Self-phase modulation
Encyclopedia
Self-phase modulation is a nonlinear optical
effect of light
-matter
interaction.
An ultrashort pulse
of light, when travelling in a medium, will induce a varying refractive index
of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase
shift in the pulse, leading to a change of the pulse's frequency spectrum
.
Self-phase modulation is an important effect in optical
systems that use short, intense pulses of light, such as laser
s and optical fibre communications
systems.
For an ultrashort pulse with a Gaussian
shape and constant phase, the intensity at time t is given by I(t):
where I0 is the peak intensity, and τ is half the pulse duration.
If the pulse is travelling in a medium, the optical Kerr effect produces a refractive index change with intensity:
where n0 is the linear refractive index, and n2 is the second-order nonlinear refractive index of the medium.
As the pulse propagates, the intensity at any one point in the medium rises and then falls as the pulse goes past. This will produce a time-varying refractive index:
This variation in refractive index produces a shift in the instantaneous phase of the pulse:
where
and 
are the carrier frequency and (vacuum) wavelength
of the pulse, and
is the distance the pulse has propagated.
The phase shift results in a frequency shift of the pulse. The instantaneous frequency ω(t) is given by:
and from the equation for dn/dt above, this is:
Plotting ω(t) shows the frequency shift of each part of the pulse. The leading edge shifts to lower frequencies ("redder" wavelengths), trailing edge to higher frequencies ("bluer") and the very peak of the pulse is not shifted. For the centre portion of the pulse (between t = ±τ/2), there is an approximately linear frequency shift (chirp
) given by:
where α is:
It is clear that the extra frequencies generated through SPM broaden the frequency spectrum of the pulse symmetrically. In the time domain, the envelope of the pulse is not changed, however in any real medium the effects of dispersion
will simultaneously act on the pulse . In regions of normal dispersion, the "redder" portions of the pulse have a higher velocity than the "blue" portions, and thus the front of the pulse moves faster than the back, broadening the pulse in time. In regions of anomalous dispersion, the opposite is true, and the pulse is compressed temporally and becomes shorter. This effect can be exploited to some degree (until it digs holes into the spectrum) to produce ultrashort pulse compression.
A similar analysis can be carried out for any pulse shape, such as the hyperbolic secant
-squared (sech2) pulse profile generated by most ultrashort pulse
lasers.
If the pulse is of sufficient intensity, the spectral broadening process of SPM can balance with the temporal compression due to anomalous dispersion and reach an equilibrium state. The resulting pulse is called an optical soliton
.
The nonlinear properties of Kerr nonlinearity has also been beneficial for various optical pulse processing techniques such as optical regeneration or wavelength conversion.
Applications of SPM :
Nonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...
effect of light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
-matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
interaction.
An ultrashort pulse
Ultrashort pulse
In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is of the order of a femtosecond . Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators...
of light, when travelling in a medium, will induce a varying refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
shift in the pulse, leading to a change of the pulse's frequency spectrum
Frequency spectrum
The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.Any signal...
.
Self-phase modulation is an important effect in optical
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
systems that use short, intense pulses of light, such as laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
s and optical fibre communications
Optical fiber
An optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...
systems.
Theory
GAUSSIAN
Gaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70. It has been continuously updated since then...
shape and constant phase, the intensity at time t is given by I(t):
where I0 is the peak intensity, and τ is half the pulse duration.
If the pulse is travelling in a medium, the optical Kerr effect produces a refractive index change with intensity:
where n0 is the linear refractive index, and n2 is the second-order nonlinear refractive index of the medium.
As the pulse propagates, the intensity at any one point in the medium rises and then falls as the pulse goes past. This will produce a time-varying refractive index:
This variation in refractive index produces a shift in the instantaneous phase of the pulse:
where
and are the carrier frequency and (vacuum) wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
of the pulse, and
is the distance the pulse has propagated.The phase shift results in a frequency shift of the pulse. The instantaneous frequency ω(t) is given by:
and from the equation for dn/dt above, this is:
Plotting ω(t) shows the frequency shift of each part of the pulse. The leading edge shifts to lower frequencies ("redder" wavelengths), trailing edge to higher frequencies ("bluer") and the very peak of the pulse is not shifted. For the centre portion of the pulse (between t = ±τ/2), there is an approximately linear frequency shift (chirp
Chirp
A chirp is a signal in which the frequency increases or decreases with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications...
) given by:
where α is:
It is clear that the extra frequencies generated through SPM broaden the frequency spectrum of the pulse symmetrically. In the time domain, the envelope of the pulse is not changed, however in any real medium the effects of dispersion
Dispersion (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...
will simultaneously act on the pulse . In regions of normal dispersion, the "redder" portions of the pulse have a higher velocity than the "blue" portions, and thus the front of the pulse moves faster than the back, broadening the pulse in time. In regions of anomalous dispersion, the opposite is true, and the pulse is compressed temporally and becomes shorter. This effect can be exploited to some degree (until it digs holes into the spectrum) to produce ultrashort pulse compression.
A similar analysis can be carried out for any pulse shape, such as the hyperbolic secant
Hyperbolic function
In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...
-squared (sech2) pulse profile generated by most ultrashort pulse
Ultrashort pulse
In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is of the order of a femtosecond . Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators...
lasers.
If the pulse is of sufficient intensity, the spectral broadening process of SPM can balance with the temporal compression due to anomalous dispersion and reach an equilibrium state. The resulting pulse is called an optical soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
.
Applications of SPM
Self-phase modulation has stimulated many applications in the field of ultrashort pulse including to cite a few :- spectral broadening and supercontinuumSupercontinuumIn optics, a supercontinuum is formed when a collection of nonlinear processes act together upon a pump beam in order to cause severe spectral broadening of the original pump beam. The result is a smooth spectral continuum...
- temporal pulse compression
- spectral pulse compression
The nonlinear properties of Kerr nonlinearity has also been beneficial for various optical pulse processing techniques such as optical regeneration or wavelength conversion.
Mitigation strategies in DWDM systems
In long-haul single-channel and DWDM systems SPM is one of the most important reach limiting nonlinear effects. It can be reduced by:- Lowering the optical power at the expense of increased noise
- Dispersion management, because dispersion can partly mitigate the SPM effect
See also
Other non-linear effects :- Cross-phase modulationCross-phase modulationCross-phase modulation is a nonlinear optical effect where one wavelength of light can affect the phase of another wavelength of light through the optical Kerr effect.- Applications of XPM :...
— XPM - Four wave mixing — FWM
- Modulational instabilityModulational instabilityIn the field of nonlinear optics, modulational instability is a phenomenon whereby deviations from an optical waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.-Initial instability and...
— MI - Stimulated Raman scatteringRaman scatteringRaman scattering or the Raman effect is the inelastic scattering of a photon. It was discovered by Sir Chandrasekhara Venkata Raman and Kariamanickam Srinivasa Krishnan in liquids, and by Grigory Landsberg and Leonid Mandelstam in crystals....
— SRS
Applications of SPM :
- SupercontinuumSupercontinuumIn optics, a supercontinuum is formed when a collection of nonlinear processes act together upon a pump beam in order to cause severe spectral broadening of the original pump beam. The result is a smooth spectral continuum...
- Mamyshev 2R regeneratorMamyshev 2R regeneratorThe Mamyshev 2R regenerator is an all-optical regenerator used in optical communications.In 1998, Pavel V. Mamyshev of Bell Labs proposed and patented the use of the self-phase modulation forsingle channel optical pulse reshaping and re-amplification....