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Diffraction - Class 12 Physics

Notes and Question Solutions for Class 12 NEB Physics on Diffraction

The spreading of light around the edge of an aperture or obstacle is called the diffraction.

Types of Diffraction

When light passes through a narrow aperture or obstacle, it spread out into the geometric shadow of the aperture or the obstacle. This spreading of light around the edge of an aperture or obstacle is called the diffraction. Diffraction of a sound wave is larger than light waves as a wavelength of sound is larger than a wavelength of light.

The light passing through a narrow slit produces a diffraction pattern consisting of a broad, intense central band called the central maximum and a series of narrower less intense bands called secondary maxima.

Diffraction pattern associates with light passing through a sharp edge of n object are shown in the figure. A similar pattern is observed when the light passes through the edges at both inside and outside the edge. As the light passes the vertical edge at left, it flares left and right and undergoes interference producing a pattern along the left edge. Actually that pattern lies within what would have been a shadow of the blade of geometric optics prevailed. The diffraction pattern of a disc in which a bridge spot called ‘poison’s spot’ is at the center and circular fringes extend outward from shadow’s edge the occurrence of the ‘poison’s spot’ is called “ Freshnel bright spot”.

Diffraction and Huygens’ Principle

The diffraction patterns, as discussed above, produce due to wave nature of light. It can be explained in terms of Huygens’ Principle which states that every point of a wavefront can be considered as a source of secondary wavelets, that spread out in all directions with a speed equal to the speed of propagation of the wave. The intensity of light at any point on the screen is obtained by superposing the individual displacements produced by these secondary wavelets, arising from the aperture or obstacle. So, diffraction is the interference produced by the secondary waves from different parts of the same wavefront.

Types of Diffraction

Diffraction patterns are classified into two categories on which source and screen are placed. When either the source or the screen is near the aperture or obstacle, the wavefronts are spherical and the pattern is quite complex. This is called the Fresnel diffraction. When both source and screen are placed at a greater distance from the aperture, the incident light planes waves and the rays leaving the opening are parallel. This is called the Fraunhofer diffraction.

Fraunhofer Diffraction at Single Slit

a)Fraunhoferdiffraction
$b) fraunhofer diffraction$
b) fraunhofer diffraction

A narrow parallel beam of light from a source is incident normally on a rectangular, vertical slit of width a. the waves propagating out of the slit diffract and produce a diffraction pattern on the screen with a central bright fringe and a number of fainter fringes on both sides of the central fringe. These fringes are images of the single slit.

Theory

Suppose a plane wave of wavelength λ, falls normally on a narrow rectangular slit of width a. now, divide the slit into two equal halves as shown in the figure. All the waves are in phase at the slit. Consider two rays 1 and 3 travelling toward the screen at an angle Ï´. Ray 1 travels farther than ray 3 by a path difference between rays 2 and 4 are also (a/2) sin Ï´, as between rays 3 and 5. If this path difference is exactly half a wavelength, two waves cancel each other and produce destructive interference. So all such pairs of rays from two halves interfere destructively and in condition,

$$ \begin{align*} \frac a2\sin \theta &= \pm \frac {\lambda }{2} \\ \text {or,} \: \sin \theta &= \pm \frac {\lambda }{a} \\ \end{align*} $$

If the slit is divided into four equal parts, dark images are obtained on the screen and we have $$ \begin{align*} \\ \sin \theta &= \pm \frac {2\lambda }{a} \\ \end{align*} $$

So, by using similar pairing process, the destructive interference of higher order is obtained and for this,

$$\sin \theta = m \frac {\lambda }{a} \dots (v) m = \pm 1, \pm 2, \pm 3, \dots$$

The $ \pm $ sign indicate the destructive interference occur in both sides of the central maxima on the screen. Above equation gives the value of Ï´ for diffraction pattern of zero intensity. The first maximum is formed when the slit is divided into three equal parts as shown in the figure. And a direction is considered in which the path difference between their ends are λ/2. Wavelets from strips in two adjacent parts cancel, and only one part remains that gives a much less bright band.

Through, the equation (v) does not give the variation of intensity, the general features of intensity distribution is shown in figure. At both sides of central maximum, secondary maxima of lower intensity lie between the minima at angle such that $ \theta = \pm \frac {3\lambda }{2a}, \pm \frac {5\lambda }{2a} $.

Width of Central Maximum

In figure, two minima lie on two sides of the sides of the central maximum. So the width of the maximum is the distance between first minimum on its both sides.

$$ \begin{align*} \text {So, far} \: 1^{st} \: \text {minimum, we have} \\ \sin \theta &= \pm \frac {2\lambda }{a} \\ \text {For small angle,} \: \sin \theta = \theta \: \text {and so} \\ \theta &= \pm \frac {\lambda }{a} \\ \text {For small angle,} \ \sin \theta = \theta \: \text {and so} \\ \theta &= \pm \frac {\lambda }{a} \\ \end{align*} $$

The angular width of the central maximum

$$ \begin{align*} \\ 2\theta &= \frac {2\lambda }{a} \: \left ( i.e. \frac {\lambda }{a} + \frac {\lambda }{a} = 2 \frac {\lambda }{a} \right ) \end{align*} $$

If y is the distance of the first minimum from the central maximum, the width of central maximum is 2y. If D is the distance between the slit and screen which is very large compared to the width of slit,

$$ \begin{align*}\ \theta &= \frac yD = \frac {\lambda }{a} \\ \text {or,} \: y &= \frac {\lambda D}{a} \\ \text {And width of the maximum,} \\ 2y &= \frac {2\lambda D}{a} \\ \end{align*} $$

Diffraction Grating

A diffraction grating consists of a large number of equally spaced, parallel slits of the same width ruled on glass or polished metal by a diamond point. Diffraction gratings are used for producing spectra and for measuring the wavelength of light accurately. These have replaced the refraction prisms as they give sharp spectra. The number of lines ruled on the grating is very, as much 5000 lines per cm.

Suppose a section GG’ of a transmission grating as shown in the figure. A plane wave of monochromatic light of wavelength λ is incident on this grating from left, normal to the plane of the grating from left, and normal to the plane of the grating. Let d be the slit separation, also called grating element or spacing.

Consider wavelets coming from corresponding points A and B on two successive slits, and travelling at an angle Ï´ to the direction of the incident beam. The path difference AC between the wavelets is d sinÏ´ and same value comes for all pairs of wavelets from other corresponding points in these two slits and in all pairs of the slit in the grating. If this path length equals one wavelength or some integral multiple of a wavelength, then waves from all slits are in phase at the screen and superpose in direction Ï´ to produce maximum. So, for the condition of maximum

$$d\sin \theta = n \lambda , n = \pm 1, \pm 2, \dots (i) $$

In this equation, n gives the order of spectrum. When n = 0, we observe in the direction of incident waves, and the central bright maximum or zero order maximum is observed.

Resolving Power

The resolving power of an optical instrument is the ability of an instrument to produce distinctly separate images of two close objects. The ability of an instrument to produce separate distinct images of two close objects is called the resolving power of the instrument. When the central maximum of one image falls on the maximum of another image, the images are said to be just resolved. The limiting condition of resolution is known as Rayleigh’s criterion.

Resolving power of microscope

Consider a point object O illuminated by a light of wavelength λ and it is observed through a microscope. If θis the semi-vertical angle as shown in the figure, the least distance between to objects which can be distinguished is given by

$$ d = \frac {\lambda }{2\mu \sin \theta } $$

where µ is the refractive index of the medium between the object and objective lens. The term µ sin Ï´ is called the numerical aperture.

The resolving power of the microscope is the reciprocal of the minimum separation between two objects between two objects seen distinctly.

$$ \therefore \text {Resolving power of microscope} = \frac 1d = \frac {2\mu \sin \theta }{\lambda } $$

From above equation, we observe that the resolving power of microscope increases when the refractive index of the medium is increased.

Resolving Power of a Telescope

It is reciprocal of the smallest angular separation between two distinct objects whose images are separated in a telescope. The angular separation is given by

$$ d\theta = 1.22\frac {\lambda }{D} $$

where dθ is the angle subtended at the objective, λ = wavelength of light used, and D is the diameter of telescope objective. The number 1.22 appears due to the diffraction through the circular aperture of the telescope. Resolving power of telescope $ = \frac {1}{d\theta } = \frac {1}{1.22} \frac {D}{\lambda } $. So resolving power of a telescope can be increased then by increasing the diameter of the objective or decreasing the wavelength of light.