#### Fresnel Diffraction, Physics tutorial

Introduction:

We are familiar from our everyday experience that we can hear persons talking in an adjoining room the door of which is open. This is due to the capability of sound waves to bend around the corners of obstacles in their manner. You are as well familiar by the capability of water waves to propagate around obstacles. You might now ask: Does light, that is an electromagnetic wave, as well bend around corners of obstacles in its path? We are as well familiar with the manifestation of wave nature of light in the form of interference: Light from the two coherent sources interferes to form the fringed pattern. However what might puzzle you is the fact that light casts shadows of objects, that is, appears to travel in straight lines instead of bending around corners. This apparent contradiction was described by Fresnel who exhibited that the ease by which a wave bends around corners is strongly affected by the size of the obstacle (that is, aperture) relative to its wavelength. Music and speech wavelengths lie in the range 1.7 cm to 17m. A door is around 1 m aperture so that long wavelength waves bend more willingly around the door way. On the other hand, wavelength of light is around 10-7m and the obstacles employed in ordinary experiments are around 105 times bigger.

Due to this reason, light appears to travel all along the straight lines and casts shadows of objects rather than bending around their corners. Though, it doesn't mean that light exhibits no bending, it does so under appropriate conditions where size of obstacles is comparable by the wavelength of light. You can attain a feel for this by closely examining the shadows cast by objects. We will view that the edges of shadows are not sharp. The deviation of waves from their original direction due to the obstacle in their path is termed as diffraction.

The phenomenon of diffraction finds out great utilization in our daily life. We will learn that the diffraction places a basic limitation on optical instruments, comprising the human eye, in respect of resolution of objects.

The phenomenon of diffraction was very first noticed by Grimaldi, an Italian mathematician. And a systematic description of diffraction was given by Fresnel on the basis of Huygen's principle. According to him, the diffraction is attributed to mutual interference of secondary wavelets from the single wave. (That is, the interference phenomenon comprises two coherent wave trains.) This signifies that in diffraction phenomenon, interference occurs between the secondary wavelets from different portions of the similar wavefront.

Fraunhofer diffraction and Fresnel diffraction are as well termed as far field diffraction and near field diffraction, correspondingly.

For mathematical ease and simplicity of understanding, diffraction is categorized in two categories: Fraunhofer diffraction and Fresnel diffraction. In Fraunhofer class of diffraction, the source of light and the observation screen (or human eye) is efficiently at infinite distance from the obstacle. This can be done most expediently by employing appropriate lenses. This is of specific practical significance in respect of the general theory of the optical instruments.

In the Fresnel class of diffraction, the source or the observation screen or both are at finite distance from the obstacle. You will be familiar with the Fresnel diffraction; the experimental arrangement is somewhat simple. However its theoretical analysis is more complex than that of Fraunhofer diffraction.

As well, Fresnel diffraction is more general; it comprises Fraunhofer diffraction as a special case. Furthermore, it has significance in historical viewpoint in that it led to the growth of the wave model of light.

Observing Diffraction: Some Simple Experiments

As we are familiar, that the wavelength of visible light is extremely small (around 10-7m).  And to view diffraction, careful observations have to be made. We will now familiarize you by several simple situations and experiments to notice diffraction of light. The fundamentals for these are:

• A source of light, rather narrow and monochromatic
• A sharp edged obstacle
• An observation screen that could be the human retina also.

1) Look at a far-away street light at night and squint. The light seems to streak out from the bulb. This is due to the reason that light has bent around the corners of your eyelids.

2) Stand in a dark-room and look at a far-away light bulb in the other room. Now move slowly till the doorway blocks half of the light bulb. The light seems to streak out into the umbra area of the dark room due to diffraction around the door-way.

3) Take a portion of fine cloth, say fine handkerchief or muslin cloth. Stretch it flat and keep it near the eye. Now focus your eye on the far-away lamp (at least 100 m away) through it. Do you notice an enlarged disc surrounded through a regular prototype of spots arranged all along a rectangle? On careful observations you will note that the spots on the outer portion of the pattern appear colored. Now rotate the handkerchief in its own plane. Does the pattern rotate? You will be eager to observe that the pattern rotates around the central disc.

Furthermore, the speed of rotation of the pattern is similar as that of the handkerchief.

We are now persuaded to ask: Do you know why this prototype of spots is obtained? You will agree that the handkerchief is a mesh (criss-cross) of fine threads in mutually perpendicular directions. Evidently, the observed pattern is made by the diffraction of light from the lamp.

4) Take a couple of razor blades and one clear glass electric bulb. Hold the blades in such a way that the edges are parallel and encompass a narrow slit in between. Keep the slit close to your eye and parallel to the filament. (Employ spectacles if you generally do.) By a little adjustment of the width of the slit, you must view a pattern of bright and dark bands that exhibit some colors.

5) Fix a small ball bearing carefully on a glass plate having a small amount of beeswax in such a way that no wax spreads beyond the rim of the ball. Place this opaque obstacle in a strong beam of light (if possible monochromatic) diverging from the pinhole. Beneath appropriate conditions, you will observe a bright spot, termed as Poisson spot at the centre of the shadow cast through the ball bearing. This exciting examination proved unchallengeable proof for the diffraction of light.

Producing a Diffraction Pattern:

In the Fresnel class of diffraction, the light source or the screen or both are, in common, at a finite distance from the diffracting obstacle. On the other hand in Fraunhofer diffraction, this distance is efficiently infinite. This condition is attained or achieved by putting an appropriate lens between the source and the screen. A huge number of workers have observed and studied Fresnel and Fraunhofer diffraction patterns. In recent times a systematic study of Fresnel diffraction pattern from obstacles of various shapes example: small spheres, discs and apertures of circular, elliptical, square, triangular or parallelograms and so on of different sizes was prepared by Indian physicist Y.V. Kathvate under the supervision of Prof. C. V, Raman. Their experimental set up for photographing these patterns is represented the figure given below. It comprises of a light tight box (almost 5 m long) having a fine pinhole at one end. The light on the pinhole from a 100 W lamp was focused by employing a convex lens; a red filter was employed to get approximately monochromatic light of wavelength 6320 Å.

The obstacle was positioned at an appropriate distance (around 2 m) from the pinhole. The photographic plate was mounted on a movable stand in such a way that its distance from the obstacle could be varied. In this, steel ball bearings of radii 1.58 mm, 1.98 mm, 2.37mm and 3.17 mm are used as spherical obstacles. These four spheres were mounted on the glass plate that was kept at a distance of around 2 m from the pinhole.

The photographic plate was kept at distances of 5 cm, 10 cm, 20 cm, 40 cm and 180 cm from the mounted glass plate (that is, obstacle). Such patterns basically characterize the distribution of the light intensity in the area of the geometrical shadow of the obstacles.

In the diffraction patterns for circular discs of the similar size, we will find out that such patterns are nearly identical to those for spheres. Furthermore, the diffraction patterns that correspond to bigger spheres and discs (that is, radii 3.17 mm and 2.37 mm); represent the geometrical shadow and a central bright spot in it. On the other hand, in the diffraction pattern corresponding to the smaller sphere (or disc) having radius 1.98 mm, the geometrical image is identifiable however consists of fringes appearing on its edges. The fringe pattern around the central spot becomes noticeably clearer for the sphere of radius 1.58 mm. The formation of the bright central spot in the shadow and the rings about the central spot are the most definite indicators of the non-rectilinear propagation of light. Rather, light bends in certain special way around opaque obstacles. Such departures from rectilinear propagation come beneath the heading of diffraction phenomenon.

Fresnel Construction:

In the study of Fresnel diffraction it is well-situated to divide the aperture into regions termed as Fresnel zones. The figure below represents a point source, 'S', illuminating an aperture a distance 'z1'away. The observation point, 'P', is a distance to the right of the aperture. Assume that the line 'SP' be normal to the plane having the aperture. Then we can write:

SQP = r1 + r2 = √ (z12 + ρ2) + √ (z22 + ρ2)

SQP = z1 + z2 + (1/2) ρ2 [(1/z1) + (1/z2)] +......

The aperture can be splitted into regions bounded through concentric circles ρ = constant defined in such a way that r1 + r2 differ by λ/2 in going from one boundary to the next. These areas are termed as Fresnel zones or half-period zones. Whenever z1and z2 are adequately large compared to the size of the aperture the higher order terms of the expansion can be neglected to yield the given result.

n (λ/2) = (1/2) ρn2 [(1/z1) + (1/z2)]

Resolving for ρn, the radius of the nth Fresnel zone, outcomes:

ρn = √ (nλL) or ρ1 = √(λL), ρ2 = √(2λL), ..... Here, L = 1/[(1/z1) + (1/z2)]

The figure shown below represents a drawing of Fresnel zones where each and every other zone is made dark. Note that in the center the zones are broadly spaced and the spacing reduces as the radius rises. As illustrated above, the radius of the zones raises as the square root of integers.

If ρn and ρn+1 are inner and outer radii of the nth zone then the area of the nth zone is represented by:

Area of nth Fresnel zone = π ρn+12 - π ρn2

= π (n + 1) λL - π (n) λL

= π λ L = πρ12, independent of n

That is, the area of all zones is equivalent. Whenever the higher order terms in the expansion for SRQ are maintained, then the area of the zones would slightly rise by increasing ρ. usually, it is supposed that z1and z2 are sufficiently large compared to ρ that the higher order terms can be neglected and the area of all zones are equivalent.

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