Linear and nonlinear optical properties of plasma-enhanced chemical-vapour deposition grown silicon nanocrystals

We provide a systematic study on the linear and nonlinear optical properties of silicon nanocrystals (Si-nc) grown by plasma-enhanced chemical vapour deposition (PECVD). Linear optical properties, namely absorption, emission and refractive indices are reported. The sign and magnitude of both real and imaginary parts of third-order nonlinear susceptibility X(³) of Si-nc are measured by the Z-scan method. Closed aperture Z-scan reveals a positive nonlinearity for all the samples. From the open aperture measurements, nonlinear absorption coefficients are evaluated and attributed to two-photon absorption. Absolute values of X(³) are in the order of 10^-9 esu and show systematic correlation with the Si-nc size, due to quantum confinement related effects. A correlation has been made between X(³), nanocrystalline size, linear refractive index and optical band gap.     

Internal symmetry groups for perfect elastoplasticity under axial‐torsional loadings

Abstract

In the material modeling of experimental axial‐torsional strain control tests, the hoop and radial strains are always unknown, a priori, and hence can not be viewed as inputs. This greatly complicates constitutive model analyses because the resulting differential equations become highly nonlinear. To tackle this problem, we demonstrate two new formulations. By using the two‐integrating factors idea we derive two Lie type systems in the product space M ¹⁺¹?M ¹⁺¹. The Lie algebra is the direct sum so(1, 1)?so(1, 1), and correspondingly the symmetry group is the direct product SO_o (1, 1) ?SO_o (1, 1). Then, by using the one‐integrating factor idea we convert the nonlinear constitutive equations to a Lie type system X=A(X, t)X with A?sl(2, 1, R), a Lie algebra of the special orthochronous pseudo‐linear group SL(2, 1, R). The underlying space is a cone in the pseudo‐Riemann manifold. Consistent numerical methods are also developed according to these Lie symmetries.     

Non-conditioned generation of Schrödinger cat states in a cavity

We investigate the dynamics of a two-level atom in a cavity filled with a nonlinear medium. We show that the atom-field detuning δ and the nonlinear parameter χ⁽³⁾ may be combined to yield a periodic dynamics, allowing the generation of almost exact superpositions of coherent states (Schrödinger cats). By analysing the atomic inversion and the field purity, we verify that any initial atom-field state is recovered at each revival time, and that a coherent field interacting with an excited atom evolves to a superposition of coherent states at each collapse time. We show that a mixed field state (statistical mixture of two coherent states) evolves towards an almost pure field state as well (Schrödinger cat). We discuss the validity of these results by using the field fidelity and the Wigner function.     

Picosecond nonlinear optical studies of gold nanoparticles synthesised using coriander leaves (Coriandrum sativum)

The results are presented from the experimental picosecond nonlinear optical (NLO) studies of gold nanoparticles synthesised using coriander leaf (Coriandrum sativum) extract. Nanoparticles with an average size of ～30?nm (distribution of 5–70?nm) were synthesised according to the procedure reported by Narayanan et al. [Mater. Lett. 2008, 62, 4588–4591]. NLO studies were carried out using the Z-scan technique using 2?ps pulses near 800?nm. Open-aperture data suggested saturation absorption as the nonlinear absorption mechanism, whereas closed-aperture data suggested a positive nonlinearity. The magnitude of third-order nonlinearity was estimated to be (3.3?±?0.6)?×?10^?13?esu. A solvent contribution to the nonlinearity was also identified and estimated. A comparison is attempted with some recently reported NLO studies of similar gold nanostructures.     

Surface plasmon-enhanced third-order optical non-linearity of silver triangular nanoplate

Abstract

We have obtained Ag triangular nanoplates with plasmon resonance bands locating from 750 to 950 nm by wet chemical method. The third-order optical non-linear response property of Ag triangular nanoplate is investigated using z-scan technique. The incident wavelength dependence of χ⁽³⁾ (the third-order susceptibility) shows the third-order nonlinear behaviour of Ag nanoplate is greatly modified by its plasmon resonance property, for the largest value of χ⁽³⁾ is observed around the wavelength of plasmon resonant peak. Also, we find the maximum value of Ag nanoplate’s χ⁽³⁾ is 7.46 × 10^?11 esu, which is about 1–3 orders of magnitude larger than that of many other nanomaterials including Au nanorod, nanobipyramid and nanocube, as well as Ag nanosphere and nanodisc. This fact indicates the Ag triangular nanoplate is a kind of useful nonlinear material with larger third-order nonlinearity, showing great potentials in the explorations of functional non-linear devices.     

The Distributed Feedback of Counter-propagating Waves in a Periodically Modulated Medium with Relaxing Cubic Nonlinearity

Abstract

An interaction of two counter-propagating waves in a periodically modulated medium with relaxing cubic nonlinearity is considered. On the basis of analytical solutions of the equations under the weak distributed feedback (DFB) approximation, we show that an increase in the reflectivity of ‘nonlinear’ DFB-structures takes place due to phase mismatching between the radiation and the light-induced grating. We carry out a numerical simulation of the obtained equations for arbitrary values of the coupling coefficient, Bragg detuning and incident pulse intensity. We also analysed the dynamics of the regular temporal modulation of initially continuous wave radiation caused by the ‘nonlinear’ grating.     

Large third-order nonlinearity of nonpolar A-plane GaN film at 800 nm determined by Z-scan technology

We report an investigation on the optical third-order nonlinear property of the nonpolar A-plane GaN film. The film sample with a thickness of ~2?μm was grown on an r-plane sapphire substrate by metal-organic chemical vapor deposition system. By performing the Z-scan method combined with a mode-locked femtosecond Ti:sapphire laser (800?nm, 50?fs), the optical nonlinearity of the nonpolar A-plane GaN film was measured with the electric vector E of the laser beam being polarized parallel (//) and perpendicular (⊥) to the c axis of the film. The results show that both the third-order nonlinear absorption coefficient β and the nonlinear refractive index n₂ of the sample film possess negative and large values, i.e. β_//?=??135?±?29?cm/GW, n_2//?=??(4.0?±?0.3)?×?10^?3?cm²/GW and β_⊥?=??234?±?29?cm/GW, n_2⊥?=??(4.9?±?0.4)?×?10^?3?cm²/GW, which are much larger than those of conventional C-plane GaN film, GaN bulk, and even the other oxide semiconductors.     

Higher-generation Schrödinger cat states in cavity QED

Abstract

Higher-generation Schrödinger cat states of the quantized electromagnetic field can be produced in a high-Q cavity, starting from a coherent state, through the passage of prepared Rydberg atoms interacting dispersively across it. These states are natural generalizations of the even and odd coherent states, the N ^th-generations corresponding to specific superpositions of 2 ^N states on a circle in phase space with well defined parity, and present very peculiar properties. Their photon statistics interchange between super- and sub-Poissonian behaviours and the nature of the photon bunching oscillates as the field intensity in the cavity is varied. For higher-generation even states, the minimum value of the Mandel factor almost reaches ?1.0 and the state represents the Fock state |2 ^N ). Squeezing properties and the Wigner function of these higher-generation Schrödinger cat states are also considered.     

Highly nonlinear balanced S-boxes with improved bound on unrestricted and generalized nonlinearity

We construct two classes of balanced S-boxes with high nonlinearity 2 ^n-1−2^(n-1)/2 for n odd. From known results, it can be deduced that for any S-box which has nonlinearity 2 ^n-1−2^(n-1)/2, the unrestricted nonlinearity is lower bounded by 2 ^n-1−2^(m+n-1)/2 while the generalized nonlinearity is lower bounded by 2 ^n-1−(2 ^m −1)2^(n-1)/2. We prove that the lower bound on the unrestricted nonlinearity of both our S-box constructions can be increased to 2 ^n-1−2^(m+n)/2-1. For the first class of S-box, the lower bound on generalized nonlinearity can be increased to 2 ^n-1−2^(n-1)/2+m-1. For the second class, the generalized nonlinearity is proven to be exactly 2 ^n-1−2^(m+n)/2-1, which is much higher than the lower bound for our first construction. The first class of S-boxes have low maximum differential while the second class corresponds to GMW sequences, whose algebraic structure allows us to construct a larger family of S-boxes. Moreover, both classes of S-boxes can attain high algebraic degree. We also compare our constructions with some known functions with high unrestricted and/or generalized nonlinearity. Parts of this paper was presented at the 2003 IEEE International Symposium on Information Theory [17].     

Self-phase Modulation in Quadratically Nonlinear Media

Abstract

We discuss self-phase modulations and related transverse effects accompanying interaction of a light beam with a second harmonic radiation generated by it. Some experimental observations are explained by a simple perturbative approach. Numerical simulations provide a further insight into the nature of transverse self-effects in quadratically nonlinear media. An effective x ⁽³⁾-like behaviour of dispersive quadratically nonlinear media is discussed. Such feature may lead to important consequences for measurements of the x ⁽³⁾ tensor in complex nonlinear media.     

Emission Spectra of a Two-level Atom in a Kerr-like Medium

Abstract

We have investigated the spectrum of light emitted by a single atom interacting with a single mode of the radiation field in an ideal cavity filled with a Kerr-like medium. It is shown that owing to the Kerr-like nonlinearity in the system the spectrum of the emitted light exhibits a single-peaked structure for sufficiently high intensities of the initial coherent field instead of the triplet structure in the case of the standard Jaynes-Cummings model.     

Frontiers and symmetries of dynamical systems

The frontiers of boundedness ? _b of the orbits of dynamical systems X defined on ? ⁿ are studied. When X is completely integrable some topological properties of ? _b are found and, in certain cases, ? _b is localized with the help of symmetries of X. Several examples in dimensions 2 and 3 are provided. In case the number of known first integrals of the vector field X is less than n ? 1, an interesting connection of ? _b with the frontier of boundedness of the level-sets of the first integrals of X is proved. This result also applies to Hamiltonian systems.     

Third-order nonlinear optical properties of 2-adamantylamino-5-nitropyridine caused by cascaded second-order nonlinearity

2-Adamantylamino-5-nitropyridine (AANP) was reported to have large nonlinear optical properties d₃₁ of 80 pm/V and have an ability of type II phase-matching by using d₁₅. The χ⁽²⁾:χ⁽²⁾ cascading effect via second-order optical nonlinearity produces the same effect of the third-order optical nonlinearity in phase-matching condition. We investigated the third-order nonlinear optical properties via cascaded second-order optical nonlinearity of AANP crystal at telecommunication wavelength range.     

On the Phenomenology of Self-organized Quasi-phase-matched Second-harmonic Generation in Optical Fibres

Abstract

A simple phenomenological theory of second-harmonic generation in centrosymmetric media is presented in this analytical paper. It is shown that in the medium without memory the energy transfer between interacting waves is negligibly small. For the medium with an optical memory (like phosphorus-germanium-doped glass fibres), the process of formation of χ ⁽²⁾ grating, owing to the third-order optical rectification and the electric-field-induced deprivation of centrosymmetric structure, is described from the phenomenological point of view. Second harmonic generation at periodical χ ⁽²⁾ grating is studied. In the case of resonant interaction, second-harmonic radiation can be systematically amplified. For the non-monochromatic and pulse-form radiations the effective length of the medium is limited largely due to the Kerr nonlinearities. The estimated effective fibre length is in good agreement with previous experiments.     

Nonlinear Refraction and Absorption in 4BCMU Planar Waveguides at 1·064 μm Wavelength

Abstract

We have measured the nonlinear refractive index, n ₂, and two-photon absorption coefficient, β, in 4BCMU planar waveguides at 1·064 μm wavelength using picosecond pulses. We use a beam propagation code, with n ₂ and β as fitting parameters, to make numerical comparison with the experimental data. Deduced values from this analysis are n₂ = ? 1·5 × 10^?13 cm²W^?1, confirming the negative sign of the nonlinearity at this wavelength, and β = 0·01 cm MW^?1.     

Evolution of Coherent States in a Dispersionless Fibre with Saturable Nonlinearity and the Generation of Macroscopic Quantum-superposition States

Abstract

We study the evolution of a initially coherent state of an electromagnetic field propagating in a Kerr medium with saturable nonlinearity. By using the quantum phase distribution formalism, we analyse the dependence of the output signal phase configuration upon input field amplitude. We observe that the saturation of the nonlinear contribution of the refractive index of the propagation medium introduces interference effects that compromise the observation of macroscopically well distinguishable components of the output state. For input amplitudes much larger than a characteristic saturation amplitude the final state differs from the input state only by an overall phase shift. Possible relevance of the present results in the experimental search of Schrödinger cat-like states using semiconductor-doped glass optical fibres is discussed.     

Efficient infrared upconversion via a ladder-type atomic configuration

We have demonstrated experimentally that infrared light at 1529.4?nm can be converted into the visible at 780?nm with 54% efficiency through a ladder-type atomic configuration in ⁸⁵Rb. Specifically we theoretically analyze that high efficiency is due to the large nonlinear dispersion of the index of refraction from the off-resonant enhancement in a four-wave mixing (FWM) process. By using two perpendicular polarized pump fields, the coherence of two FWM processes in this configuration is verified.     

Nonlinear Dispersion Coefficients of a Cylindrical Optical Waveguide

Abstract

The nonlinear carrier wave dispersion relation at weak nonlinearity is derived for the linearly polarized (LP) modes of a step-index optical fibre that has both a nonlinear core and a nonlinear cladding. The calculation begins with the exact equations for the nonlinear fibre and the nonlinear shift (coefficient), from its linear value, of the propagation wave-number is given in closed analytical form. The nonlinear coefficient is completely general and accounts both for the nonlinearity and the structure of the guided mode. Some numerical results for the LP₀₁ mode are presented which show that significant deviations occur from the conventionally accepted (averaged) nonlinear coefficient.     

Adaptive modification of objective function for lagrange multiplier method in constrained optimization problems

Abstract

This paper proposes an adaptive modification method to transform the objective function with a stationary point to an objective function with a minima point, such that search methods can be used to find the stationary point. The stationary point can be a saddle point in addition to a minima or a maxima. Therefore, this method can be used to transform a constrained optimization by applying Lagrange multipliers to an unconstrained optimization problem. A quadratic term, ½(X — X_N ) ^{T D} (X—X_N ), is added to the original function such that the modified function is a minima at the Newton point X_N of the original function, where D is a diagonal matrix to make the modified Hessian matrix H_O + D positive definite, and H_O is the original Hessian matrix at the initial point X_O .