Fresnel diffraction.pdf

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9/12/13 Fresnel diffraction - Wikipedia, the free encyclopedia

en.wikipedia.org/wiki/Fresnel_diffraction

Fresnel diffraction showing central Arago

spot

Fresnel diffractionFrom Wikipedia, the free encyclopedia

In optics, the Fresnel diffraction equation fornear-field diffraction, is an approximation of Kirchhoff-Fresnel

diffraction that can be applied to the propagation of waves in the near field.[1]

The near field can be specified by the Fresnel number,Fof the optical arrangement, which is defined, for a waveincident on an aperture, as:

where

is the characteristic size of the aperture

is the distance of the observation point from the aperture

is the wavelength of the wave.

When the diffracted wave is considered to be in the near field, and the Fresnel diffraction equation can b

used to calculate its form.

The multiple Fresnel diffraction at nearly placed periodical ridges

(ridged mirror) causes the specular reflection; this effect can be used

for atomic mirrors.[2]

Contents1 Early treatments of this phenomenon

2 The Fresnel diffraction integral

2.1 The Fresnel approximation

2.2 Fresnel diffraction

3 Alternative forms

3.1 Convolution

3.2 Fourier transform

3.3 Linear canonical transformation

4 See also5 Notes

6 References

Early treatments of this phenomenon
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Diffraction geometry, showing aperture (or diffracting

object) plane and image plane, with coordinate system.

Some of the earliest work on what would become known as Fresnel diffraction was carried out by Francesco

Maria Grimaldi in Italy in the 17th century. In his monograph entitled "Light,"[3] Richard C. MacLaurin explains

Fresnel diffraction by asking what happens when light propagates, and how that process is affected when a barrier

with a slit or hole in it is interposed in the beam produced by a distant source of light. He uses the Principle of

Huygens to investigate, in classical terms, what transpires. The wave front that proceeds from the slit and on to a

detection screen some distance away very closely approximates a wave front originating across the area of the gap

without regard to any minute interactions with the actual physical edge.

The result is that if the gap is very narrow only diffraction patterns with bright centers can occur. If the gap is made

progressively wider, then diffraction patterns with dark centers will alternate with diffraction patterns with bright

centers. As the gap becomes larger, the differentials between dark and light bands decrease until a diffraction effec

can no longer be detected.

MacLaurin does not mention the possibility that the center of the series of diffraction rings produced when light is

shone through a small hole may be black, but he does point to the inverse situation wherein the shadow produced

by a small circular object can paradoxically have a bright center. (p. 219)

In his Optics,,[4] Francis Weston Sears offers a mathematical approximation suggested by Fresnel that predicts th

main features of diffraction patterns and uses only simple mathematics. By considering the perpendicular distancefrom the hole in a barrier screen to a nearby detection screen along with the wavelength of the incident light, it is

possible to compute a number of regions called half-period elements or Fresnel zones. The inner zone will be a

circle and each succeeding zone will be a concentric annular ring. If the diameter of the circular hole in the screen i

sufficient to expose the first or central Fresnel zone, the amplitude of light at the center of the detection screen will

be double what it would be if the detection screen were not obstructed. If the diameter of the circular hole in the

screen is sufficient to expose two Fresnel zones, then the amplitude at the center is almost zero. That means that a

Fresnel diffraction pattern can have a dark center. These patterns can be seen and measured, and correspond wel

to the values calculated for them. Figure 9-5, following p. 222, in Sears shows four Fraunhofer patterns in the top

row followed by sixteen Fresnel diffraction patterns. Three of them have dark centers. (See the photograph above

and check the book by Sears for very much nicer photographs.

The Fresnel diffraction integral

The electric field diffraction pattern at a point

(x,y,z) is given by:
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where

is the aperture,

, 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 ofr. First, we can simplify the algebra by introducing th

substitution:

Substituting into the expression forr, we find:

Next, using the Taylor series expansion

we can express ras

If we consider all the terms of Taylor series, then there is no approximation.[5] 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

term 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. :
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expressing kin terms of the wavelength,

we get the following relationship:

Multiplying both sides by , we have

or, substituting the earlier expression for 2,

If this condition holds true for all values ofx,x',y andy', 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 ca

ignore them as well.

For applications involving optical wavelengths, the wavelength 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.


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Fresnel diffraction occurs when:

Fraunhofer diffraction occurs when:

aperture or slit size,

wavelength, distance from the apertur

Fresnel diffraction

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 rin 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 ofx andy are muchsmaller thanz. 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 alongz. 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 relativ

phase of interfering waves.

Alternative forms

Convolution

The integral can be expressed in other ways in order to calculate it using some mathematical properties. If we defin

the following function:

then the integral can be expressed in terms of a convolution:

in other words we are representing the propagation using a linear-filter modeling. That is why we might call the

function h(x,y,z) the impulse response of free space propagation.

Fourier transform

Another possible way is through the Fourier transform. If in the integral we express kin terms of the wavelength,
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and we expand each component of the transverse displacement,

then we can express the integral in terms of the two dimensional Fourier transform. Let us use the following

definition:

wherep and q are spatial frequencies (in units of lines/meter). The Fresnel integral can be expressed as:

where

i.e. first multiply the field to be propagated for a complex exponential, calculate its two dimensional Fourier

transform, replace (p,q) with and multiply it by another factor. This expression is better than the

others when the process leads to a known Fourier transform, and the connection with the Fourier transform is

tightened in the linear canonical transformation, discussed below.

Linear canonical transformation

Main article: Linear canonical transformation

From the point of view of the linear canonical transformation, Fresnel diffraction can be seen as a shear in the time-

frequency domain, corresponding to how the Fourier transform is a rotation in the time-frequency domain.

See also

Fraunhofer diffraction

Fresnel integral

Fresnel zone

Fresnel number

Augustin-Jean Fresnel

Ridged mirror
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Fresnel imager

Euler spiral

Notes

1. ^ M. Born & E. Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge

2. ^ http://www.ils.uec.ac.jp/~dima/PhysRevLett_94_013203.pdf H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita,

Shimizu. Fresnel diffraction mirror for atomic wave, Physical Review Letters, 94, 013203 (2005).3. ^Light," by Richard C. MacLaurin, 1909, Columbia University Press

4. ^Optics, Francis Weston Sears, p. 248ff, Addison-Wesley, 1948

5. ^ There was actually an approximation in a prior step, when assuming is a real wave. In fact this is not

real solution to the vector Helmholtz equation, but to the scalar one. See scalar wave approximation

References

Goodman, Joseph W. (1996).Introduction to Fourier optics. New York: McGraw-Hill. ISBN 0-07-

024254-2.

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