Spectral Edges Research Articles

The spectrum of non-local discrete Schrödinger operators with a δ-potential

The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schrodinger operators with a δ-potential. These operators arise by replacing the discrete Laplacian by a strictly increasing C1-function of the discrete Laplacian. The dependence of the results on this function and the lattice dimension are explicitly derived. It is found that while in the case of the discrete Schrodinger operator these behaviours are the same no matter which end of the continuous spectrum is considered, an asymmetry occurs for the non-local cases. A classification with respect to the spectral edge behaviour is also offered.

980 nm all-fiber NPR mode-locking Yb-doped phosphate fiber oscillator and its amplifier

We report on a 980nm all-fiber passively mode-locking Yb-doped phosphate fiber oscillator with the nonlinear polarization rotation (NPR) technique and its amplifier. In order to obtaining the stable self-starting mode-locking oscillator at 980nm, a bandpass filter with 30nm transmission bandwidth around 980nm is inserted into the cavity. The oscillator generates the average output power of 26.1mW with the repetition rate of 20.38MHz, corresponding to the single pulse energy of 1.28nJ. The pulse width is 159.48ps. The output spectrum of the pulses is centered at 977nm with a full width half maximum (FWHM) of 10nm and has the characteristic steep spectral edges of dissipative soliton. No undesired ASE and harmful oscillation around 1030nm is observed. Moreover, through two stage all-fiber-integrated amplifier by using the 980nm oscillator as seed source, an amplified output power of 205mW at 980nm and pulse duration of 178.10ps is achieved.

Dissipative soliton operation in diode pumped ultrafast Yb:GdYSiO5 oscillator

We demonstrate the dissipative soliton mode locking in a diode pumped Yb:GdYSiO5(Yb:GYSO) laser operating in the positive dispersion regime. We obtain stable passively mode-locked pulses with strong positive chirp and with very steep spectral edges. The central wavelength is 1050 nm with bandwidth of about 4 nm, autocorrelation trace shown the typical pulse duration is about 3.5 ps. We obtain the maximum average power of 558 mW for a 3.3-W absorbed pump power, with 22% slope efficiency, and a 78-MHz pulse repetition frequency.

Laser-based assessment of optical interference filters with sharp spectral edges and high optical density

Interference filters have improved over the years. Sharp spectral edges (>1db/nm) reach high optical density (OD>8) in a few nm. In-situ optical monitoring to within 0.1% error enables these levels of performance. Due to limitations related to f-number and resolution bandwidth, post-deposition testing in typical spectrophotometers cannot reveal the quality of today's filters. Laser based measurements at selected wavelengths prove that blocking above OD8 to OD9 is manufacturable with high yield. This paper compares modeled spectra and laser based measurements.

Covariance estimation for distributions with ${2+\varepsilon}$ moments

We study the minimal sample size N=N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. We establish the optimal bound N=O(n) for every distribution whose k-dimensional marginals have uniformly bounded $2+\varepsilon$ moments outside the sphere of radius $O(\sqrt{k})$. In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan-Lovasz-Simonovits problem, which was recently solved by Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535-561]. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption - uniformly bounded $2+\varepsilon$ moments of one-dimensional marginals. Our argument consists of randomizing the spectral sparsifier, a deterministic tool developed recently by Batson, Spielman and Srivastava [SIAM J. Comput. 41 (2012) 1704-1721]. The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points.

Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

We consider the ensemble of adjacency matrices of Erd\H{o}s-R\'{e}nyi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erd\H{o}s-R\'{e}nyi ensemble is given by the Wigner semicircle law for spectral windows of length larger than $N^{-1}$ (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the $\ell^{\infty}$-norms of the $\ell^2$-normalized eigenvectors are at most of order $N^{-1/2}$ with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erd\H{o}s-R\'{e}nyi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that $pN\gg N^{2/3}$.

Optimal volume Wegner estimate for random magnetic Laplacians on ${\mathbb {Z}}^2$Z2

We consider a two dimensional magnetic Schrödinger operator on a square lattice with a spatially stationary random magnetic field. We prove a Wegner estimate with optimal volume dependence. The Wegner estimate holds around the spectral edges, and it implies Hölder continuity of the integrated density of states in this region. The proof is based on the Wegner estimate obtained in Erdős and Hasler [“Wegner estimate for random magnetic Laplacians on ${\mathbb {Z}}^2$Z2,” Ann. Henri Poincaré 12, 1719–1731 (2012)]10.1007/s00023-012-0177-9.

High Pulse Energy All-Fiber F-P Cavity Dissipative Soliton Laser

We report on high energy dissipative soliton pulses with large positive chirp and steep spectral edges from an all-fiber 1.5 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu{\rm m}$</tex> </formula> passively mode-locked laser with a F-P cavity configuration. Passive mode-locking is achieved employing a fiber-pigtailed semiconductor saturable absorber mirror as one of the cavity mirrors. The output pulses from two output ports of the cavity have pulse energies of 3.5 and 1.1 nJ, respectively. Their corresponding pulse durations are 32.7 and 32.1 ps, respectively. The fundamental repetition rate is at 1-MHz level, which makes this laser suitable to directly seed an all-fiber amplifier.

Enhanced representation of spectral contrasts in the primary auditory cortex

The role of early auditory processing may be to extract some elementary features from an acoustic mixture in order to organize the auditory scene. To accomplish this task, the central auditory system may rely on the fact that sensory objects are often composed of spectral edges, i.e., regions where the stimulus energy changes abruptly over frequency. The processing of acoustic stimuli may benefit from a mechanism enhancing the internal representation of spectral edges. While the visual system is thought to rely heavily on this mechanism (enhancing spatial edges), it is still unclear whether a related process plays a significant role in audition. We investigated the cortical representation of spectral edges, using acoustic stimuli composed of multi-tone pips whose time-averaged spectral envelope contained suppressed or enhanced regions. Importantly, the stimuli were designed such that neural responses properties could be assessed as a function of stimulus frequency during stimulus presentation. Our results suggest that the representation of acoustic spectral edges is enhanced in the auditory cortex, and that this enhancement is sensitive to the characteristics of the spectral contrast profile, such as depth, sharpness and width. Spectral edges are maximally enhanced for sharp contrast and large depth. Cortical activity was also suppressed at frequencies within the suppressed region. To note, the suppression of firing was larger at frequencies nearby the lower edge of the suppressed region than at the upper edge. Overall, the present study gives critical insights into the processing of spectral contrasts in the auditory system.

Effects of passive, moderate-level sound exposure on the mature auditory cortex: Spectral edges, spectrotemporal density, and real-world noise

Persistent, passive exposure of adult cats to bandlimited tone pip ensembles or sharply-filtered white noise at moderate levels (∼70 dB SPL) leads to a long-term suppression of spontaneous and sound-evoked activity in the region(s) of primary auditory cortex (AI) normally tuned to the exposure spectrum, and to an enhancement of activity in one or more neighboring regions of AI, all in the apparent absence of hearing loss. Here, we first examined the effects of passive exposure to a more structured, real-world noise, consisting of a mix of power tool and construction sounds. This “factory noise” had less pronounced effects on adult cat AI than our previous random tone pip ensembles and white noise, and these effects appeared limited to the region of AI tuned to frequencies near the sharp factory noise cutoff at 16 kHz. To further investigate the role of sharp spectral edges in passive exposure-induced cortical plasticity, a second group of adult cats was exposed to a tone pip ensemble with a flat spectrum between 2 and 4 kHz and shallow cutoff slopes (12 dB/oct) on either side. Compared to our previous ensemble with the same power in the 2–4 kHz band but very steep slopes, exposure to the overall more intense, sloped stimulus had much weaker effects on AI. Finally, we explored the issue of exposure stimulus spectrotemporal density and found that low aggregate tone pip presentation rates of about one per second sufficed to induce changes in the adult AI similar to those characteristic of our previous, much denser exposures. These results are discussed in light of the putative mechanisms underlying exposure-induced auditory cortical plasticity, and the potential adverse consequences of working or living in moderately noisy environments.

Optimum PCF tapers for blue-enhanced supercontinuum sources

Tapering of photonic crystal fibers has proven to be an effective way of blueshifting the dispersive wavelength edge of a supercontinuum spectrum down in the deep-blue. In this article we will review the state-of-the-art in fiber tapers, and discuss the underlying mechanisms of supercontinuum generation in tapers. We show, by introducing the concept of a group-acceleration mismatch, that for a given taper length, the downtapering section should be as long as possible to enhance the amount of blueshifted light. We also discuss the noise properties of supercontinuum generation in uniform and tapered fibers, and we demonstrate that the intensity noise at the spectral edges of the generated supercontinuum is at a constant level independent on the pump power in both tapered and uniform fibers.

Wegner Estimate for Random Magnetic Laplacian on $${\mathbb{Z}^{2}}$$

We consider a two-dimensional magnetic Schrodinger operator on a square lattice with a spatially stationary random magnetic field. We prove Anderson localization near the spectral edges. We use a new approach to establish a Wegner estimate that does not rely on the monotonicity of the energy on the random parameters.

Power dependence of supercontinuum noise in uniform and tapered PCFs

We experimentally investigate the noise properties of picosecond supercontinuum spectra generated at different power levels in uniform and tapered photonic crystal fibers. We show that the noise at the spectral edges of the generated supercontinuum is at a constant level independent on the pump power in both tapered and uniform fibers. At high input power the spectral bandwidth is limited by the infrared loss edge, this however has no effect on the noise properties.

Direct Generation of 4.6-nJ 78.9-fs Dissipative Solitons in an All-Fiber Net-Normal-Dispersion Er-Doped Laser

We have demonstrated the direct generation of 4.6-nJ 78.9-fs ultrashort pulses with 58 kW peak power in an all-fiber net-normal-dispersion oscillator. Such high peak power is a record in an all-fiber oscillator and is comparable to chirped-pulse amplification systems. The corresponding spectrum has steep spectral edges, which is the typical shape of dissipative solitons (DS). Also, we find that the spectral width is tunable from 40 to 58 nm by adjusting polarization controllers, and the pulse duration can be tuned from 1.7 ps to 120 fs through increasing the pump power, which is attractive to applications. Besides, we demonstrate experimentally the relationship between spectral width and the pump power of DS laser for the first time.

Dissipative Soliton (12 nJ) From an All-Fiber Passively Mode-Locked Laser With Large Normal Dispersion

We experimentally demonstrate an all-fiber erbium-doped fiber laser operating in the large normal dispersion regime. The laser is mode-locked by use of nonlinear polarization rotation and generates dissipative solitons (DSs) with large positive chirp and steep spectral edges. Within the cavity, there is no additional spectral filter used for self-starting and stabilizing the mode-locking of the laser. The output DSs have a duration of 30.5 ps with pulse energies up to 12 nJ, which can be compressed externally to 240 fs.

Evidence of dissipative solitons in Yb^3+:CaYAlO_4

Operation of an end-pumped Yb³⁺:CaYAlO₄ laser operating in the positive dispersion regime is experimentally investigated. The laser emitted strongly chirped pulses with extremely steep spectral edges, resembling the characteristics of dissipative solitons observed in fiber lasers. The results show that dissipative soliton emission constitutes another operating regime for mode locked Yb³⁺-doped solid state lasers, which can be explored for the generation of stable large energy femtosecond pulses.

Defect Modes and Homogenization of Periodic Schrödinger Operators

We consider the discrete eigenvalues of the operator $H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$ is localized on $\R^d,\ \ d\ge1$. For $\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schr\"odinger operator, $H_0 = -\Delta_\x+V(\x)$, into spectral gaps. The nature of the bifurcation depends on the homogenized Schr\"odinger operator $L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y)$. Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation.

On the location of spectral edges in -periodic media**This paper is dedicated to the memory of P Duclos.

Periodic second-order ordinary differential operators on are known to have the edges of their spectra to occur only at the spectra of periodic and anti-periodic boundary value problems. The multi-dimensional analog of this property is false, as was shown in a 2007 paper by some of the authors of this paper. However, one sometimes encounters the claims that in the case of a single periodicity (i.e., with respect to the lattice ), the 1D property still holds, and spectral edges occur at the periodic and anti-periodic spectra only. In this work, we show that even in the simplest case of quantum graphs this is not true. It is shown that this is true if the graph consists of a 1D chain of finite graphs connected by single edges, while if the connections are formed by at least two edges, the spectral edges can already occur away from the periodic and anti-periodic spectra.

The spectral edge of some random band matrices

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the band is sufficiently wide (W ≫ N 5/6 ). Otherwise, a different limiting distribution appears.

Harmonic pitch: Dependence on resolved partials, spectral edges, and combination tones

Perceptual weights were estimated in a pitch-comparison experiment to assess the relative influences of individual partial tones on listeners’ pitch judgments. The stimuli were harmonic sounds ( F 0 = 200 Hz ) with partials up to the 12th. Low-numbered partials were removed step-by-step, so that the remaining higher-numbered partials would have a better chance of showing any effect. The individual frequencies of the partials were perturbed randomly on each stimulus presentation, and weights were estimated as the correlation coefficients between the frequency perturbations and the listeners’ responses. When the harmonic sounds contained all twelve partials, the listeners depended mostly on the low-numbered, resolved partials within the well-established dominance region. As the low-numbered partials were taken out of the dominance region, the listeners mostly listened to the lowest and highest partials at the spectral edges. For one listener, such an edge-listening strategy took the form of relying on nonlinear combination tones. Overall, there was no indication of any influence on pitch from unresolved partials, thus no evidence of contribution to pitch from temporal cues carried by this group of partials. The estimated patterns of weights were well described by the predictions of Goldstein’s optimal-processor model. The predicted weights were inversely proportional to the amount of error for estimating the individual frequencies of the partials. The agreement between the predicted and measured weights suggests that, for harmonic sounds, partials whose frequencies are perceived with the best precision will likely have the greatest influence on perceived pitch.