Now we go to the next step—the possible polarization of the amplitude vector ﬁeld (3.4). Recently, this problem was discussed intensively [4, 5] and the diﬀerence between the polarization of the optical waves with spectral bandwidth (slowly varying amplitudes) and the polarization of monochromatic ﬁelds was pointed out. In the case of monochro- matic and quasimonochromatic ﬁelds, the Stokes parameters can be constructed from transverse components of the wave ﬁeld . This leads to two-component vector ﬁelds in a plane, transverse to the direction of propagation. For electromagnetic ﬁelds with spectral bandwidth (our case), the two-dimensional coherency tensor cannot be used and the Stokes parameters cannot be found directly. As it was shown by Carozzi et al. in , using a high order of symmetry (SU ( 3 ) ), in this case, six independent Stokes param- eters can be found. This corresponds to a three-component vector ﬁeld. We investigate this case here. The increase of the spectral bandwidth of the vector wave increases also the depolarization term (a component normal to the standard Stokes coherent polar- ization plane). The amplitude vector function of electrical ﬁeld A is represented as the sum of three orthogonal linearly polarized amplitudes:
The exploitation of a symbolic computation package will make it realistic to pro- pose a number of direct analytical methods. The research of traveling wave solutions of some nonlinear evolution equations derived from such ﬁelds played an important role in the analysis of some phenomena such as the exp(–ϕ(ξ ) method, Bernoulli’s sub-ODE method, the homogeneous balance method, the modiﬁed simple equation method, the modiﬁed extended direct algebraic method, the modiﬁed extended mapping method, the Kudryashov method, the extended sinh–cosh and sin–cos methods, the Lie symmetry method, the soliton ansatz method and many more methods [15–40].
The solar corona is a highly structured medium. Observa- tions of the corona during eclipses using white light allowed the detection of rapidly propagating quasi-periodic waves ( Williams et al. 2001, 2002; Katsiyannis et al. 2003 ) . The spatial and temporal resolution provided by the Solar Dynamics Observatory Atmospheric Imaging Assembly now allow their detection in extreme ultraviolet ( EUV ) light ( Liu et al. 2011, 2012 ) . These disturbances are interpreted as fast magnetoacoustic waves ( e.g., Cooper et al. 2003; Ofman et al. 2011 ) that are highly dispersive in coronal waveguides if their wavelength is comparable to the local width of the waveguide. An impulsive driver that generates a wide range of wavenumbers can therefore generate a quasi-periodic wave train some distance from the initial perturbation due to each wavenumber arriving at different times ( Roberts et al. 1983, 1984 ) . Numerical simulations ( e.g., review by Pascoe 2014 ) have demonstrated that this behavior is a robust feature for coronal structures, for example, also being detected in models of current sheets ( Jelínek & Karlický 2012; Jelínek et al. 2012 ) and observed above a fan structure ( Mészárosová et al. 2013 ) . Van Doorsselaere et al. ( 2016b ) recently produced a review of other mechanisms that may be responsible for quasi-periodic pulsations ( QPPs ) in solar and stellar ﬂ ares, while the statistical comparison of stellar and solar X-ray QPPs by Cho et al. ( 2016 ) supports a shared MHD wave mechanism for the modulation of emission.
The first chapter of this paper establishes Strichartz estimates for (0.1), which depend upon the dispersive estimates obtained by Anker and Pierfelice in . These results were first proven in  and ; we state them here for completeness and for later use in proving our main results. The Strichartz estimates obtained will hold for admissible triples (p, q, γ), where γ dictates the regularity of the Cauchy data and (p, q) are the indices of the mixed-norm L p t L q
Optical microcavities confine light to small mode volume due to resonant recirculation . In whispering-gallery-mode (WGM) optical microcavities, light circulates around the boundary of a dielectric microcavity due to total internal reflection, in analogy to acoustic waves circulating around a round enclosure (e.g., the St Paul’s Cathedral in London and the Echo Wall of Temple of Heaven in Beijing). Maintaining a large photon storage time, or equivalently high Q factors in optical cavities, are critical in many scientific and technological applications of microcavities, including cavity QED , cavity optomechanics , biosensing [4, 5], microresonator-based frequency combs , and narrow-linewidth laser sources [7, 8]. To achieve high Q factor in WGM cavities, it relies on the use of low-absorption dielectrics (to reduce material absorption loss) and the creation of very smooth dielectric surfaces (to reduce surface scattering loss). Although crystalline resonators currently have the highest Q factors on the order of 10 10 and 10 11 [9–11], for silicon-chip-based devices, microtoroid silica cavities provide the highest Q factors on the order of several hundred million . Microtoroid resonators combine low material loss of silica with a reflow technique in which surface tension is used to smooth lithographic and etch-related blemishes. However, reflow smoothing makes it very challenging to fabricate larger-diameter ultrahigh-Q (UHQ) resonators or to leverage the full range of integration tools and devices available on silicon.
For future research, we are looking into telecommunication sector, the nano– technology and also optical solitons. In 1973 Hasegawa and Tappert had proposed that soliton pulses could be used in optical communications through the balance of non- linearity and dispersion. They showed that these solitons would propagate according to the nonlinear Schrodinger equation (NLS), which had been solved by the inverse scattering method a year earlier by Zakharov and Shabat. At that time there was no capability to produce the fibers with the proper characteristics for doing this and the dispersive properties of optical fibers were not known. Also, the system required a laser which could produce very small wavelengths, which also was unavailable. It wasn’t until seven years later, when Mollenauer, Stollen and Gordon at AT&T Bell Laboratories had experimentally demonstrated the propagation of solitons in optical fibers. The original communications systems employed pulse trains with widths of about one nanosecond. However, there was still some distortion due to fiber loss. This was corrected by placing repeaters every several of tens of kilometers. As the width of the available pulses was decreased, the spacing of the repeaters was increased. In the mid 1980’s it was proposed that by sending in an additional pump wave along the fiber, the dispersion of a soliton could be halted through a process known as Raman scattering. In 1988 Mollenauer and his group had shown that this could be done by propagating a soliton over 6000 km without the need for repeaters.
In this paper the nonlinear interfacial gravity wave transfor- mation is studied in the Boussinesq assumption, and when the dispersive effects are neglected. Explicit formulation of the evolution equation in terms of the Riemann invariants al- lows us to obtain analytical results characterizing strongly nonlinear wave steepening, including the spectral evolution. This dynamics is considered to be the first stage resulting in wave breaking. Effects showing the action of highly nonlin- ear corrections of the model are highlighted. It is shown, in particular, that the breaking points on the wave profile may shift from the zero-crossing level; wave steepening occurs differently in the case when the density jump is placed near the middle of the water column: then the wave deformation is almost symmetrical and two breaking wave phases exist. Acknowledgements. This work was supported by grants INTAS 03- 51-4286 and 06-1000013-9236 (E.P., T.T., A.K., O.P., A.S.); RFBR 05-05-64333, 06-05-64087 (O.P., A.K.), and 06-05-64232 (T.T., A.S.); grant MK-798.2007.5 and Russian Science Support Foun- dation (A.S.).
We then analyze a simple one-dimensional Lagrangian averaged model, a subcase of the general models derived above. We prove the existence of a large family of traveling waves; these solutions correspond to homoclinic orbits in the phase plane. Computing the dispersion relation for this model, we find it is nonlinear, implying that the equation is dispersive. Since the amount of dispersion in the model is controlled by α, we expect that the zero-α limit will be highly oscillatory. We carry out numerical experiments that show that the model possesses smooth, bounded solutions that display interesting pattern formation. Finally, we relate the mathematical features of this model to other models for solitary waves in compressible fluids, and to other solitary wave phenomena that occurs in air.
The SCISSOR conguration has two types of PBGs, one is called the Resonator Band (RB) and the other the Bragg Band (BB) . The RB is due to the response of each resonator of the chain. Indeed, it occurs when the input wavelength satises the condition in Eq.1.10. The BB is due to the coherent, constructive superposition, of all waves which are back reected by each resonator. Infact, even when the input wavelength is o resonance, a small portion of electric eld is coupled into the Drop port (back reected). If the resonator distance Λ satises Λβ = qπ , in which q is an integer number, then all the reected waves superimpose contructively, and signicant power can be build up into the drop port. If the SCISSOR is designed to have Λ = πR , in which R is the radius of the rings, the RB and the BB overlap for all orders. From Fig.1.20, it is possible to notice that as the number of resonators increase, the bandwith of the lter both attens and increases, becoming very similar to a box-like response. One may wonder what happens if the resonators which compose the sequence are not all equal, i.e., they have resonant frequencies which are slightly detuned between each other. This situation can be intentionally induced already at the design stage , or it can be an undesired consequence of defects in the fabrication process . What happens is that defect states, in which the energy is mosty localized in the neighborhood of the defects, are created. These manifest in the spectral response as narrow transparency peaks inside the PBGs of the Through port, or equivalently in dips into the Drop one. The phenomenon is called Coupled Resonator Induced Transparency (CRIT) and is widely discussed in Refs.[87, 78, 82]. By virtue of the intrinsic link between CRIT and defects, a device which quanties the amount of fabrication disorder in a SOI wafer by exploiting the CRIT eect has been proposed .
In this section, we use the split-step method  to discretize the nonlinear partial differential equations (17) and (28) and to propagate their corresponding solutions. Thus, the constraint relations between the coefficients of the terms of the nonlinear partial differential equation allowed choosing the values of the parameters. We organized this numerical study in two cases.
While phase-locked states and pulse generation have been demonstrated in earlier works on microcombs, microcombs typically were not phase-locked and the noise characteristics were difficult to control. Recently, temporal soliton mode-locking[39, 40, 41, 42, 43], a major advancement in microcomb research, has been realized. Optical solitons are propa- gating pulses of light that retain their shape, and soliton microcombs are temporal solitons because the pulse shape coming out of the cavity is same at each time period. Soliton microcombs feature dissipative Kerr solitons that leverage the Kerr nonlinearity to both compensate dispersion and to overcome cavity loss by way of parametric gain. Un- like earlier microcombs, this new device provides phase-locked femtosecond pulses with well-defined, repeatable spectral envelopes. Their pulse repetition rate is typically several GHz to THz, and has excellent phase noise characteristics. Different from conven- tional mode-locked lasers, microresonator-based solitons don’t require active gain or sat- urable absorber. Self-referencing of microcombs has also been demonstrated via external broadening[44, 45] or generation of dispersive wave.
The ability to generate complex optical photon states involving entanglement between multiple optical modes is not only critical to advancing our understanding of quantum mechanics but will play a key role in generating many applications in quantum technologies. These include quantum communications, computation, imaging, microscopy and many other novel technologies that are constantly being proposed. However, approaches to generating parallel multiple, customisable bi- and multi-entangled quantum bits (qubits) on a chip are still in the early stages of development. Here, we review recent developments in the realisation of integrated sources of photonic quantum states, focusing on approaches based on nonlinearoptics that are compatible with contemporary optical fibre telecommunications and quantum memory infrastructures as well as with chip-scale semiconductor technology. These new and exciting platforms hold the promise of compact, low-cost, scalable and practical implementations of sources for the generation and manipulation of complex quantum optical states on a chip, which will play a major role in bringing quantum technologies out of the laboratory and into the real world.
near the resonnance of the coresponding linear frequency of the equation x + = x 0 . We present the numerical solutions in time and frequency domains and demonstrate the use of the frequency sweep method in detecting the nonlinear resonnance of the system.We solve Equation (20) using Matlab solver to obtain numerical results in the time domain. FFT algorithm is then applied to the time signal to find the frequencies of the solutions. The expected frequency corresponds to the excitation ω , the nonlinear resonance and some harmonics. The double scales method can be used to find analytical approximate solution of Equation (20) similar to the autonomeous system case of Section 4. We present our numerical results in Figure 6.
This work considers initiation of nonlinearwaves, their propagation, reflection, and their interac- tions in thermoelastic solids and thermoviscoelastic solids with and without memory. The con- servation and balance laws constituting the mathematical models as well as the constitutive theo- ries are derived for finite deformation and finite strain using second Piola-Kirchoff stress tensor and Green’s strain tensor and their material derivatives . Fourier heat conduction law with constant conductivity is used as the constitutive theory for heat vector. Numerical studies are performed using space-time variationally consistent finite element formulations derived using space-time residual functionals and the non-linear equations resulting from the first variation of the residual functional are solved using Newton’s Linear Method with line search. Space-time local approximations are considered in higher order scalar product spaces that permit desired order of global differentiability in space and time. Computed results for non-linear wave propagation, ref- lection, and interaction are compared with linear wave propagation to demonstrate significant differences between the two, the importance of the nonlinear wave propagation over linear wave propagation as well as to illustrate the meritorious features of the mathematical models and the space-time variationally consistent space-time finite element process with time marching in ob- taining the numerical solutions of the evolutions.
Quantum photonics is a rapidly developing platform for future quantum network applications. Waveguide-based architectures, in which embedded quantum emitters act as both nonlinear elements to mediate photon–photon inter- actions and as highly coherent single-photon sources, offer a highly promising route to realize such networks. A key requirement for the scale-up of the waveguide architecture is local control and tunability of individual quantum emitters. Here, we demonstrate electrical control, tuning, and switching of the nonlinear photon–photon interaction arising due to a quantum dot embedded in a single-mode nano-photonic waveguide. A power-dependent waveguide transmission extinction as large as 40 2% is observed on resonance. Photon statistics measurements show clear, voltage-controlled bunching of the transmitted light and antibunching of the reflected light, demonstrating the single- photon, quantum character of the nonlinearity. Importantly, the same architecture is also shown to act as a source of highly coherent, electrically tunable single photons. Overall, the platform presented addresses the essential requirements for the implementation of photonic gates for scalable nano-photonic-based quantum information processing.
NONLINEAR DISPERSIVE WAVE PROBLEMS Thesis by Jon Christian Luke In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, Californi[.]
Abstract. A computational model is presented which will help guide and interpret an upcoming series of experiments on nonlinear compressional waves in marine sediments. The model includes propagation physics of nonlinear acoustics augmented with granular Hertzian stress of order 3/2 in the strain rate. The model is a variant of the time domain NPE (McDonald and Kuperman, 1987) supplemented with a causal algorithm for frequency-linear attenuation. When at- tenuation is absent, the model equations are used to construct analytic solutions for nonlinear plane waves. The results im- ply that Hertzian stress causes a unique nonlinear behavior near zero stress. A fluid, in contrast, exhibits nonlinear be- havior under high stress. A numerical experiment with nom- inal values for attenuation coefficient implies that in a wa- ter saturated Hertzian chain, the nonlinearity near zero stress may be experimentally observable.