We demonstrate a cascaded Raman fibre laser (RFL) oscillation up to the fifth order Stokes wave, from ∼1550 nm to mid-IR at 2334 nm, with silica fibres. The pumping was done with an amplified spontaneous emission (ASE) light of erbium-doped fibre (EDF) in the 1550 nm wavelength regime. The low orders (first to third) could be easily obtained in continuous wave (CW) mode, while the fourth and fifth orders were obtained in a pulse mode. We used highly nonlinear fibres that allowed relatively short fibre cavity lengths of (10-100)m.

We demonstrate a cascaded Raman fiber laser (RFL) oscillation up to the 5-th order Stokes wave, from ~1550nm to mid-ir at 2334 nm, with silica fibers. The pumping was done with an amplified spontaneous emission (ASE) light of erbium-doped fiber (edf) in the 1550 nm wavelength regime. The low orders (1st to 3rd) could be easily obtained in continuous wave (CW) mode, while the 4th and 5th orders were obtained in a pulse mode. We used highly nonlinear fibers that allowed relatively short fiber cavity lengths of (10-100) m.

It is shown that in the problem of cardinal interpolation, spline interpolants of various degrees are R -minimax, with respect to corresponding Sobolev and Hardy functional classes, under restrictions determined by the interference between their oscillating variance and bias. The results raise a natural question: what degrees of interpolating splines are more appropriate, for given Sobolev or Hardy classes? It turns out that the scales of such functional classes can be divided into “very smooth” and “not-so-smooth” subfamilies, whereby “very smooth” classes can benefit from higher degrees of cardinal splines, and vice versa.

We demonstrate photon Bose-Einstein condensation (photon-BEC) at a broad temperature range that is valid also in the long 1D fiber cavity limit. It is done with an erbium-ytterbium co-doped fiber (EYDF) cavity by overcoming the challenging requirement of sublinear light dispersion for BEC in 1D using a chirped-gratings Fabry-Perot. We experimentally show with a square-root mode-dispersion, a quadratic temperature dependence of the critical power for condensation (compared to a linear dependence in finite regular fiber-cavities) between 90 K and 382 K, as the theory predicts.

We demonstrate a nonlinear light mode dispersion and a nonuniform frequency mode comb by a chirped fiber Bragg gratings (CFBG) Fabry-Perot (FP) at the 1550 nm wavelength regime. We give analytical expressions for the general chirp case, and an experimental demonstration with a linear chirp, showing a square-root dependence of the dispersion as a function of the FP mode number. Such sublinear dispersion is required, for example, for photon Bose-Einstein condensation (BEC) in a one-dimensional (1D) system like fiber cavities.

Bose–Einstein condensation (BEC) is a special many-boson phenomenon that was observed in atomic particles at ultra-low temperatures. Later, BEC was also shown for non-atomic bosons, such as photons. Those experiments were usually done in micron-size cavities, where the power (particle number) was varied, and not the temperature, until condensation was reached. Here we demonstrate BEC of photons in a few-meters-long one-dimensional (1D) erbium–ytterbium co-doped fiber cavity at, below and above room temperature, between 100 K and 415 K. The experiments were done at about the 1550 nm wavelength regime having a few to tens of μW intra-cavity light power (10 7 −10 8 photons). By varying the power and also the temperature, we found linear dependence of the condensation on power for various temperatures and of the critical power (for condensation) on temperature. These findings agree, functionally and quantitatively, with the theoretical BEC prediction without any adjustable parameter.

For the Hardy classes of functions analytic in the strip around real axis of a size 2 β , an optimal method of cardinal interpolation has been proposed within the framework of Optimal Recovery [12]. Below this method, based on the Jacobi elliptic functions, is shown to be optimal according to the criteria of Nonparametric Regression and Optimal Design. In a stochastic non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from 0. A pivotal role is played by the interference effect, in which the oscillations exhibited by the interpolant’s bias and variance mutually cancel each other. In the limiting case β → ∞, the optimal interpolant converges to the well-knownNyquist–Shannon cardinal series.

We study wavelength selectable and switchable lasers with two kinds of fiber Bragg grating (FBG) reflectors, based on high harmonic mode-locking frequency addressing near 10 GHz and 25 GHz. We used fiber lasers and also show results with a cavity extended semiconductor laser. The first implementation of the tunable laser uses a chirped FBG. Different wavelengths define slightly different cavity lengths and an effective dispersion and a wavelength is selected by applying a corresponding mode-locking frequency. We obtained a similar to 17 nm wavelength tuning range at the 1550 nm regime. We also summarize results with a cavity extended semiconductor laser by a chirped fiber grating that gave stable 28 addressable and switchable wavelength channels with a spacing of 100 GHz (approximate to 0.8 nm). Here we obtained a tuning range of similar to 22.4 nm at the 1550 nm wavelength regime. The second implementation uses multiple sampled FBGs with different reflecting wavelengths, each determines a different cavity length and therefore a different modulation frequency.

We present a wavelength selectable actively mode-locked ring fiber laser. We used for the reflection sampled fiber Bragg gratings (SFBG) with 25GHz spacing and high harmonic mode-locking frequency modulation. The method includes successive SFBGs mirrors with different reflecting wavelengths, each SFBG determines a different cavity length and therefore a different modulation frequency. The selection of the wavelengths is performed by applying the corresponding frequency.

We show thermal-equilibrium (TE) and Bose-Einstein distribution of photons in standard erbium — doped fibers. We also find a coexistence of TE with oscillation without an overall inversion that can be attributed to lasing or BEC.

We demonstrate thermalization and Bose-Einstein (BE) distribution of photons in standard erbium-doped fibers (edf) in a broad spectral range up to ~200nm at the 1550nm wavelength regime. Our measurements were done at a room temperature ~300K and 77K. It is a special demonstration of thermalization of photons in fiber cavities and even in open fibers. They are one-dimensional (1D), meters-long, with low finesse, high loss and small capture fraction of the spontaneous emission. Moreover, we find in the edf cavities coexistence of thermal-equilibrium (TE) and thermal lasing without an overall inversion (T-LWI). The experimental results are supported by a theoretical analysis based on the rate equations.

We show and identify the role of the backward amplified spontaneous emission (ASE) in inducing bistability and hysteresis in a unidirectional ring erbium-doped fiber (edf) laser. It results from the interplay between the signal and the backward ASE in the gain medium that is ejected from the fiber loop by an isolator. A removal of the isolator eliminates the bistability. Another important factor is the strong wavelength dependence of the absorption and emission coefficients and their ratio.

Within the framework of Optimal Recovery, optimal methods of interpolation, based on the Abel–Jacobi elliptic functions, have been found for some Hardy classes of analytic functions [9]. It will be shown that these methods are also optimal according to criteria of Optimal Design and Nonparametric Regression.

Adaptive pointwise estimation of an unknown regression function f(x), x ? R corrupted by additive Gaussian noise is considered in the equidistant design setting. The function f is assumed to belong to the class A(?) of functions whose Fourier transform are rapidly decreasing in the weighted L2-sense. The rate of decrease is described by a weight function that depends on the vector of parameters ? which, in the adaptive setting, is typically unknown. For any of the classes A(?) , ? fixed, we describe minimax estimators up to a constant as the bin-width goes to zero. Conditions under which an adaptive study is suitable are presented and a notion of adaptive asymptotic optimality is introduced based on distinguishing, among all possible functional scales, between the so-called non-parametric (NP) and pseudo-parametric (PP) scales. We propose adaptive estimators which ‘tune up’ point-wisely to the unknown smoothness of f. We prove them to be asymptotically adaptively minimax for large collections of NP functional scales, subject to being rate efficient for any of the PP functional scales.

We present a first experimental demonstration of classical CW laser condensation in the frequency (mode) domain. It also sheds light on the general question of photon-BEC (BoseEinstein condensation) in laser cavities.

Conditions are found under which d-dimensional linear interpolating spaces \(\mathcal{L}_d \) generated by the classical Abel-Jacobi elliptic functions contain constants and are variance optimal with respect to the equidistant d-design χ d . The so-called modulus parameter k = k(d) of the Abel-Jacobi functions is assumed to belong to (−1, 1) ∪ i R. The spaces \(\mathcal{L}_d \) are optimal if k(d) is restricted to i R when d is odd, with no such restriction needed when d is even.

A new notion of universally optimal experimental design is introduced, relevant from the perspective of adaptive nonparametric estimation. It is demonstrated that both discrete and continuous Chebyshev designs are universally optimal in the problem of fitting properly weighted algebraic polynomials to random data. The result is a direct consequence of the well-known relation between Chebyshev’s polynomials and the trigonometric functions.

We present a realization and first experimental results of a conceptual d-dimensional laser mode lattices (mode hyper-combs). They are constructed from regular 1-dimensional combs by multi-frequency modulation in active mode-locking (AML). The hyper-comb, with near neighbor mode interaction and noise functioning as temperature, is mapped to interacting magnetic spin-lattices in the spherical-model, which is one of the few statistical-mechanics systems soluble in all dimensions. Such systems have in d>;2 dimensions, a phase-transition to a global mode-phase-ordered hyper-comb. Such lasers can have unique properties for generating short pulses compared to regular AML, by capturing very broad frequency bandwidths. Additionally, the hyper-combs can serve as a rare physical realization of the spherical-model in any dimension. We report on preliminary experimental results on the pulsation transition in "three-dimensional" AML.

We demonstrate a switchable multi-wavelength ring fiber laser based on cavity-resonance-activation. It is obtained by addressing the suitable mode-locking frequency of a multi-length laser cavity formed by successive fiber Bragg grating mirrors with different reflecting wavelengths. In the experiment, we used for the reflection sampled fiber Bragg gratings (SFBG). Each SFBG has several equally spaced reflection lines, which determine the lasing wavelengths. The RF modulation frequencies that correspond to the various cavity lengths for the different SFBGs are slightly different. At each wavelength, the pulses consist of several sidebands spaced by the frequency modulation or its harmonics. The laser was shown to operate either in a single switchable wavelength, with its sidebands, or simultaneously at several wavelengths. Such lasers can be useful in optical communication systems and fiber sensors.

The global lower bound for the minimax risk proposed in Part I [12] is applied to the pointwise estimation of functions in the white Gaussian noise, under the squared losses. Some general ellipsoidal and cuboidal functional classes are discussed, including classes of entire functions of exponential type, Paley-Wiener classes of analytic functions, Sobolev classes and their modifications. Based on the proposed risk bounds, a numerical comparison of the minimax risks and the linear minimax risks is made. A nonasymptotic comparison of different types of functional classes is facilitated by their respective embeddings provided the classes are properly calibrated. This discussion demonstrates that the commonly perceived notion of a close connection between the smoothness of an unknown function and the accuracy of estimation can be misleading in a nonasymptotic setting. In particular, the notion of optimal rates of convergence, which has dominated nonparametric statistics for the last three decades, may no longer be productive.