9/12/13Fresnel diffraction - Wikipedia, the free encyclopedia Fresnel diffraction From Wikipedia, the free encyclopedia In optics, the Fresnel diffraction equation for near-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, F of the optical arrangement, which is defined, for a wave incident 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 be 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] Contents 1 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 also 5 Notes 6 References Fresnel diffraction showing central Arago spot Early treatments of this phenomenon en.wikipedia.org/wiki/Fresnel_diffraction 1/7 wikipedia.[4] Francis Weston Sears offers a mathematical approximation suggested by Fresnel that predicts the main features of diffraction patterns and uses only simple mathematics. 222."[3] Richard C. the differentials between dark and light bands decrease until a diffraction effect can no longer be detected. If the diameter of the circular hole in the screen is sufficient to expose the first or central Fresnel zone. with coordinate system. showing aperture (or diffracting object) plane and image plane.y. Three of them have dark centers. 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. and check the book by Sears for very much nicer photographs. 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.org/wiki/Fresnel_diffraction 2/7 . If the gap is made progressively wider. 219) In his Optics.. the free encyclopedia 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. following p. what transpires.z) is given by: Diffraction geometry. He uses the Principle of Huygens to investigate. Figure 9-5. As the gap becomes larger.Wikipedia. en. By considering the perpendicular distance from 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. That means that a Fresnel diffraction pattern can have a dark center. but he does point to the inverse situation wherein the shadow produced by a small circular object can paradoxically have a bright center. MacLaurin explains Fresnel diffraction by asking what happens when light propagates. If the diameter of the circular hole in the screen is sufficient to expose two Fresnel zones. then diffraction patterns with dark centers will alternate with diffraction patterns with bright centers. The Fresnel diffraction integral The electric field diffraction pattern at a point (x. The result is that if the gap is very narrow only diffraction patterns with bright centers can occur. 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. in classical terms. In his monograph entitled "Light.9/12/13 Fresnel diffraction . then the amplitude at the center is almost zero. in Sears shows four Fraunhofer patterns in the top row followed by sixteen Fresnel diffraction patterns. These patterns can be seen and measured. (See the photograph above. The inner zone will be a circle and each succeeding zone will be a concentric annular ring. and correspond well to the values calculated for them. 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. (p. we find: Next. it has to contribute to the variation of the exponential for an almost null term.[5] Let us substitute this expression in the argument of the exponential within the integral. Therefore. it has to be much smaller than the period of the complex exponential.9/12/13 Fresnel diffraction . . then there is no approximation. Analytical solution of this integral is impossible for all but the simplest diffraction geometries. In other words. we can simplify the algebra by introducing the substitution: Substituting into the expression for r.e. i. the free encyclopedia where is the aperture. it is usually calculated numerically. In order to make this possible. the key to the Fresnel approximation is to assume that the third term is very small and can be ignored. and is the imaginary unit.wikipedia.org/wiki/Fresnel_diffraction 3/7 . using the Taylor series expansion we can express r as If we consider all the terms of Taylor series. First.Wikipedia. The Fresnel approximation The main problem for solving the integral is the expression of r. : en. the free encyclopedia expressing k in terms of the wavelength. so we can ignore them as well. the required inequality will always hold true as long as We can then approximate the expression with only the first two terms: This equation. y and y' . the wavelength λ is typically many orders of magnitude smaller than the relevant physical dimensions. as a practical matter.wikipedia. we have or. and the inequality stated above is a condition for the approximation's validity.Wikipedia. then. For applications involving optical wavelengths. In particular: and Thus.org/wiki/Fresnel_diffraction 4/7 . then all terms of higher order will be even smaller. substituting the earlier expression for ρ2.9/12/13 Fresnel diffraction . is the Fresnel approximation. we get the following relationship: Multiplying both sides by . then we can ignore the third term in the Taylor expression. x' . en. if the third term is negligible. Furthermore. If this condition holds true for all values of x . the free encyclopedia Fresnel diffraction The condition for validity is fairly weak. provided the aperture is small compared to the path length.9/12/13 Fresnel diffraction . For Fresnel diffraction the electric field at point (x. and it allows all length parameters to take comparable values. Alternative forms Convolution The integral can be expressed in other ways in order to calculate it using some mathematical properties. where L is the distance between the aperture and the field point. That is why we might call the function h(x. Analytical solution of this expression is still only possible in rare cases.Wikipedia. it is always valid if as well as the Fresnel condition. originating at the aperture and moving along z. where the values of x and y are much smaller than z. The integral modulates the amplitude and phase of the spherical wave. In addition. in order to correctly calculate the relative phase of interfering waves.z) the impulse response of free space propagation.z) is then given by: Fresnel diffraction occurs when: Fraunhofer diffraction occurs when: – aperture or slit size.y. Fresnel diffraction accounts for the curvature of the wavefront. the propagating field is a spherical wave. we have .org/wiki/Fresnel_diffraction 5/7 . valid only for much larger distances from the diffraction source see Fraunhofer diffraction. If in the integral we express k in terms of the wavelength. – wavelength. For the r in the denominator we go one step further. For a further simplified case. and approximate it with only the first term. it means that.y. en.wikipedia. . Fourier transform Another possible way is through the Fourier transform. Unlike Fraunhofer diffraction. – distance from the aperture This is the Fresnel diffraction integral. This is valid in particular if we are interested in the behaviour of the field only in a small area close to the origin. if the Fresnel approximation is valid. If we define 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. discussed below. See also Fraunhofer diffraction Fresnel integral Fresnel zone Fresnel number Augustin-Jean Fresnel Ridged mirror en.e. This expression is better than the others when the process leads to a known Fourier transform. corresponding to how the Fourier transform is a rotation in the time-frequency domain. Let us use the following definition: where p and q are spatial frequencies (in units of lines/meter). the free encyclopedia and we expand each component of the transverse displacement.Wikipedia.wikipedia. and the connection with the Fourier transform is tightened in the linear canonical transformation. calculate its two dimensional Fourier transform. Fresnel diffraction can be seen as a shear in the timefrequency domain.q) with and multiply it by another factor. then we can express the integral in terms of the two dimensional Fourier transform.org/wiki/Fresnel_diffraction 6/7 . Linear canonical transformation Main article: Linear canonical transformation From the point of view of the linear canonical transformation.9/12/13 Fresnel diffraction . first multiply the field to be propagated for a complex exponential. replace (p. The Fresnel integral can be expressed as: where i. ^ There was actually an approximation in a prior step. Addison-Wesley.pdf H. Born & E. Retrieved from "http://en. 013203 (2005). en. Inc.org/w/index. but to the scalar one. By using this site.org/wiki/Fresnel_diffraction 7/7 . Kouznetsov. ^ M.jp/~dima/PhysRevLett_94_013203. Text is available under the Creative Commons Attribution-ShareAlike License. Wikipedia® is a registered trademark of the Wikimedia Foundation. when assuming is a real wave. Francis Weston Sears. Oberst. p. See scalar wave approximation References Goodman. the free encyclopedia Fresnel imager Euler spiral Notes 1. ^ Optics. Shimizu.Wikipedia..uec. 1909. D. 248ff. Columbia University Press 4.ils. Principles of Optics. (1996).php?title=Fresnel_diffraction&oldid=564662007" Categories: Diffraction This page was last modified on 17 July 2013 at 15:01. Cambridge University Press. F. you agree to the Terms of Use and Privacy Policy. Wolf. 94. Fresnel diffraction mirror for atomic wave. additional terms may apply. Introduction to Fourier optics.ac. K. In fact this is not a real solution to the vector Helmholtz equation.wikipedia. a non-profit organization. J. Joseph W.9/12/13 Fresnel diffraction . MacLaurin. 1948 5. New York: McGraw-Hill. Shimizu. Fujita. ^ http://www. Physical Review Letters." by Richard C. ^ Light. 1999. ISBN 0-07024254-2.wikipedia. 3. Cambridge 2.