The Concepts Of Fresnel And Fraunhofer Diffraction Environmental Sciences Essay

Diffraction refers to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects are observed when light waves travel through a medium with a varying refractive index or a sound wave through one with varying acoustic impedance. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. As physical objects have wave-like properties (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics.

HISTORY OF DIFFRACTION

Diffraction was first observed by Francesco Grimaldi in 1665. He noticed that light waves spread out when made to pass through a slit. Later it was observed that diffraction not only occurs in small slits or holes but in every case where light waves bend round a corner.

One of the most common examples of difraction in nature is the tiny specks or hair-like transparent structures, known as “floaters” that we can see when we look up at the sky. This illusion is produced within the eye-ball, when light passes through tiny bits in the vitreous humour. They are more prominently observed when one half-closes his eyes and peeps through them.

The phenomenon of diffraction can be readily explained using Huygens’ principle:

When the wavefront of a light ray is partially obstructed, only those wavelets which belong to the exposed parts superpose, in such a way that the resulting wavefront has a different shape. This permits bending of light around the edges. Colourful fringe patterns are observed on a screen due to diffraction.

In the early 1800s, most of the people who wrote and submitted papers on diffraction of light were believers of the wave-theory of light. However, their views contradicted those of Newton’s supporters’ and their would be regular discussions between these two sides. One such person, who believed in the wave theory was Augustin Fresnel, whoin 1819, handed a paper to the French Academy of Sciences, about the phenomenon of diffraction. However, the Academy mainly consisting of Newton’s supporters, tried to challenge Fresnel’s point of view by saying that if light was indeed a wave, these waves, which were diffracted from the edges of a sphere, would cause a bright area to occur within the shadow of the sphere. This was indeed oberved later, and the area is today known as the Fresnel Bright Spot.

WHAT IS DIFFRACTION?

Diffraction is a loss of sharpness or resolution caused by photographing with small f/stops. The same softening effect happens when photographing through diffusion cloth or window screens.

Diffraction is the slight bending of light as it passes around the edge of an object. The amount of bending depends on the relative size of the wavelength of light to the size of the opening. If the opening is much larger than the light’s wavelength, the bending will be almost unnoticeable. However, if the two are closer in size or equal, the amount of bending is considerable, and easily seen with the naked eye.

In the atmosphere, diffracted light is actually bent around atmospheric particles — most commonly, the atmospheric particles are tiny water droplets found in clouds. Diffracted light can produce fringes of light, dark or colored bands. An optical effect that results from the diffraction of light is the silver lining sometimes found around the edges of clouds or coronas surrounding the sun or moon. The illustration above shows how light (from either the sun or the moon) is bent around small droplets in the cloud.

Optical effects resulting from diffraction are produced through the interference of light waves. To visualize this, imagine light waves as water waves. If water waves were incident upon a float residing on the water surface, the float would bounce up and down in response to the incident waves, producing waves of its own. As these waves spread outward in all directions from the float, they interact with other water waves. If the crests of two waves combine, an amplified wave is produced (constructive interference). However, if a crest of one wave and a trough of another wave combine, they cancel each other out to produce no vertical displacement (destructive interference).

This concept also applies to light waves. When sunlight (or moonlight) encounters a cloud droplet, light waves are altered and interact with one another in a similar manner as the water waves described above. If there is constructive interference, (the crests of two light waves combining), the light will appear brighter. If there is destructive interference, (the trough of one light wave meeting the crest of another), the light will either appear darker or disappear entirely.

TYPES OF DIFFRACTION

There are basically 2 different types of diffraction.They are:

1.Fresnel diffraction

2.Fraunhofer diffraction

FRESNEL DIFFRACTION

In optics, Fresnel diffraction or near-field diffraction is a process of diffraction which occurs when a wave passes through an aperture and diffracts in the near field, causing any diffraction pattern observed to differ in size and shape, relative to the distance. It occurs due to the short distance in which the diffracted waves propagate, which results in a fresnel number greater than 1. When the distance is increased, outgoing diffracted waves become planar and Fraunhofer diffraction occurs. The multiple Fresnel diffraction at nearly placed periodical ridges (ridged mirror) causes the specular reflection; this effect can be used for atomic mirrors.

Fresnel diffraction refers to the general case where those restrictions are relaxed. This makes it much more complex mathematically. Some cases can be treated in a reasonable empirical and graphical manner to explain some observed phenomena.

In optics, Fresnel diffraction or near-field diffraction is a process of diffraction that occurs when a wave passes through an aperture and diffracts in the near field, causing any diffraction pattern observed to differ in size and shape, depending on the distance between the aperture and the projection. It occurs due to the short distance in which the diffracted waves propagate, which results in a Fresnel number greater than 1 (). When the distance is increased, outgoing diffracted waves become planar and Fraunhofer diffraction occurs.

Fresnel diffraction showing center black spot

The multiple Fresnel diffraction at nearly placed periodical ridges (ridged mirror) causes the specular reflection; this effect can be used for atomic mirrors.

The Fresnel diffraction integral

Diffraction geometry, showing aperture (or diffracting object) plane and image plane, with coordinate system.

The electric field diffraction pattern at a point (x,y,z) is given by:

where

, and

is the imaginary unit.

Analytical solution of this integral is impossible for all but the simplest diffraction geometries. Therefore, it is usually calculated numerically.

The Fresnel approximation

The main problem for solving the integral is the expression of r. First, we can simplify the algebra by introducing the substitution:

Substituting into the expression for r, we find:

Next, using the Taylor series expansion

we can express r as

If we consider all the terms of Taylor series, then there is no approximation.[4] Let us substitute this expression in the argument of the exponential within the integral; the key to the Fresnel approximation is to assume that the third element is very small and can be ignored. In order to make this possible, it has to contribute to the variation of the exponential for an almost null term. In other words, it has to be much smaller than the period of the complex exponential, i.e. 2Ï€:

expressing k in terms of the wavelength,

we get the following relationship:

Multiplying both sides by z3 / λ3, we have

or, substituting the earlier expression for ρ2 ,

If this condition holds true for all values of x, x’ , y and y’ , then we can ignore the third term in the Taylor expression. Furthermore, if the third term is negligible, then all terms of higher order will be even smaller, so we can ignore them as well.

For applications involving optical wavelengths, the wavelength λ is typically many orders of magnitude smaller than the relevant physical dimensions. In particular:

and

Thus, as a practical matter, the required inequality will always hold true as long as

We can then approximate the expression with only the first two terms:

This equation, then, is the Fresnel approximation, and the inequality stated above is a condition for the approximation’s validity.

The condition for validity is fairly weak, and it allows all length parameters to take comparable values, provided the aperture is small compared to the path length. For the r in the denominator we go one step further, and approximate it with only the first term, . This is valid in particular if we are interested in the behaviour of the field only in a small area close to the origin, where the values of x and y are much smaller than z. In addition, it is always valid if as well as the Fresnel condition, we have , where L is the distance between the aperture and the field point.

For Fresnel diffraction the electric field at point (x,y,z) is then given by:

This is the Fresnel diffraction integral; it means that, if the Fresnel approximation is valid, the propagating field is a spherical wave, originating at the aperture and moving along z. The integral modulates the amplitude and phase of the spherical wave. Analytical solution of this expression is still only possible in rare cases. For a further simplified case, valid only for much larger distances from the diffraction source see Fraunhofer diffraction. Unlike Fraunhofer diffraction, Fresnel diffraction accounts for the curvature of the wavefront, in order to correctly calculate the relative phase of interfering waves.

FRAUNHOFER DIFFRACTION

In optics, Fraunhofer diffraction (named after Joseph von Fraunhofer), or far-field diffraction, is a form of wave diffraction that occurs when field waves are passed through an aperture or slit causing only the size of an observed aperture image to change[1][2] due to the far-field location of observation and the increasingly planar nature of outgoing diffracted waves passing through the aperture.

It is observed at distances beyond the near-field distance of Fresnel diffraction, which affects both the size and shape of the observed aperture image, and occurs only when the Fresnel number , wherein the parallel rays approximation can be applied.

An example of an optical setup that displays Fresnel diffraction occurring in the near-field. On this diagram, a wave is diffracted and observed at point σ. As this point is moved further back, beyond the Fresnel threshold or in the far-field, Fraunhofer diffraction occurs.

The Fraunhofer approximation

In scalar diffraction theory, the Fraunhofer approximation is a far field approximation made to the Fresnel diffraction integral,

[3]

Explanation

Fraunhofer diffraction employs the Huygens-Fresnel principle, whereby a wave is split into several outgoing waves when passed through an aperture, slit or hole, and is usually described through the use of observational experiments using lenses to purposefully diffract light. When waves pass through, the wave is split into two diffracted waves traveling at parallel angles to each other along with the continuing incoming wave, and are often used in methods of observation by placing a screen in its path in order to view the image-pattern observed.[4]

When a diffracted wave is observed parallel to the other at an initial near-field distance, Fresnel diffraction is seen to occur due to the distance between the aperture and the observed canvas σ being more than 1 when calculated with the Fresnel number equation,[4] which can be used to observe the extent of diffraction in the parallel waves through the calculation of the aperture or slit size a, wavelength λ and distance from the aperture L. When the distance or wavelength is increased,[2] Fraunhofer diffraction occurs due to the waves going towards becoming planar, over the extent of diffracting apertures or objects.[5]

Aperture form

When observed, the image of the aperture from Fresnel diffraction will change in terms of size and shape, namely, the edges become more or less ‘jagged’, whereas the aperture image observed when Fraunhofer diffraction is in effect only alters in terms of size due to the more collimated or planar nature of the waves.

The far-field diffraction pattern of a source may also be observed (except for scale) in the focal plane of a well-corrected lens. The far-field pattern of a diffracting screen illuminated by a point source may be observed in the image plane of the source.

If a light source and an observation screen are effectively far enough from a diffraction aperture (for example a slit), then the wavefronts arriving at the aperture and the screen can be considered to be collimated, or plane. Fresnel diffraction, or near-field diffraction occurs when this is not the case and the curvature of the incident wavefronts is taken into account.

In far-field diffraction, if the observation screen is moved relative to the aperture, the diffraction pattern produced changes uniformly in size. This is not the case in near-field diffraction, where the diffraction pattern changes both in size and shape.

Slit form

Fraunhofer diffraction through a slit can be achieved with two lenses and a screen. Using a point-like source for light and a collimating lens it is possible to make parallel light, which will then be passed through the slit. After the slit there is another lens that will focus the parallel light onto a screen for observation. The same setup with multiple slits can also be used, creating a different diffraction pattern.

Since this type of diffraction is mathematically simple, this experimental setup can be used to find the wavelength of the incident monochromatic light with high accuracy.

THE FRAUNHOFER AND FRESNEL APPROXIMATIONS

Whenever all the phase threads are effectively parallel to one another, then we refer to the resulting diffraction pattern as a Fraunhofer, or Fourier domain, or far-field diffraction pattern. We’ve already discussed one type of Fraunhofer pattern with our Young’s slits experiment. The diagram looked like this:

Well, the threads are not perfectly parallel here. But if we were to make the hemi-sphere very, very large, then all the threads would be parallel. The pattern we see would exist purely as a function of angle around the hemi-sphere. The co-ordinates of Frauhofer diffraction are therefore angles (or, more precisely, direction cosines). For all threads to be parallel, the object of interest (in the case above, the separation of the slits) must be small and the radius of the hemi-sphere must be large. How small and how large these dimensions are allowed to be depends on the wavelength, which determines the allowable error caused by the threads not being quite parallel.

We have an easy way of making a Fraunhofer diffraction pattern in the electron microscope. We just press the ‘diffraction’ button. Remember, we are imaging the back-focal plane, which by definition is where all parallel beams emerging from the specimen come to a focus:

On the contrary, Fresnel diffraction is the term used whenever we cannot make this ‘parallel thread’ approximation, in other words when we want to calculate a wave near a source of scattering.

IMPORTANT DIFFERENCES BETWEEN FRAUNHOFER AND FRESNEL DIFFRACTION:

In optics, Fraunhofer diffraction (named after Joseph von Fraunhofer), or far-field diffraction, is a form of wave diffraction that occurs when field waves are passed through an aperture or slit causing only the size of an observed aperture image to change due to the far-field location of observation and the increasingly planar nature of outgoing diffracted waves passing through the aperture.

It is observed at distances beyond the near-field distance of Fresnel diffraction, which affects both the size and shape of the observed aperture image, and occurs only when the Fresnel number , wherein the parallel rays approximation can be applied.

On the other hand, Fresnel diffraction or near-field diffraction is a process of diffraction that occurs when a wave passes through an aperture and diffracts in the near field, causing any diffraction pattern observed to differ in size and shape, depending on the distance between the aperture and the projection. It occurs due to the short distance in which the diffracted waves propagate, which results in a Fresnel number greater than 1 (). When the distance is increased, outgoing diffracted waves become planar and Fraunhofer diffraction occurs.