Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. Very similar effects are observed when there is an alteration in the properties of the medium in which the wave is travelling, for example a variation in refractive index for light waves or in acoustic impedance for sound waves and these can also be referred to as diffraction effects. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium The acoustic impedance Z (or sound impedance) is a frequency f dependent parameter and is very useful for example for describing the behaviour of musical Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies Water is a common Chemical substance that is essential for the survival of all known forms of Life. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. X-radiation (composed of X-rays) is a form of Electromagnetic radiation. Radio waves are electromagnetic waves occurring on the Radio frequency portion of the Electromagnetic spectrum. As physical objects have wave-like properties, diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons
While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves where the wavelength is on the order of the size of the diffracting objects. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. The complex patterns resulting from the intensity of a diffracted wave are a result of the superpostion, or interference of different parts of a wave that traveled to the observer by different paths. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern
The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer are all explained by diffraction theory.
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Examples of diffraction in everyday life
The effects of diffraction can be readily seen in everyday life. The most colorful examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern we see when looking at a disk. In Optics, a Diffraction grating is an optical component with a regular pattern which splits ( diffracts) light into several beams travelling in different This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Holography (from the Greek, ὅλος - hólos whole + γραφή - grafē writing drawing is a technique that allows the Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. Atmospheric diffraction is manifested in the following principal ways Fourier optics is the bending of light rays in the Atmosphere A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The speckle pattern which is observed when laser light falls on an optically rough service is also a diffraction phenomenon. A speckle pattern is a random intensity pattern produced by the mutual Interference of a set of Wavefronts This phenomenon has been investigated by scientists All these effects are a consequence of the fact that light is a wave.
Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, this is the reason we can still hear someone calling us even if we are hiding behind a tree. Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope. The resolution of an optical imaging system like a Microscope or Telescope or Camera can be limited by multiple factors like imperfections in the lenses or misalignment
History
The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The Royal Society of London for the Improvement of Natural Knowledge, known simply as The Royal Society, is a Learned society for science that was founded in 1660 Francesco Maria Grimaldi ( April 2, 1618 - December 28, 1663) was an Italian Mathematician and Physicist who The results of Grimaldi's observations were published posthumously in 1665. [2][3] Isaac Newton studied these effects and attributed them to inflexion of light rays. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating. James Gregory (November 1638 &ndash October 1675 was a Scottish Mathematician and Astronomer. In Optics, a Diffraction grating is an optical component with a regular pattern which splits ( diffracts) light into several beams travelling in different In 1803 Thomas Young did his famous experiment observing interference from two closely spaced slits. Thomas Young (13 June 1773 &ndash 10 May 1829 was an English Polymath who contributed to the scientific understanding of vision, Light Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory. Christiaan Huygens (ˈhaɪgənz in English ˈhœyɣəns in Dutch) ( April 14, 1629 &ndash July 8, 1695) was a Dutch
The mechanism of diffraction
Diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle. In Physics and Engineering, a ripple tank is a shallow glass tank of water used in schools and colleges to demonstrate the basic properties of Waves It The Huygens–Fresnel principle (named for Dutch Physicist Christiaan Huygens, and French physicist Augustin-Jean Fresnel The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and addition of all these radial waves form the new wavefront. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves, an effect which is often known as wave interference. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern The summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. hence, diffraction patterns usually have a series of maxima and minima.
To determine the form of a diffraction pattern, we must determine the phase and amplitude of each of the Huygens wavelets at each point in space and then find the sum of these waves. There are various analytical models which can be used to do this including the Fraunhoffer diffraction equation for the far field and the Fresnel Diffraction equation for the near-field. In Optics, Fraunhofer diffraction is a form of wave Diffraction, which occurs when field waves are passed through an Aperture or slit causing only the In Optics, Fresnel diffraction or near-field diffraction is a process of diffraction which occurs when a wave passes through an Aperture and diffracts Most configurations cannot be solved analytically; solutions can be found using various numerical analytical methods including Finite element and boundary element methods
Diffraction systems
It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out. The finite element method (FEM (sometimes referred to as finite element analysis) is a numerical technique for finding approximate solutions of Partial differential The boundary element method is a numerical computational method of solving linear Partial differential equations which have been formulated as Integral equations (i
The simplest descriptions of diffraction are those in which the situation can be reduced to a two dimensional problem. For water waves, this is already the case, water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem.
Some of the simpler cases of diffraction are considered below.
Single-slit diffraction
A long slit of infinitesmal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.
A slit which is wider than a wavelength has a large number of point sources spaced evenly across the width of the slit. The light at a given angle is made up contributions from each of these point sources and if the relative phases of these contributions vary by more than 2π, we expect to find minima and maxima in the diffracted light.
We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interferes destructively with the source located just to below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is given by (d sinθ)/2 so that the minimum intensity occurs at an angle θmin given by
where d is the width of the slit.
A similar argument can be used to show that if we imagine the slit to be divided into four, six eight parts, etc, minima are obtained at angles θn given by
where n is an integer greater than zero.
There is no such simple argument to enable us to to find the maxima of the diffraction pattern. The intensity profile can be calculated using the Fraunhofer diffraction integral as
 | ![= I_0 {\left[ \mathrm{sinc} \left( \frac{\pi d}{\lambda} \sin \theta \right) \right] }^2](../../../../math/b/9/6/b96599de19033c8543943fa807362fcd.png) |
where the sinc function is given by sinc(x)=sin(x)/x. See also Diffraction Quantitative description and analysis Because diffraction is the result of addition of all waves (of given wavelength along all unobstructed In Optics, Fraunhofer diffraction is a form of wave Diffraction, which occurs when field waves are passed through an Aperture or slit causing only the In Mathematics, the sinc function, denoted by \scriptstyle\mathrm{sinc}(x\ and sometimes as \scriptstyle\mathrm{Sa}(x\ has two definitions sometimes
It should be noted that this analysis applies only to the far field, i. The near field and far field of an antenna or other isolated source of Electromagnetic radiation are regions around the source where different parts of the field e a significant distance from the diffracting slit.
Diffraction Grating
A diffraction grating is an optical component with a regular pattern. In Optics, a Diffraction grating is an optical component with a regular pattern which splits ( diffracts) light into several beams travelling in different The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation
where θi is the angle at which the light is incident, d is the separation of grating elements and m is an integer which can be positive or negative.
The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and
The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.
Diffraction by a circular aperture
The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the Airy Disk. The Airy disk (or Airy disc) is a phenomenon in Optics. Owing to the wave nature of light, light passing through an Aperture is diffracted The Airy disk (or Airy disc) is a phenomenon in Optics. Owing to the wave nature of light, light passing through an Aperture is diffracted The variation in intensity with angle is given by
where a is the radius of the circular aperture, k is equal to 2π/λ and J1 is a Bessel function. The Airy disk (or Airy disc) is a phenomenon in Optics. Owing to the wave nature of light, light passing through an Aperture is diffracted In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.
Propagation of a laser beam
The way in which the profile of a laser beam changes as it propagates is determined by diffraction. In Optics, a Gaussian beam is a Beam of Electromagnetic radiation whose transverse Electric field and Intensity ( Irradiance A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission. The output mirror of the laser is an aperture, and the subsequent beam shape is determined by that aperture. Hence, the smaller the output beam, the quicker it diverges. Diode lasers have much greater divergence than He-Ne lasers for this reason.
Paradoxically, it is possible to reduce the divergence of a laser beam by first expanding it with one convex lens, and then collimating it with a second convex lens whose focal point is co-incident with that of the first lens. A lens is an optical device with perfect or approximate Axial symmetry which transmits and refracts Light, converging or diverging The resulting beam has a larger aperture, and hence a lower divergence.
Diffraction limited imaging
The ability of an imaging system to resolve detail is ultimately limited by diffraction. The resolution of an optical imaging system like a Microscope or Telescope or Camera can be limited by multiple factors like imperfections in the lenses or misalignment This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy pattern with a central spot of diameter
where λ is the wavelength of the light, f is the focal length of the lens, and a is the diameter of the beam of light, or (if the beam is filling the lens) the diameter of the lens.
This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see.
Speckle patterns
The speckle pattern which is seen when using a laser pointer is another diffraction phenomenon. A speckle pattern is a random intensity pattern produced by the mutual Interference of a set of Wavefronts This phenomenon has been investigated by scientists A speckle pattern is a random intensity pattern produced by the mutual Interference of a set of Wavefronts This phenomenon has been investigated by scientists It is a result of the superpostion of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity varies randomly.
Common features of diffraction patterns
Several qualitative observations can be made of diffraction in general:
- The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction, in other words: the smaller the diffracting object the 'wider' the resulting diffraction pattern and vice versa. (More precisely, this is true of the sines of the angles. )
- The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.
- When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.
Particle diffraction
- See also: neutron diffraction and electron diffraction
Quantum theory tells us that every particle exhibits wave properties. Neutron diffraction is a crystallographic method for the determination of the atomic and/or magnetic structure of a material Electron diffraction is a technique used to study matter by firing Electrons at a sample and observing the resulting Interference pattern In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The wavelength associated with a particle is the de Broglie wavelength
where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles) . In Physics, the de Broglie hypothesis (pronounced /brœj/ as French breuil close to "broy" is the statement that all Matter (any object has a Wave The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A Sodium atom traveling at about 3000 m/s would have a De Broglie wavelength of about 5 pico meters.
Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.
Relatively recently, larger molecules like buckyballs,[4] have been shown to diffract. "C60" and "C-60" redirect here For other uses see C60 (disambiguation. Currently, research is underway into the diffraction of viruses, which, being huge relative to electrons and other more commonly diffracted particles, have tiny wavelengths so must be made to travel very slowly through an extremely narrow slit in order to diffract. A virus (from the Latin virus meaning Toxin or Poison) is a sub-microscopic infectious agent that is unable The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J
Bragg diffraction
- For more details on this topic, see Bragg diffraction. Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg
Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg It is similar to what occurs when waves are scattered from a diffraction grating. In Optics, a Diffraction grating is an optical component with a regular pattern which splits ( diffracts) light into several beams travelling in different Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law:
where
- λ is the wavelength,
- d is the distance between crystal planes,
- θ is the angle of the diffracted wave. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency.
- and m is an integer known as the order of the diffracted beam.
Bragg diffraction may be carried out using either light of very short wavelength like x-rays or matter waves like neutrons (and electrons) whose wavelength is on the order of (or much smaller than) the atomic spacing[5]. X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystallographic structure chemical composition Neutron diffraction is a crystallographic method for the determination of the atomic and/or magnetic structure of a material Electron diffraction is a technique used to study matter by firing Electrons at a sample and observing the resulting Interference pattern The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. Diffraction contrast, in electron microscopes and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals. An electron microscope is a type of Microscope that uses Electrons to illuminate a specimen and create an enlarged image Diffraction topography (short "topography") is an X-ray imaging technique based on Bragg diffraction.
Coherence
The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In Physics, coherence is a property of waves that enables stationary (i In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.
The length over which the phase in a beam of light is correlated, is called the coherence length. In Physics, coherence is a property of waves that enables stationary (i In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence as it is related to the presence of different frequency components in the wave. In the case light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition. A quantum mechanical system or particle that is bound, confined spacially can only take on certain discrete values of energy as opposed to classical particles which
If waves are emitted from an extended source this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment this would mean that if the transverse coherence length is smaller than the spacing between the two slits the resulting pattern on a screen would look like two single slit diffraction patterns.
In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spacial extent of the wave function that describes the particle.
References
- ^ Dietrich Zawischa. Optical effects on spider webs. Retrieved on 2007-09-21. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1217 - The Estonian tribal leader Lembitu of Lehola was killed in a battle against Teutonic Knights.
- ^ Jean Louis Aubert (1760). Memoires pour l'histoire des sciences et des beaux arts. Paris: Impr. de S. A. S. ; Chez E. Ganeau, 149.
- ^ Sir David Brewster (1831). A Treatise on Optics. London: Longman, Rees, Orme, Brown & Green and John Taylor, 95.
- ^ Brezger, B. ; Hackermüller, L. ; Uttenthaler, S. ; Petschinka, J. ; Arndt, M. ; Zeilinger, A. (February 2002). "Matter-Wave Interferometer for Large Molecules" (reprint). Physical Review Letters 88 (10): 100404. doi:. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
- ^ John M. Cowley (1975) Diffraction physics (North-Holland, Amsterdam) ISBN 0 444 10791 6
See also
- Atmospheric diffraction
- Bragg diffraction
- Diffraction formalism
- Diffractometer
- Dynamical theory of diffraction
- Diffraction grating
- Electron diffraction
- Fraunhofer diffraction
- Fresnel diffraction
- Fresnel number
- Fresnel zone
- Iridescent Cloud
- Neutron diffraction
- Prism
- Powder diffraction
- Refraction
- Schaefer-Bergmann diffraction
- Thinned array curse
- X-ray diffraction
External links
- How to build a diffraction spectrometer
- Diffraction and acoustics.
- On Diffraction at MathPages. Atmospheric diffraction is manifested in the following principal ways Fourier optics is the bending of light rays in the Atmosphere Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg See also Diffraction Quantitative description and analysis Because diffraction is the result of addition of all waves (of given wavelength along all unobstructed A Diffractometer (Main Entry dif·frac·tom·e·ter Pronunciation di-"frak-'tä-m&-t&r Function nounis a Measuring instrument for analyzing the structure of a The dynamical theory of diffraction describes the interaction of Waves with a regular lattice In Optics, a Diffraction grating is an optical component with a regular pattern which splits ( diffracts) light into several beams travelling in different Electron diffraction is a technique used to study matter by firing Electrons at a sample and observing the resulting Interference pattern In Optics, Fraunhofer diffraction is a form of wave Diffraction, which occurs when field waves are passed through an Aperture or slit causing only the In Optics, Fresnel diffraction or near-field diffraction is a process of diffraction which occurs when a wave passes through an Aperture and diffracts The Fresnel number F, named after the physicist Augustin-Jean Fresnel, is a Dimensionless number occurring in Optics, in particular in diffraction In Optics and Radio communications, a Fresnel zone (pronounced FRA-nel Zone) named for physicist Augustin-Jean Fresnel, is one of Cloud iridescence is a common phenomenon in which a Cloud shows vivid unusual colors or an entire spectrum at once Neutron diffraction is a crystallographic method for the determination of the atomic and/or magnetic structure of a material In Optics, a prism is a transparent optical element with flat polished surfaces that refract Light. Powder diffraction is a scientific technique using X-ray, Neutron, or Electron Diffraction on powder or Microcrystalline samples for Refraction is the change in direction of a Wave due to a change in its Speed. Schaefer-Bergmann diffraction is the resulting diffraction pattern of light interacting with sound waves in transparent crystals or glasses The thinned array curse (sometimes sparse array curse) is a theorem in electromagnetic theory of Transmitters It states that a transmitting aperture X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystallographic structure chemical composition
- Wave Optics - A chapter of an online textbook.
- 2-D wave Java applet - Displays diffraction patterns of various slit configurations.
- Diffraction Java applet - Displays diffraction patterns of various 2-D apertures.
- Diffraction approximations illustrated - MIT site that illustrates the various approximations in diffraction and intuitively explains the Fraunhofer regime from the perspective of linear system theory.
- Gap Obstacle Corner - Java simulation of diffraction of water wave.
- Google Maps - Satellite image of Panama Canal entry ocean wave diffraction.
Dictionary
diffraction
-noun
- (quantum mechanics) The breaking up of an electromagnetic wave as it passes a geometric structure (e.g. a slit), followed by reconstruction of the wave by interference.
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