescopes that are currently in use have a cage at their prime focus that permits the observer to sit inside the telescope tube while operating the instrument. The five-meter reflector at Palomar Observatory, near San Diego, Calif., is so equipped. Reflectors, like refractors, usually have small guide telescopes mounted parallel to their main optical axis to facilitate locating the desired object. These guide telescopes have low magnification and a wide field of view, the latter being a desirable attribute for finding stars or other remote cosmic objects.

primary mirror eyepiece secondary mirror telescope tube

• For a circular aperture, lens, or mirror, the Rayleigh criterion 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.
• This occurs for two point objects separated by the angle $\theta =1\text{.}\text{22}\frac{\lambda }{D}$, where $\lambda$ is the wavelength of light (or other electromagnetic radiation) and is the diameter of the aperture, lens, mirror, etc. This equation also gives the angular spreading of a source of light having a diameter .

For single slit diffraction phenomena Width slit with a mth For order dare fringe, a sinom = ma [mth destructive fringe Om = 10° for m=2 a = 740 nm 740X10-9 a= ma Sin Om 2 x 740 X10-9 Sin (100) a= 8523 X10-9 a= 8.523 um Shit width a=8.523 um Ans