Snell's law

Last updated
Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction th2 is less than the angle of incidence th1; that is, the ray in the higher-index medium is closer to the normal. Snells law2.svg
Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Snell's law (also known as Snell–Descartes law and ibn-Sahllaw and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air. This law was named after the Dutch astronomer and mathematician Willebrord Snellius (also called Snell).

Contents

In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material. The law is also satisfied in meta-materials, which allow light to be bent "backward" at a negative angle of refraction with a negative refractive index.

Snell's law states that, for a given pair of media, the ratio of the sines of angle of incidence () and angle of refraction () is equal to the refractive index of the second medium w.r.t the first (n21) which is equal to the ratio of the refractive indices (n2/n1) of the two media, or equivalently, to the ratio of the phase velocities (v1/v2) in the two media. [1]

The law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.

History

Reproduction of a page of Ibn Sahl's manuscript showing his ray diagrams relating to the law of refraction. Ibn Sahl manuscript.jpg
Reproduction of a page of Ibn Sahl's manuscript showing his ray diagrams relating to the law of refraction.

Ptolemy, in Alexandria, Egypt, [2] had found a relationship regarding refraction angles, but it was inaccurate for angles that were not small. Ptolemy was confident he had found an accurate empirical law, partially as a result of slightly altering his data to fit theory (see: confirmation bias). [3] Alhazen, in his Book of Optics (1021), came closer to discovering the law of refraction, though he did not take this step. [4]

An 1837 view of the history of "the Law of the Sines" Snell Law of Sines 1837.png
An 1837 view of the history of "the Law of the Sines"

The Persian scientist Ibn Sahl, at the Baghdad court in 984, recorded ray diagrams but made no record of material properties relating to refractive index, so cannot lay claim to have discovered the law of refraction. In the manuscript On Burning Mirrors and Lenses, Sahl used ray diagrams to derive lens shapes that focus light with no geometric aberrations. [6]

The law was first discovered by Thomas Harriot in 1602, [7] who however did not publish his results although he had corresponded with Kepler on this very subject. In 1621, the Dutch astronomer Willebrord Snellius (1580–1626)—Snell—derived a mathematically equivalent form, that remained unpublished during his lifetime. René Descartes independently derived the law using heuristic momentum conservation arguments in terms of sines in his 1637 essay Dioptrique , and used it to solve a range of optical problems. Rejecting Descartes' solution, Pierre de Fermat arrived at the same solution based solely on his principle of least time. Descartes assumed the speed of light was infinite, yet in his derivation of Snell's law he also assumed the denser the medium, the greater the speed of light. Fermat supported the opposing assumptions, i.e., the speed of light is finite, and his derivation depended upon the speed of light being slower in a denser medium. [8] [9] Fermat's derivation also utilized his invention of adequality, a mathematical procedure equivalent to differential calculus, for finding maxima, minima, and tangents. [10] [11]

In his influential mathematics book Geometry, Descartes solves a problem that was worked on by Apollonius of Perga and Pappus of Alexandria. Given n lines L and a point P(L) on each line, find the locus of points Q such that the lengths of the line segments QP(L) satisfy certain conditions. For example, when n = 4, given the lines a, b, c, and d and a point A on a, B on b, and so on, find the locus of points Q such that the product QA*QB equals the product QC*QD. When the lines are not all parallel, Pappus showed that the loci are conics, but when Descartes considered larger n, he obtained cubic and higher degree curves. To show that the cubic curves were interesting, he showed that they arose naturally in optics from Snell's law. [12]

According to Dijksterhuis, [13] "In De natura lucis et proprietate (1662) Isaac Vossius said that Descartes had seen Snell's paper and concocted his own proof. We now know this charge to be undeserved but it has been adopted many times since." Both Fermat and Huygens repeated this accusation that Descartes had copied Snell. In French, Snell's Law is called "la loi de Descartes" or "loi de Snell-Descartes."

Christiaan Huygens' construction Huygens Refracted Waves.png
Christiaan Huygens' construction

In his 1678 Traité de la Lumière , Christiaan Huygens showed how Snell's law of sines could be explained by, or derived from, the wave nature of light, using what we have come to call the Huygens–Fresnel principle.

With the development of modern optical and electromagnetic theory, the ancient Snell's law was brought into a new stage. In 1962, Bloembergen showed that at the boundary of nonlinear medium, the Snell's law should be written in a general form. [14] In 2008 and 2011, plasmonic metasurfaces were also demonstrated to change the reflection and refraction directions of light beam. [15] [16]

Explanation

Snell's law on a wall in Leiden SnelliusLeiden1.jpg
Snell's law on a wall in Leiden

Snell's law is used to determine the direction of light rays through refractive media with varying indices of refraction. The indices of refraction of the media, labeled , and so on, are used to represent the factor by which a light ray's speed decreases when traveling through a refractive medium, such as glass or water, as opposed to its velocity in a vacuum.

As light passes the border between media, depending upon the relative refractive indices of the two media, the light will either be refracted to a lesser angle, or a greater one. These angles are measured with respect to the normal line, represented perpendicular to the boundary. In the case of light traveling from air into water, light would be refracted towards the normal line, because the light is slowed down in water; light traveling from water to air would refract away from the normal line.

Refraction between two surfaces is also referred to as reversible because if all conditions were identical, the angles would be the same for light propagating in the opposite direction.

Snell's law is generally true only for isotropic or specular media (such as glass). In anisotropic media such as some crystals, birefringence may split the refracted ray into two rays, the ordinary or o-ray which follows Snell's law, and the other extraordinary or e-ray which may not be co-planar with the incident ray.

When the light or other wave involved is monochromatic, that is, of a single frequency, Snell's law can also be expressed in terms of a ratio of wavelengths in the two media, and :

Derivations and formula

Wavefronts from a point source in the context of Snell's law. The region below the grey line has a higher index of refraction, and proportionally lower speed of light, than the region above it. Snells law wavefronts.gif
Wavefronts from a point source in the context of Snell's law. The region below the grey line has a higher index of refraction, and proportionally lower speed of light, than the region above it.

Snell's law can be derived in various ways.

Derivation from Fermat's principle

Snell's law can be derived from Fermat's principle, which states that the light travels the path which takes the least time. By taking the derivative of the optical path length, the stationary point is found giving the path taken by the light. (There are situations of light violating Fermat's principle by not taking the least time path, as in reflection in a (spherical) mirror.) In a classic analogy, the area of lower refractive index is replaced by a beach, the area of higher refractive index by the sea, and the fastest way for a rescuer on the beach to get to a drowning person in the sea is to run along a path that follows Snell's law.

Light from medium 1, point Q, enters medium 2, refraction occurs, and reaches point P finally. Snells law Diagram B vector.svg
Light from medium 1, point Q, enters medium 2, refraction occurs, and reaches point P finally.

As shown in the figure to the right, assume the refractive index of medium 1 and medium 2 are and respectively. Light enters medium 2 from medium 1 via point O.

is the angle of incidence, is the angle of refraction with respect to the normal.

The phase velocities of light in medium 1 and medium 2 are

and
respectively.

is the speed of light in vacuum.

Let T be the time required for the light to travel from point Q through point O to point P.

where a, b, l and x are as denoted in the right-hand figure, x being the varying parameter.

To minimize it, one can differentiate :

(stationary point)

Note that

and

Therefore,

Derivation from Huygens's principle

Alternatively, Snell's law can be derived using interference of all possible paths of light wave from source to observer—it results in destructive interference everywhere except extrema of phase (where interference is constructive)—which become actual paths.

Derivation from Maxwell's equations

Another way to derive Snell's Law involves an application of the general boundary conditions of Maxwell equations for electromagnetic radiation and induction.

Derivation from conservation of energy and momentum

Yet another way to derive Snell's law is based on translation symmetry considerations. [17] For example, a homogeneous surface perpendicular to the z direction cannot change the transverse momentum. Since the propagation vector is proportional to the photon's momentum, the transverse propagation direction must remain the same in both regions. Assume without loss of generality a plane of incidence in the plane . Using the well known dependence of the wavenumber on the refractive index of the medium, we derive Snell's law immediately.

where is the wavenumber in vacuum. Although no surface is truly homogeneous at the atomic scale, full translational symmetry is an excellent approximation whenever the region is homogeneous on the scale of the light wavelength.

Vector form

Given a normalized light vector (pointing from the light source toward the surface) and a normalized plane normal vector , one can work out the normalized reflected and refracted rays, via the cosines of the angle of incidence and angle of refraction , without explicitly using the sine values or any trigonometric functions or angles: [18]

Note: must be positive, which it will be if is the normal vector that points from the surface toward the side where the light is coming from, the region with index . If is negative, then points to the side without the light, so start over with replaced by its negative.

This reflected direction vector points back toward the side of the surface where the light came from.

Now apply Snell's law to the ratio of sines to derive the formula for the refracted ray's direction vector:

The formula may appear simpler in terms of renamed simple values and , avoiding any appearance of trig function names or angle names:

Example:

The cosine values may be saved and used in the Fresnel equations for working out the intensity of the resulting rays.

Total internal reflection is indicated by a negative radicand in the equation for , which can only happen for rays crossing into a less-dense medium ().

Total internal reflection and critical angle

Demonstration of no refraction at angles greater than the critical angle. Refraction internal reflection diagram.svg
Demonstration of no refraction at angles greater than the critical angle.

When light travels from a medium with a higher refractive index to one with a lower refractive index, Snell's law seems to require in some cases (whenever the angle of incidence is large enough) that the sine of the angle of refraction be greater than one. This of course is impossible, and the light in such cases is completely reflected by the boundary, a phenomenon known as total internal reflection. The largest possible angle of incidence which still results in a refracted ray is called the critical angle; in this case the refracted ray travels along the boundary between the two media.

Refraction of light at the interface between two media. RefractionReflextion.svg
Refraction of light at the interface between two media.

For example, consider a ray of light moving from water to air with an angle of incidence of 50°. The refractive indices of water and air are approximately 1.333 and 1, respectively, so Snell's law gives us the relation

which is impossible to satisfy. The critical angle θcrit is the value of θ1 for which θ2 equals 90°:

Dispersion

In many wave-propagation media, wave velocity changes with frequency or wavelength of the waves; this is true of light propagation in most transparent substances other than a vacuum. These media are called dispersive. The result is that the angles determined by Snell's law also depend on frequency or wavelength, so that a ray of mixed wavelengths, such as white light, will spread or disperse. Such dispersion of light in glass or water underlies the origin of rainbows and other optical phenomena, in which different wavelengths appear as different colors.

In optical instruments, dispersion leads to chromatic aberration; a color-dependent blurring that sometimes is the resolution-limiting effect. This was especially true in refracting telescopes, before the invention of achromatic objective lenses.

Lossy, absorbing, or conducting media

In a conducting medium, permittivity and index of refraction are complex-valued. Consequently, so are the angle of refraction and the wave-vector. This implies that, while the surfaces of constant real phase are planes whose normals make an angle equal to the angle of refraction with the interface normal, the surfaces of constant amplitude, in contrast, are planes parallel to the interface itself. Since these two planes do not in general coincide with each other, the wave is said to be inhomogeneous. [19] The refracted wave is exponentially attenuated, with exponent proportional to the imaginary component of the index of refraction. [20] [21]

See also

Related Research Articles

<span class="mw-page-title-main">Ellipse</span> Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

<span class="mw-page-title-main">Fresnel equations</span> Equations of light transmission and reflection

The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

<span class="mw-page-title-main">Refractive index</span> Ratio of the speed of light in vacuum to that in the medium

In optics, the refractive index of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.

<span class="mw-page-title-main">Refraction</span> Physical phenomenon relating to the direction of waves

In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.

<span class="mw-page-title-main">Total internal reflection</span> Total Internal Refraction

Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another are not refracted into the second ("external") medium, but completely reflected back into the first ("internal") medium. It occurs when the second medium has a higher wave speed than the first, and the waves are incident at a sufficiently oblique angle on the interface. For example, the water-to-air surface in a typical fish tank, when viewed obliquely from below, reflects the underwater scene like a mirror with no loss of brightness (Fig. 1).

<span class="mw-page-title-main">Brewster's angle</span> Angle of incidence for which all reflected light will be polarized

Brewster's angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist Sir David Brewster (1781–1868).

In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.

<span class="mw-page-title-main">Numerical aperture</span> Characteristic of an optical system

In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another, provided there is no refractive power at the interface. The exact definition of the term varies slightly between different areas of optics. Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective, and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.

In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.

In physics and chemistry, Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave fronts scattered by lattice planes, leading to a strict relation between wavelength and scattering angle, or else to the wavevector transfer with respect to the crystal lattice. Such law had initially been formulated for X-rays upon crystals. However, It applies to all sorts of quantum beams, including neutron and electron waves at atomic distances, as well as visible light at artificial periodic microscale lattices.

Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.

Etendue or étendue is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. Other names for etendue include acceptance, throughput, light grasp, light-gathering power, optical extent, and the AΩ product. Throughput and AΩ product are especially used in radiometry and radiative transfer where it is related to the view factor. It is a central concept in nonimaging optics.

<span class="mw-page-title-main">Strophoid</span> Geometric curve

In geometry, a strophoid is a curve generated from a given curve C and points A and O as follows: Let L be a variable line passing through O and intersecting C at K. Now let P1 and P2 be the two points on L whose distance from K is the same as the distance from A to K. The locus of such points P1 and P2 is then the strophoid of C with respect to the pole O and fixed point A. Note that AP1 and AP2 are at right angles in this construction.

<span class="mw-page-title-main">Dispersive prism</span> Device used to disperse light

In optics, a dispersive prism is an optical prism that is used to disperse light, that is, to separate light into its spectral components. Different wavelengths (colors) of light will be deflected by the prism at different angles. This is a result of the prism material's index of refraction varying with wavelength (dispersion). Generally, longer wavelengths (red) undergo a smaller deviation than shorter wavelengths (blue). The dispersion of white light into colors by a prism led Sir Isaac Newton to conclude that white light consisted of a mixture of different colors.

<span class="mw-page-title-main">Acousto-optics</span> The study of sound and light interaction

Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound through an ultrasonic grating.

<span class="mw-page-title-main">Radiative transfer equation and diffusion theory for photon transport in biological tissue</span>

Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation (RTE). However, the RTE is difficult to solve without introducing approximations. A common approximation summarized here is the diffusion approximation. Overall, solutions to the diffusion equation for photon transport are more computationally efficient, but less accurate than Monte Carlo simulations.

<span class="mw-page-title-main">Gravitational lensing formalism</span>

In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to

The direct-quadrature-zerotransformation or zero-direct-quadraturetransformation is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix in an effort to simplify analysis. The DQZ transform is the product of the Clarke transform and the Park transform, first proposed in 1929 by Robert H. Park.

<span class="mw-page-title-main">Thin-film interference</span> Optical phenomenon

Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, either enhancing or reducing the reflected light. When the thickness of the film is an odd multiple of one quarter-wavelength of the light on it, the reflected waves from both surfaces interfere to cancel each other. Since the wave cannot be reflected, it is completely transmitted instead. When the thickness is a multiple of a half-wavelength of the light, the two reflected waves reinforce each other, increasing the reflection and reducing the transmission. Thus when white light, which consists of a range of wavelengths, is incident on the film, certain wavelengths (colors) are intensified while others are attenuated. Thin-film interference explains the multiple colors seen in light reflected from soap bubbles and oil films on water. It is also the mechanism behind the action of antireflection coatings used on glasses and camera lenses.

References

  1. Born and Wolf (1959). Principles of Optics . New York, NY: Pergamon Press INC. p. 37.
  2. David Michael Harland (2007). " Cassini at Saturn: Huygens results ". p.1. ISBN   0-387-26129-X
  3. "Ptolemy (ca. 100-ca. 170)". Eric Weinstein's World of Scientific Biography.
  4. A. I. Sabra (1981), Theories of Light from Descartes to Newton, Cambridge University Press. (cf. Pavlos Mihas, Use of History in Developing ideas of refraction, lenses and rainbow, p. 5, Demokritus University, Thrace, Greece.)
  5. William Whewell, History of the Inductive Science from the Earliest to the Present Times, London: John H. Parker, 1837.
  6. Rashed, Roshdi (1990). "A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses". Isis . 81 (3): 464–491. doi:10.1086/355456. S2CID   144361526.[ disputed ][ clarification needed ]
  7. Kwan, A.; Dudley, J.; Lantz, E. (2002). "Who really discovered Snell's law?". Physics World . 15 (4): 64. doi:10.1088/2058-7058/15/4/44.
  8. Florian Cajori, A History of Physics in its Elementary Branches: Including the Evolution of Physical Laboratories (1922)
  9. Ferdinand Rosenberger, Geschichte der Physik (1882) Part. II, p.114
  10. Carl Benjamin Boyer, The Rainbow: From Myth to Mathematics (1959)
  11. Florian Cajori, "Who was the First Inventor of Calculus" The American Mathematical Monthly (1919) Vol.26
  12. The Geometry of Rene Descartes (Dover Books on Mathematics) by Rene Descartes, David Eugene Smith and Marcia L. Latham (Jun 1, 1954).
  13. Dijksterhuis, Fokko Jan (2004). Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century. Springer. ISBN   1-4020-2697-8.
  14. Bloembergen, N.; Pershan, P. S. (1962). "Light waves at the boundary of nonlinear media" (PDF). Physical Review. 128 (2): 606. Bibcode:1962PhRv..128..606B. doi:10.1103/PhysRev.128.606. hdl:1874/7432. Archived (PDF) from the original on 2022-10-09.
  15. Xu, T.; et al. (2008). "Plasmonic deflector". Opt. Express. 16 (7): 4753–9. Bibcode:2008OExpr..16.4753X. doi: 10.1364/oe.16.004753 . PMID   18542573.
  16. Yu, Nanfang; Genevet, Patrice; Kats, Mikhail A.; Aieta, Francesco; Tetienne, Jean-Philippe; Capasso, Federico; Gaburro, Zeno (October 2011). "Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction". Science. 334 (6054): 333–7. Bibcode:2011Sci...334..333Y. doi:10.1126/science.1210713. PMID   21885733. S2CID   10156200.
  17. Joannopoulos, John D; Johnson, SG; Winn, JN; Meade, RD (2008). Photonic Crystals: Molding the Flow of Light (2nd ed.). Princeton NJ: Princeton University Press. ISBN   978-0-691-12456-8.
  18. Glassner, Andrew S. (1989). An Introduction to Ray Tracing. Morgan Kaufmann. ISBN   0-12-286160-4.
  19. Born and Wolf, sec.13.2, "Refraction and reflection at a metal surface"
  20. Hecht, Optics, sec. 4.8, Optical properties of metals.
  21. S. J. Orfanidis, Electromagnetic Waves & Antennas, sec. 7.9, Oblique Incidence on a Lossy Medium,