Fourier optics is the study of classical optics using Fourier transforms, in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium. A curved phasefront may be synthesized from an infinite number of these "natural modes" i.e. from plane wave phasefronts oriented in different directions in space. Far from its sources, an expanding spherical wave is locally tangent to a planar phase front, transverse to the radial direction of propagation. In this case, a Fraunhofer diffraction pattern is created, which emanates from a single spherical wave phase center.
In the near field, no single well-defined spherical wave phase center exists, so the wavefront isn't locally tangent to a spherical ball. In this case, a Fresnel diffraction pattern would be created, which emanates from an extended source, consisting of a distribution of spherical wave sources in space. In the near field, a full spectrum of plane waves is necessary to represent the Fresnel near-field wave locally. A "wide" wave moving forward can be regarded as an infinite number of "plane wave modes", all of which could scatter independently of one other; these mathematical simplifications and calculations are the realm of Fourier analysis and synthesis – together, they can describe what happens when light passes through various slits, lenses or mirrors curved one way or the other, or is or reflected. Fourier optics forms much of the theory behind image processing techniques, as well as finding applications where information needs to be extracted from optical sources such as in quantum optics.
To put it in a more complex way, similar to the concept of frequency and time used in traditional Fourier transform theory, Fourier optics makes use of the spatial frequency domain as the conjugate of the spatial domain. Terms and concepts such as transform theory, bandwidth, window functions and sampling from one-dimensional signal processing are used. Light can be described as a waveform propagating through a material medium. Mathematically, the amplitude of one wave component is represented by a scalar wave function u that depends on both space and time: u = u where r = represents position in three dimensional space, t represents time. Fourier optics begins with the homogeneous, scalar wave equation: u = 0. Where u is a real valued Cartesian component of an electromagnetic wave propagating through free space. If light of a fixed frequency/wavelength/color is assumed the time-harmonic form of the optical field is given as: u = R e.where j is the imaginary unit, ω = 2 π f is the angular frequency of the light waves, ψ = a e j ϕ is, in general, a complex quantity, with separate amplitude a and phase ϕ.
Substituting this expression into the wave equation yields the time-independent form of the wave equation known as the Helmholtz equation: ψ = 0 where k = ω c = 2 π λ is the wave number, ψ is the time-independent, complex-valued component of the propagating wave. Note that the propagation constant, k, the frequency, ω, are linearly related to one another, a typical characteristic of transverse electromagnetic waves in homogeneous media. Solutions to the Helmholtz equation may be found in rectangular coordinate
The Netherlands is a country located in Northwestern Europe. The European portion of the Netherlands consists of twelve separate provinces that border Germany to the east, Belgium to the south, the North Sea to the northwest, with maritime borders in the North Sea with Belgium and the United Kingdom. Together with three island territories in the Caribbean Sea—Bonaire, Sint Eustatius and Saba— it forms a constituent country of the Kingdom of the Netherlands; the official language is Dutch, but a secondary official language in the province of Friesland is West Frisian. The six largest cities in the Netherlands are Amsterdam, The Hague, Utrecht and Tilburg. Amsterdam is the country's capital, while The Hague holds the seat of the States General and Supreme Court; the Port of Rotterdam is the largest port in Europe, the largest in any country outside Asia. The country is a founding member of the EU, Eurozone, G10, NATO, OECD and WTO, as well as a part of the Schengen Area and the trilateral Benelux Union.
It hosts several intergovernmental organisations and international courts, many of which are centered in The Hague, dubbed'the world's legal capital'. Netherlands means'lower countries' in reference to its low elevation and flat topography, with only about 50% of its land exceeding 1 metre above sea level, nearly 17% falling below sea level. Most of the areas below sea level, known as polders, are the result of land reclamation that began in the 16th century. With a population of 17.30 million people, all living within a total area of 41,500 square kilometres —of which the land area is 33,700 square kilometres —the Netherlands is one of the most densely populated countries in the world. It is the world's second-largest exporter of food and agricultural products, owing to its fertile soil, mild climate, intensive agriculture; the Netherlands was the third country in the world to have representative government, it has been a parliamentary constitutional monarchy with a unitary structure since 1848.
The country has a tradition of pillarisation and a long record of social tolerance, having legalised abortion and human euthanasia, along with maintaining a progressive drug policy. The Netherlands abolished the death penalty in 1870, allowed women's suffrage in 1917, became the world's first country to legalise same-sex marriage in 2001, its mixed-market advanced economy had the thirteenth-highest per capita income globally. The Netherlands ranks among the highest in international indexes of press freedom, economic freedom, human development, quality of life, as well as happiness; the Netherlands' turbulent history and shifts of power resulted in exceptionally many and varying names in different languages. There is diversity within languages; this holds for English, where Dutch is the adjective form and the misnomer Holland a synonym for the country "Netherlands". Dutch comes from Theodiscus and in the past centuries, the hub of Dutch culture is found in its most populous region, home to the capital city of Amsterdam.
Referring to the Netherlands as Holland in the English language is similar to calling the United Kingdom "Britain" by people outside the UK. The term is so pervasive among potential investors and tourists, that the Dutch government's international websites for tourism and trade are "holland.com" and "hollandtradeandinvest.com". The region of Holland consists of North and South Holland, two of the nation's twelve provinces a single province, earlier still, the County of Holland, a remnant of the dissolved Frisian Kingdom. Following the decline of the Duchy of Brabant and the County of Flanders, Holland became the most economically and politically important county in the Low Countries region; the emphasis on Holland during the formation of the Dutch Republic, the Eighty Years' War and the Anglo-Dutch Wars in the 16th, 17th and 18th century, made Holland serve as a pars pro toto for the entire country, now considered either incorrect, informal, or, depending on context, opprobrious. Nonetheless, Holland is used in reference to the Netherlands national football team.
The region called the Low Countries and the Country of the Netherlands. Place names with Neder, Nieder and Nedre and Bas or Inferior are in use in places all over Europe, they are sometimes used in a deictic relation to a higher ground that consecutively is indicated as Upper, Oben, Superior or Haut. In the case of the Low Countries / Netherlands the geographical location of the lower region has been more or less downstream and near the sea; the geographical location of the upper region, changed tremendously over time, depending on the location of the economic and military power governing the Low Countries area. The Romans made a distinction between the Roman provinces of downstream Germania Inferior and upstream Germania Superior; the designation'Low' to refer to the region returns again in the 10th century Duchy of Lower Lorraine, that covered much of the Low Countries. But this time the corresponding Upper region is Upper Lorraine, in nowadays Northern France; the Dukes of Burgundy, who ruled the Low Countries in the 15th century, used the term les pays de par deçà for the Low Countries as opposed to les pays de par delà for their original
Wave field synthesis
Wave field synthesis is a spatial audio rendering technique, characterized by creation of virtual acoustic environments. It produces artificial wavefronts synthesized by a large number of individually driven loudspeakers; such wavefronts seem to originate from a virtual starting point, the virtual source or notional source. Contrary to traditional spatialization techniques such as stereo or surround sound, the localization of virtual sources in WFS does not depend on or change with the listener's position. WFS is based on the Huygens–Fresnel principle, which states that any wavefront can be regarded as a superposition of elementary spherical waves. Therefore, any wavefront can be synthesized from such elementary waves. In practice, a computer controls a large array of individual loudspeakers and actuates each one at the time when the desired virtual wavefront would pass through it; the basic procedure was developed in 1988 by Professor A. J. Berkhout at the Delft University of Technology, its mathematical basis is the Kirchhoff–Helmholtz integral.
It states that the sound pressure is determined within a volume free of sources, if sound pressure and velocity are determined in all points on its surface. P = ∬ d A d z ′ Therefore, any sound field can be reconstructed, if sound pressure and acoustic velocity are restored on all points of the surface of its volume; this approach is the underlying principle of holophony. For reproduction, the entire surface of the volume would have to be covered with spaced loudspeakers, each individually driven with its own signal. Moreover, the listening area would have to be anechoic, in order to avoid sound reflections that would violate source-free volume assumption. In practice, this is hardly feasible; because our acoustic perception is most exact in the horizontal plane, practical approaches reduce the problem to a horizontal loudspeaker line, circle or rectangle around the listener. The origin of the synthesized wavefront can be at any point on the horizontal plane of the loudspeakers. For sources behind the loudspeakers, the array will produce convex wavefronts.
Sources in front of the speakers can be rendered by concave wavefronts that focus in the virtual source and diverge again. Hence the reproduction inside the volume is incomplete - it breaks down if the listener sits between speakers and inner virtual source; the origin represents the virtual acoustic source, which approximates an acoustic source at the same position. Unlike conventional reproduction, the perceived position of the virtual sources is independent of listener position allowing the listener to move or giving an entire audience consistent perception of audio source location. A sound field with stable position of the acoustic sources can be established using wave field synthesis. In principle, it is possible to establish a virtual copy of a genuine sound field indistinguishable from the real sound. Changes of the listener position in the rendition area can produce the same impression as an appropriate change of location in the recording room. Listeners are no longer relegated to a sweet spot area within the room.
The Moving Picture Expert Group standardized the object-oriented transmission standard MPEG-4 which allows a separate transmission of content and form. Each virtual acoustic source needs its own audio channel; the spatial sound field in the recording room consists of the direct wave of the acoustic source and a spatially distributed pattern of mirror acoustic sources caused by the reflections by the room surfaces. Reducing that spatial mirror source distribution onto a few transmitting channels causes a significant loss of spatial information; this spatial distribution can be synthesized much more by the rendition side. Compared to conventional channel-orientated rendition procedures, WFS provides a clear advantage: Virtual acoustic sources guided by the signal content of the associated channels can be positioned far beyond the conventional material rendition area; this reduces the influence of the listener position because the relative changes in angles and levels are smaller compared to conventional loudspeakers located within the rendition area.
This extends the sweet spot considerably. WFS thus is not only compatible with, but improves the reproduction for conventional channel-oriented methods. Since WFS attempts to simulate the acoustic characteristics of the recording space, the acoustics of the rendition area must be suppressed. One possible solution is use of acoustic damping or to otherwise arrange the walls in an absorbing and non-reflective configuration. A second possibility is playback within the near field. For this to work the loudspeakers must couple closely at the hearing zone or the diaphragm surface must be la
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is designated by the Greek letter lambda; the term wavelength is sometimes applied to modulated waves, to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, lower frequencies have longer wavelengths. Wavelength depends on the medium. Examples of wave-like phenomena are sound waves, water waves and periodic electrical signals in a conductor.
A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary. Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength; the range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components; the wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, where v is called the phase speed of the wave and f is the wave's frequency.
In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s, thus the wavelength of a 100 MHz electromagnetic wave is about: 3×108 m/s divided by 108 Hz = 3 metres. The wavelength of visible light ranges from deep red 700 nm, to violet 400 nm. For sound waves in air, the speed of sound is 343 m/s; the wavelengths of sound frequencies audible to the human ear are thus between 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light. A standing wave is an undulatory motion. A sinusoidal standing wave includes stationary points of no motion, called nodes, the wavelength is twice the distance between nodes; the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed.
For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall. The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Wavelength and wave velocity are related just as for a traveling wave. For example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. Traveling sinusoidal waves are represented mathematically in terms of their velocity v, frequency f and wavelength λ as: y = A cos = A cos where y is the value of the wave at any position x and time t, A is the amplitude of the wave, they are commonly expressed in terms of wavenumber k and angular frequency ω as: y = A cos = A cos in which wavelength and wavenumber are related to velocity and frequency as: k = 2 π λ = 2 π f v = ω
The International Solvay Institutes for Physics and Chemistry, located in Brussels, were founded by the Belgian industrialist Ernest Solvay in 1912, following the historic invitation-only 1911 Conseil Solvay, considered a turning point in the world of physics. The Institutes coordinate conferences, workshops and colloquia. Following the initial success of 1911, the Solvay Conferences have been devoted to outstanding preeminent open problems in both physics and chemistry; the usual schedule is every three years. Hendrik A. Lorentz was chairman of the first Solvay Conference held in Brussels from October 30th to November 3th, 1911; the subject was the Quanta. This conference looked at the problems of having two approaches, namely the classical physics and quantum theory. Albert Einstein was the second youngest physicist present. Other members of the Solvay Congress included such luminaries as Marie Curie and Henri Poincaré; the Third Solvay Conference was held in April 1921, soon after World War I.
Most German scientists were barred from attending. In protest at this action, Albert Einstein, himself a citizen and a vocal supporter of the infant Weimar Republic, declined his invitation to attend the conference where most of his countrymen were barred. However, the real reason of Einstein's absence is because he accepted the invitation by Dr. Chaim Weizmann for a trip to the United States; the most famous conference was the October 1927 Fifth Solvay International Conference on Electrons and Photons, where the world's most notable physicists met to discuss the newly formulated quantum theory. The leading figures were Niels Bohr. 17 of the 29 attendees were or became Nobel Prize winners, including Marie Curie, who alone among them, had won Nobel Prizes in two separate scientific disciplines. This conference was the culmination of the struggle between, on one side and other scientific realists—who wanted strict rules of scientific method as laid out by Charles Peirce and Karl Popper—and on the other and other instrumentalists, who wanted looser rules based on outcomes.
Starting at this point, the instrumentalists won, instrumentalism having been seen as the norm since, although the debate has been continued by the likes of Alan Musgrave. A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. de Donder, E. Schrödinger, J. E. Verschaffelt, W. Pauli, W. Heisenberg, R. H. Fowler, L. Brillouin. L. Bragg, H. A. Kramers, P. A. M. Dirac, A. H. Compton, L. de Broglie, M. Born, N. Bohr. Langmuir, M. Planck, M. Curie, H. A. Lorentz, A. Einstein, P. Langevin, Ch.-E. Guye, C. T. R. Wilson, O. W. Richardson Fifth conference participants, 1927. Institut International de Physique Solvay in Leopold Park. Straumann, N.. "On the first Solvay Congress in 1911". European Physical Journal H. arXiv:1109.3785. Bibcode:2011EPJH...36..379S. Doi:10.1140/epjh/e2011-20043-9. International Solvay Institutes The Solvay Science Project Previous Solvay Conferences on Physics Previous Solvay Conferences on Chemistry Proceedings 1911 Proceedings 1913 Proceedings 1933 Overview of the transcript of the famous Fifth Conference — American Institute of Physics Bacciagaluppi G. Valentini A.
Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, Cambridge University Press, Cambridge, U. K
Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle. In classical physics, the diffraction phenomenon is described as the interference of waves according to the Huygens–Fresnel principle that treats each point in the wave-front as a collection of individual spherical wavelets; these characteristic behaviors are exhibited when a wave encounters an obstacle or a slit, comparable in size to its wavelength. Similar effects occur when a light wave travels through a medium with a varying refractive index, or when a sound wave travels through a medium with varying acoustic impedance. Diffraction has an impact on the acoustic space. Diffraction occurs with all waves, including sound waves, water waves, electromagnetic waves such as visible light, X-rays and radio waves. Since physical objects have wave-like properties, diffraction occurs with matter and can be studied according to the principles of quantum mechanics.
Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660. While diffraction occurs whenever propagating waves encounter such changes, its effects are most pronounced for waves whose wavelength is comparable to the dimensions of the diffracting object or slit. If the obstructing object provides multiple spaced openings, a complex pattern of varying intensity can result; this is due to the addition, or interference, of different parts of a wave that travel to the observer by different paths, where different path lengths result in different phases. The formalism of diffraction can 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 can all be analyzed using diffraction equations; the effects of diffraction are seen in everyday life. The most striking examples of diffraction are those.
This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired. 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. A shadow of a solid object, using light from a compact source, shows small fringes near its edges; the speckle pattern, observed when laser light falls on an optically rough surface is a diffraction phenomenon. When deli meat appears to be iridescent, diffraction off the meat fibers. All these effects are a consequence of the fact. Diffraction can occur with any kind of wave. Ocean waves diffract around other obstacles. Sound waves can diffract around objects, why one can still hear someone calling when hiding behind a tree. Diffraction can be a concern in some technical applications; the effects of diffraction of light were first observed and characterized by Francesco Maria Grimaldi, who coined the term diffraction, from the Latin diffringere,'to break into pieces', referring to light breaking up into different directions.
The results of Grimaldi's observations were published posthumously in 1665. Isaac Newton attributed them to inflexion of light rays. James Gregory observed the diffraction patterns caused by a bird feather, the first diffraction grating to be discovered. Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two spaced slits. 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, made public in 1815 and 1818, thereby gave great support to the wave theory of light, advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory. In traditional classical physics diffraction arises because of the way; the propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a point source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves.
When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns have a series of maxima and minima. In the modern quantum mechanical understanding of light propagation through a slit every photon has what is known as a wavefunction which describes its path from the emitter through the slit to the screen; the wavefunction is determined by the physical surroundings such as slit geometry, screen distance and initial conditions when the photon is created. In important experiments the existence of the photon's wavef
Augustin-Jean Fresnel was a French civil engineer and physicist whose research in optics led to the unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular theory, from the late 1830s until the end of the 19th century. But he is better known for inventing the catadioptric Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea; the simpler dioptric stepped lens, first proposed by Count Buffon and independently reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors. By expressing Huygens' principle of secondary waves and Young's principle of interference in quantitative terms, supposing that simple colors consist of sinusoidal waves, Fresnel gave the first satisfactory explanation of diffraction by straight edges, including the first satisfactory wave-based explanation of rectilinear propagation. Part of his argument was a proof that the addition of sinusoidal functions of the same frequency but different phases is analogous to the addition of forces with different directions.
By further supposing that light waves are purely transverse, Fresnel explained the nature of polarization and lack thereof, the mechanism of chromatic polarization, the transmission and reflection coefficients at the interface between two transparent isotropic media. By generalizing the direction-speed-polarization relation for calcite, he accounted for the directions and polarizations of the refracted rays in doubly-refractive crystals of the biaxial class; the period between the first publication of his pure-transverse-wave hypothesis and the submission of his first correct solution to the biaxial problem was less than a year. He coined the terms linear polarization, circular polarization, elliptical polarization, explained how optical rotation could be understood as a difference in propagation speeds for the two directions of circular polarization, accounted for the change in polarization due to total internal reflection, as exploited in the Fresnel rhomb. Defenders of the established corpuscular theory could not match his quantitative explanations of so many phenomena on so few assumptions.
Fresnel's legacy is the more remarkable in view of his lifelong battle with tuberculosis, to which he succumbed at the age of 39. Although he did not become a public celebrity in his short lifetime, he lived just long enough to receive due recognition from his peers, including the Rumford Medal of the Royal Society of London, his name is ubiquitous in the modern terminology of optics and waves. After the wave theory of light was subsumed by Maxwell's electromagnetic theory in the 1860s, some attention was diverted from the magnitude of Fresnel's contribution. In the period between Fresnel's unification of physical optics and Maxwell's wider unification, a contemporary authority, Professor Humphrey Lloyd, described Fresnel's transverse-wave theory as "the noblest fabric which has adorned the domain of physical science, Newton's system of the universe alone excepted." Augustin-Jean Fresnel, born in Broglie, Normandy, on 10 May 1788, was the second of four sons of the architect Jacques Fresnel and his wife Augustine, née Mérimée.
In 1790, following the Revolution, Broglie became part of the département of Eure. The family moved at least twice — in 1790 to Cherbourg, in 1794 to Jacques' home town of Mathieu, where Madame Fresnel would spend 25 years as a widow, outliving two of her sons; the first son, was admitted to the École Polytechnique, became a lieutenant in the artillery, was killed in action at Jaca, the day before his 23rd birthday. The third, Léonor, followed Augustin into civil engineering, succeeded him as Secretary of the Lighthouse Commission, helped to edit his collected works; the fourth, Fulgence Fresnel, became a noted linguist and orientalist, assisted Augustin with negotiations. Léonor was the only one of the four who married, their mother's younger brother, Jean François "Léonor" Mérimée, father of the writer Prosper Mérimée, was a paint artist who turned his attention to the chemistry of painting. He became the Permanent Secretary of the École des Beaux-Arts and a professor at the École Polytechnique, was the initial point of contact between Augustin and the leading optical physicists of the day.
The Fresnel brothers were home-schooled by their mother. The sickly Augustin was considered the slow one. At nine and ten he was undistinguished except for his ability to turn tree-branches into toy bows and guns that worked far too well, earning himself the title l'homme de génie from his accomplices, a united crackdown from their elders. In 1801, Augustin was sent to the École Centrale as company for Louis, but Augustin lifted his performance: in late 1804 he was accepted into the École Polytechnique, being placed 17th in the entrance examination, in which his solutions to geometry problems impressed the examiner, Adrien-Marie Legendre. As the surviving records of the École Polytechnique begin in 1808, we know little of Augustin's time there, except that