Snell's law
sin θ 2 sin θ 1 = v 2 v 1 = n 1 n 2 displaystyle frac sin theta _ 2 sin theta _ 1 = frac v_ 2 v_ 1 = frac n_ 1 n_ 2 with each θ displaystyle theta as the angle measured from the normal of the boundary, v displaystyle v as the velocity of light in the respective medium (SI units are meters per second, or m/s), λ displaystyle lambda as the wavelength of light in the respective medium and n displaystyle n as the refractive index (which is unitless) of the respective medium.
The law follows from
Fermat's principle
Contents 1 History 2 Explanation 3 Derivations and formula 3.1 Vector form 4
Total internal reflection
History[edit] Reproduction of a page of Ibn Sahl's manuscript showing his discovery of the law of refraction. Ptolemy, a Greek living in Alexandria, Egypt,[1] had found a
relationship regarding refraction angles, but it was inaccurate for
angles that were not small.
Ptolemy
An 1837 view of the history of "the Law of the Sines"[4] Although named after Dutch astronomer Willebrord Snellius
(1580–1626), the law was first accurately described by the Persian
scientist Ibn Sahl[disputed – discuss] at the
Baghdad
Christiaan Huygens' construction In his 1678 Traité de la Lumière,
Christiaan Huygens
Explanation[edit]
Snell's law
n 1 displaystyle n_ 1 , n 2 displaystyle n_ 2 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
sin θ 1 sin θ 2 = v 1 v 2 = λ 1 λ 2 displaystyle frac sin theta _ 1 sin theta _ 2 = frac v_ 1 v_ 2 = frac lambda _ 1 lambda _ 2 Derivations and formula[edit]
Wavefronts
Snell's law
Light from medium 1, point Q, enters medium 2, refraction occurs, and reaches point P finally. As shown in the figure on the right, assume the refractive index of medium 1 and medium 2 are n 1 displaystyle n_ 1 and n 2 displaystyle n_ 2 respectively. Light enters medium 2 from medium 1 via point O. θ 1 displaystyle theta _ 1 is the angle of incidence, θ 2 displaystyle theta _ 2 is the angle of refraction. The traveling velocities of light in medium 1 and medium 2 are v 1 = c / n 1 displaystyle v_ 1 =c/n_ 1 and v 2 = c / n 2 displaystyle v_ 2 =c/n_ 2 respectively. c displaystyle c is the speed of light in vacuum. Let T be the time required for the light to travel from point Q to point P. T = x 2 + a 2 v 1 + b 2 + ( l − x ) 2 v 2 displaystyle T= frac sqrt x^ 2 +a^ 2 v_ 1 + frac sqrt b^ 2 +(lx)^ 2 v_ 2 d T d x = x v 1 x 2 + a 2 + − ( l − x ) v 2 ( l − x ) 2 + b 2 = 0 displaystyle frac dT dx = frac x v_ 1 sqrt x^ 2 +a^ 2 + frac (lx) v_ 2 sqrt (lx)^ 2 +b^ 2 =0 (stationary point) Note that x x 2 + a 2 = sin θ 1 displaystyle frac x sqrt x^ 2 +a^ 2 =sin theta _ 1 l − x ( l − x ) 2 + b 2 = sin θ 2 displaystyle frac lx sqrt (lx)^ 2 +b^ 2 =sin theta _ 2 d T d x = sin θ 1 v 1 − sin θ 2 v 2 = 0 displaystyle frac dT dx = frac sin theta _ 1 v_ 1  frac sin theta _ 2 v_ 2 =0 sin θ 1 v 1 = sin θ 2 v 2 displaystyle frac sin theta _ 1 v_ 1 = frac sin theta _ 2 v_ 2 n 1 sin θ 1 c = n 2 sin θ 2 c displaystyle frac n_ 1 sin theta _ 1 c = frac n_ 2 sin theta _ 2 c n 1 sin θ 1 = n 2 sin θ 2 displaystyle n_ 1 sin theta _ 1 =n_ 2 sin theta _ 2 Alternatively,
Snell's law
k → displaystyle vec k is proportional to the photon's momentum, the transverse propagation direction ( k x , k y , 0 ) displaystyle (k_ x ,k_ y ,0) must remain the same in both regions. Assume without loss of generality a plane of incidence in the z , x displaystyle z,x plane k x Region 1 = k x Region 2 displaystyle k_ x text Region _ 1 =k_ x text Region _ 2 . Using the well known dependence of the wavenumber on the refractive
index of the medium, we derive
Snell's law
k x Region 1 = k x Region 2 displaystyle k_ x text Region _ 1 =k_ x text Region _ 2 , n 1 k 0 sin θ 1 = n 2 k 0 sin θ 2 displaystyle n_ 1 k_ 0 sin theta _ 1 =n_ 2 k_ 0 sin theta _ 2 , n 1 sin θ 1 = n 2 sin θ 2 displaystyle n_ 1 sin theta _ 1 =n_ 2 sin theta _ 2 , where k 0 = 2 π λ 0 = ω c displaystyle k_ 0 = frac 2pi lambda _ 0 = frac omega c 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[edit] See also: Specular reflection § Direction of reflection Given a normalized light vector l (pointing from the light source toward the surface) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays, via the cosines of the angle of incidence θ 1 displaystyle theta _ 1 and angle of refraction θ 2 displaystyle theta _ 2 , without explicitly using the sine values or any trigonometric functions or angles:[17] cos θ 1 = − n ⋅ l displaystyle cos theta _ 1 =mathbf n cdot mathbf l Note: cos θ 1 displaystyle cos theta _ 1 must be positive, which it will be if n is the normal vector that points from the surface toward the side where the light is coming from, the region with index n 1 displaystyle n_ 1 . If cos θ 1 displaystyle cos theta _ 1 is negative, then n points to the side without the light, so start over with n replaced by its negative. v r e f l e c t = l + 2 cos θ 1 n displaystyle mathbf v _ mathrm reflect =mathbf l +2cos theta _ 1 mathbf n This reflected direction vector points back toward the side of the
surface where the light came from.
Now apply
Snell's law
sin θ 2 = ( n 1 n 2 ) sin θ 1 = ( n 1 n 2 ) 1 − ( cos θ 1 ) 2 displaystyle sin theta _ 2 =left( frac n_ 1 n_ 2 right)sin theta _ 1 =left( frac n_ 1 n_ 2 right) sqrt 1left(cos theta _ 1 right)^ 2 cos θ 2 = 1 − ( sin θ 2 ) 2 = 1 − ( n 1 n 2 ) 2 ( 1 − ( cos θ 1 ) 2 ) displaystyle cos theta _ 2 = sqrt 1(sin theta _ 2 )^ 2 = sqrt 1left( frac n_ 1 n_ 2 right)^ 2 left(1left(cos theta _ 1 right)^ 2 right) v r e f r a c t = ( n 1 n 2 ) l + ( n 1 n 2 cos θ 1 − cos θ 2 ) n displaystyle mathbf v _ mathrm refract =left( frac n_ 1 n_ 2 right)mathbf l +left( frac n_ 1 n_ 2 cos theta _ 1 cos theta _ 2 right)mathbf n The formula may appear simpler in terms of renamed simple values r = n 1 / n 2 displaystyle r=n_ 1 /n_ 2 and c = − n ⋅ l displaystyle c=mathbf n cdot mathbf l , avoiding any appearance of trig function names or angle names: v r e f r a c t = r l + ( r c − 1 − r 2 ( 1 − c 2 ) ) n displaystyle mathbf v _ mathrm refract =rmathbf l +left(rc sqrt 1r^ 2 left(1c^ 2 right) right)mathbf n Example: l = 0.707107 , − 0.707107 , n = 0 , 1 , r = n 1 n 2 = 0.9 displaystyle mathbf l = 0.707107,0.707107 ,~mathbf n = 0,1 ,~r= frac n_ 1 n_ 2 =0.9 c = cos θ 1 = 0.707107 , 1 − r 2 ( 1 − c 2 ) = cos θ 2 = 0.771362 displaystyle c=cos theta _ 1 =0.707107,~ sqrt 1r^ 2 left(1c^ 2 right) =cos theta _ 2 =0.771362 v r e f l e c t = 0.707107 , 0.707107 , v r e f r a c t = 0.636396 , − 0.771362 displaystyle mathbf v _ mathrm reflect = 0.707107,0.707107 ,~mathbf v _ mathrm refract = 0.636396,0.771362 The cosine values may be saved and used in the
Fresnel equations
cos θ 2 displaystyle cos theta _ 2 , which can only happen for rays crossing into a lessdense medium ( n 2 < n 1 displaystyle n_ 2 <n_ 1 ).
Total internal reflection
Demonstration of no refraction at angles greater than the critical angle. Main article: Total internal reflection
When light travels from a medium with a higher refractive index to one
with a lower refractive index,
Snell's law
Refraction
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
sin θ 2 = n 1 n 2 sin θ 1 = 1.333 1 ⋅ sin ( 50 ∘ ) = 1.333 ⋅ 0.766 = 1.021 , displaystyle sin theta _ 2 = frac n_ 1 n_ 2 sin theta _ 1 = frac 1.333 1 cdot sin left(50^ circ right)=1.333cdot 0.766=1.021, which is impossible to satisfy. The critical angle θcrit is the value of θ1 for which θ2 equals 90°: θ crit = arcsin ( n 2 n 1 sin θ 2 ) = arcsin n 2 n 1 = 48.6 ∘ . displaystyle theta _ text crit =arcsin left( frac n_ 2 n_ 1 sin theta _ 2 right)=arcsin frac n_ 2 n_ 1 =48.6^ circ . Dispersion[edit] Main article: Dispersion (optics) In many wavepropagation 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 colordependent blurring that sometimes is the resolutionlimiting effect. This was especially true in refracting telescopes, before the invention of achromatic objective lenses. Lossy, absorbing, or conducting media[edit] See also: Mathematical descriptions of opacity In a conducting medium, permittivity and index of refraction are complexvalued. Consequently, so are the angle of refraction and the wavevector. 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.[18] The refracted wave is exponentially attenuated, with exponent proportional to the imaginary component of the index of refraction.[19][20] See also[edit] List of refractive indices
The refractive index vs wavelength of light
Evanescent wave
Reflection (physics)
Snell's window
Calculus of variations
Brachistochrone curve
References[edit] ^ David Michael Harland (2007). "Cassini at Saturn: Huygens results".
p.1. ISBN 038726129X
^ "
Ptolemy
External links[edit] Ibn Sahl and Snell's Law
Discovery of the law of refraction
Snell's Law of
Refraction
