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Refraction

Refraction is the change in direction of a wave, such as or , as it passes obliquely from one medium to another in which its speed is different, resulting in a of the wave's path. This occurs because the wave's changes at the between the two media, causing the to alter its orientation. In , refraction primarily refers to the behavior of electromagnetic waves, particularly visible , when transitioning between materials with different optical densities. The extent of bending in refraction is quantitatively described by , which states that the product of the of the first medium and the sine of the angle of incidence equals the product of the of the second medium and the sine of the . The (n) of a medium is defined as the ratio of the in a to its speed in that medium, serving as a measure of how much the material slows down light propagation. For example, air has a close to 1, while is approximately 1.33 and around 1.5, leading to noticeable bending when light enters these substances from air. When light travels from a medium with a higher to one with a lower index at an angle greater than the , occurs instead of refraction, which is fundamental to devices like fiber optics. Refraction plays a crucial role in numerous optical applications, enabling the design of lenses that focus light for vision correction in eyeglasses and contact lenses, as well as in cameras, microscopes, and telescopes. In telecommunications, it underpins the guiding of light signals through optical fibers, where total internal reflection maintains signal integrity over long distances. Atmospheric refraction, caused by varying air density, affects astronomical observations by bending starlight, influencing the apparent positions of celestial bodies. Additionally, refraction is essential in medical imaging techniques, such as ophthalmology for assessing eye refraction errors, and in scientific instruments for precise light manipulation.

Fundamentals

Definition

Refraction is the change in direction of a propagating wave that results from a change in its transmission speed upon passing from one medium to another. This phenomenon occurs for various types of waves, including electromagnetic waves such as light and mechanical waves like sound or water waves, provided the wave encounters a boundary between media with different wave speeds. The bending happens only when the wave approaches the interface at an oblique angle; if incident normally, the wave continues straight without deviation. For wave propagation across such boundaries, the frequency f of the wave remains invariant, as it is determined by the source and conserved across the interface. However, the speed v changes due to the medium's properties, such as density or elasticity for mechanical waves, or permittivity and permeability for electromagnetic waves. Consequently, the wavelength \lambda adjusts to maintain the fundamental relation v = f \lambda, resulting in a shorter wavelength in a medium where the wave slows down and a longer one where it speeds up. The phenomenon of refraction was first observed and qualitatively described by scholars, with Claudius providing detailed accounts in his work around 150 AD, including experimental attempts to measure the bending of light rays. 's descriptions focused on visual effects and rudimentary tables of angles, marking an early systematic study. Although earlier attempts existed, including approximate tables by , the precise quantitative relationship was first established by Ibn Sahl in the and rediscovered in the through experiments by Willebrord Snell, later known as . A basic ray diagram for refraction depicts an incident striking the boundary between two media at an \theta_i to (a line perpendicular to the ), bending to a refracted at \theta_r, while remains unchanged. The diagram typically shows the incident approaching from the first medium, the straight at the point of incidence, and the refracted departing into the second medium, illustrating the directional change without altering the wave's overall path if normal incidence occurs.

Snell's Law

Snell's law, also known as the law of refraction, describes the relationship between the angles of incidence and refraction for a light passing from one medium to another. It states that the product of the of the first medium and the sine of the angle of incidence equals the product of the of the second medium and the sine of of refraction:
n_1 \sin \theta_1 = n_2 \sin \theta_2
where n_1 and n_2 are the of the respective media, \theta_1 is the angle between the incident and to the interface, and \theta_2 is between the refracted and . The refractive index n of a medium is defined as the ratio of the in c to the in that medium v:
n = \frac{c}{v}.
This quantity is dimensionless and greater than or equal to 1 for all media, with air having an approximate value of n \approx 1 under standard conditions. Snell's law can be derived from Fermat's principle, which posits that light travels along the path that minimizes the travel time between two points. Consider a light ray crossing a planar from medium 1 (speed v_1 = c/n_1) to medium 2 (speed v_2 = c/n_2). The time of travel t for a path parameterized by the point of incidence is t = \frac{l_1}{v_1} + \frac{l_2}{v_2}, where l_1 and l_2 are the path lengths in each medium. Minimizing t with respect to the lateral displacement at the yields \frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}, or equivalently n_1 \sin \theta_1 = n_2 \sin \theta_2. Alternatively, the law follows from the continuity of wave fronts across the using Huygens' principle, ensuring phase matching at the . When light travels from a medium with higher n_1 to one with lower n_2 < n_1, a critical angle \theta_c exists beyond which no refraction occurs into the second medium. This critical angle is given by
\sin \theta_c = \frac{n_2}{n_1},
derived by setting \theta_2 = 90^\circ in Snell's law. For incident angles \theta_1 > \theta_c, occurs, with the light entirely reflected back into the first medium at an angle equal to the incident angle, following the law of reflection. A common example is light passing from air (n_1 \approx 1) to glass (n_2 \approx 1.5). For an incident angle of \theta_1 = 30^\circ, the refracted angle is \theta_2 = \arcsin\left(\frac{\sin 30^\circ}{1.5}\right) \approx 19.5^\circ. Conversely, from glass to air, the critical angle is \theta_c = \arcsin\left(\frac{1}{1.5}\right) \approx 41.8^\circ, above which total internal reflection takes place.

Light Refraction

Mechanism of Bending

Refraction occurs when a wave passes from one medium to another with a different , causing the wave front to change direction due to a variation in propagation speed. According to Huygens' principle, every point on an advancing wave front serves as a source of secondary spherical wavelets, and the new wave front is the tangent envelope of these wavelets. In refraction, the differential speed between media leads to an asymmetric expansion of these wavelets: those entering the slower medium lag behind, resulting in an oblique propagation of the overall wave front and bending of the ray. Light travels more slowly in denser media, such as glass compared to air, because the electromagnetic wave interacts with the atoms or molecules, inducing oscillations that re-radiate secondary waves. These re-radiated waves interfere constructively with the incident wave in the forward direction but introduce a phase delay, effectively reducing the phase velocity of the light. The refractive index n = c / v, where c is the speed in vacuum and v is the speed in the medium, quantifies this slowdown, with higher n indicating stronger interactions and greater delay. In ray optics, the incident ray strikes the interface at an angle, and upon entering a medium with higher refractive index, the refracted ray bends toward the normal (the perpendicular to the interface), while in a lower index medium, it bends away from the normal. This directional change maintains the continuity of the wave front across the boundary. At normal incidence, where the ray is perpendicular to the interface, no bending occurs despite the speed change, as the wave front advances uniformly. The resulting angular relationship, described by Snell's law, emerges directly from this mechanism of speed variation and wave front reconstruction. From a quantum perspective, consists of , each carrying E = h f (where h is Planck's and f is ) and p = E / v, with v the speed in the medium. During refraction, a photon's —and thus —remains to conserve at the , but its magnitude changes with the speed alteration. The component of parallel to the is conserved, ensuring the photon adjusts its direction to satisfy this condition, analogous to a particle while preserving parallel .

Dispersion

Dispersion refers to the variation of the of a medium with the of , causing different s to refract at slightly different angles. In most transparent media, such as or , the refractive index n(\lambda) decreases as the \lambda increases, a behavior known as normal dispersion. This dependence arises from the interaction of electromagnetic waves with the electrons in the material, leading to greater bending for shorter s (like ) compared to longer ones (like red light). As a result, white passing through such media separates into its constituent colors, producing a . An empirical relation often used to approximate this normal dispersion in the visible range is Cauchy's equation: n(\lambda) = A + \frac{B}{\lambda^2}, where A and B are material-specific constants determined experimentally, with A representing the refractive index at infinite wavelength and B accounting for the dispersive contribution. This simple two-term form provides a good fit for many optical glasses away from absorption regions, though more complex models like the Sellmeier equation are used for broader spectral ranges. For example, in crown glass, typical values are A \approx 1.51 and B \approx 0.0068 \, \mu \mathrm{m}^2, yielding refractive indices around 1.52 for yellow light. In prisms, this wavelength dependence manifests as angular dispersion, the spread in deviation angles between different colors. For a thin prism with apex angle A, the deviation for a given wavelength is approximately \delta(\lambda) = (n(\lambda) - 1) A, so the angular separation between blue and red light is \delta_b - \delta_r = (n_b - n_r) A. The dispersive power \omega, a material property measuring the relative dispersion, is defined as \omega = \frac{n_b - n_r}{\mu - 1}, where \mu is the mean refractive index (often for yellow light), typically ranging from 0.008 for low-dispersion crown glass to 0.017 for flint glass. This quantifies how effectively the material separates colors without excessive mean deviation. A prominent natural example of dispersion is the formation of rainbows, where refracts, reflects, and disperses within spherical droplets in the atmosphere. In the primary rainbow, each droplet causes one internal , with dispersion separating the colors such that appears on the outer edge (deviation about 42°) and on the inner edge (about 40°), forming an centered on the . The secondary rainbow, fainter and higher in the sky, results from two internal reflections, reversing the color order ( inner, violet outer) with a larger deviation of about 51°. These phenomena require droplets of roughly 0.1–1 mm diameter and are observable when is low. Anomalous dispersion occurs in rare cases near regions of strong , such as lines or molecular bands, where the increases with (dn/d\lambda > 0). Here, the usual order of colors reverses, with longer wavelengths bending more than shorter ones, though this regime is accompanied by high and limited . This is described by the Kramers-Kronig relations linking real and imaginary parts of the dielectric function and is observed in gases like sodium vapor near the yellow D-lines or in certain dyed glasses.

Natural Phenomena

Atmospheric Refraction

Atmospheric refraction occurs due to the gradual variation in the of air with altitude, primarily caused by the decrease in air as increases. The of air near is approximately 1.0003 under standard conditions, but it diminishes toward 1 at higher altitudes because lower allows to travel faster. This creates a vertical in refractivity, bending rays concave toward the Earth's surface and making objects appear higher in the sky than their true positions. One common everyday phenomenon is the apparent flattening of the near the horizon at sunrise or sunset. As passes through the denser lower atmosphere, the rays from the Sun's upper edge bend less than those from the lower edge due to the increasing gradient near the ground, compressing the solar disk vertically and giving it an oblate shape. Mirages represent more dramatic effects of this refraction, arising from horizontal temperature gradients that invert the usual profile. An inferior mirage occurs when hot ground air creates a layer less dense than the air above, bending light rays upward and producing illusory images like puddles on a hot road, where the sky appears reflected below the actual object. In contrast, a superior mirage forms under a temperature inversion with colder air near the surface overlaid by warmer air, causing rays to bend downward and elevate distant objects, such as ships appearing to float above the horizon. , a related , magnifies and raises objects when the refractive gradient lifts the apparent horizon, often seen over cold water bodies. In astronomy, necessitates corrections for the observed positions of stars and other celestial bodies, as it displaces their apparent altitude toward the by up to about 35 arcminutes at the horizon. The angular correction [R](/page/R) can be approximated by the formula [R](/page/R) \approx (n-1) \tan [z](/page/Z), where n is the at and [z](/page/Z) is the true zenith distance; for small angles, this simplifies to [R](/page/R) \approx 60'' \tan [z](/page/Z), with the result in arcseconds. This effect is most pronounced for objects low on the horizon and requires adjustment in precise observations, such as or . The of stars, or , results from refraction through turbulent pockets of air with varying densities and temperatures, causing rapid fluctuations in the light's arrival angle and intensity. These small-scale atmospheric cells, driven by wind and , refract starlight irregularly, producing the characteristic shimmer, while appear steadier due to their larger apparent disks averaging out the distortions. Temperature inversions, where warmer air traps cooler air below, exacerbate mirage formation by steepening the gradient and confining light paths to unusual trajectories. Such inversions are common in stable climatic conditions, like over polar seas or calm nights, enhancing superior s and contributing to optical illusions in regions with persistent cold surface layers.

Aquatic Refraction

Aquatic refraction occurs at the interface between water and air, where light bends due to the difference in refractive indices, leading to various optical illusions and adaptations in marine environments. The refractive index of water is approximately 1.33 for visible light, though it varies slightly with temperature, salinity, and wavelength; for instance, freshwater at 20°C has n \approx 1.333, while seawater can reach up to 1.34 due to dissolved salts. This index governs how light rays from air enter water or vice versa, causing phenomena observable in everyday settings and underwater ecosystems. A classic example is the apparent depth of submerged objects, where an item at actual depth d appears closer to the surface at d' = d / n, making pools or lakes seem shallower than they are; for water, this reduces perceived depth by about 25%. Similarly, a straw or pencil partially immersed in a glass of water appears bent at the water's surface because light rays from the submerged portion refract toward the normal upon entering air, creating a discontinuity in the perceived straight line. From underwater, this refraction culminates in Snell's window, a circular field of view spanning approximately 97° above the observer, beyond which total internal reflection occurs at the critical angle of \theta_c = \sin^{-1}(1/n) \approx 48.6^\circ, reflecting the underwater scene instead of the surface world. In , refraction influences visual adaptations, particularly in fish vision, where spherical lenses with higher refractive indices compensate for the medium's properties to focus effectively , unlike the corneal-dominated focusing in air-breathing animals. Deep-sea species further adapt to low-light conditions involving , with specialized photoreceptors and lens pigments enhancing detection of narrow-bandwidth blue-green emissions that propagate through water with minimal , aided by the refractive environment.

Optical Applications

In Human Vision

Refraction plays a central role in human vision by bending rays to focus them on the , primarily through the optical properties of the and crystalline . The , the eye's transparent front surface, has a of approximately 1.376 and contributes about 43 diopters to the total refractive power. The crystalline , with a of around 1.41, adds roughly 17 diopters, resulting in an overall refractive power of about 60 diopters for the relaxed eye. This combined refraction, governed by at the air- and cornea-aqueous humor interfaces, accounts for approximately two-thirds of the eye's focusing ability from the alone. In a normal eye, known as , parallel light rays from distant objects converge precisely on the , producing clear vision without . Refractive errors disrupt this process: , or nearsightedness, occurs when the falls in front of the due to an elongated eyeball or excessive corneal curvature, making distant objects blurry while near vision remains clear. , or farsightedness, results in the behind the from a shortened eyeball or insufficient refractive power, often causing strain during near tasks. arises from irregular curvature of the or , leading to multiple focal points and distorted vision at all distances. Corrective measures address these errors by altering the light path before it enters the eye or by modifying the eye's structure. Spectacles and contact lenses use thin designed via the lensmaker's formula, \frac{1}{f} = (n-1)\left( \frac{1}{R_1} - \frac{1}{R_2} \right), where f is the , n is the lens material's , and R_1, R_2 are the radii of of the lens surfaces, to provide diverging power for or converging power for hyperopia and . LASIK surgery corrects refractive errors by using an to reshape the , flattening it for or steepening it for hyperopia, thereby adjusting its refractive power without altering the . The eye's ability to focus on near objects, called , involves the ciliary muscles contracting to relax zonular fibers, allowing the lens to become more spherical and increase its refractive power by up to 10-12 diopters in . This dynamic refraction enables clear vision from infinity to about 25 cm. , an age-related condition typically onset after age 40, diminishes this capacity as the lens stiffens and loses elasticity, reducing accommodative amplitude to near zero by age 60 and necessitating reading glasses for close work. The development of corrective lenses traces back to the 13th century in Italy, where monks and scholars first crafted convex glass lenses to aid presbyopic reading, marking the earliest known optical correction for refractive errors.

In Lenses and Instruments

Lenses are optical devices that exploit refraction to converge or diverge light rays, forming images through the bending of light at curved surfaces. Converging lenses, often convex in shape, have a positive focal length and focus parallel rays to a single point, while diverging lenses, typically concave, possess a negative focal length and cause parallel rays to spread apart. These effects arise from the variation in refractive index between the lens material and the surrounding medium, governed by Snell's law at each surface./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/02%3A_Geometric_Optics_and_Image_Formation/2.05%3A_Thin_Lenses) The behavior of thin lenses, where thickness is negligible compared to , is described by the thin lens equation: \frac{1}{f} = \frac{1}{u} + \frac{1}{v} Here, f is the , u is the object distance, and v is the image distance. This equation derives from applying iteratively at the two refracting surfaces of the lens, combined with the paraxial approximation for small angles. The lensmaker's formula further relates f to the lens's radii of curvature R_1 and R_2, and n: \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) For a biconvex converging lens, R_1 > 0 and R_2 < 0, yielding a positive f./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/02%3A_Geometric_Optics_and_Image_Formation/2.05%3A_Thin_Lenses) Imperfections in lens performance, known as aberrations, degrade image quality. Chromatic aberration occurs because dispersion causes different wavelengths to refract by varying amounts, focusing them at different points along the optical axis. This is corrected in achromatic doublets, which combine a convex crown glass element (low dispersion) with a concave flint glass element (high dispersion) to bring two wavelengths to the same focus. Spherical aberration, meanwhile, results from the stronger refraction at the lens periphery compared to the center for spherical surfaces, blurring the image. Aspheric lenses, with non-spherical surfaces, mitigate this by tailoring the curvature to equalize focal points across the aperture. In microscopes, refraction through and lenses enables high linear . The forms a real, magnified intermediate image with m = v/u, where v and u are the image and object distances from the thin lens equation. The total is then the product of the 's linear and the 's angular , often achieving hundreds to thousands of times enlargement for detailed specimen viewing. Telescopes utilize refraction to achieve angular , expanding the apparent size of distant objects. In a , the angular is m = f_{\text{obj}} / f_{\text{eyepiece}}, where f_{\text{obj}} and f_{\text{eyepiece}} are the focal lengths of the and lenses, respectively, allowing resolution of fine celestial details./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/02%3A_Geometric_Optics_and_Image_Formation/2.09%3A_Microscopes_and_Telescopes) Fiber optics rely on refraction via total internal reflection to guide light signals over long distances. The fiber core, with a higher refractive index n_{\text{core}} than the surrounding cladding (n_{\text{clad}} < n_{\text{core}}), confines light within the core when the incidence angle exceeds the critical angle, defined by Snell's law as \sin \theta_c = n_{\text{clad}} / n_{\text{core}}. This principle enables low-loss transmission in telecommunications, with typical index differences of about 0.01 for single-mode fibers.

Mechanical Waves

Water Waves

Refraction of surface occurs when propagate from regions of one water depth to another, causing a change in wave direction due to variations in wave speed. This phenomenon is prominent in , coastal areas, and even laboratory ripple tanks, where encounter gradually changing depths. The bending of wave crests aligns them more to the depth , altering the of wave along the shore./05:_Coastal_hydrodynamics/5.02:_Wave_transformation/5.2.3:_Refraction) The speed of surface gravity waves depends on water depth relative to the wavelength. In deep water, where the depth h is greater than half the wavelength \lambda, the phase speed v is given by v = \sqrt{\frac{g \lambda}{2\pi}}, where g is the acceleration due to gravity; this speed is independent of depth. In shallow water, where h < \lambda / 20, the speed simplifies to v = \sqrt{g h}, which depends solely on depth and decreases as water shallows. These relations arise from the dispersion relation for gravity waves, describing how wave propagation varies with depth. As approach shallower regions, such as near a , their speed reduces in those areas, causing the part of the in shallower to lag behind the part in deeper water. This differential slowing results in the or refracting toward to the depth lines, effectively turning the more directly toward the shore. For instance, oblique approaching a straight will refract such that their crests become nearly parallel to the shoreline, concentrating perpendicularly onto the . This mechanism is analogous to the observed in light but applied to mechanical ./05:_Coastal_hydrodynamics/5.02:_Wave_transformation/5.2.3:_Refraction) The refraction process can be understood through Huygens' principle, which posits that every point on a wave front acts as a source of secondary spherical wavelets that propagate forward at the local wave speed, with the new wave front forming as the to these wavelets. In varying depths, wavelets in shallower water expand more slowly than those in deeper water, causing the overall front to pivot and adjust its direction. This adjustment ensures continuity of the wave phase across the front, leading to the observed bending without altering the wave frequency./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/01:_The_Nature_of_Light/1.07:_Huygenss_Principle) One notable phenomenon resulting from water refraction is harbor , where incoming at frequencies matching the harbor's natural modes amplify inside the enclosed or semi-enclosed , potentially causing significant fluctuations and structural . Refraction also contributes to focusing along irregular coastlines, where energy converges on protruding headlands, enhancing local rates while dispersing energy in adjacent bays. For example, in headland-bay systems, refracted increase and power at headlands, accelerating cliff retreat and sediment removal./13:_Coastal_Oceanography/13.03:_Landforms_of_Coastal_Erosion) During refraction, the remains constant as it is determined by the distant source, such as or a wave maker, but the shortens in slower (shallower) regions to maintain the v = f \lambda, where f is . This compression of wavefronts in shallow areas further intensifies local wave steepness and breaking, contributing to and patterns.

Sound Waves

Refraction of sound waves occurs when acoustic waves propagate through media with spatial variations in sound speed, causing the waves to bend according to gradients in temperature, density, or wind velocity. The speed of sound in air is primarily dependent on temperature, approximated by the formula v \approx 331 + 0.6T m/s, where T is the temperature in degrees Celsius; this speed is higher in warmer media due to increased molecular kinetic energy facilitating faster pressure wave propagation. In seawater, sound speed typically ranges from about 1500–1530 m/s in surface waters, decreasing to a minimum of around 1480 m/s in the SOFAR channel at approximately 1000 m depth due to temperature gradients in the thermocline, and then increasing to higher values in deeper layers primarily due to pressure effects. These speed variations lead to curved propagation paths for sound rays, analogous to light rays in optics. In atmospheric conditions with a decreasing temperature gradient with height—such as over hot ground during the day—the sound speed decreases upward, causing rays to refract upward and away from the surface, creating regions of reduced audibility known as shadow zones. Conversely, in temperature inversions where temperature (and thus sound speed) increases with height—often over cold surfaces at night—rays bend downward toward the ground, extending the range of sound propagation. Wind gradients can further influence refraction by adding an effective speed component, bending rays downwind or upwind depending on the shear. Notable phenomena arise from these refractions. Auditory mirages occur when or gradients distort the apparent of a sound source, similar to optical mirages but for hearing; for instance, cover can refract vole calls, misleading predators like about prey position. In ocean acoustics, shadow zones form where rays are refracted away from receivers due to thermoclines—layers of rapid change—preventing from reaching certain areas and aiding concealment. The SOFAR (Sound Fixing and Ranging) channel, a deep-ocean layer around 1000 m where speed reaches a minimum due to pressure and effects, traps low-frequency , allowing them to propagate thousands of kilometers with minimal loss by refracting rays back toward the axis. In the ray acoustics approximation, valid for high frequencies where wavelengths are short compared to inhomogeneities, sound propagation follows paths governed by an analog of : \frac{\sin \theta}{v} = \constant, where \theta is the ray angle to the gradient normal and v is the local sound speed; this is equivalent to an effective n = \frac{v_\text{ref}}{v}, with v_\text{ref} a reference speed, mirroring optical behavior but scaled by acoustic velocities. This framework predicts ray tracing in varying , essential for modeling complex environments. Such refractions have practical applications in systems, where beam bending in layers like thermoclines or the affects detection ranges, convergence zones (focusing sound energy), and shadow avoidance strategies for underwater navigation and target tracking.