In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, the Fresnel diffraction equation for near-field diffraction is an approximation of the
Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the
near field. It is used to calculate the
diffraction pattern
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the
far field
The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the ant ...
region is given by the
Fraunhofer diffraction
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer ...
equation.
The near field can be specified by the
Fresnel number
The Fresnel number (''F''), named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in diffraction, scalar diffraction theory.
Definition
For an electromagnetic wave passing through an apertu ...
, , of the optical arrangement. When
the diffracted wave is considered to be in the near field. However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below. Specifically, the phase terms of third order and higher must be negligible, a condition that may be written as
where
is the maximal angle described by
, and the same as in the definition of the
Fresnel number
The Fresnel number (''F''), named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in diffraction, scalar diffraction theory.
Definition
For an electromagnetic wave passing through an apertu ...
.
The multiple Fresnel diffraction at closely spaced periodical ridges (
ridged mirror
In atomic physics, a ridged mirror (or ridged atomic mirror, or Fresnel diffraction mirror) is a kind of atomic mirror, designed for the specular reflection of neutral particles (atoms) coming at a grazing incidence angle. In order to reduce the ...
) causes the
specular reflection
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
; this effect can be used for
atomic mirror In physics, an atomic mirror is a device which reflects neutral atoms in the similar way as a conventional mirror reflects visible light. Atomic mirrors can be made of electric fields or magnetic fields, electromagnetic waves or just silicon wafer; ...
s. The example of Fresnel Diffraction is Nearfield Diffraction
Early treatments of this phenomenon
Some of the earliest work on what would become known as Fresnel diffraction was carried out by
Francesco Maria Grimaldi
Francesco Maria Grimaldi, SJ (2 April 1618 – 28 December 1663) was an Italian Jesuit priest, mathematician and physicist who taught at the Jesuit college in Bologna. He was born in Bologna to Paride Grimaldi and Anna Cattani.
Work
Between 1 ...
in Italy in the 17th century. In his monograph entitled "Light",
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 effect 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'', Francis Weston Sears offers a mathematical approximation suggested by Fresnel that predicts the main features of diffraction patterns and uses only simple mathematics. 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 zone
A Fresnel zone ( ), named after physicist Augustin-Jean Fresnel, is one of a series of confocal prolate ellipsoidal regions of space between and around a transmitter and a receiver. The primary wave will travel in a relative straight line fro ...
s. The inner zone is a circle and each succeeding zone will be a concentric annular ring. If the diameter of the circular hole in the screen is 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 well to the values calculated for them.
The Fresnel diffraction integral
The electric field
diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
pattern at a point ''(x, y, z)'' is given by:
where
*
is the electric field at the aperture;
*
;
*
is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
; and
*
is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.
The analytical solution of this integral quickly becomes impractically complex for all but the simplest diffraction geometries. Therefore, it is usually calculated numerically.
The Fresnel approximation
![Comparison Sommerfled-Fresnel-Fraunhofer](https://upload.wikimedia.org/wikipedia/commons/6/61/Comparison_Sommerfled-Fresnel-Fraunhofer.gif)
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, by the binomial expansion,
We can express
as
If we consider all the terms of binomial series, then there is no approximation.
[There was actually an approximation in a prior step, when assuming is a real wave. In fact this is not a real solution to the vector ]Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
, but to the scalar one. See scalar wave approximation
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
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 and henceforth any higher orders. 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.,
:
expressing ''k'' in terms of the wavelength,
we get the following relationship:
Multiplying both sides by
, we have
or, substituting the earlier expression for
,
If this condition holds true for all values of , , and , 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 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.
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 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 and are much smaller than . In general, Fresnel diffraction is valid if the
Fresnel number
The Fresnel number (''F''), named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in diffraction, scalar diffraction theory.
Definition
For an electromagnetic wave passing through an apertu ...
is approximately 1.
For Fresnel diffraction the electric field at point
is then given by:
![Circular Aperture Fresnel Diffraction high res](https://upload.wikimedia.org/wikipedia/commons/d/d3/Circular_Aperture_Fresnel_Diffraction_high_res.jpg)
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 . 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
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer ...
. Unlike Fraunhofer diffraction, Fresnel diffraction accounts for the curvature of the
wavefront
In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
, in order to correctly calculate the relative
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
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 define the following function:
then the integral can be expressed in terms of a
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
:
in other words we are representing the propagation using a linear-filter modeling. That is why we might call the function
the impulse response of free space propagation.
Fourier transform
Another possible way is through the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
. If in the integral we express in terms of the wavelength:
and 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:
where and are spatial frequencies (
wave number
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
s). The Fresnel integral can be expressed as
That is, first multiply the field to be propagated by a complex exponential, calculate its two-dimensional Fourier transform, replace
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 In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the ac ...
, discussed below.
Linear canonical transformation
From the point of view of the
linear canonical transformation In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the ac ...
, Fresnel diffraction can be seen as a
shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
in the
time-frequency domain, corresponding to how the Fourier transform is a rotation in the time-frequency domain.
See also
*
Fraunhofer diffraction
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer ...
*
Fresnel integral
250px,
Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below).
The Fresnel integrals and are two transcendental functions n ...
*
Fresnel zone
A Fresnel zone ( ), named after physicist Augustin-Jean Fresnel, is one of a series of confocal prolate ellipsoidal regions of space between and around a transmitter and a receiver. The primary wave will travel in a relative straight line fro ...
*
Fresnel number
The Fresnel number (''F''), named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in diffraction, scalar diffraction theory.
Definition
For an electromagnetic wave passing through an apertu ...
*
Augustin-Jean Fresnel
Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Isaac Newton, Newton's co ...
*
Ridged mirror
In atomic physics, a ridged mirror (or ridged atomic mirror, or Fresnel diffraction mirror) is a kind of atomic mirror, designed for the specular reflection of neutral particles (atoms) coming at a grazing incidence angle. In order to reduce the ...
*
Fresnel imager
A Fresnel imager is a proposed ultra-lightweight design for a space telescope that uses a Fresnel lens, Fresnel array as primary optics instead of a typical lens. It focuses light with a thin opaque foil sheet punched with specially shaped holes, t ...
*
Euler spiral
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals.
E ...
Notes
References
*
{{DEFAULTSORT:Fresnel Diffraction
Diffraction