Wikipedia: Fresnel zone plates
Vibrations (19:33)
by Sir Lawrence Bragg (RI, 1965).
Waves and Vibrations (20:04)
by Sir Lawrence Bragg (RI, 1965).
Young and the Wave Theory of Light (20:58)
Sir Lawrence Bragg (RI, 1965).
MIT OpenCourseWare Vibrations & Waves (8.03)
by Walter Lewin.
Originally, the principle was formulated strictly from a geometrical standoint by noting that a wavefront can be seen as the envelope of circular (or spherical) wavelets centered on a previous wavefront.
Christiaan Huygens himself noticed that this principle was sufficient to derive the optical laws of reflexion and refraction in the wave theory of light which he was pioneering.
The principle was given a more precise form by Augustin Fresnel (1816) who applied it to the computation of diffraction patterns.
Huygens-Fresnel principle | Christiaan Huygens (1629-1695)
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In 1816, Augustin Fresnel (1788-1827; X1804) had the idea to combine this with Huygens' principle to compute the diffraction patterns formed by light when it encounters obstacles...
[ This ] may be repeated with great ease, whenever the sun shines,
and without any other apparatus than is at hand to everyone.
Thomas Young
(1773-1829) Nov. 24, 1803
The celerity u of a wave is always equal to the product l n of its wavelength by its frequency.
When this celerity is constant, the medium is said to be nondispersive. In a nondispersive medium, a planar wave would retain its shape as it propagates. This is not generally true in a dispersive medium.
By definition, a diopter is the surface (usually a plane or a sphere) which separates two regions where specific waves (light, sound, etc.) travel at different celerities (celerity = phase velocity).
Incidentally, the name "diopter" also denotes a unit of curvature equal to the reciprocal of a meter (m) which is used to rate an optical element by specifying the reciprocal of its focal length.
Snell's Law applies not only to waves but also to other objects at a boundary between two domains where the travelling speeds are proportional to different values of a so-called index of refraction n.
n1 sin q1 = n2 sin q2
This law of refraction was stated by Ibn Sahl (AD 984). It was re-discovered by Thomas Harriot in July 1601 and independently by Willebrord Snell (1621) and René Descartes (1637) who was the first to publish it. The Dutchman Christiaan Huygens (1629-1695) was instrumental in naming the Law after Snell (1678).
[画像: Snell's Law ][画像: Come back later, we're still working on this one... ]
1/f = (n-1) ( 1/R1 + 1/R2 )
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Ibn Sahl discovers the law of refraction (AD 984) | Fermat's Principle of Least Time (c. 1655)
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Fermat's Principle of Least Time (c. 1655)
Some transparent minerals like Iceland spar exhibit a strange optical property known as birefringence.
Surprisingly enough, this was the phenomenon which prompted Huygens to fomulate the principle bearing his name (1678). (Fresnel would only apply the principle to diffraction much later, in 1816).
The full polarization of light by optical reflection at a particular angle of incidence qB was discovered by Etienne-Louis Malus (1775-1812) in 1808.
In 1815, Sir David Brewster (1781-1868) found that angle to be a simple function of the ratio of the two refractive indices involved:
This expression can be derived from Fresnel equations by imposing Rp = 0.
Snell's Law makes this equivalent to the observation that Brewster's angle of incidence is such that the reflected and the refracted beam are perpendicular.
Apparently, Malus did not come up with this relation experimentally because he focused on just the two cases of water and glass. It turns out that the type of glass available at that time could have surface properties unrelated to the index of refraction of its bulk. Brewster was faced with the same difficulty but he could establish the above general law by considering a variety of transparent minerals and (ultimately) disgarding the peculiarities of glass.
[画像: Light transmission ] Unlike Snell's Law, the Fresnel Equations apply specifically to light and involve the different polarizations of light which Augustin Fresnel (1788-1827) firmly established himself in 1821.
Fresnel determined that light consists entirely of transversal vibrations without any longitudinal component whatsoever. This went against the opinion of Thomas Young (1773-1829) who held that light was mostly a longitudinal phenomenon with only small transversal components.
The Fresnel equations describe how the strength of an incident light beam is split between a reflected (r) and a refracted (t) beam.
These coefficients are the ratios of the amplitudes of the emergent beams (either reflected or refracted) to the amplitude of the incident one. The coefficients depend on the polarization of the incident beam.
Traditionally, the linear polarization of an electromagnetic wave (light) where the electric field is parallel to the plane of incidence is denoted by the subscript "p" whereas the perpendicular polarization is denoted by the subscript "s" (the word senkrecht means "perpendicular" in German).
A coherent incident beam whose polarization is neither "p" nor "s" can be viewed as a superposition of two such beams. The above formulas give the amplitudes of the emergent beams corresponding to both components. Each emergent beam (transmitted or reflected) has two components of orthogonal polarization (p or s) corresponding to a superposition obtained by adding those emerging amplitudes.
These are the ratios of the intensities of the emergent beams to the intensity of the incident one. They are respectively equal to the squares of the above coefficients of reflection and transmission, for both polarizations:
Rs = | rs |2 Ts = | ts |2 Rp = | rp |2 Tp = | tp |2
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Maxwell's equations in matter...
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The Discovery of Polarization
by J. Alcoz.
Fresnel Equations
(8.03
Vibrations and Waves) by Walter Lewin, MIT.
Fresnel
Relations (531
Optics) by Cass Sackett, University of Virginia.
"Fresnel Equations" by Bob Eagle (DrPhysicsA) :
Boundary Conditions
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Deriving the Equations
Wikipedia
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Weisstein
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HyperPhysics
(calculator) by Rod Nave
Why does light bend when it enters glass?
(13:36) by Don Lincoln (Fermilab, 2019年05月01日).
On a rope of linear density m stretched along the x-axis with constant tension F, let's consider the behavior of small transverse perturbations of amplitude h(x,t) in the direction of the y-axis.
If the rope has no rigidity, the forces exerted on each other by adjacent pieces of the rope are strictly tangential to it. Thus, they have the same slope j with respect to the x-axis as the rope itself. The usual "small angle" approximation holds:
j = sin j = tg j = ¶h / ¶x
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The resonnant frequencies of a string stretched between two fixed points are:
The first law is attributed to Phythagoras (c.569-475 BC) and was the earliest result in acoustics. The other laws are direct consequences of the above expression for the celerity of a transverse wave in a stretched string.
Mersenne's laws
|
Harmonie universelle (1636)
by Marin Mersenne (1588-1648)
Video :
Deriving the wave equation
for a rope by Walter Lewin
(MIT, Fall 2004)
Ernst Chladni (1756-1827) was a German physicist whose family hailed from the medieval mining town of Kremnica (Kingdom of Hungary, now in central Slovakia). Chladni has been called the father of acoustics. He obtained the speed of sound for several gases and experimented with vibrating plates peppered with sand to visualize node lines (the sand accumulates wherever the motion of the plate is minimal). Similar experiments now go by the name of Chladni plate experiments and the intriguing patterns so obtained are dubbed Chladni patterns. Ernst Chladni was also an avid meteorite collector and he successfully argued in favor of the celestial origin of meteorites. (In English and in French, at least, his name is usually pronounced like clad-knee.)
In 1808, Chladni visited Paris and caused quite a stir with the demonstration of his patterns: The Institut de France set up a prize competition (including a 1 kg solid gold medal ) with the following challenge, to be met within two years (deadline in 1811).
Formulate a mathematical theory of elastic surfaces
and indicate just how it agrees with empirical evidence.
Lagrange himself went on record to state that all available mathematical methods were inadequate to solve that problem.
Arms of Sophie Germain 1776-1831 In 1811, the only entrant was Sophie Germain (1776-1831) who could not justify her hypothesis from physical principles because, at the time, she lacked the proper knowledge of the calculus of variations (a brainchild of Lagrange's, by the way). She did not get the prize. Instead, Lagrange (who was one of the judges) suggested a new approach and the contest was extended for another two years.
In 1813, Lagrange died. Germain, still the only entrant, showed that the approach of Lagrange accurately described Chladni's patterns in several special cases. She didn't provide a physical justification for it. She just received an honorable mention and the contest was extended, again, for another two years.
In 1815, the third attempt of Sophie Germain was deemed worthy of the prize (in spite of a few deficiencies in the rigor of her mathematics).
However, she didn't show up at the award ceremony... It's been suggested that she was protesting the lack of appreciation of her work by some of the judges, including her younger rival on the subject of elasticity (that she had arguably founded in the process) Siméon Poisson (1781-1840; X1798) who ignored her in public...
In 1825, Sophie Germain sent an extension of her research to a commission whose members included Poisson, Laplace and de Prony. The paper was ignored until it was retrieved from de Prony's personal archives (long after the death of everyone involved) and belatedly published, in 1880.
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Chladni Patterns for Violin
Plates by Joe Wolfe (UNSW, Sydney, Australia)
Videos :
Chladni Patterns
on a Square Plate
|
Holding a Chladni Plate
(and ruining a bow)
Fantastic DIY Speakers for less than 30ドル
in New England (Tech Ingredients, 2018年01月29日).
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How to point a Space Telescope (7:53)
by Meghan Gray (Sixty Symbols by Brady Haran, 2017年06月21日)
The Hemispherical Resonator Gyro
by David M. Rozelle (Northrop Grumman Co, 2011).
Wikipedia :
Hemispherical resonator gyroscope (HRG)
|
George Hartley Bryan (1864-1928; FRS 1895)
