Fraunhofer diffraction
Encyclopedia
In optics
, the Fraunhofer diffraction equation is used to model the diffraction
of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens
.
The equation was named in honour of Joseph von Fraunhofer
although he was not actually involved in the development of the theory.
This article explains where the Fraunhofer equation can be applied, and shows the form of the Fraunhofer diffraction pattern for various apertures. A detailed mathematical treatment of Fraunhofer diffraction is given in this article
.
The Fraunhofer diffraction equation=
When a beam of light is partly blocked by an obstacle, some of the light is scattered around the object, and light and dark bands are often seen at the edge of the shadow - this effect is known as diffraction. These effects can be modelled using the Huygens–Fresnel principle. Huygens postulated that every point on a primary wavefront acts as a source of spherical secondary wavelets and the sum of these secondary waves determines the form of the wave at any subsequent time. Fresnel developed an equation using the Huygens wavelets together with the principle of superposition of waves, which models these diffraction effects quite well.
It is not a straightforward matter to calculate the displacement given by the sum of the secondary wavelets, each of which has its own amplitude and phase, since this involves addition of many waves of varying phase and amplitude. When two waves are added together, the total displacement depends on both the amplitude
and the phase
of the individual waves: two waves of equal amplitude
which are in phase give a displacement whose amplitude is double the individual wave amplitudes, while two waves which are in opposite phases give a zero displacement. Generally, a two-dimensional integral over complex variables has to be solved and in many cases, an analytic solution is not available.
The Fraunhofer diffraction equation
is a simplified version of the Kirchhoff's diffraction formula
and it can be used to model the light diffracted when both the light source and the viewing plane are effectively at infinity with respect to the diffracting aperture. In this case, the incident light is a plane wave
so that the phase of the light at each point in the aperture is the same. The phase of the contributions of the individual wavelets in the aperture varies linearly with position in the aperture, making the calculation of the sum of the contributions relatively straightforward in many cases.
Strictly speaking, the Fraunhofer approximation only applies when the diffracted pattern is viewed at infinity, but in practice it can be applied in the far field, and also in the focal plane of a positive lens.
When the distance between the aperture and the plane in which the pattern is observed is large enough that the difference in phase between the light from the extremes of the aperture is much less than the wavelength, then individual contributions can be treated as though they are parallel. This is often known as the far field and is defined as being located at a distance which is greater than , where is the wavelength and is the largest dimension in the aperture. The Fraunhofer equation can be used to model the diffraction in this case.
For example, if a 0.5mm diameter circular hole is illuminated by a laser with 0.6μm wavelength, the Fraunhofer diffraction equation can be employed if the viewing distance is greater than 100mm.
incident on a positive lens is focused at a point by the lens; all the 'rays' have the same phase at the point of focus, so that this is equivalent to viewing the plane wave at infinity. Thus, if the diffracted light is focused with a lens, the observed diffraction pattern can be modelled using Fraunhofer diffraction. The diffracted light can be considered to be made up of a set of plane waves of varying orientation. When a lens is located in front of the diffracting aperture, each plane wave is brought to a focus at a different point in the focal plane with the point of focus being proportional to the x- and y-direction cosines, so that the variation in intensity as a function of direction is mapped into a positional varation in intensity.
Examples of Fraunhofer diffraction=
In each of these examples, the aperture is illuminated by a monochromatic plave wave at normal incidence.
The width of the slit is . The Fraunhofer diffraction pattern is shown in the image together with a plot of the intensity vs. angle . . The pattern has maximum intensity at , and a series of peaks of decreasing intensity. Most of the diffracted light falls between the first minima. The angle,, subtended by these two minima is given by:
Thus, the smaller the aperture, the larger the angle, subtended by the diffraction bands. The size of the central band at a distance is given by
For example, when a slit of width 0.5mm is illuminated by light of wavelength 0.6um, and viewed at a distance of 1000mm, the width of the central band in the diffraction pattern is 2.4mm.
The fringes extend to infinity in the direction since the slit and illumination also extend to infinity.
If , the intensity of the diffracted light does not fall to zero, and if , the diffracted wave is cylindrical.
The angle subtended by the first minima on either side of the centre is then, as above:
There is no such simple argument to enable us to find the maxima of the diffraction pattern.
If the illuminating beam does not illuminate the whole length of the slit, the spacing of the vertical fringes is determined by the dimensions of the illuminating beam. Close examination of the double-slit diffraction pattern below shows that there are very fine horizontal diffraction fringes above and below the main spot, as well as the more obvious horizontal fringes.
where is the diameter of the aperture.
The Airy disk can be an important parameter in limiting the ability
of an imaging system to resolve closely located objects.
It should be noted that in some textbooks, this formula is defined in terms of the radius of the aperture rather than its diameter, and the multiple in that case will be 2.44.
- the aperture is covered by a filter whose transmission varies as a Gaussian function, giving a diffraction pattern with no secondary rings.
The output profile of single mode laser beam may have a Gaussian
intensity profile and the diffraction equation can be used to show that it maintains that profile however far away is propagates from the source.
.
The angular spacing of the fringes is given by
The spacing of the fringes at a distance from the slits is given by
where is the separation of the slits.
The fringes in the picture were obtained using the yellow light from a sodium light (wavelength = .589μm), with slits separated by 0.25mm, and projected directly onto the image plane of a digital camera.
Double slit interference fringes can be observed by cutting two slits in a piece of card, illuminating with a laser pointer, and observing the diffracted light at a distance of 1m. If the slit separation is 0.5mm, and the wavelength of the laser is 0.6μm the spacing of the fringes viewed at a distance of 1m would be 1.2mm.
If the viewing distance is large compared with the separation of the slits (the far field), the phase difference can be found using the geometry shown in the figure below right. The path difference between two waves travelling at an angle is given by:
When the two waves are in phase, i.e. the path difference is equal to an integral number of wavelengths, the summed amplitude, and therefore the summed intensity is maximum, and when they are in anti-phase, i.e. the path difference is equal to half a wavelength, one and a half wavelengths, etc, then the two waves cancel and the summed intensity is zero. This effect is known as interference.
The interference fringe maxima occur at angles
where λ is the wavelength
of the light. The angular spacing of the fringes is is given by
When the distance between the slits and the viewing plane is , the spacing of the fringes is equal to and is the same as above:
A grating whose elements are separated by diffracts a normally incident beam of light into a set of beams, at angles given by:
This is known as the grating equation
The finer the grating spacing, the greater the angular separation of the diffracted beams.
If the light is incident at an angle , the grating equation is:
The detailed structure of the repeating pattern determines the form of the indvidual diffracted beams, as well as their relative intensity while the grating spacing always determines the angles of the diffracted beams.
The image on the right shows a laser beam diffracted by a grating into = 0, and ±1 beams. The angles of the first order beams is about 20o; if we assume the wavelength fo the laser beam is 0.6μm, we can infer that the grating spacing is about 1.8μm.
This is the same relationship that given above.
See also =
Reference sources=
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, the Fraunhofer diffraction equation is used to model the diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...
.
The equation was named in honour of Joseph von Fraunhofer
Joseph von Fraunhofer
Joseph von Fraunhofer was a German optician. He is known for the discovery of the dark absorption lines known as Fraunhofer lines in the Sun's spectrum, and for making excellent optical glass and achromatic telescope objectives.-Biography:Fraunhofer was born in Straubing, Bavaria...
although he was not actually involved in the development of the theory.
This article explains where the Fraunhofer equation can be applied, and shows the form of the Fraunhofer diffraction pattern for various apertures. A detailed mathematical treatment of Fraunhofer diffraction is given in this article
Fraunhofer diffraction (mathematics)
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens....
.
The Fraunhofer diffraction equation=
When a beam of light is partly blocked by an obstacle, some of the light is scattered around the object, and light and dark bands are often seen at the edge of the shadow - this effect is known as diffraction. These effects can be modelled using the Huygens–Fresnel principle. Huygens postulated that every point on a primary wavefront acts as a source of spherical secondary wavelets and the sum of these secondary waves determines the form of the wave at any subsequent time. Fresnel developed an equation using the Huygens wavelets together with the principle of superposition of waves, which models these diffraction effects quite well.
It is not a straightforward matter to calculate the displacement given by the sum of the secondary wavelets, each of which has its own amplitude and phase, since this involves addition of many waves of varying phase and amplitude. When two waves are added together, the total displacement depends on both the amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
and the phase
Phase
-In physics:*Phase , a physically distinctive form of a substance, such as the solid, liquid, and gaseous states of ordinary matter**Phase transition is the transformation of a thermodynamic system from one phase to another*Phase...
of the individual waves: two waves of equal amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
which are in phase give a displacement whose amplitude is double the individual wave amplitudes, while two waves which are in opposite phases give a zero displacement. Generally, a two-dimensional integral over complex variables has to be solved and in many cases, an analytic solution is not available.
The Fraunhofer diffraction equation
Fraunhofer diffraction (mathematics)
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens....
is a simplified version of the Kirchhoff's diffraction formula
Kirchhoff's diffraction formula
Kirchhoff's diffraction formula can be used to model the propagation of light in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave passes through an opening in an opaque screen...
and it can be used to model the light diffracted when both the light source and the viewing plane are effectively at infinity with respect to the diffracting aperture. In this case, the incident light is a plane wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
so that the phase of the light at each point in the aperture is the same. The phase of the contributions of the individual wavelets in the aperture varies linearly with position in the aperture, making the calculation of the sum of the contributions relatively straightforward in many cases.
Strictly speaking, the Fraunhofer approximation only applies when the diffracted pattern is viewed at infinity, but in practice it can be applied in the far field, and also in the focal plane of a positive lens.
Far field
| Fresnel diffraction Fresnel diffraction 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.... occurs when:  |
Fraunhofer diffraction occurs when: |
 - aperture or slit size, - wavelength, - distance from the aperture |
When the distance between the aperture and the plane in which the pattern is observed is large enough that the difference in phase between the light from the extremes of the aperture is much less than the wavelength, then individual contributions can be treated as though they are parallel. This is often known as the far field and is defined as being located at a distance which is greater than , where is the wavelength and is the largest dimension in the aperture. The Fraunhofer equation can be used to model the diffraction in this case.
For example, if a 0.5mm diameter circular hole is illuminated by a laser with 0.6μm wavelength, the Fraunhofer diffraction equation can be employed if the viewing distance is greater than 100mm.
Focal plane of a positive lens
A plane wavePlane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
incident on a positive lens is focused at a point by the lens; all the 'rays' have the same phase at the point of focus, so that this is equivalent to viewing the plane wave at infinity. Thus, if the diffracted light is focused with a lens, the observed diffraction pattern can be modelled using Fraunhofer diffraction. The diffracted light can be considered to be made up of a set of plane waves of varying orientation. When a lens is located in front of the diffracting aperture, each plane wave is brought to a focus at a different point in the focal plane with the point of focus being proportional to the x- and y-direction cosines, so that the variation in intensity as a function of direction is mapped into a positional varation in intensity.
Examples of Fraunhofer diffraction=
In each of these examples, the aperture is illuminated by a monochromatic plave wave at normal incidence.
Diffraction by a slit of infinite depth
Thus, the smaller the aperture, the larger the angle, subtended by the diffraction bands. The size of the central band at a distance is given by
For example, when a slit of width 0.5mm is illuminated by light of wavelength 0.6um, and viewed at a distance of 1000mm, the width of the central band in the diffraction pattern is 2.4mm.
The fringes extend to infinity in the direction since the slit and illumination also extend to infinity.
If , the intensity of the diffracted light does not fall to zero, and if , the diffracted wave is cylindrical.
Semi-quantitative analysis of single slit diffraction
We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. Consider the light diffracted at an angle where the distance is equal to the wavelength of the illuminating light. The component of the wavelet emitted from the point A which is travelling in the direction is in anti-phase with the wave from the point at middle of the slit, so that the net contribution at the angle from these two waves is zero. The same applies to the posints just below and , and so on. Therefore, the amplitude of the total wave travelling in the direction is zero. We have:The angle subtended by the first minima on either side of the centre is then, as above:
There is no such simple argument to enable us to find the maxima of the diffraction pattern.
Diffraction by a rectangular aperture
The form of the diffraction pattern given by a rectangular aperture is shown in the figure on the right. There is a central semi-rectangular peak, with a series of horizontal and vertical fringes. The dimensions of the central band are related to the dimensions of the slit by the same relationship as for a single slit so that the larger dimension in the diffracted image corresponds to the smaller dimension in the slit. The spacing of the fringes is also inversely proportional to the slit dimension.If the illuminating beam does not illuminate the whole length of the slit, the spacing of the vertical fringes is determined by the dimensions of the illuminating beam. Close examination of the double-slit diffraction pattern below shows that there are very fine horizontal diffraction fringes above and below the main spot, as well as the more obvious horizontal fringes.
Diffraction by a circular aperture
The diffraction pattern given by a circular aperture is shown in the figure on the right. This is known as the Airy diffraction pattern. It can be seen that most of the light is in the central disk. The angle subtended by this disk, known as the Airy disk, iswhere is the diameter of the aperture.
The Airy disk can be an important parameter in limiting the ability
Angular resolution
Angular resolution, or spatial resolution, describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object...
of an imaging system to resolve closely located objects.
It should be noted that in some textbooks, this formula is defined in terms of the radius of the aperture rather than its diameter, and the multiple in that case will be 2.44.
Diffraction by an aperture with a gaussian profile
The diffraction pattern obtained given by an aperture with a Gaussian profile, for example, a photographic slide whose transmission has a Gaussian variation is also a Gaussian function. The form of the function is plotted on the right, and it can be seen that, unlike the diffraction patterns produced by rectangular or circular apertures, it has no secondary rings. This technique can be used in a process called apodizationApodization
Apodization literally means "removing the foot". It is the technical term for changing the shape of a mathematical function, an electrical signal, an optical transmission or a mechanical structure.- Apodization in signal processing :...
- the aperture is covered by a filter whose transmission varies as a Gaussian function, giving a diffraction pattern with no secondary rings.
The output profile of single mode laser beam may have a Gaussian
Gaussian beam
In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity distributions are well approximated by Gaussian functions. Many lasers emit beams that approximate a Gaussian profile, in which case the laser is said to be operating on the fundamental...
intensity profile and the diffraction equation can be used to show that it maintains that profile however far away is propagates from the source.
Diffraction by a double slit
In the double-slit experiment, the two slits are illuminated by a single light beam. If the width of the slits is small enough (less than the wavelength of the light), the slits diffract the light into cylindrical waves. These two cylindrical wavefronts are superimposed, and the amplitude, and therefore the intensity, at any point in the combined wavefronts depends on both the magnitude and the phase of the two wavefronts. These fringes are often known as Young's fringesYoung's interference experiment
Young's interference experiment, also called Young's double-slit interferometer, was the original version of the modern double-slit experiment, performed at the beginning of the nineteenth century by Thomas Young. This experiment played a major role in the general acceptance of the wave theory of...
.
The angular spacing of the fringes is given by
The spacing of the fringes at a distance from the slits is given by
where is the separation of the slits.
The fringes in the picture were obtained using the yellow light from a sodium light (wavelength = .589μm), with slits separated by 0.25mm, and projected directly onto the image plane of a digital camera.
Double slit interference fringes can be observed by cutting two slits in a piece of card, illuminating with a laser pointer, and observing the diffracted light at a distance of 1m. If the slit separation is 0.5mm, and the wavelength of the laser is 0.6μm the spacing of the fringes viewed at a distance of 1m would be 1.2mm.
Semi quantitative explanation of double-slit fringes
The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves.If the viewing distance is large compared with the separation of the slits (the far field), the phase difference can be found using the geometry shown in the figure below right. The path difference between two waves travelling at an angle is given by:
When the two waves are in phase, i.e. the path difference is equal to an integral number of wavelengths, the summed amplitude, and therefore the summed intensity is maximum, and when they are in anti-phase, i.e. the path difference is equal to half a wavelength, one and a half wavelengths, etc, then the two waves cancel and the summed intensity is zero. This effect is known as interference.
The interference fringe maxima occur at angles
where λ is the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
of the light. The angular spacing of the fringes is is given by
When the distance between the slits and the viewing plane is , the spacing of the fringes is equal to and is the same as above:
Diffraction by a grating
A grating is defined in Born and Wolf as "any arrangement which imposes on an incident wave a periodic variation of amplitude or phase, or both".A grating whose elements are separated by diffracts a normally incident beam of light into a set of beams, at angles given by:
This is known as the grating equation
Diffraction grating
In optics, a diffraction grating is an optical component with a periodic structure, which splits and diffracts light into several beams travelling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as...
The finer the grating spacing, the greater the angular separation of the diffracted beams.
If the light is incident at an angle , the grating equation is:
The detailed structure of the repeating pattern determines the form of the indvidual diffracted beams, as well as their relative intensity while the grating spacing always determines the angles of the diffracted beams.
The image on the right shows a laser beam diffracted by a grating into = 0, and ±1 beams. The angles of the first order beams is about 20o; if we assume the wavelength fo the laser beam is 0.6μm, we can infer that the grating spacing is about 1.8μm.
Semi-quantitative explanation
A simple grating consists of a series of slits in a screen. If the light travelling at an angle from each slit has a path difference of one wavelength with respect to the adjacent slit, all these waves will add together, so that the maximum intensity of the diffracted light is obtained when:This is the same relationship that given above.
See also =
- Fraunhofer diffraction (mathematics)Fraunhofer diffraction (mathematics)In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens....
- DiffractionDiffractionDiffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
- Huygens-Fresnel principleHuygens-Fresnel principleThe Huygens–Fresnel principle is a method of analysis applied to problems of wave propagation both in the far-field limit and in near-field diffraction.-History:...
- Kirchhoff's diffraction formulaKirchhoff's diffraction formulaKirchhoff's diffraction formula can be used to model the propagation of light in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave passes through an opening in an opaque screen...
- Fresnel diffractionFresnel diffractionIn 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....
- Airy discAiry discIn optics, the Airy disk and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light....
- Fourier opticsFourier opticsFourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...
Reference sources=
- Born MMax BornMax Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
& Wolf E, Principles of Optics, 1999, 7th Edition, Cambridge University Press, ISBN 13 978-0-521-64222-4 - Heavens OS and Ditchburn W, Insight into Optics, 1991, Longman and Sons, Chichester ISBN 978-0471927693
- Hecht Eugene, Optics, 2002, Addison Wesley, ISBN-0-321-18878-0
- Jenkins FA & White HE, Fundamentals of Optics, 1957, 3rd Edition, McGraw Hill, New York
- Lipson A., Lipson SG, Lipson HHenry LipsonHenry Lipson CBE was a British physicist. He was Professor of Physics, University of Manchester Institute of Science and Technology, 1954-77, then Professor Emeritus....
, Optical Physics, 2011, Cambridge University Press, ISBN 978-0-521-40345-1 - Longhurst RS, Geometrical and Physical Optics,1967, 2nd Edition, Longmans, London
External links
- Fraunhofer diffraction on ScienceWorldScienceWorldScienceWorld, also known as Eric Weisstein's World of Science, is a Web site that opened to the general public in January 2002. As of November 2007, ScienceWorld includes more than 4,000 entries in fields of science including astronomy, chemistry, physics, as well as biographies of many scientists...
- Fraunhofer diffraction on HyperPhysicsHyperPhysicsHyperPhysics is an educational resource about physics topics. The information architecture of the website is based on trees that organize topics from general to specific...