Diffraction

If an opaque obstacle (or aperture) is placed between a source of light and screen, a sufficiently distinct shadow of opaque (or an illuminated aperture) is obtained on the screen .This shows that the light travels approximately in straight lines. If, however, the size of obstacle (or aperture) is small (comparable with the wavelength of light), there is a departure from straight line propagation and the light bends round the corners of the obstacles (or aperture) and enters the geometrical shadow. The bending of light at sharp corners/edges is called diffraction. As a result of diffraction, the edges of the shadow (or illuminated region) are not sharp, but the intensity is distributed in a certain way depending upon the nature of obstacle (or aperture).

Let us first explain how the light bends around a sharp corner. According to Huygen's principle, when a wave propagates, each point on its wave front serves as the source of spherical secondary wavelets having same frequency as that of original wave [Fig. 1a]. The resultant at any point afterward is the envelope of these secondary wavelets. However, this picture does not explain the diffraction of light through small apertures.

If we assume that, as shown in fig.1b, each unobstructed point of a wavefront, at a given instant, serves as a source of spherical secondary wavefront, the amplitude of the optical field at any point beyond is the superposition of all these wavefronts. The maximum path difference between these secondary wave fronts at any point P is equal to AB. (Path difference will be equal to AB if point P merges with either point A or point B)

Fig: 1a.

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Fig: 1b.

Now if , at all the points on far away screen, the waves originating from different parts of the slit will be partially in phase, and we observe finite intensity of wave there. This automatically explains the bending of waves at sharp obstacles.

For , the secondary wavelets originating from the slit, when reach the screen, will have different phases. At points, where path difference is zero (point on the screen for which the line joining center of slit to screen is perpendicular to the plane of slit), the intensity will be maximum and will slowly diminish as we move further away from this point. At point where path difference become the minima will be observed and as we move further again the intensity will increase.

The difftraction phenomenon is conveniently divided into two general classes:

1. Fraunhoffer Diffraction: When both the source of light and the screen where diffraction pattern is observed are at very large distances (distances >> ) from the obstacle (aperture) .

2. Freshnel Diffraction: When, either or both, source of light and screen on which the diffraction pattern is observed are at finite distance from the obstacle (aperture).

In the following lectures we shall be studying the Fraunhoffer diffraction phenomenon occuring due to narrow slit, double slit and grating (composed on large number of narrow slits). We shall also be discussing the resolving power of a grating and other image forming systems.

Fraunhoffer Diffraction : In the case of Fraunhoffer diffraction, both the source and slit are very far from the diffracting aperture. In other words, this is equivalent to saying that the light originating from the source (where it has a spherical wavefront) when reaches at the aperture, the

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wavefront is nearly plane. The intensity of light at each point of aperture is thus nearly same. (If the distance between source and slit is not large, the wavefront at the aperture will be spherical and intensity of light at each point of aperture will be diffrent ).

After diffraction, each point of wavefront at aperture, will act as a new source of light and will spread as spherical wavefront .When the screen is very far from the aperture, the wavefront reaching the screen wiil again be planar and the wavefronts converging on any point on the screen will have same amplitude, and thus we need not worry about difference in field strength from different wavefronts.

Diffraction by a narrow single slit

Consider a single slit Fig. 2a(a rectangular aperture, whose length is larger compared to its breadth, and breadth is quite narrow, comparable to the wavelength of light 0.1 mm for visible light) placed in front of a monochromatic light source as shown in figure 2(b).

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When a plane wave front is incident on the slit, each point on the wave front acts as the source of spherical secondary wave fronts. Let’s divide the wave front incident on the slit in a large number of elements, each of width ds, (infinitesimaly small).

The part of each secondary wave, originating from these small elements and travelling normal to the plane of slit will be focussed at P0 while those travelling at an angle will reach P (Fig.2b). Considering first the wavelet emitted by the element ds situated at the center of slit (let us call it origin). Its amplitude will be directly proportional to length ds and inversely proportional to the distance x (it will produce spherical wave front). At P, it will produce an infinitesimal amplitude (electric field) which is expressed as

where 'a' is the amplitude of incident wave, and are its frequency and wave vector respectively.

The wave fronts reaching P from other elements ds will vary in phase due to extra path travelled by them. The displacements produced by another element ds at a distance s from the center(origin) will be given by

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Where is the path difference.

The net amplitude at P will be sum of effect due to all the elements ds and can be obtained by integrating dEs from s=-d/2 to d/2 where d is the width of the slit. In doing so, we can first sum the amplitude produced by the symmetrically placed elements ds and then integrate it from s=0 to d/2. The contribution due to a pair of symmetrically placed element ds is

Using the trigonomerical identity We have

Net effect at P is now

We can treat constant,

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Where

The resultant wave reaching at P , is therefore, a simple harmonic one. The amplitude of this varies as position P is varied ( varies). The resultant amplitude is given by

Where

The intensity of light at any point on screen is, thus given by

An intensity pattern as shown in fig. 3. is observed on screen

The intensity on the screen for diffraction due to single slit is

Analysis of Diffraction pattern due to single slit

where .

The intensity will be minimum (or zero) when

or m=1,2,.......................................

or

If slit is far from the screen, is small such that where y is distance of point P from center point P0

and D is distance between slit and screen.

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Thus knowing m(order of minima), , D and y (separation of m'th minima from principal maxima ) the slit width 'd' can be determined.

Now, let us see, when we will observe maxima in diffraction pattern.

Fig. 3:

At the center of screen, the path traveled by the secondary waves originating from one point of slit is same as that traveled by another wave on the opposite side of center of slit at the same distance and hence path difference is zero, ie all the waves meet in phase thus we observe a maxima at this point (Fig 3). This can also be seen mathematically for the intensity distribution

equation for point , and (for small , the series expansion of will contain only first term ). Thus the intensity at the center will be

Now we will determine the position of the maxima.

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or

for corresponds to minima as discussed earlier. The second condition : gives the position of maxima.

Solution of transcendental equation can be determined graphically, where two

functions and are plotted against (Fig.4). The point where these two curves intersects, satisfies and these values of corresponds to secondary maxima in the diffraction pattern. The corresponding values of

.This clearly shows that the secondary maximas are not exactly midway between the two minimas but are slightly displaced towards the principle

maxima. The intensity at the mid way between two minima is where n=1,2,3..... and is slightly less than the intensity at the secondary maxima. The intensity of the first

secondary maxima is times the intensity of principal maxima.

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Width of central (principal) maxima

The first minima in single slit diffraction pattern occurs for . The width of central

maxima at a distance D from the slit is thus given by , where is the distance between the position of peak in central maxima and first minima.

Now for equation

Width of central maxima

This is the extent to which the light is distributed before diminishing first time. For a given D, ( ). 2y1 varies as . Further for a given , y is small if 'd' is large and vice versa. Also

if 'd' is fixed, y is small if is small.

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Diffraction by double slit

Let us see, what happens to the intensity distribution on screen when the light is allowed to pass through two narrow slits close to each other (Fig.12.2.1). In Young's double slit experiment we have seen that this arrangement will give an interference pattern comprising alternate dark and bright fringes on screen. However, in that experiment, we have assumed each of slit as behaving like point source (we did not consider the width of slit). This was equivalent to the condition that

. Now from our earlier discussion in this chapter, we have seen that if , the ray bends around the sharp edge but everywhere (within the area of view) the waves superpose constructively and we see illumination all over.( No diffraction pattern due to either slit).

Now if the width of each slit is finite, (No longer a point source), we shall observe a combined effect of diffraction (through each slit) and interference (between light from two slits) on the screen.

At any point on screen the rays reaching from the first and second slit will have a path difference, where ‘b' is the separation between the center of two slits.

Fig.12.2.1

Let both slits have same width d and their separation is b ( b is distance between AA' or BB' or CC'. C & C' are midpoint of each slit). Once again we wish to find the intensity at point P, which makes angle with the midpoint O . Now the picture is similar to the case of single slit except that some portion (AB') of the slit BA' is blocked. Thus the secondary wave front originating from the section AB' will be blocked and will not reach screen. Let ds be small element at a distance s from the center point O. There will be a similar element at a distance - s from the point O.

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In the previous lecture we had discussed the intensity distribution on screen when there was only one narrow slit in the path of light rays.

Under small angle approximation and for distance between slit and screen to be large, the amplitude of the waves reaching point P and originating from s and –s distance from O will be.

The resultant amplitude at point P for all the wavelets originating from entire slits.

(In first term is integration for all the waves from B to 0 and second term is for all the waves from A to 0 , which is blocked, the contribution for lower slit is already included in the expression for dAs)

Upon rearranging,

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Where,

(The eqn. reduces to single slit expression if b=0,d=2d).

The intensity at point P is therefore,

where

The maximum intensity at point is

Which is 4 times the intensity due to individual slit and appears at the point on screen midway between the slits.

Distribution of intensity on screen

The equation for intensity on screen for double slit contains two terms. One of these term is

(Fig.12.2.2) ,which represents the distribution due to pure interference effects (fig 12.2.2 )

and the other (Fig.12.2.3) is due to diffraction.

The intensity at screen will be zero whenever either of or is zero i.e. whenever

or

By definition

hence first series of minima occurs, when (interference minima)

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Fig.12.2.2

Fig.12.2.3

or ,

The second series of minima occur when

(Diffraction minima)

or,

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As in the case of single slit diffraction, here also exact position of maxima is not defined(except the principal maxima). Approximate position of these maxima can be determined by neglecting

term and getting the condition of maxima by making ie. for

or (interference maxima)

Here p represent the order of interference. The resultant intensity distribution in the case of diffraction by double slit is given by that in fig.12.2.4 . We can see that resultant intensity distribution in the case comprises on several peaks within the envelope formed by the diffraction

term .

There are several interesting features in this intensity distribution pattern. The intensity goes to zero when either of the term is zero or both are zero. The second condition is satisfied when

or,

or, the two minima coincide.

Width of principal maxima : The width of principal maxima in this case is given by the distance between two points on screen when first order diffraction minima occurs i.e. if D is separation between slit and screen and z is the linear separation between first diffraction minima and central maxima, then for small

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Fig.12.2.4

Width of principal maxima

Fringe width The fringe width in the double slit diffraction pattern is same as the fringe width in the case of double slit interference ie the fringe width

Missing Orders In the intensity distribution curve in double slit diffraction pattern : Any particular order interference maxima may be absent if it coincides with the diffraction minima. These missing orders ( p values) in interference will be given by the condition.

or

Thus if the separation between slit is 10 times the slit width, the 10 order interference maxima will coincide with the first order diffraction minima and 20th order interference maxima will coincide with 2nd order diffraction minima and thus 10th , 20th, 30th , ---- order interference maxima will be missing.

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Diffraction by multiple slits: Diffraction Grating

In earlier lecture, we have seen the effect on intensity distribution when the light waves passe through two nearby narrow slits. As a result of this, the broad principal maximum produced by single slit actually consists of alternate dark and bright region. Now we will see the combined effect of interference and diffraction on intensity distribution, when the light waves are allowed to pass through large number of narrow slits, very close to each other. This arrangement of slits(Fig.12.3.1 ) is known as diffraction grating and finds lots of application in optics. Once again, each of these slits acts as source of secondary wavelengths and we will see again combined effect of interference and diffraction on screen. In this case, the broad principal maxima corresponding to single slit pattern, consists of several principal maxima and very low intensity secondary maxima separated by minima. We assume that width of each slit is ‘d' and the separation between the centers of any two adjacent slits is ‘b'. We will calculate the resultant intensity at the point P on the screen due to N such slits.

In order to find intensity distribution, we have to calculate the resultant amplitude of all waves reaching the point P . These wavelets are now originating from N slits and the calculation of the amplitude, as we did in case of single slit and double slit, is complicated. However, we can use the complex amplitude method. Here we note that the phase difference between the waves originating from similar position of two consecutive slits is ( b is separation

between the two slits). The amplitude part from each wave is Thus if we have two slits

Fig.12.3.1

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Where , ,

and resultant intensity

(where ) This expression is same as the one obtained for two slits in the previous section.

Now in case on N slits, the resultant amplitude can be written as

which is a geometrical series

the resultant intensity at P due to N slits is

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Intensity distribution in diffraction pattern by Grating:

The intensity at point P0 in diffraction pattern by Grating is given by

in this expression term corresponds to diffraction, whereas is the term due to interference

Central Maxima (Fig.12.3.2)

The intensity of diffracted pattern is maximum at the point where . For this value of , , (this also correspond to m=0 in ) and

Thus the intensity of central maxima is N2 the intensity due to individual wave front [Remember

for double slit case; N=2 and ].

Fig.12.3.2

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Principal Maxima (Fig.12.3.2): In addition to the central maxima, there are other higher order maxima (m=1,2,............), before the first order diffraction maxima( ) occurs .

Intensity of principal maxima

The intensity of principal maxima decreases as the order (m) increases due to term. The envelope covering the central maxima in the case of single slit contains all these principal maxima.

The intensity will be maximum when Also intensity will be maximum when ,

since for these value of , . Thus the values of for which corresponds to principal maxima of order m.

Position of minima (Fig.12.3.2)

Let us now find the position of minima in the resultant intensity distribution. The intensity goes,

to zero when (diffraction condition) or (interference term) .

minima will occur at

or

or,

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or,

more generally,

, p=1,2,............. N-1

= (minima)

Thus we see that between two principal consecutive pricipal maxima (say corresponding to (central maxima) and (first order principal maxima), there are a total

minima (zero intensity point) appear.

Secondary Maxima (Fig.12.3.2)

We have seen that between two adjacent principal maxima, there are points where intensity goes to zero. This implies that between two adjacent principal maxima there are at least N-2 points for which intensity goes to a maximum value. These are called secondary maximum and their intensity is not same but falls off as we move away from the principal maxima. Similar to single slit diffraction, these secondary maxima are not equally spaced and are shifted slightly towards the nearest principal maxima.

The intensity of principal maxima and secondary maxima depends on number of slits. When N is large , principal maxima become more intense and narrow and secondary maxima almost disappear (Fig.12.3.2 ),(Fig.12.3.3)..

Fig.12.3.3

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Formation of spectra by Grating Let us now see, what happens if white light (comprising many wavelengths) falls on a grating. The resultant intensity distribution on the screen will be

due to all the wavelengths. The conditions for central principal maxima will be satisfied by all the wavelengths simultaneously and hence a white slit will appear at the centers. However, the other principal maxima for these wavelengths will occur for different values of such that

(order of principal maxima) thus for smaller wavelengths, principal maxima will occur at smaller value and for larger wavelengths, corresponding will be more. The angular separation (20m) between the same order principal maxima on both side of central maxima can thus be used to determine the wavelength of spectral lines in the source.

(Fig.Grating 1), (Grating 2), (Grating 3).

Width of principal maxima

The angular separartion between the m'th order principal maxima and the next first minima will follow the following conditions

(maxima)

and (minima)

Fig.Grating 1

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Fig.Grating 2

Fig.Grating 3

The second equation can be written as

(minima)

for small

The above two equations, therefore give

or

This is also the half angular width of central maxima.

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