In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one (scalar) polarization component, while vector solitons have two distinct polarization components. Among all the types of solitons, optical vector solitons draw the most attention due to their wide range of applications, particularly in generating ultrafast pulses and light control technology. Optical vector solitons can be classified into temporal vector solitons and spatial vector solitons. During the propagation of both temporal solitons and spatial solitons, despite being in a medium with birefringence, the orthogonal polarizations can copropagate as one unit without splitting due to the strong cross-phase modulation and coherent energy exchange between the two polarizations of the vector soliton which may induce intensity differences between these two polarizations. Thus vector solitons are no longer linearly polarized but rather elliptically polarized.
C.R. Menyuk first derived the nonlinear pulse propagation equation in a single-mode optical fiber (SMF) under weak birefringence. Then, Menyuk described vector solitons as two solitons (more accurately called solitary waves) with orthogonal polarizations which co-propagate together without dispersing their energy and while retaining their shapes. Because of nonlinear interaction among these two polarizations, despite the existence of birefringence between these two polarization modes, they could still adjust their group velocity and be trapped together.[1]
Vector solitons can be spatial or temporal, and are formed by two orthogonally polarized components of a single optical field or two fields of different frequencies but the same polarization.
History
In 1987 Menyuk first derived the nonlinear pulse propagation equation in SMF under weak birefringence. This seminal equation opened up the new field of "scalar" solitons to researchers. His equation concerns the nonlinear interaction (cross-phase modulation and coherent energy exchange) between the two orthogonal polarization components of the vector soliton. Researchers have obtained both analytical and numerical solutions of this equation under weak, moderate and even strong birefringence.
In 1988 Christodoulides and Joseph first theoretically predicted a novel form of phase-locked vector soliton in birefringent dispersive media, which is now known as a high-order phase-locked vector soliton in SMFs. It has two orthogonal polarization components with comparable intensity. Despite of the existence of birefringence, these two polarizations could propagate with the same group velocity as they shift their central frequencies.[2]
In 2000, Cundiff and Akhmediev found that these two polarizations could form not only a so-called group-velocity-locked vector soliton but also a polarization-locked vector soliton. They reported that the intensity ratio of these two polarizations can be about 0.25–1.00.[3]
However, recently, another type of vector soliton, "induced vector soliton" has been observed. Such a vector soliton is novel in that the intensity difference between the two orthogonal polarizations is extremely large (20dB). It seems that weak polarizations are ordinarily unable to form a component of a vector soliton. However, due to the cross-polarization modulation between strong and weak polarization components, a "weak soliton" could also be formed. It thus demonstrates that the soliton obtained is not a "scalar" soliton with a linear polarization mode, but rather a vector soliton with a large ellipticity. This expands the scope of the vector soliton so that the intensity ratio between the strong and weak components of the vector soliton is not limited to 0.25–1.0 but can now extend to 20dB.[4]
Based on the classic work by Christodoulides and Joseph,[5] which concerns a high-order phase-locked vector soliton in SMFs, a stable high-order phase-locked vector soliton has recently been created in a fiber laser. It has the characteristic that not only are the two orthogonally polarized soliton components phase-locked, but also one of the components has a double-humped intensity profile.[6]
The following pictures show that, when the fiber birefringence is taken into consideration, a single nonlinear Schrödinger equation (NLSE) fails to describe the soliton dynamics but instead two coupled NLSEs are required. Then, solitons with two polarization modes can be numerically obtained.
FWM spectral sideband in vector soliton
A new pattern of spectral sidebands was first experimentally observed on the polarization-resolved soliton spectra of the polarization-locked vector solitons of fiber lasers. The new spectral sidebands are characterized by the fact that their positions on the soliton's spectrum vary with the strength of the linear cavity birefringence, and while one polarization component's sideband has a spectral peak, the orthogonal polarization component has a spectral dip, indicating the energy exchange between the two orthogonal polarization components of the vector solitons. Numeric simulations also confirmed that the formation of the new type of spectral sidebands was caused by the FWM between the two polarization components.[7]
Bound vector soliton
Two adjacent vector solitons could form a bound state. Compared with scalar bound solitons, the polarization state of this soliton is more complex. Because of the cross interactions, the bound vector solitons could have much stronger interaction forces than can exist between scalar solitons.[8]
Vector dark soliton
Dark solitons[9] are characterized by being formed from a localized reduction of intensity compared to a more intense continuous wave background. Scalar dark solitons (linearly polarized dark solitons) can be formed in all normal dispersion fiber lasers mode-locked by the nonlinear polarization rotation method and can be rather stable. Vector dark solitons[10] are much less stable due to the cross-interaction between the two polarization components. Therefore, it is interesting to investigate how the polarization state of these two polarization components evolves.
In 2009, the first dark soliton fiber laser has been successfully achieved in an all-normal dispersion erbium-doped fiber laser with a polarizer in cavity. Experimentally finding that apart from the bright pulse emission, under appropriate conditions the fiber laser could also emit single or multiple dark pulses. Based on numerical simulations we interpret the dark pulse formation in the laser as a result of dark soliton shaping.[11]
Vector dark bright soliton
A "bright soliton" is characterized as a localized intensity peak above a continuous wave (CW) background while a dark soliton is featured as a localized intensity dip below a continuous wave (CW) background. "Vector dark bright soliton" means that one polarization state is a bright soliton while the other polarization is a dark soliton.[12] Vector dark bright solitons have been reported in incoherently coupled spatial DBVSs in a self-defocusing medium and matter-wave DBVS in two-species condensates with repulsive scattering interactions,[13][14][15] but never verified in the field of optical fiber.
Induced vector soliton
Using a birefringent cavity fiber laser, an induced vector soliton may be formed due to the cross-coupling between the two orthogonal polarization components. If a strong soliton is formed along one principal polarization axis, then a weak soliton will be induced along the orthogonal polarization axis. The intensity of the weak component in an induced vector soliton may be so weak that by itself it could not form a soliton in the SPM. The characteristics of this type of soliton have been modeled numerically and confirmed by experiment.[16]
Vector dissipative soliton
A vector dissipative soliton could be formed in a laser cavity with net positive dispersion, and its formation mechanism is a natural result of the mutual nonlinear interaction among the normal cavity dispersion, cavity fiber nonlinear Kerr effect, laser gain saturation and gain bandwidth filtering. For a conventional soliton, it is a balance between only the dispersion and nonlinearity. Differing from a conventional soliton, a Vector dissipative soliton is strongly frequency chirped. It is unknown whether or not a phase-locked gain-guided vector soliton could be formed in a fiber laser: either the polarization-rotating or the phase-locked dissipative vector soliton can be formed in a fiber laser with large net normal cavity group velocity dispersion. In addition, multiple vector dissipative solitons with identical soliton parameters and harmonic mode-locking to the conventional dissipative vector soliton can also be formed in a passively mode-locked fiber laser with a SESAM.[17]
Multiwavelength dissipative soliton
Recently, multiwavelength dissipative soliton in an all normal dispersion fiber laser passively mode-locked with a SESAM has been generated. It is found that depending on the cavity birefringence, stable single-, dual- and triple-wavelength dissipative soliton can be formed in the laser. Its generation mechanism can be traced back to the nature of dissipative soliton.[18]
Polarization rotation of vector soliton
In scalar solitons, the output polarization is always linear due to the existence of an in-cavity polarizer. But for vector solitons, the polarization state can be rotating arbitrarily but still locked to the cavity round-trip time or an integer multiple thereof.[19]
Higher-order vector soliton
In higher-order vector solitons, not only are the two orthogonally polarized soliton components phase-locked, but also one of the components has a double-humped intensity profile. Multiple such phase-locked high-order vector solitons with identical soliton parameters and harmonic mode-locking of the vector solitons have also been obtained in lasers. Numerical simulations confirmed the existence of stable high-order vector solitons in fiber lasers.[6]
Optical domain wall soliton
Recently, a phase-locked dark-dark vector soliton was only observed in fiber lasers of positive dispersion, a phase-locked dark-bright vector soliton was obtained in fiber lasers of either positive or negative dispersion. Numerical simulations confirmed the experimental observations, and further showed that the observed vector solitons are the two types of phase-locked polarization domain-wall solitons theoretically predicted.[20]
Vector soliton fiber laser with atomic layer graphene
Except the conventional semiconductor saturable absorber mirrors (SESAMs), which use III–V semiconductor multiple quantum wells grown on distributed Bragg reflectors (DBRs), many researchers have turned their attention onto other materials as saturable absorbers. Especially because there are a number of drawbacks associated with SESAMs. For example, SESAMs require complex and costly clean-room-based fabrication systems such as Metal-Organic Chemical Vapor Deposition (MOCVD) or Molecular Beam Epitaxy (MBE), and an additional substrate removal process is needed in some cases; high-energy heavy-ion implantation is required to introduce defect sites in order to reduce the device recovery time (typically a few nanoseconds) to the picosecond regime required for short-pulse laser mode-locking applications; since the SESAM is a reflective device, its use is restricted to only certain types of linear cavity topologies.
Other laser cavity topologies such as the ring-cavity design, which requires a transmission-mode device, which offers advantages such as doubling the repetition rate for a given cavity length, and which is less sensitive to reflection-induced instability with the use of optical isolators, is not possible unless an optical circulator is employed, which increases cavity loss and laser complexity; SESAMs also suffer from a low optical damage threshold. But there had been no alternative saturable absorbing materials to compete with SESAMs for the passive mode-locking of fiber lasers.
Recently, by the virtue of the saturable absorption properties in single wall carbon nanotubes (SWCNTs) in the near-infrared region with ultrafast saturation recovery times of ~1 picosecond, researchers have successfully produced a new type of effective saturable absorber quite different from SESAMs in structure and fabrication, and has, in fact, led to the demonstration of pico- or subpicosecond erbium-doped fiber (EDF) lasers. In these lasers, solid SWCNT saturable absorbers have been formed by direct deposition of SWCNT films onto flat glass substrates, mirror substrates, or end facets of optical fibers. However, the non-uniform chiral properties of SWNTs present inherent problems for precise control of the properties of the saturable absorber. Furthermore, the presence of bundled and entangled SWNTs, catalyst particles, and the formation of bubbles cause high nonsaturable losses in the cavity, despite the fact that the polymer host can circumvent some of these problems to some extent and afford ease of device integration. In addition, under large energy ultrashort pulses multi-photon effect induced oxidation occurs, which degrades the long term stability of the absorber.
Graphene is a single two-dimensional (2D) atomic layer of carbon atom arranged in a hexagonal lattice. Although as an isolated film it is a zero bandgap semiconductor, it is found that like the SWCNTs, graphene also possesses saturable absorption. In particular, as it has no bandgap, its saturable absorption is wavelength independent. It is potentially possible to use graphene or graphene-polymer composite to make a wideband saturable absorber for laser mode locking. Furthermore, comparing with the SWCNTs, as graphene has a 2D structure it should have much smaller non-saturable loss and much higher damage threshold. Indeed, with an erbium-doped fiber laser we self-started mode locking and stable soliton pulse emission with high energy have been achieved.
Due to the perfect isotropic absorption properties of graphene, the generated solitons could be regarded as vector solitons. How the evolution of vector soliton under the interaction of graphene was still unclear but interesting, particularly because it involved the mutual interaction of nonlinear optical wave with the atoms.,[21][22][23] which was highlighted in Nature Asia Materials[24] and nanowerk.[25]
Furthermore, atomic layer graphene possesses wavelength-insensitive ultrafast saturable absorption, which can be exploited as a "full-band" mode locker. With an erbium-doped dissipative soliton fiber laser mode locked with few layer graphene, it has been experimentally shown that dissipative solitons with continuous wavelength tuning as large as 30nm (1570nm-1600nm) can be obtained.[26]
Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (when the electric field of the light is >108 V/m and thus comparable to the atomic electric field of ~1011 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.
An optical amplifier is a device that amplifies an optical signal directly, without the need to first convert it to an electrical signal. An optical amplifier may be thought of as a laser without an optical cavity, or one in which feedback from the cavity is suppressed. Optical amplifiers are important in optical communication and laser physics. They are used as optical repeaters in the long distance fiber-optic cables which carry much of the world's telecommunication links.
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is strongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a balanced cancellation of nonlinear and dispersive effects in the medium. Solitons were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersive partial differential equations describing physical systems.
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent. The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress.
Mode locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds (10−12 s) or femtoseconds (10−15 s). A laser operated in this way is sometimes referred to as a femtosecond laser, for example, in modern refractive surgery. The basis of the technique is to induce a fixed phase relationship between the longitudinal modes of the laser's resonant cavity. Constructive interference between these modes can cause the laser light to be produced as a train of pulses. The laser is then said to be "phase-locked" or "mode-locked".
Kerr-lens mode-locking (KLM) is a method of mode-locking lasers via the nonlinear optical Kerr effect. This method allows the generation of pulses of light with a duration as short as a few femtoseconds.
Saturable absorption is a property of materials where the absorption of light decreases with increasing light intensity. Most materials show some saturable absorption, but often only at very high optical intensities. At sufficiently high incident light intensity, the ground state of a saturable absorber material is excited into an upper energy state at such a rate that there is insufficient time for it to decay back to the ground state before the ground state becomes depleted, causing the absorption to saturate. The key parameters for a saturable absorber are its wavelength range, its dynamic response, and its saturation intensity and fluence.
A solid-state laser is a laser that uses a gain medium that is a solid, rather than a liquid as in dye lasers or a gas as in gas lasers. Semiconductor-based lasers are also in the solid state, but are generally considered as a separate class from solid-state lasers, called laser diodes.
A fiber laser is a laser in which the active gain medium is an optical fiber doped with rare-earth elements such as erbium, ytterbium, neodymium, dysprosium, praseodymium, thulium and holmium. They are related to doped fiber amplifiers, which provide light amplification without lasing.
A frequency comb or spectral comb is a spectrum made of discrete and regularly spaced spectral lines. In optics, a frequency comb can be generated by certain laser sources.
In 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, for example using a microstructured optical fiber. The result is a smooth spectral continuum. There is no consensus on how much broadening constitutes a supercontinuum; however researchers have published work claiming as little as 60 nm of broadening as a supercontinuum. There is also no agreement on the spectral flatness required to define the bandwidth of the source, with authors using anything from 5 dB to 40 dB or more. In addition the term supercontinuum itself did not gain widespread acceptance until this century, with many authors using alternative phrases to describe their continua during the 1970s, 1980s and 1990s.
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and dispersive effects in the medium. There are two main kinds of solitons:
Dissipative solitons (DSs) are stable solitary localized structures that arise in nonlinear spatially extended dissipative systems due to mechanisms of self-organization. They can be considered as an extension of the classical soliton concept in conservative systems. An alternative terminology includes autosolitons, spots and pulses.
In physics, a pulse is a generic term describing a single disturbance that moves through a transmission medium. This medium may be vacuum or matter, and may be indefinitely large or finite.
A domain wall is a term used in physics which can have similar meanings in optics, magnetism, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.
Optical rogue waves are rare pulses of light analogous to rogue or freak ocean waves. The term optical rogue waves was coined to describe rare pulses of broadband light arising during the process of supercontinuum generation—a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input waveform—in nonlinear optical fiber. In this context, optical rogue waves are characterized by an anomalous surplus in energy at particular wavelengths or an unexpected peak power. These anomalous events have been shown to follow heavy-tailed statistics, also known as L-shaped statistics, fat-tailed statistics, or extreme-value statistics. These probability distributions are characterized by long tails: large outliers occur rarely, yet much more frequently than expected from Gaussian statistics and intuition. Such distributions also describe the probabilities of freak ocean waves and various phenomena in both the man-made and natural worlds. Despite their infrequency, rare events wield significant influence in many systems. Aside from the statistical similarities, light waves traveling in optical fibers are known to obey the similar mathematics as water waves traveling in the open ocean, supporting the analogy between oceanic rogue waves and their optical counterparts. More generally, research has exposed a number of different analogies between extreme events in optics and hydrodynamic systems. A key practical difference is that most optical experiments can be done with a table-top apparatus, offer a high degree of experimental control, and allow data to be acquired extremely rapidly. Consequently, optical rogue waves are attractive for experimental and theoretical research and have become a highly studied phenomenon. The particulars of the analogy between extreme waves in optics and hydrodynamics may vary depending on the context, but the existence of rare events and extreme statistics in wave-related phenomena are common ground.
Kerr frequency combs are optical frequency combs which are generated from a continuous wave pump laser by the Kerr nonlinearity. This coherent conversion of the pump laser to a frequency comb takes place inside an optical resonator which is typically of micrometer to millimeter in size and is therefore termed a microresonator. The coherent generation of the frequency comb from a continuous wave laser with the optical nonlinearity as a gain sets Kerr frequency combs apart from today's most common optical frequency combs. These frequency combs are generated by mode-locked lasers where the dominating gain stems from a conventional laser gain medium, which is pumped incoherently. Because Kerr frequency combs only rely on the nonlinear properties of the medium inside the microresonator and do not require a broadband laser gain medium, broad Kerr frequency combs can in principle be generated around any pump frequency.
The numerical models of lasers and the most of nonlinear optical systems stem from Maxwell–Bloch equations (MBE). This full set of Partial Differential Equations includes Maxwell equations for electromagnetic field and semiclassical equations of the two-level atoms. For this reason the simplified theoretical approaches were developed for numerical simulation of laser beams formation and their propagation since the early years of laser era. The Slowly varying envelope approximation of MBE follows from the standard nonlinear wave equation with nonlinear polarization as a source:
Semiconductor saturable-absorber mirrors (SESAMs) are a type of saturable absorber used in mode locking lasers.
Baruch Fischer is an Israeli optical physicist and Professor Emeritus in the Andrew and Erna Viterbi Faculty of Electrical and Computer Engineering of the Technion, where he was the Max Knoll Chair in Electro-Optics and Electronics.
References
↑ C.R. Menyuk, Optics Letters, 12, 614 (1987); J. Opt. Soc. Am. B 5, 392(1988); "Nonlinear Pulse-Propagation in Birefringent Optical Fibers", IEEE J. Quantum Electron. QE-23, 174–176 (1987).
↑ D.N. Christodoulides and R.I. Joseph, Opt. Lett., 13, 53(1988).
↑ S.T. Cundiff et al., Phys. Rev. Lett., 82, 3988(1999); N.N. Akhmediev et al., Opt. Lett., 23, 852(1998); B.C. Collings et al., J. Opt. Soc. Am, B 17, 354(2000).
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