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83-468: Reflectometry is a general term for the use of the reflection of waves or pulses at surfaces and interfaces to detect or characterize objects, sometimes to detect anomalies as in fault detection and medical diagnosis . There are many different forms of reflectometry. They can be classified in several ways: by the used radiation (electromagnetic, ultrasound, particle beams), by the geometry of wave propagation (unguided versus wave guides or cables), by

166-403: A doubly-refractive calcite crystal. He later coined the term polarization to describe this behavior.  In 1815, the dependence of the polarizing angle on the refractive index was determined experimentally by David Brewster . But the reason for that dependence was such a deep mystery that in late 1817, Thomas Young was moved to write: [T]he great difficulty of all, which

249-471: A torus . Note that these are theoretical ideals, requiring perfect alignment of perfectly smooth, perfectly flat perfect reflectors that absorb none of the light. In practice, these situations can only be approached but not achieved because the effects of any surface imperfections in the reflectors propagate and magnify, absorption gradually extinguishes the image, and any observing equipment (biological or technological) will interfere. In this process (which

332-532: A "postscript" to the work in which Fresnel first revealed his theory that light waves, including "unpolarized" waves, were purely transverse. Details of Fresnel's derivation, including the modern forms of the sine law and tangent law, were given later, in a memoir read to the French Academy of Sciences in January 1823. That derivation combined conservation of energy with continuity of the tangential vibration at

415-405: A complex conjugating mirror, it would be black because only the photons which left the pupil would reach the pupil. Materials that reflect neutrons , for example beryllium , are used in nuclear reactors and nuclear weapons . In the physical and biological sciences, the reflection of neutrons off of atoms within a material is commonly used to determine the material's internal structure. When

498-490: A denser medium strikes the surface of a less dense medium (i.e., n 1 > n 2 ), beyond a particular incidence angle known as the critical angle , all light is reflected and R s = R p = 1 . This phenomenon, known as total internal reflection , occurs at incidence angles for which Snell's law predicts that the sine of the angle of refraction would exceed unity (whereas in fact sin  θ ≤ 1 for all real θ ). For glass with n = 1.5 surrounded by air,

581-402: A flat surface forms a mirror image , which appears to be reversed from left to right because we compare the image we see to what we would see if we were rotated into the position of the image. Specular reflection at a curved surface forms an image which may be magnified or demagnified; curved mirrors have optical power . Such mirrors may have surfaces that are spherical or parabolic . If

664-400: A longitudinal sound wave strikes a flat surface, sound is reflected in a coherent manner provided that the dimension of the reflective surface is large compared to the wavelength of the sound. Note that audible sound has a very wide frequency range (from 20 to about 17000 Hz), and thus a very wide range of wavelengths (from about 20 mm to 17 m). As a result, the overall nature of

747-493: A second medium with refractive index n 2 , both reflection and refraction of the light may occur. The Fresnel equations give the ratio of the reflected wave's electric field to the incident wave's electric field, and the ratio of the transmitted wave's electric field to the incident wave's electric field, for each of two components of polarization. (The magnetic fields can also be related using similar coefficients.) These ratios are generally complex, describing not only

830-417: A species of birefringence : linearly-polarized light can be resolved into two circularly-polarized components rotating in opposite directions, and if these propagate at different speeds, the phase difference between them — hence the orientation of their linearly-polarized resultant — will vary continuously with distance. Thus Fresnel's interpretation of the complex values of his reflection coefficients marked

913-702: A wave in an absorbing medium following transmission or reflection. The reflectance for s-polarized light is R s = | Z 2 cos ⁡ θ i − Z 1 cos ⁡ θ t Z 2 cos ⁡ θ i + Z 1 cos ⁡ θ t | 2 , {\displaystyle R_{\mathrm {s} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {i} }-Z_{1}\cos \theta _{\mathrm {t} }}{Z_{2}\cos \theta _{\mathrm {i} }+Z_{1}\cos \theta _{\mathrm {t} }}}\right|^{2},} while

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996-533: A wave is generally proportional to the square of the electric (or magnetic) field amplitude. We call the fraction of the incident power that is reflected from the interface the reflectance (or reflectivity , or power reflection coefficient ) R , and the fraction that is refracted into the second medium is called the transmittance (or transmissivity , or power transmission coefficient ) T . Note that these are what would be measured right at each side of an interface and do not account for attenuation of

1079-413: A wave reflected at angle θ r = θ i {\displaystyle \theta _{\mathrm {r} }=\theta _{\mathrm {i} }} , and a wave transmitted at angle θ t {\displaystyle \theta _{\mathrm {t} }} . In the case of an interface into an absorbing material (where n is complex) or total internal reflection,

1162-474: A wave's irradiance in the direction normal to the interface; this is also what is measured in typical experiments. That number could be obtained from irradiances in the direction of an incident or reflected wave (given by the magnitude of a wave's Poynting vector ) multiplied by cos  θ for a wave at an angle θ to the normal direction (or equivalently, taking the dot product of the Poynting vector with

1245-423: Is also known as phase conjugation), light bounces exactly back in the direction from which it came due to a nonlinear optical process. Not only the direction of the light is reversed, but the actual wavefronts are reversed as well. A conjugate reflector can be used to remove aberrations from a beam by reflecting it and then passing the reflection through the aberrating optics a second time. If one were to look into

1328-531: Is an equal amount of power in the s and p polarizations, so that the effective reflectivity of the material is just the average of the two reflectivities: R e f f = 1 2 ( R s + R p ) . {\displaystyle R_{\mathrm {eff} }={\frac {1}{2}}\left(R_{\mathrm {s} }+R_{\mathrm {p} }\right).} For low-precision applications involving unpolarized light, such as computer graphics , rather than rigorously computing

1411-453: Is based on the Fresnel equations, but with additional calculations to account for interference. The transfer-matrix method , or the recursive Rouard method  can be used to solve multiple-surface problems. In 1808, Étienne-Louis Malus discovered that when a ray of light was reflected off a non-metallic surface at the appropriate angle, it behaved like one of the two rays emerging from

1494-659: Is derived from the first by eliminating θ t using Snell's law and trigonometric identities . As a consequence of conservation of energy , one can find the transmitted power (or more correctly, irradiance : power per unit area) simply as the portion of the incident power that isn't reflected:  T s = 1 − R s {\displaystyle T_{\mathrm {s} }=1-R_{\mathrm {s} }} and T p = 1 − R p {\displaystyle T_{\mathrm {p} }=1-R_{\mathrm {p} }} Note that all such intensities are measured in terms of

1577-403: Is important for radio transmission and for radar . Even hard X-rays and gamma rays can be reflected at shallow angles with special "grazing" mirrors. Reflection of light is either specular (mirror-like) or diffuse (retaining the energy , but losing the image) depending on the nature of the interface. In specular reflection the phase of the reflected waves depends on the choice of

1660-421: Is indeed very close to 1; that is, μ ≈ μ 0 . In optics, one usually knows the refractive index n of the medium, which is the ratio of the speed of light in vacuum ( c ) to the speed of light in the medium. In the analysis of partial reflection and transmission, one is also interested in the electromagnetic wave impedance Z , which is the ratio of the amplitude of E to the amplitude of H . It

1743-404: Is located at the imaginary intersection of the mirrors. A square of four mirrors placed face to face give the appearance of an infinite number of images arranged in a plane. The multiple images seen between four mirrors assembling a pyramid, in which each pair of mirrors sits an angle to each other, lie over a sphere. If the base of the pyramid is rectangle shaped, the images spread over a section of

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1826-430: Is not desired, since the light would then be directed back into the headlights of an oncoming car rather than to the driver's eyes. When light reflects off a mirror , one image appears. Two mirrors placed exactly face to face give the appearance of an infinite number of images along a straight line. The multiple images seen between two mirrors that sit at an angle to each other lie over a circle. The center of that circle

1909-498: Is returned in the direction from which it came. When flying over clouds illuminated by sunlight the region seen around the aircraft's shadow will appear brighter, and a similar effect may be seen from dew on grass. This partial retro-reflection is created by the refractive properties of the curved droplet's surface and reflective properties at the backside of the droplet. Some animals' retinas act as retroreflectors (see tapetum lucidum for more detail), as this effectively improves

1992-2935: Is the impedance of free space and i = 1, 2 . Making this substitution, we obtain equations using the refractive indices: R s = | n 1 cos ⁡ θ i − n 2 cos ⁡ θ t n 1 cos ⁡ θ i + n 2 cos ⁡ θ t | 2 = | n 1 cos ⁡ θ i − n 2 1 − ( n 1 n 2 sin ⁡ θ i ) 2 n 1 cos ⁡ θ i + n 2 1 − ( n 1 n 2 sin ⁡ θ i ) 2 | 2 , {\displaystyle R_{\mathrm {s} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}\cos \theta _{\mathrm {t} }}{n_{1}\cos \theta _{\mathrm {i} }+n_{2}\cos \theta _{\mathrm {t} }}}\right|^{2}=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}}{n_{1}\cos \theta _{\mathrm {i} }+n_{2}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}}}\right|^{2}\!,} R p = | n 1 cos ⁡ θ t − n 2 cos ⁡ θ i n 1 cos ⁡ θ t + n 2 cos ⁡ θ i | 2 = | n 1 1 − ( n 1 n 2 sin ⁡ θ i ) 2 − n 2 cos ⁡ θ i n 1 1 − ( n 1 n 2 sin ⁡ θ i ) 2 + n 2 cos ⁡ θ i | 2 . {\displaystyle R_{\mathrm {p} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {t} }-n_{2}\cos \theta _{\mathrm {i} }}{n_{1}\cos \theta _{\mathrm {t} }+n_{2}\cos \theta _{\mathrm {i} }}}\right|^{2}=\left|{\frac {n_{1}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}-n_{2}\cos \theta _{\mathrm {i} }}{n_{1}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}+n_{2}\cos \theta _{\mathrm {i} }}}\right|^{2}\!.} The second form of each equation

2075-468: Is the inverse of one produced by a single mirror. A surface can be made partially retroreflective by depositing a layer of tiny refractive spheres on it or by creating small pyramid like structures. In both cases internal reflection causes the light to be reflected back to where it originated. This is used to make traffic signs and automobile license plates reflect light mostly back in the direction from which it came. In this application perfect retroreflection

2158-445: Is the reciprocal of the ratio of the media's wave impedances. The cos( θ ) factors adjust the waves' powers so they are reckoned in the direction normal to the interface, for both the incident and transmitted waves, so that full power transmission corresponds to T = 1 . In the case of total internal reflection where the power transmission T is zero, t nevertheless describes the electric field (including its phase) just beyond

2241-428: Is the well-known principle by which total internal reflection is used to effect polarization transformations . In the above formula for r s , if we put n 2 = n 1 sin ⁡ θ i / sin ⁡ θ t {\displaystyle n_{2}=n_{1}\sin \theta _{\text{i}}/\sin \theta _{\text{t}}} (Snell's law) and multiply

2324-430: Is to assign a sufficient reason for the reflection or nonreflection of a polarised ray, will probably long remain, to mortify the vanity of an ambitious philosophy, completely unresolved by any theory. In 1821, however, Augustin-Jean Fresnel derived results equivalent to his sine and tangent laws (above), by modeling light waves as transverse elastic waves with vibrations perpendicular to what had previously been called

2407-441: Is used as a means of focusing waves that cannot effectively be reflected by common means. X-ray telescopes are constructed by creating a converging "tunnel" for the waves. As the waves interact at low angle with the surface of this tunnel they are reflected toward the focus point (or toward another interaction with the tunnel surface, eventually being directed to the detector at the focus). A conventional reflector would be useless as

2490-482: The law of reflection : θ i = θ r , {\displaystyle \theta _{\mathrm {i} }=\theta _{\mathrm {r} },} and Snell's law : n 1 sin ⁡ θ i = n 2 sin ⁡ θ t . {\displaystyle n_{1}\sin \theta _{\mathrm {i} }=n_{2}\sin \theta _{\mathrm {t} }.} The behavior of light striking

2573-407: The plane of polarization . Fresnel promptly confirmed by experiment that the equations correctly predicted the direction of polarization of the reflected beam when the incident beam was polarized at 45° to the plane of incidence, for light incident from air onto glass or water; in particular, the equations gave the correct polarization at Brewster's angle. The experimental confirmation was reported in

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2656-502: The s and p polarizations, and even at normal incidence (where the designations s and p do not even apply!) the sign of r is reversed depending on whether the wave is considered to be s or p polarized, an artifact of the adopted sign convention (see graph for an air-glass interface at 0° incidence). The equations consider a plane wave incident on a plane interface at angle of incidence θ i {\displaystyle \theta _{\mathrm {i} }} ,

2739-514: The wave impedances of media 1 and 2, respectively. We assume that the media are non-magnetic (i.e., μ 1 = μ 2 = μ 0 ), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies). Then the wave impedances are determined solely by the refractive indices n 1 and n 2 : Z i = Z 0 n i , {\displaystyle Z_{i}={\frac {Z_{0}}{n_{i}}}\,,} where Z 0

2822-667: The (electric) permittivity and the (magnetic) permeability of the medium. For vacuum, these have the values ϵ 0 and μ 0 , respectively. Hence we define the relative permittivity (or dielectric constant ) ϵ rel = ϵ / ϵ 0 , and the relative permeability μ rel = μ / μ 0 . In optics it is common to assume that the medium is non-magnetic, so that μ rel = 1 . For ferromagnetic materials at radio/microwave frequencies, larger values of μ rel must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies (except possible metamaterials ), μ rel

2905-513: The X-rays would simply pass through the intended reflector. When light reflects off of a material with higher refractive index than the medium in which is traveling, it undergoes a 180° phase shift . In contrast, when light reflects off of a material with lower refractive index the reflected light is in phase with the incident light. This is an important principle in the field of thin-film optics . Specular reflection forms images . Reflection from

2988-425: The angle at which the wave is incident on the surface equals the angle at which it is reflected. In acoustics , reflection causes echoes and is used in sonar . In geology, it is important in the study of seismic waves . Reflection is observed with surface waves in bodies of water. Reflection is observed with many types of electromagnetic wave , besides visible light . Reflection of VHF and higher frequencies

3071-411: The angle of incidence equals the angle of reflection. In fact, reflection of light may occur whenever light travels from a medium of a given refractive index into a medium with a different refractive index. In the most general case, a certain fraction of the light is reflected from the interface, and the remainder is refracted . Solving Maxwell's equations for a light ray striking a boundary allows

3154-2356: The angle of transmission does not generally evaluate to a real number. In that case, however, meaningful results can be obtained using formulations of these relationships in which trigonometric functions and geometric angles are avoided; the inhomogeneous waves launched into the second medium cannot be described using a single propagation angle. Using this convention, r s = n 1 cos ⁡ θ i − n 2 cos ⁡ θ t n 1 cos ⁡ θ i + n 2 cos ⁡ θ t , t s = 2 n 1 cos ⁡ θ i n 1 cos ⁡ θ i + n 2 cos ⁡ θ t , r p = n 2 cos ⁡ θ i − n 1 cos ⁡ θ t n 2 cos ⁡ θ i + n 1 cos ⁡ θ t , t p = 2 n 1 cos ⁡ θ i n 2 cos ⁡ θ i + n 1 cos ⁡ θ t . {\displaystyle {\begin{aligned}r_{\text{s}}&={\frac {n_{1}\cos \theta _{\text{i}}-n_{2}\cos \theta _{\text{t}}}{n_{1}\cos \theta _{\text{i}}+n_{2}\cos \theta _{\text{t}}}},\\[3pt]t_{\text{s}}&={\frac {2n_{1}\cos \theta _{\text{i}}}{n_{1}\cos \theta _{\text{i}}+n_{2}\cos \theta _{\text{t}}}},\\[3pt]r_{\text{p}}&={\frac {n_{2}\cos \theta _{\text{i}}-n_{1}\cos \theta _{\text{t}}}{n_{2}\cos \theta _{\text{i}}+n_{1}\cos \theta _{\text{t}}}},\\[3pt]t_{\text{p}}&={\frac {2n_{1}\cos \theta _{\text{i}}}{n_{2}\cos \theta _{\text{i}}+n_{1}\cos \theta _{\text{t}}}}.\end{aligned}}} One can see that t s = r s + 1 and ⁠ n 2 / n 1 ⁠ t p = r p + 1 . One can write very similar equations applying to

3237-404: The animals' night vision. Since the lenses of their eyes modify reciprocally the paths of the incoming and outgoing light the effect is that the eyes act as a strong retroreflector, sometimes seen at night when walking in wildlands with a flashlight. A simple retroreflector can be made by placing three ordinary mirrors mutually perpendicular to one another (a corner reflector ). The image produced

3320-429: The arithmetic as well as the geometric average of R s and R p , and then averaging these two averages again arithmetically, gives a value for R 0 with an error of less than about 3% for most common optical materials. This is useful because measurements at normal incidence can be difficult to achieve in an experimental setup since the incoming beam and the detector will obstruct each other. However, since

3403-453: The auditory feel of a space. In the theory of exterior noise mitigation , reflective surface size mildly detracts from the concept of a noise barrier by reflecting some of the sound into the opposite direction. Sound reflection can affect the acoustic space . Seismic waves produced by earthquakes or other sources (such as explosions ) may be reflected by layers within the Earth . Study of

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3486-458: The characteristic impedance). This results in: T = n 2 cos ⁡ θ t n 1 cos ⁡ θ i | t | 2 {\displaystyle T={\frac {n_{2}\cos \theta _{\text{t}}}{n_{1}\cos \theta _{\text{i}}}}|t|^{2}} using the above definition of t . The introduced factor of ⁠ n 2 / n 1 ⁠

3569-518: The complex reflection coefficient inspired James MacCullagh and Augustin-Louis Cauchy , beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a complex refractive index . Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoir  in which he introduced the needed terms linear polarization , circular polarization , and elliptical polarization , and in which he explained optical rotation as

3652-416: The confluence of several streams of his research and, arguably, the essential completion of his reconstruction of physical optics on the transverse-wave hypothesis (see Augustin-Jean Fresnel ). Here we systematically derive the above relations from electromagnetic premises. In order to compute meaningful Fresnel coefficients, we must assume that the medium is (approximately) linear and homogeneous . If

3735-472: The consistency of the measurements of R s and R p , or to derive one of them when the other is known. This relationship is only valid for the simple case of a single plane interface between two homogeneous materials, not for films on substrates, where a more complex analysis is required. Measurements of R s and R p at 45° can be used to estimate the reflectivity at normal incidence. The "average of averages" obtained by calculating first

3818-400: The correct results in the limit as θ i → 0 . When light makes multiple reflections between two or more parallel surfaces, the multiple beams of light generally interfere with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than

3901-475: The critical angle is approximately 42°. Reflection at 45° incidence is very commonly used for making 90° turns. For the case of light traversing from a less dense medium into a denser one at 45° incidence ( θ = 45° ), it follows algebraically from the above equations that R p equals the square of R s : R p = R s 2 {\displaystyle R_{\text{p}}=R_{\text{s}}^{2}} This can be used to either verify

3984-472: The deep reflections of waves generated by earthquakes has allowed seismologists to determine the layered structure of the Earth . Shallower reflections are used in reflection seismology to study the Earth's crust generally, and in particular to prospect for petroleum and natural gas deposits. Fresnel equations When light strikes the interface between a medium with refractive index n 1 and

4067-406: The dependence of R s and R p on the angle of incidence for angles below 10° is very small, a measurement at about 5° will usually be a good approximation for normal incidence, while allowing for a separation of the incoming and reflected beam. The above equations relating powers (which could be measured with a photometer for instance) are derived from the Fresnel equations which solve

4150-411: The derivation below); then the magnetic field is in the plane of incidence. The p polarization refers to polarization of the electric field in the plane of incidence (the xy plane in the derivation below); then the magnetic field is normal to the plane of incidence. The names "s" and "p" for the polarization components refer to German "senkrecht" (perpendicular or normal) and "parallel" (parallel to

4233-465: The derivation of the Fresnel equations , which can be used to predict how much of the light is reflected, and how much is refracted in a given situation. This is analogous to the way impedance mismatch in an electric circuit causes reflection of signals. Total internal reflection of light from a denser medium occurs if the angle of incidence is greater than the critical angle . Total internal reflection

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4316-589: The effective reflection coefficient for each angle, Schlick's approximation is often used. For the case of normal incidence , θ i = θ t = 0 , and there is no distinction between s and p polarization. Thus, the reflectance simplifies to R 0 = | n 1 − n 2 n 1 + n 2 | 2 . {\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\,.} For common glass ( n 2 ≈ 1.5 ) surrounded by air ( n 1 = 1 ),

4399-654: The formula for r p , the result is easily shown to be equivalent to  r p = tan ⁡ ( θ i − θ t ) tan ⁡ ( θ i + θ t ) . {\displaystyle r_{\text{p}}={\frac {\tan(\theta _{\text{i}}-\theta _{\text{t}})}{\tan(\theta _{\text{i}}+\theta _{\text{t}})}}.} These formulas  are known respectively as Fresnel's sine law and Fresnel's tangent law . Although at normal incidence these expressions reduce to 0/0, one can see that they yield

4482-432: The forward radiation cancels the incident light, and backward radiation is just the reflected light. Light–matter interaction in terms of photons is a topic of quantum electrodynamics , and is described in detail by Richard Feynman in his popular book QED: The Strange Theory of Light and Matter . When light strikes the surface of a (non-metallic) material it bounces off in all directions due to multiple reflections by

4565-403: The glass is the combination of the forward radiation of the electrons and the incident light. The reflected light is the combination of the backward radiation of all of the electrons. In metals, electrons with no binding energy are called free electrons. When these electrons oscillate with the incident light, the phase difference between their radiation field and the incident field is π (180°), so

4648-425: The incident wave, whereas for p polarization r is the ratio of the waves complex magnetic field amplitudes (or equivalently, the negative of the ratio of their electric field amplitudes). The transmission coefficient t is the ratio of the transmitted wave's complex electric field amplitude to that of the incident wave, for either polarization. The coefficients r and t are generally different between

4731-414: The incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations. The s polarization refers to polarization of a wave's electric field normal to the plane of incidence (the z direction in

4814-537: The individual atoms (or oscillation of electrons, in metals), causing each particle to radiate a small secondary wave in all directions, like a dipole antenna . All these waves add up to give specular reflection and refraction, according to the Huygens–Fresnel principle . In the case of dielectrics such as glass, the electric field of the light acts on the electrons in the material, and the moving electrons generate fields and become new radiators. The refracted light in

4897-408: The interface between two media of refractive indices n 1 and n 2 at point O . Part of the wave is reflected in the direction OR , and part refracted in the direction OT . The angles that the incident, reflected and refracted rays make to the normal of the interface are given as θ i , θ r and θ t , respectively. The relationship between these angles is given by

4980-446: The interface is explained by considering the electric and magnetic fields that constitute an electromagnetic wave , and the laws of electromagnetism , as shown below . The ratio of waves' electric field (or magnetic field) amplitudes are obtained, but in practice one is more often interested in formulae which determine power coefficients, since power (or irradiance ) is what can be directly measured at optical frequencies. The power of

5063-431: The interface, but failed to allow for any condition on the normal component of vibration. The first derivation from electromagnetic principles was given by Hendrik Lorentz in 1875. In the same memoir of January 1823, Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients ( r s and r p ) gave complex values with unit magnitudes. Noting that

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5146-400: The interface. This is an evanescent field which does not propagate as a wave (thus T = 0 ) but has nonzero values very close to the interface. The phase shift of the reflected wave on total internal reflection can similarly be obtained from the phase angles of r p and r s (whose magnitudes are unity in this case). These phase shifts are different for s and p waves, which

5229-423: The involved length scales (wavelength and penetration depth in relation to size of the investigated object), by the method of measurement (continuous versus pulsed, polarization resolved, ...), and by the application domain. Many techniques are based on the principle of reflectometry and are distinguished by the type of waves used and the analysis of the reflected signal. Among all these techniques, we can classify

5312-414: The light is reflected with equal luminance (in photometry) or radiance (in radiometry) in all directions, as defined by Lambert's cosine law . The light sent to our eyes by most of the objects we see is due to diffuse reflection from their surface, so that this is our primary mechanism of physical observation. Some surfaces exhibit retroreflection . The structure of these surfaces is such that light

5395-405: The light's coherence length , which for ordinary white light is few micrometers; it can be much larger for light from a laser . An example of interference between reflections is the iridescent colours seen in a soap bubble or in thin oil films on water. Applications include Fabry–Pérot interferometers , antireflection coatings , and optical filters . A quantitative analysis of these effects

5478-427: The magnitude, as usual, represented the ratio of peak amplitudes, he guessed that the argument represented the phase shift, and verified the hypothesis experimentally. The verification involved Thus he finally had a quantitative theory for what we now call the Fresnel rhomb — a device that he had been using in experiments, in one form or another, since 1817 (see Fresnel rhomb §   History ). The success of

5561-404: The main but not limited to: Reflection (physics) Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light , sound and water waves . The law of reflection says that for specular reflection (for example at a mirror )

5644-442: The medium is also isotropic , the four field vectors E ,  B ,  D ,  H   are related by D = ϵ E B = μ H , {\displaystyle {\begin{aligned}\mathbf {D} &=\epsilon \mathbf {E} \\\mathbf {B} &=\mu \mathbf {H} \,,\end{aligned}}} where ϵ and μ are scalars, known respectively as

5727-434: The microscopic irregularities inside the material (e.g. the grain boundaries of a polycrystalline material, or the cell or fiber boundaries of an organic material) and by its surface, if it is rough. Thus, an 'image' is not formed. This is called diffuse reflection . The exact form of the reflection depends on the structure of the material. One common model for diffuse reflection is Lambertian reflectance , in which

5810-539: The numerator and denominator by ⁠ 1 / n 1 ⁠   sin   θ t , we obtain  r s = − sin ⁡ ( θ i − θ t ) sin ⁡ ( θ i + θ t ) . {\displaystyle r_{\text{s}}=-{\frac {\sin(\theta _{\text{i}}-\theta _{\text{t}})}{\sin(\theta _{\text{i}}+\theta _{\text{t}})}}.} If we do likewise with

5893-468: The origin of coordinates, but the relative phase between s and p (TE and TM) polarizations is fixed by the properties of the media and of the interface between them. A mirror provides the most common model for specular light reflection, and typically consists of a glass sheet with a metallic coating where the significant reflection occurs. Reflection is enhanced in metals by suppression of wave propagation beyond their skin depths . Reflection also occurs at

5976-407: The other hand, calculation of the power transmission coefficient T is less straightforward, since the light travels in different directions in the two media. What's more, the wave impedances in the two media differ; power ( irradiance ) is given by the square of the electric field amplitude divided by the characteristic impedance of the medium (or by the square of the magnetic field multiplied by

6059-434: The physical problem in terms of electromagnetic field complex amplitudes , i.e., considering phase shifts in addition to their amplitudes . Those underlying equations supply generally complex-valued ratios of those EM fields and may take several different forms, depending on the formalism used. The complex amplitude coefficients for reflection and transmission are usually represented by lower case r and t (whereas

6142-417: The plane of incidence). Although the reflection and transmission are dependent on polarization, at normal incidence ( θ = 0 ) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned below in which that is true). In the diagram on the right, an incident plane wave in the direction of the ray IO strikes

6225-461: The power coefficients are capitalized). As before, we are assuming the magnetic permeability, µ of both media to be equal to the permeability of free space µ 0 as is essentially true of all dielectrics at optical frequencies. In the following equations and graphs, we adopt the following conventions. For s polarization, the reflection coefficient r is defined as the ratio of the reflected wave's complex electric field amplitude to that of

6308-491: The power reflectance at normal incidence can be seen to be about 4%, or 8% accounting for both sides of a glass pane. At a dielectric interface from n 1 to n 2 , there is a particular angle of incidence at which R p goes to zero and a p-polarised incident wave is purely refracted, thus all reflected light is s-polarised. This angle is known as Brewster's angle , and is around 56° for n 1 = 1 and n 2 = 1.5 (typical glass). When light travelling in

6391-420: The ratio of the waves' magnetic fields, but comparison of the electric fields is more conventional. Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the power reflection coefficient R is just the squared magnitude of r :  R = | r | 2 . {\displaystyle R=|r|^{2}.} On

6474-659: The reflectance for p-polarized light is R p = | Z 2 cos ⁡ θ t − Z 1 cos ⁡ θ i Z 2 cos ⁡ θ t + Z 1 cos ⁡ θ i | 2 , {\displaystyle R_{\mathrm {p} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {t} }-Z_{1}\cos \theta _{\mathrm {i} }}{Z_{2}\cos \theta _{\mathrm {t} }+Z_{1}\cos \theta _{\mathrm {i} }}}\right|^{2},} where Z 1 and Z 2 are

6557-432: The reflecting surface is very smooth, the reflection of light that occurs is called specular or regular reflection. The laws of reflection are as follows: These three laws can all be derived from the Fresnel equations . In classical electrodynamics , light is considered as an electromagnetic wave, which is described by Maxwell's equations . Light waves incident on a material induce small oscillations of polarisation in

6640-402: The reflection varies according to the texture and structure of the surface. For example, porous materials will absorb some energy, and rough materials (where rough is relative to the wavelength) tend to reflect in many directions—to scatter the energy, rather than to reflect it coherently. This leads into the field of architectural acoustics , because the nature of these reflections is critical to

6723-463: The relative amplitudes but also the phase shifts at the interface. The equations assume the interface between the media is flat and that the media are homogeneous and isotropic . The incident light is assumed to be a plane wave , which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations. There are two sets of Fresnel coefficients for two different linear polarization components of

6806-439: The surface of transparent media, such as water or glass . In the diagram, a light ray PO strikes a vertical mirror at point O , and the reflected ray is OQ . By projecting an imaginary line through point O perpendicular to the mirror, known as the normal , we can measure the angle of incidence , θ i and the angle of reflection , θ r . The law of reflection states that θ i = θ r , or in other words,

6889-489: The unit vector normal to the interface). This complication can be ignored in the case of the reflection coefficient, since cos  θ i = cos  θ r , so that the ratio of reflected to incident irradiance in the wave's direction is the same as in the direction normal to the interface. Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be described as unpolarized. That means that there

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