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  Fraunhofer Diffraction, Physics tutorial

Introduction:

The term Diffraction signifies to the spreading of waves and appearance of the fringes that take place if a wave front is constricted through an aperture in a screen which is otherwise opaque. The light pattern varies as we move away from the aperture, being characterized through three areas or regions.

1165_Diffraction of plane wave at aperture.jpg

1) In the shadow area, close to the aperture, the boundary of the transmitted light is sharp and looks like the aperture in shape.

2) As you move away into the Fresnel area or region, the beam width remains comparable to that in the aperture; however narrow fringes come out at the edges.

3) Far away, in the Fraunhofer area or region, the beam spreads to a width much greater than that of the aperture and is bordered through numerous weaker fringes.

The Fraunhofer area or region is selected for the experiment as the broader fringes are simpler to evaluate or measure by an optical detector of finite aperture and the computations are more straightforward.

The complete information of the distribution of light over the diffraction pattern depends on the distances of both the source and the screen from the diffracting aperture. The common situation of Fresnel diffraction can be mathematically extremely complex. Though, computations are greatly simplified when both the source and the screen are at extremely large distances from the aperture (that is, when such distances are much greater than the diameter of the aperture), which is termed as the Fraunhofer diffraction.

Fraunhofer conditions can be attained in practice by employing two lenses. The first lens makes sure that the wave-fronts arriving at the aperture will be plane (having parallel rays) and the second lens brings beams of diffracted light altogether to form an interference pattern on the screen.

1219_Fraunhofer diffraction.jpg

Diffraction at a Single Slit:

You will remember that light comprises of a superposition of lots of elementary waves, having a broad range of frequencies. In the simple theory of diffraction we mainly deal by one frequency at a time. The diffraction of a complex beam of light is then illustrated in terms of what happens to each of the component waves having different frequencies. The simplest case of Fraunhofer diffraction is that for a long narrow slit in an opaque screen. The interference pattern comprises of a set of light and dark parallel fringes.

2089_Fraunhofer diffraction-single slit.jpg

The figure above represents how the diffraction via a slit is studied by using the Fraunhofer conditions. The first lens makes sure that the wave-fronts coming out at the aperture will be plane and the second lens focuses the light onto the screen.

The wave-fronts arriving at the slit are plane, however due to diffraction; the wave-fronts on the other side will not be plane. Light arriving at any point in the aperture consists of a fixed phase relationship to light from the similar portion of the source arriving at any other point in the aperture. The simplest case is to assume an elementary plane wavefront arriving parallel to the slit as represented in the figure above. As all the points on a wavefront encompass the similar phase we can assume that the aperture filled with numerous tiny coherent sources. Whenever the light from all of such sources comes altogether at different points on the screen an interference pattern will be observed. The brightness at any point will fully dependent on the phase differences among all the secondary waves arriving there and such phase differences will based on the optical paths travelled through various waves. The path differences can be computed by using rays that leave the slit parallel to one other, in such a way that they would ultimately meet up at infinity were it not for the presence of the second lens. As the lens itself introduces no extra optical path difference, computations can be done on the supposition that the rays meet up at infinity.

122_formation of central bright fringe.jpg

The entire rays parallel to the axis will be focused at P0. However the geometrical path lengths of the rays are clearly different, the optical paths from various points across the slit to the point P0 are all equivalent. That is so because the longer paths outside the lens are compensated via the shorter optical paths of the rays within the lens. The rays from the outside of slit go through more air however less glass than rays closes to the middle of the lens. Therefore all the light arriving at P0 is in phase, providing a bright region there.

Observed Pattern:

1406_Fraunhofer diffraction-Observed pattern.jpg

Take a moment and think how would diffraction pattern of the vertical slit appear? Or what would be the distribution of intensity in this pattern? You might suppose that the diffraction pattern would be a single vertical line or a sequence of vertical lines on the observation screen. This line of thought is wildly off-target. The authentic diffraction pattern is amazingly different; it comprises of a horizontal streak of light composed of bright lengthened spots joined by faint streaks. In another word, after passing via the vertical slit, light spreads all along a horizontal line. This signifies that the diffraction pattern is all along a line perpendicular to the length of the diffracting slit. You might interpret this horizontal diffraction as a spread out image of the point source. The degree of horizontal spreading is controlled through the width of the slit; as the width rises, the spreading reduces. And in the extreme case of a very broad slit, the (horizontal) diffraction streak decreases to a bright point. Physically, extremely wide slit signifies that the slit has efficiently been removed.

The significant features of the observed Fraunhofer diffraction pattern of a single vertical slit from a point source are represented in the figure above. These are concluded below:

1) The diffraction pattern comprises of a horizontal streak of light all along a line perpendicular to the length of the slit.

2) The horizontal pattern is a sequence of bright spots. The spot at the central point P0 that lies at the intersection of the axis of L1 and L2 by means of the observation screen is the brightest. On either side of the brightest spot we view many more bright spots symmetrically positioned with respect to P0.

3) The intensity of the central spot is highest or maximum. The peak intensities of other spots, on either side of the central spot, reduce rapidly as we move away from P0. The central maximum is termed as principal maximum and the others as the secondary maxima.

4) The width of the central spot is two times of the width of the other spots.

5) A cautious observation of the diffraction pattern represents that the central peak is symmetrical. However on either side of the central maximum, secondary maxima are asymmetrical. However, the positions of the maxima are slightly shifted in the direction of the observation point Po.

Diffraction by a circular aperture:

2099_Diffraction by circular aperture.jpg

Half-angle beam spread to first minimum, θ1/2 is:

θ1/2 = 1.22 λ/D

Radius of the central bright disk (airy disk), R is:

R = 1.22 λ Z′/D

Here,

λ = wavelength of the light

D = diameter of pinhole

Z′ = aperture-to-screen distance

Diffraction by a rectangular aperture:

2475_Diffraction by rectangular aperture.jpg

Half-angle beam divergences to the first minimum in x and y directions:

(θ1/2)x = (λ/dx) and (θ1/2)y = λ/dy

Half-widths of central bright fringe in the x and y directions:

x1 = Z'λ/dx and y1 = Z'λ/dy

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