Solitons in Optical Fibers: Fundamentals and Applications
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Since its discovery, optical solitons have had widespread impact in optics, in particular to telecommunications and ultrafast science. Dianov et al.
In , Mitschke and Mollenauer reported a increasing redshift of the center frequency of a subpicosecond soliton pulse with increasing power in standard single-mode, polarization maintaining PM fiber [ 6 ]. They named the phenomenon soliton self-frequency shift SSFS. Due to Raman gain, the blue portion of the soliton spectrum pumps the red portion of the spectrum, causing a continuous redshift in the soliton spectrum.
Handbook of Optical Fibers
This wavelength shift was observed to increase with both input power and fiber length. The mathematical basis of SSFS is described in [ 7 ]. SSFS is not significant for telecom-scale at least tens or hundreds of picoseconds long pulses. The long pulsewidth leads to only a fraction of gigahertz shift over tens of thousands of kilometers.
The delay between the discovery of the optical soliton and the observation of SSFS was primarily because of the lack of a reliable subpicosecond high-power soliton source. These fundamental solitons would then individually shift in frequency. Beaud et al. These Stokes pulses underwent frequency shift, resulting in an output spectrum with spectrally separated soliton peaks.
The remaining energy not converted to a soliton was dissipated in a dispersive wave at the source wavelength. The region of anomalous dispersion for silica limits the SSFS in standard step-index silica-based fiber to wavelengths longer than1. Microstructured optical fibers and other new fiber designs, on the other hand, have nonconventional mode propagation characteristics [ 17 ], [ 18 ], which give rise to fiber dispersion profiles previously unattainable by step-index silica-based fibers. These novel fiber structures have extended the accessible region down to the near IR, and even visible wavelengths [ 18 ], [ 19 ].
Since the frequency shift from SSFS is deterministic, the SSFS has seen much application over the last decade in fabricating fiber-delivered, widely frequency-tunable, femtosecond pulse sources [ 20 ], [ 21 ]. Other applications of SSFS include analog-to-digital conversion [ 22 ], [ 23 ] and telecom applications such as signal processing [ 24 ], [ 25 ], tunable time delays [ 26 ], [ 27 ], switching and demultiplexing [ 28 ]—[ 30 ]. In addition, we will review applications of SSFS for making tunable and multiwavelength sources, analog-to-digital conversion, and slow light.
We now briefly outline the mathematical description of SSFS [ 31 ], [ 32 ]. Light propagation in fiber is governed by the NLSE, which takes the form. This particular form of the NLSE does not include loss, higher order dispersion, or other nonlinear terms. This solution has a length invariant temporal profile and is called the fundamental soliton. The variables for time t , propagation distance z , and electric field u are in soliton units.
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Higher order solitons, characterized by soliton order N , also exist. When pulses are short on the order of picoseconds or less , higher order dispersive and nonlinear terms must be considered. If we rearrange the NLSE such that all the higher order terms appear on the right-hand side, the equation becomes:. The three terms on the right-hand side of 3 are for third-order dispersion, self-steepening, and intrapulse Raman scattering, respectively.
The constant T R , the Raman response, is a reasonable approximation for the full integral form when pulses are longer than a few optical cycles. By treating the Raman effect as a perturbation on the standard NLSE 1 , Gordon analytically derived a dependence of the frequency shift in terahertz per kilometer on pulsewidth [ 7 ],.
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More generally, the higher order NLSE 3 can be solved numerically by a split-step Fourier method [ 31 ]. SSFS has been observed in many types of fibers. The behavior of SSFS in these fiber platforms is dictated by the dispersion and nonlinearity of the fibers. Total dispersion of a fiber is the sum of material dispersion D m and waveguide dispersion D w. We can understand this by considering the mode evolution of the LP 01 mode.
As wavelength increases, the LP 01 mode monotonically transitions from the high-index central core to the surrounding lower index regions, yielding an effective index that decreases with wavelength. Since the velocity of light is inversely related to the index, the LP 01 mode experiences smaller group delays as wavelength increases.
D w , which is the derivative of group delay with respect to wavelength, is thus negative normal for the LP 01 mode.
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Mitschke and Mollenauer [ 6 ] observed up to 10 THz shift of an input at 1. Other efforts include demonstrations in doped fiber amplifiers [ 37 ], highly nonlinear fiber [ 38 ], birefringent fiber [ 39 ], [ 40 ], fiber with length-variable dispersion [ 41 ], as well as non-PM fibers [ 34 ], [ 42 ], [ 43 ]. The propagation of solitons near the zero dispersion wavelength is particularly interesting due to the generation of the Cerenkov radiation. With the introduction of perturbations such as higher order dispersion, the stable soliton solution breaks down, allowing the transfer of energy between the soliton in the anomalous dispersion regime and newly shed dispersive radiation in the normal dispersion regime.
Such energy transfer occurs most efficiently in fibers for solitons near the zero-dispersion wavelength. The phenomenon of Cerenkov radiation in fibers is often associated with SSFS as it allows a convenient mechanism for more efficient energy transfer between the soliton and the Cerenkov band. When the third-order dispersion is negative, SSFS will shift the center frequency of the soliton toward the zero-dispersion wavelength, resulting in efficient energy transfer into the Cerenkov radiation in the normal dispersion regime.
A more rigorous description and analytical derivation of Cerenkov radiation in fibers can be found in various theoretical works [ 46 ]—[ 48 ]. Newer fiber designs that exhibit different mode propagation characteristics allow for positive waveguide dispersion values. Positive waveguide dispersion larger than the magnitude of negative material dispersion can then achieve anomalous dispersion at previously unattainable wavelengths.
In addition, these newer fiber platforms can be engineered to have different dispersion profiles by dimensional scaling or tuning of index parameters. The light guiding mechanism in PCF is index guiding total internal reflection , as with conventional solid silica SMF. Due to the high index contrast between the silica core and airhole clad, PCF can be thought of as a thin silica strand in air. Such a structure can have large anomalous dispersion over the wavelength regions where silica is normally dispersive [ 19 ].
sotecepgeco.gq In addition, the photonic crystal lattice that forms the cladding can be designed to give a wide range of dispersion profiles [ 17 ]. The small core size, and thus, small effective area of PCF enable observation of nonlinear effects at low pulse energies. PBGFs, on the other hand, with its low-index core, does not guide with total internal reflection. Instead, it guides light by the bandgap effect created by the periodic lattice that surrounds the central defect [ 49 ].
The appeal of PBGF lies in its hollow core: low nonlinearities enable high-energy pulse delivery [ 15 ]. Later, Washburn et al. Since these early demonstrations, groups have shown wavelength shifts within the wavelength range of 0. In addition to the demonstrations in index-guided PCFs, Ouzounov et al. The wavelength access provided by dispersion engineering in these microstructured fibers is exciting; however, each novel fiber design has limitations in allowed pulse energies.
The pulse energy required to support stable Raman-shifted solitons in index-guided PCFs and hollow-core PBGFs is either very low, a fraction of a nanojoule for silica-core PCFs [ 13 ], [ 54 ], or very high, greater than nJ requiring an input from an amplified optical system for hollow-core PBGFs [ 58 ].
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The lowenergy limit is due to high nonlinearity in the PCF. In order to generate large positive waveguide dispersion to overcome the negative dispersion of the material, the effective area of the fiber core must be reduced.
For positive total dispersion at wavelengths less than 1. The high-energy requirement for the PBGF is due to low nonlinearity in the air core where the n 2 of air is roughly times less than that of silica.
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In fact, most PCF and tapered fibers with positive dispersion are intentionallydesigned to demonstrate nonlinear optical effects at the lowest possible pulse energy. On the other hand, hollow-core PBGFs are often used for applications that require linear propagation, such as pulse delivery. For these reasons, previous work showing SSFS below 1. HOM fiber, in contrast with microstructured optical fibers, with its large A eff and the moderate nonlinearity both comparable to conventional SMF , shows promise for supporting self-frequency-shifted solitons of pulse energies 1—10 nJ.
This all-solid silica fiber structure is index-guided as with SMF. This represents a major breakthrough in fiber design because it was previously considered impossible to obtain anomalous dispersion at wavelength shorter than1.