The
refractive index (or
index of refraction) of a medium is a measure of
how much the speed of light (or other waves such as sound waves) is
reduced inside the medium. For example, typical
soda-lime glass has a refractive index close
to 1.5, which means that in glass, light travels at
1 / 1.5 = 2/3 the speed of light in a vacuum.
Two common properties of glass and other transparent materials are
directly related to their refractive index. First, light rays
change direction when they cross the interface from air to the
material, an effect that is used in
lenses. Second, light reflects partially from
surfaces that have a refractive index different from that of their
surroundings.
Definitions
The refractive index,
n, of a medium is defined as the
ratio of the velocity,
c, of a
wave
phenomenon such as
light or
sound in a reference medium to the
phase velocity,
vp in the
medium itself:
- n = \frac{c}{v_{\mathrm {p}}}.
It is most commonly used in the context of
light with
vacuum as a reference
medium, although historically other reference media (e.g.
air at a standardized
pressure
and
temperature) have been common. It is
usually given the symbol
n. In the case of light, it
equals
- n=\sqrt{\epsilon_r\mu_r},
where
εr is the material's relative
permittivity, and
μr is its
relative
permeability. For most
materials,
μr is very close to 1 at optical
frequencies, therefore
n is approximately
\sqrt{\epsilon_r}. Contrary to a widespread misconception,
n may be less than 1, for example for
X-rays. This has practical technical applications,
such as effective mirrors for X-rays based on
total external reflection. Another
example is that the
n of electromagnetic waves in plasmas
is less than 1.
The
phase velocity is defined as the
rate at which the crests of the
waveform
propagate; that is, the rate at which the
phase of the waveform is moving. The
group velocity is the rate
at which the
envelope of the waveform is propagating; that
is, the rate of variation of the
amplitude
of the waveform. Provided the waveform is not distorted
significantly during propagation, it is the group velocity that
represents the rate at which information (and energy) may be
transmitted by the wave, for example the velocity at which a pulse
of light travels down an
optical
fiber.
A closely related quantity is
refractivity, which in
atmospheric applications is denoted
N and defined as
N
= 106(n - 1) the 10
6 factor is used
because for air,
n deviates from unity at most a few parts
per thousand.
Speed of light
The speed of all electromagnetic radiation in vacuum is the same,
approximately 3×10
8 m/s, and is denoted by
c.Therefore, if
v is the
phase velocity of radiation of a
specific frequency in a specific material, the refractive index is
given by
- n =\frac{c}{v}
or inversely
- v =\frac{c}{n}.
This number is typically greater than one: the higher the index of
the material, the more the light is slowed down (see also
Cherenkov radiation). However, at
certain frequencies (e.g. near
absorption resonances, and for
X-rays),
n will actually be smaller than one.
This does not contradict the
theory
of relativity, which holds that no
information-carrying signal
can ever propagate faster than
c, because the
phase velocity is not the same as the
group velocity or the
signal velocity.
Sometimes, a "group velocity refractive index", usually called the
group index is defined:
- n_g=\frac{c}{v_g}
where
vg is the group velocity. This value
should not be confused with
n, which is always defined
with respect to the phase velocity. The group index can be written
in terms of the wavelength dependence of the refractive index as
- n_g = n - \lambda\frac{dn}{d\lambda},
where \lambda is the wavelength in vacuum.At the microscale, an
electromagnetic wave's phase velocity is slowed in a material
because the
electric field creates a
disturbance in the charges of each atom (primarily the
electrons) proportional to the
permittivity of the medium. The charges will,
in general, oscillate slightly out of
phase with respect to the driving electric
field. The charges thus radiate their own electromagnetic wave that
is at the same frequency but with a phase delay. The macroscopic
sum of all such contributions in the material is a wave with the
same frequency but shorter wavelength than the original, leading to
a slowing of the wave's phase velocity. Most of the radiation from
oscillating material charges will modify the incoming wave,
changing its velocity. However, some net energy will be radiated in
other directions (see
scattering).
If the refractive indices of two materials are known for a given
frequency, then one can compute the angle by which radiation of
that frequency will be
refracted as it
moves from the first into the second material from
Snell's law.
If in a given region the values of refractive indices
n or
ng were found to differ from unity (whether
homogeneously, or isotropically, or not), then this region was
distinct from vacuum in the above sense for lacking
Poincaré symmetry.
Negative refractive index
Recent research has also demonstrated the existence of
negative refractive index which
can occur if the real parts of both \epsilon_r and \mu_r are
simultaneously negative, although such is a necessary but
not sufficient condition. Not thought to occur naturally, this can
be achieved with so-called
metamaterials and the resulting
negative refraction (i.e. a reversal of
Snell's law) offers the possibility of
perfect lenses and other exotic
phenomena.
Dispersion and absorption
The variation of refractive index with wavelength for various
glasses.
In real materials, the
polarization does not respond
instantaneously to an applied field. This causes
dielectric loss, which can be expressed by a
permittivity that is both
complex and
frequency dependent. Real materials are not
perfect
insulator either, i.e.
they have non-zero
direct current
conductivity. Taking both
aspects into consideration, we can define a complex index of
refraction:
- \tilde{n}=n+i\kappa.
Here,
n is the refractive index indicating the phase
velocity as above, while
κ is called the
extinction coefficient, which
indicates the amount of
absorption loss when the electromagnetic
wave propagates through the material. (See
Mathematical descriptions
of opacity.) Both
n and
κ are dependent on
the frequency (
wavelength). Note that the
sign of the complex part is a matter of convention, which is
important due to possible confusion between loss and gain. The
notation above, which is usually used by physicists, corresponds to
waves with time evolution given by e^{-i\omega t}.
The effect that
n varies with
frequency (except in vacuum, where all frequencies
travel at the same speed,
c) is known as
dispersion, and it is what causes a
prism to divide white
light into its constituent spectral
colors,
explains
rainbows, and is the cause of
chromatic aberration in
lenses. In regions of the spectrum
where the material does not absorb, the real part of the refractive
index tends to increase with frequency. Near absorption peaks, the
curve of the refractive index is a complex form given by the
Kramers–Kronig
relations, and can decrease with frequency.
Since the refractive index of a material varies with the frequency
(and thus wavelength) of light, it is usual to specify the
corresponding vacuum wavelength at which the refractive index is
measured. Typically, this is done at various well-defined spectral
emission lines; for example,
nD is the refractive index at the
Fraunhofer "D" line, the centre of the
yellow
sodium double emission at 589.29
nm wavelength.
The
Sellmeier equation is an
empirical formula that works well in describing dispersion, and
Sellmeier coefficients are often quoted instead of the refractive
index in tables. For some representative refractive indices at
different wavelengths, see
list of indices of
refraction.
As shown above, dielectric loss and non-zero DC conductivity in
materials cause absorption. Good dielectric materials such as glass
have extremely low DC conductivity, and at low frequencies the
dielectric loss is also negligible, resulting in almost no
absorption (κ ≈ 0). However, at higher frequencies (such as visible
light), dielectric loss may increase absorption significantly,
reducing the material's
transparency to these
frequencies.
The real and imaginary parts of the complex refractive index are
related through use of the
Kramers–Kronig relations.
For example, one can determine a material's full complex refractive
index as a function of wavelength from an absorption spectrum of
the material.
Relation to dielectric constant
The
dielectric constant (which
is often dependent on wavelength) is simply the square of the
(complex) refractive index in a non-magnetic medium (one with a
relative
permeability of unity). The
refractive index is used for optics in
Fresnel equations and
Snell's law; while the dielectric constant is
used in
Maxwell's equations and
electronics.
Where \tilde\epsilon, \ \epsilon_1, \ \epsilon_2, n, and \ \kappa
are functions of wavelength:
- \tilde\epsilon=\epsilon_1+i\epsilon_2= (n+i\kappa)^2.
Conversion between refractive index and dielectric constant is done
by:
- \ \epsilon_1= n^2 - \kappa^2
- \ \epsilon_2 = 2n\kappa
- \ n =
\sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}}
- \kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}-
\epsilon_1}{2}}.
Anisotropy
The refractive index of certain media may be different depending on
the
polarization and direction of
propagation of the light through the medium. This is known as
birefringence or anisotropy and is
described by the field of
crystal
optics. In the most general case, the
dielectric constant is a rank-2
tensor (a 3 by 3 matrix), which cannot simply
be described by refractive indices except for polarizations along
principal axes.
In magneto-optic (gyro-magnetic) and
optically active materials, the principal
axes are complex (corresponding to elliptical polarizations), and
the dielectric tensor is complex-
Hermitian
(for lossless media); such materials break time-reversal symmetry
and are used e.g. to construct
Faraday
isolators.
In crystalline calcium carbonate (calcite), the birefringent
(uniaxial) optics depend on directional differences in the
structure. The index of refraction also depends on composition, and
can be calculated using the
Gladstone-Dale relation.
Nonlinearity
The strong
electric field of high
intensity light (such as output of a
laser)
may cause a medium's refractive index to vary as the light passes
through it, giving rise to
nonlinear
optics. If the index varies quadratically with the field
(linearly with the intensity), it is called the
optical Kerr effect and causes phenomena such as
self-focusing and
self-phase modulation. If the index
varies linearly with the field (which is only possible in materials
that do not possess
inversion
symmetry), it is known as the
Pockels
effect.
Inhomogeneity
A gradient-index lens with a parabolic
variation of refractive index (
n) with radial distance
(
x).
The lens focuses light in the same way as a conventional
lens.
If the refractive index of a medium is not constant, but varies
gradually with position, the material is known as a gradient-index
medium and is described by
gradient index optics. Light traveling
through such a medium can be bent or focused, and this effect can
be exploited to produce
lenses, some
optical fibers and other devices. Some
common
mirages are caused by a
spatially-varying refractive index of
air.
Relation to density
Relation between the refractive index
and the density of silicate and borosilicate glasses.
In general, the refractive index of a glass increases with its
density. However, there does not exist an overall linear relation
between the refractive index and the density for all silicate and
borosilicate glasses. A relatively high refractive index and low
density can be obtained with glasses containing light metal oxides
such as
Li2O and
MgO, while the opposite trend is observed
with glasses containing
PbO and
BaO as seen in the diagram at the
right.
Momentum paradox
In 1908,
Hermann Minkowski
calculated the momentum of a refracted ray,
p, where
E is energy of the photon,
c is the speed of
light in vacuum and
n is the refractive index of the
medium as follows:
- p=\frac{nE}{c} .
In 1909,
Max Abraham proposed the
following formula for this calculation:
- p=\frac{E}{nc}.
Rudolf Peierls raises this
inconsistency in
More Surprises in Theoretical Physics.
Ulf
Leonhardt, Chair in Theoretical Physics at the University of St
Andrews
, has discussed this problem, including experiments
to resolve it.
Applications
The refractive index of a material is the most important property
of any
optical system that uses
refraction. It is used to calculate the focusing
power of lenses, and the dispersive power of prisms.
Since refractive index is a fundamental physical property of a
substance, it is often used to identify a particular substance,
confirm its purity, or measure its concentration. Refractive index
is used to measure solids (glasses and gemstones), liquids, and
gases. Most commonly it is used to measure the concentration of a
solute in an
aqueous solution. A
refractometer is the instrument used to
measure refractive index. For a solution of sugar, the refractive
index can be used to determine the sugar content (see
Brix).
In
GPS, the index of
refraction is utilized in
ray-tracing to account
for the
radio propagation delay
due to the Earth's electrically neutral atmosphere.
See also
References
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