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Light Group Velocity Dispersion LGVD.com
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Light GVD: Light group velocity dispersion (GVD), which causes a short pulse of
light to spread in time as a result of different frequency components of the
pulse travelling at different velocities. GVD is often quantified as the group
delay dispersion parameter
In optics, dispersion is the
phenomenon in which the phase velocity of a wave depends on its frequency. Media
having such a property are termed dispersive media.
The most familiar example of dispersion is probably a rainbow, in which
dispersion causes the spatial separation of a white light into components of
different wavelengths (different colors). However, dispersion also has an effect
in many other circumstances: for example, it causes pulses to spread in optical
fibers, degrading signals over long distances; also, a cancellation between
dispersion and nonlinear effects leads to soliton waves. Dispersion is most
often described for light waves, but it may occur for any kind of wave that
interacts with a medium or passes through an inhomogeneous geometry (e.g. a
waveguide), such as sound waves. Dispersion is sometimes called chromatic
dispersion to emphasize its wavelength-dependent nature.
There are generally two sources of dispersion: material dispersion and waveguide
dispersion. Material dispersion comes from a frequency-dependent response of a
material to waves. For example, material dispersion leads to undesired chromatic
aberration in a lens or the separation of colors in a prism. Waveguide
dispersion occurs when the speed of a wave in a waveguide (such as an optical
fiber) depends on its frequency for geometric reasons, independent of any
frequency dependence of the materials from which it is constructed. More
generally, "waveguide" dispersion can occur for waves propagating through any
inhomogeneous structure (e.g. a photonic crystal), whether or not the waves are
confined to some region. In general, both types of dispersion may be present,
although they are not strictly additive. Their combination leads to signal
degradation in optical fibers for telecommunications, because the varying delay
in arrival time between different components of a signal "smears out" the signal
in time.
Material dispersion in optics
The variation of refractive index vs. wavelength for various glasses. The
wavelengths of visible light are shaded in red.
The variation of refractive index vs. wavelength for various glasses. The
wavelengths of visible light are shaded in red.
Influences of selected glass component additions on the mean dispersion of a
specific base glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm
(red))
Influences of selected glass component additions on the mean dispersion of a
specific base glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm
(red))
Material dispersion can be a desirable or undesirable effect in optical
applications. The dispersion of light by glass prisms is used to construct
spectrometers and spectroradiometers. Holographic gratings are also used, as
they allow more accurate discrimination of wavelengths. However, in lenses,
dispersion causes chromatic aberration, an undesired effect that may degrade
images in microscopes, telescopes and photographic objectives.
The phase velocity, v, of a wave in a given uniform medium is given by
v = \frac{c}{n}
where c is the speed of light in a vacuum and n is the refractive index of the
medium.
In general, the refractive index is some function of the frequency f of the
light, thus n = n(f), or alternately, with respect to the wave's wavelength n =
n(λ). The wavelength dependency of a material's refractive index is usually
quantified by an empirical formula, the Cauchy or Sellmeier equations.
The most commonly seen consequence of dispersion in optics is the separation of
white light into a color spectrum by a prism. From Snell's law it can be seen
that the angle of refraction of light in a prism depends on the refractive index
of the prism material. Since that refractive index varies with wavelength, it
follows that the angle that the light is refracted will also vary with
wavelength, causing an angular separation of the colors known as angular
dispersion.
For visible light, most transparent materials (e.g. glasses) have:
1 < n(\lambda_{\rm red}) < n(\lambda_{\rm yellow}) < n(\lambda_{\rm blue})\ ,
or alternatively:
\frac{{\rm d}n}{{\rm d}\lambda} < 0,
that is, refractive index n decreases with increasing wavelength λ. In this
case, the medium is said to have normal dispersion. Whereas, if the index
increases with increasing wavelength the medium has anomalous dispersion.
At the interface of such a material with air or vacuum (index of ~1), Snell's
law predicts that light incident at an angle θ to the normal will be refracted
at an angle arcsin( sin (θ) / n) . Thus, blue light, with a higher refractive
index, will be bent more strongly than red light, resulting in the well-known
rainbow pattern.
Group and phase velocity
Another consequence of dispersion manifests itself as a temporal effect. The
formula above, v = c / n calculates the phase velocity of a wave; this is the
velocity at which the phase of any one frequency component of the wave will
propagate. This is not the same as the group velocity of the wave, which is the
rate that changes in amplitude (known as the envelope of the wave) will
propagate. The group velocity vg is related to the phase velocity by, for a
homogeneous medium (here λ is the wavelength in vacuum, not in the medium):
v_g = c \left( n - \lambda \frac{dn}{d\lambda} \right)^{-1}.
The group velocity vg is often thought of as the velocity at which energy or
information is conveyed along the wave. In most cases this is true, and the
group velocity can be thought of as the signal velocity of the waveform. In some
unusual circumstances, where the wavelength of the light is close to an
absorption resonance of the medium, it is possible for the group velocity to
exceed the speed of light (vg > c), leading to the conclusion that superluminal
(faster than light) communication is possible. In practice, in such situations
the distortion and absorption of the wave is such that the value of the group
velocity essentially becomes meaningless, and does not represent the true signal
velocity of the wave, which stays less than c.
The group velocity itself is usually a function of the wave's frequency. This
results in group velocity dispersion (GVD), which causes a short pulse of light
to spread in time as a result of different frequency components of the pulse
travelling at different velocities. GVD is often quantified as the group delay
dispersion parameter (again, this formula is for a uniform medium only):
D = - \frac{\lambda}{c} \, \frac{d^2 n}{d \lambda^2}.
If D is less than zero, the medium is said to have positive dispersion. If D is
greater than zero, the medium has negative dispersion.If a light pulse is
propagated through a normally dispersive medium, the result is the higher
frequency components travel slower than the lower frequency components. The
pulse therefore becomes positively chirped, or up-chirped, increasing in
frequency with time. Conversely, if a pulse travels through an anomalously
dispersive medium, high frequency components travel faster than the lower ones,
and the pulse becomes negatively chirped, or down-chirped, decreasing in
frequency with time.
The result of GVD, whether negative or positive, is ultimately temporal
spreading of the pulse. This makes dispersion management extremely important in
optical communications systems based on optical fiber, since if dispersion is
too high, a group of pulses representing a bit-stream will spread in time and
merge together, rendering the bit-stream unintelligible. This limits the length
of fiber that a signal can be sent down without regeneration. One possible
answer to this problem is to send signals down the optical fibre at a wavelength
where the GVD is zero (e.g. around ~1.3-1.5 μm in silica fibres), so pulses at
this wavelength suffer minimal spreading from dispersion—in practice, however,
this approach causes more problems than it solves because zero GVD unacceptably
amplifies other nonlinear effects (such as four wave mixing). Another possible
option is to use soliton pulses in the regime of anomalous dispersion, a form of
optical pulse which uses a nonlinear optical effect to self-maintain its
shape—solitons have the practical problem, however, that they require a certain
power level to be maintained in the pulse for the nonlinear effect to be of the
correct strength. Instead, the solution that is currently used in practice is to
perform dispersion compensation, typically by matching the fiber with another
fiber of opposite-sign dispersion so that the dispersion effects cancel; such
compensation is ultimately limited by nonlinear effects such as self-phase
modulation, which interact with dispersion to make it very difficult to undo.
Dispersion control is also important in lasers that produce short pulses. The
overall dispersion of the optical resonator is a major factor in determining the
duration of the pulses emitted by the laser. A pair of prisms can be arranged to
produce net negative dispersion, which can be used to balance the usually
positive dispersion of the laser medium. Diffraction gratings can also be used
to produce dispersive effects; these are often used in high-power laser
amplifier systems. Recently, an alternative to prisms and gratings has been
developed: chirped mirrors. These dielectric mirrors are coated so that
different wavelengths have different penetration lengths, and therefore
different group delays. The coating layers can be tailored to achieve a net
negative dispersion.
Dispersion in waveguides
Optical fibers, which are used in telecommunications, are among the most
abundant types of waveguides. Dispersion in these fibers is one of the limiting
factors that determine how much data can be transported on a single fiber.
The transverse modes for waves confined laterally within a waveguide generally
have different speeds (and field patterns) depending upon their frequency (that
is, on the relative size of the wave, the wavelength) compared to the size of
the waveguide.
In general, for a waveguide mode with an angular frequency ω(β) at a propagation
constant β (so that the electromagnetic fields in the propagation direction z
oscillate proportional to ei(βz ? ωt)), the group-velocity dispersion parameter
D is defined as:
D = -\frac{2\pi c}{\lambda^2} \frac{d^2 \beta}{d\omega^2} = \frac{2\pi c}{v_g^2
\lambda^2} \frac{dv_g}{d\omega}
where λ = 2πc / ω is the vacuum wavelength and vg = dω / dβ is the group
velocity. This formula generalizes the one in the previous section for
homogeneous media, and includes both waveguide dispersion and material
dispersion. The reason for defining the dispersion in this way is that |D| is
the (asymptotic) temporal pulse spreading Δt per unit bandwidth Δλ per unit
distance travelled, commonly reported in ps / nm km for optical fibers.
A similar effect due to a somewhat different phenomenon is modal dispersion,
caused by a waveguide having multiple modes at a given frequency, each with a
different speed. A special case of this is polarization mode dispersion (PMD),
which comes from a superposition of two modes that travel at different speeds
due to random imperfections that break the symmetry of the waveguide.
Dispersion in gemology
In the technical terminology of gemology, dispersion is the difference in the
refractive index of a material at the B and G Fraunhofer wavelengths of 686.7 nm
and 430.8 nm and is meant to express the degree to which a prism cut from the
gemstone shows "fire", or color. Dispersion is a material property. Fire depends
on the dispersion, the cut angles, the lighting environment, the refractive
index, and the viewer.
Dispersion in imaging
In photographic and microscopic lenses, dispersion causes chromatic aberration,
distorting the image, and various techniques have been developed to counteract
it.
Dispersion in pulsar timing
Pulsars are spinning neutron stars that emit pulses at very regular intervals
ranging from milliseconds to seconds. It is believed that the pulses are emitted
simultaneously over a wide range of frequencies. However, as observed on Earth,
the components of each pulse emitted at higher radio frequencies arrive before
those emitted at lower frequencies. This dispersion occurs because of the
ionised component of the interstellar medium, which makes the group velocity
frequency dependent. The extra delay added at frequency ν is
D = 4.15 ms (\frac{\nu}{GHz})^{-2} \times (\frac{DM}{cm^{-3} pc})
where the dispersion measure DM is
DM = \int_0^d{n_e\;dl}
is the integrated free electron column density ne out to the pulsar at a
distance d .
Of course, this delay cannot be measured directly, since the emission time is
unknown. What can be measured is the difference in arrival times at two
different frequencies. The delay ΔT between a high frequency νhi and a low
frequency νlo component of a pulse will be
\Delta T = 4.15 ms [(\frac{\nu_{lo}}{GHz})^{-2} - (\frac{\nu_{hi}}{GHz})^{-2} ]
\times (\frac{DM}{cm^{-3} pc})
and so DM is normally computed from measurements at two different frequencies.
This allows computation of the absolute delay at any frequency, which is used
when combining many different pulsar observations into an integrated timing
solution.
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