Snell's law (also known as Snell–Descartes
law 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.
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 metamaterials, which allow light to be
bent "backward" at a negative angle of refraction
with a negative refractive index.
Snell's law states that the ratio of the sines
of the angles of incidence and refraction
is equivalent to the ratio of phase velocities
in the two media, or equivalent to the reciprocal
of the ratio of the indices of refraction:
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 of
least time, which in turn follows from the
propagation of light as waves.
== History ==
Ptolemy, in Alexandria, Egypt, 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 fudging his data to fit theory (see:
confirmation bias). Alhazen, in his Book of
Optics (1021), came closer to discovering
the law of refraction, though he did not take
this step.
The law eventually named after Snell was first
accurately described by the Persian scientist
Ibn Sahl at the Baghdad court in 984. In the
manuscript On Burning Mirrors and Lenses,
Sahl used the law to derive lens shapes that
focus light with no geometric aberrations.The
law was rediscovered by Thomas Harriot in
1602, 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unpublished during his lifetime.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 Dioptrics, 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. Fermat's
derivation also utilized his invention of
adequality, a mathematical procedure equivalent
to differential calculus, for finding maxima,
minima, and tangents.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.According to Dijksterhuis, "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."
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. In 2008 and 2011,
plasmonic metasurfaces were also demonstrated
to change the reflection and refraction directions
of light beam.
== Explanation ==
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
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 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,
λ
1
{\displaystyle \lambda _{1}}
and
λ
2
{\displaystyle \lambda _{2}}
:
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 ==
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 (though the result does
not show light taking the least time path,
but rather one that is stationary with respect
to small variations as there are cases where
light actually takes the greatest time path,
as 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.
As shown in the figure to 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}+(l-x)^{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 {-(l-x)}{v_{2}{\sqrt
{(l-x)^{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 {l-x}{\sqrt {(l-x)^{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}}
=== 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.
=== Derivation from conservation of energy
and momentum ===
Yet another way to derive Snell's law is based
on translation symmetry considerations. For
example, a homogeneous surface perpendicular
to the z direction cannot change the transverse
momentum. Since the propagation vector
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 immediately.
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 {2\pi }{\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 ===
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:
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} +2\cos \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 to the ratio of sines
to derive the formula for the refracted ray's
direction vector:
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 {1-\left(\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 {1-\left({\frac
{n_{1}}{n_{2}}}\right)^{2}\left(1-\left(\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}
}=r\mathbf {l} +\left(rc-{\sqrt {1-r^{2}\left(1-c^{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
{1-r^{2}\left(1-c^{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 for working out the
intensity of the resulting rays.
Total internal reflection is indicated by
a negative radicand in the equation for
cos
⁡
θ
2
{\displaystyle \cos \theta _{2}}
, which can only happen for rays crossing
into a less-dense medium (
n
2
<
n
1
{\displaystyle n_{2}<n_{1}}
).
== Total internal reflection and 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.
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
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)=1.333\cdot 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 ==
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. The
refracted wave is exponentially attenuated,
with exponent proportional to the imaginary
component of the index of refraction.
== See also ==
List of refractive indices
The refractive index vs wavelength of light
Evanescent wave
Reflection (physics)
Snell's window
Calculus of variations
Brachistochrone curve for a simple proof by
Jacob Bernoulli
Hamiltonian optics
Computation of radiowave attenuation in the
atmosphere
Shore line effect
N-slit interferometric equation
