Efficient algorithms for computing the 2-D hexagonal Fouriertransforms

In this paper, representations of the two-dimensional (2-D) signals are presented that reduce the computation of 2-D discrete hexagonal Fourier transforms (2-D DHFTs). The representations are based on the concept of the covering that reveals the mathematical structure of the transforms. Specifically, a set of unitary paired transforms is derived that splits the 2-D DHFT into a number of smaller one-dimensional (1-D) DFTs. Examples for the 8×4 and 16×8 hexagonal lattices are described in detail. The number of multiplications required for computing the 8×4- and 16×8-point DHFTs are equal 20 and 136, respectively. In the general N⩾8 case, the number of multiplications required to compute the 2N×N-point DHFT by the paired transforms equals N² (log N-1)+N     

First-order approximation of the ordered binary-symmetric channel

Soft-decision decoding algorithms of binary linear block codes require reordering of the received symbols within each block in decreasing reliability. Efficient decoding algorithms based on reordering have been devised. This paper presents different results related to the ordering of a sequence of N received symbols with respect to their reliability measure, for BPSK transmission over the AWGN channel model. First, a tight approximation of Pe(i; N), the probability that the hard decision associated with the ith symbol of the ordered sequence is in error, is derived. Then, it is shown that despite the fact that the random variables representing the noise at positions n₁, n₂,...,n_j of the ordering are no longer independent, the events of having a hard decision decoding error at these positions remain almost independent. Pe(n₁,n₂ ,...,n_j; N), the probability that the hard decisions associated with the symbols at positions n₁, n₂,...n_j in the ordered sequence are in error, is thus well approximated from each of the Pe (n_i; N), for i∈[1,j]. Finally, based on the independence of these events, the fully connected 2^N-state BSC representing the channel after ordering is simplified by N independent time-shared 2-state BSCss. This new model allows one to easily and tightly approximate the capacity of the channel after ordering     

Superimposed codes based on punctured convolutional codes

Punctured convolutional codes of rates k₁/n and k₂ /n are applied to |u|u+v construction, and then a superimposed code of rate (k₁+k₂)/(2n) is constructed. A suboptimal decoding procedure is presented for the superimposed codes, and it reduces the decoding complexity as compared with maximum likelihood decoding for the known convolutional codes     

Dopant dependence of effective nonlinear refractive index in GeO2- and F-doped core single-mode fibers

We describe the dopant dependence of the effective nonlinear refractive index n_2eff in GeO₂- or F-doped core fibers theoretically and experimentally. We show that the dopant dependence of the nonlinear refractive index n₂ of F-doped bulk glass is the inverse of that of GeO₂-doped bulk glass. We also show that the effective nonlinear refractive index n_2eff in F-doped core fibers, estimated by using the dopant content dependence of n₂, is in good agreement with our experimental results. Moreover, we confirm that n_2eff in an optical fiber strongly depends on the refractive index profile of its core     

A nonseparable VLSI architecture for two-dimensional discreteperiodized wavelet transform

A modified two-dimensional (2-D) discrete periodized wavelet transform (DPWT) based on the homeomorphic high-pass filter and the 2-D operator correlation algorithm is developed in this paper. The advantages of this modified 2-D DPWT are that it can reduce the multiplication counts and the complexity of boundary data processing in comparison to other conventional 2-D DPWT for perfect reconstruction. In addition, a parallel-pipeline architecture of the nonseparable computation algorithm is also proposed to implement this modified 2-D DPWT. This architecture has properties of noninterleaving input data, short bus width request, and short latency. The analysis of the finite precision performance shows that nearly half of the bit length can be saved by using this nonseparable computation algorithm. The operation of the boundary data processing is also described in detail. In the three-stage decomposition of an N×N image, the latency is found to be N²+2N+18     

Fast algorithms for 2-D circular convolutions and number theoretictransforms based on polynomial transforms over finite rings

We develop new fast algorithms for 2-D integer circular convolutions and 2-D number theoretic transforms (NTT). These new algorithms, which offer improved computational complexity, are constructed based on polynomial transforms over Z_p; these transforms are Fourier-like transforms over Z_p, which is the integral domain of polynomial forms over Z_p[x]. Having defined such polynomial transforms over Z_p we prove several necessary and sufficient conditions for their existence. We then apply the existence conditions to recognize two applicable polynomial transforms over Z_p. One is for p equal to Mersenne numbers and the other for Fermat numbers. Based on these two transforms, referred to as Mersenne number polynomial transforms (MNPT) and Fermat number polynomial transforms (FNPT), we develop fast algorithms for 2-D integer circular convolutions, 2-D Mersenne number transforms, and 2-D Fermat number transforms. As compared to the conventional row-column computation of 2-D NTT for 2-D integer circular convolutions and 2-D NTT, the new algorithms give rise to reduced computational complexities by saving more than 25 or 42% in numbers of operations for multiplying 2 ⁱ, i⩾1; these percentages of savings also grow with the size of the 2-D integer circular convolutions or the 2-D NTT     

Nondegenerate optical Kerr effect in semiconductors

We calculate the nondegenerate bound electronic nonlinear refractive index n₂(ω₁;ω₂) (i.e., an index change at frequency ω₁ due to the presence of a beam at frequency ω₂) in semiconductors. We calculate this nonlinearity and its dispersion using a Kramers-Kronig transformation on the calculated nondegenerate nonlinear absorption spectrum due to two-photon absorption, electronic Raman and optical Stark effects. The calculated n₂ values and their dispersion are compared to new experimental values for ZnSe and ZnS obtained using a 2-color Z-scan     

Half-Delay B-Spline Filter for Construction of Shift-Invariant Wavelet Transform

We introduce a general framework for the shift-invariant biorthogonal wavelet transform. The method is based on the two parallel wavelet transforms, where the wavelets form a Hilbert transform pair. This condition requires that the impulse responses of the scaling filters are half-delayed versions of each other:h_o [n_] and h_o [n-1/2_]. The ideal half-delay operator is constructed by the interpolation and decimation procedure based on the polyphase decomposition of the two-scale B-spline equation. The present method yields linear phase and shift-invariant wavelet transform coefficients and can be adapted to any of the existing biorthogonal DWT filter bank.     

Quadtree and symmetry in FFT computation of digital images

The discrete Fourier transform (DFT) of a real sequence f[x, y] of size N×N, where N=2ⁿ, can be computed by a two-dimensional (2-D) FFT of size N/4, or smaller if f[x, y] is known to have certain symmetries. This paper presents theorems that identify the symmetry in f[x, y] based on the depth of the quadtree to expedite 2-D FFT computation of coherent digital images. In principle, it establishes that if the quadtree of f[x, y] has maximum depth kn , then the DFT can be computed by a 2-D FFT of size K/2. An algorithm is given, and its performance analyzed. Finally, applications are considered in transform coding systems and lossy compression of images     

Universal conditions for estimating the nonlinear refractive indexn2 of dispersion-compensating fibers by the CW-SPMmethod

We investigated universal conditions for measuring the nonlinear refractive index n₂ of dispersion-compensating fibers by the continuous wave self-phase modulation (CW-SPM) method both theoretically and experimentally. We showed that the required fiber length and the minimum power for n₂ estimation can be expressed as simple functions in terms of chromatic dispersion D. Moreover, we confirmed experimentally that the present conditions for n₂ estimation are reasonable     

H2+和D2+谐波强度与波长的关系

为了了解H₂+及其同位素分子谐波光谱效率与激光波长之间的关系,采用求解2维薛定谔方程的方法,理论研究了600nm~1600nm激光波长下H₂+和D₂+谐波光谱强度随波长的变化关系。结果表明,光谱强度随波长增大而减小;在短波长区间,H₂+光谱强度减小的倍率要大于D₂+,在长波长区间,H₂+光谱强度减小的倍率要小于D₂+;此外,在弱光强下,H₂+光谱强度总是大于D₂+, 在强光强下,H₂+光谱强度在短波长区间小于D₂+, 而其在长波长区间大于D₂+; 核间距延伸和电荷共振增强电离在H₂+和D₂+谐波光谱强度变化上起到主要作用。这一结果对分子谐波调控是有帮助的。     

The nonlinear optical properties of AlGaAs at the half band gap

We report experimental values for the nonlinear optical coefficients of AlGaAs, in the half-band-gap spectral region. The dispersion of the nonlinear refractive-index coefficient, n₂, is measured for both TE- and TM-polarized light. We observe n₂(TE)>n₂(TM) and a ratio of cross-phase modulation to self-phase modulation (TE) of ~0.95, as predicted from band structure calculations. The spectral dependence of the two- and three-photon absorption coefficients are also measured. Finally, the implications for all-optical switching and spatial soliton propagation are discussed     

Optical bistability in reflective fiber gratings

Optical bistability in a nonlinearly reflective fiber grating through the mode coupling between the LP₀₁ and counter propagation LP₀₂ modes (i,e., the reflective LP₀₁&rlhar2;LP₀₂ mode converter) is investigated by using the coupled-mode theory. Both the transmissive and the reflective properties of this nonlinear device are analyzed, which show that the output-versus-input relation exhibits the optical bistability. The switching power and the bistable hysteresis performances are also discussed. In addition, the comparison between the nonlinearly reflective mode converter (LP₀₁&rlhar2;LP₀₂) and the nonlinear fiber Bragg grating reflector (i,e., the LP₀₁ and counter-propagation LP₀₁ modes) is also presented. It is shown that the former has much lower switching power than the latter     

Radix-3$,times,$3 Algorithm for The 2-D Discrete Hartley Transform

In this correspondence, we propose a vector-radix algorithm for the fast computation of a 2-D discrete Hartley transform (DHT). For data sequences whose length is a power of three, a radix-3 times 3 decimation in frequency algorithm is developed. It decomposes a length-N times N DHT into nine length-(N/3) times N (N/3) DHTs. Comparison of the computational complexity with known algorithms shows that the proposed algorithm, in some cases, reduces significantly the number of arithmetic operations.     

Scalable and modular memory-based systolic architectures for discrete Hartley transform

In this paper, we present a design framework for scalable memory-based implementation of the discrete Hartley transform (DHT) using simple and efficient systolic and systolic-like structures for short and prime transform lengths, as well as, for lengths 4 and 8. We have used the proposed short-length structures to construct highly modular architectures for higher transform lengths by a new prime-factor implementation approach. The structures proposed for the prime-factor DHT, interestingly, do not involve any transposition hardware/time. Besides, it is shown here that an N-point DHT can be computed efficiently from two (N/2)-point DHTs of its even- and odd-indexed input subsequences in a recursive manner using a ROM-based multiplication stage. Apart from flexibility of implementation, the proposed structures offer significantly lower area-time complexity compared with the existing structures. The proposed schemes of computation of the DHT can conveniently be scaled not only for higher transform lengths but also according to the hardware constraint or the throughput requirement of the application.     

Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids

The bound electronic nonlinear refractive index, n₂, and two-photon absorption (2PA) coefficient, β, are measured in a variety of inorganic dielectric solids at the four harmonics of the Nd:YAG laser using Z scan. The specific materials studied are: barium fluoride (BaF₂), calcite (CaCO₃), potassium bromide (KBr), lithium fluoride (LiF), magnesium fluoride (MgF₂ ), sapphire (Al₂O₃), a tellurite glass (75%TeO₂+20%ZnO+5%Na₂O) and fused silica (SiO₂). We also report n₂ and β in three second-order, χ⁽²⁾, nonlinear crystals: potassium titanyl phosphate (KTiOPO₄ or KTP), lithium niobate (LiNbO₃), and β-barium berate (β-BaB₂O ₄ or BBO). Nonlinear absorption or refraction can alter the wavelength conversion efficiency in these materials. The results of this study are compared to a simple two-parabolic band model originally developed to describe zincblende semiconductors. This model gives the bandgap energy (E_g) scaling and spectrum of the change in absorption. The dispersion of nl as obtained from a Kramers-Kronig transformation of this absorption change scales as E_g^-1 . The agreement of this theory to data for semiconductors was excellent. However, as could be expected, the agreement for these wide bandgap materials is not as good, although general trends such as increasing nonlinearity with decreasing bandgap energy can be seen     

Fast splitting alpha-rooting method of image enhancement: tensor representation.

In the tensor representation, a two-dimensional (2-D) image is represented uniquely by a set of one-dimensional (1-D) signals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific sets that cover the whole domain of frequencies. The image enhancement is thus reduced to processing splitting-signals and such process requires a modification of only a few spectral components of the image, for each signal. For instance, the alpha-rooting method of image enhancement can be fulfilled through processing separately a maximum of 3N/2 splitting-signals of an image (N x N), where N is a power of two. In this paper, we propose a fast implementation of the a-rooting method by using one splitting-signal of the tensor representation with respect to the discrete Fourier transform (DFT). The implementation is described in the frequency and spatial domains. As a result, the proposed algorithms for image enhancement use two 1-D N-point DFTs instead of two 2-D N x N-point DFTs in the traditional method of alpha-rooting.     

Intersubband χ3 in coupled InGaAs-AlGaAs multiplequantum wells

The third-order nonlinearity, χ³(ω,ω,-ω), is measured for a mid-infrared intersubband transition in strained InGaAs-AlGaAs multiple quantum wells (MQW's). The high conduction band offset of this system allows an intersubband transition at 3.1 μm. The level structure of the quantum well is designed to include a meta-stable trapping level, resulting in a peak saturation intensity of 6 MW/cm² at Brewster's angle, approximately 20 times lower than would be found in a square quantum well with similar linewidth. A near-resonant n₂ of 8.4×10^-7 cm²/W at 3.1 μm is calculated. The off-resonant n₂ is also calculated and shown to be attractive at wavelengths as short as 1.55 μm     

Stimulated coherent Smith-Purcell radiation from a metallic grating

A relativistic electron beam with velocity ν_bz passing over a metallic grating, with deep periodic depressions of wavenumber k₀z, excites a slow wave (ω, k_z) via Cerenkov interaction, ω=(k_z+k₀)ν_b. The beam bunches induce currents in the fins of the grating that act as a phased radiating dipole array, thereby generating coherent radiation at a wavelength λ=(2π/k₀l)(c/ν_b-sin&thetas;) where l is an integer and &thetas; is the angle of emission measured from the normal of the grating. The current threshold for the instability and radiation power scaling with beam current are discussed     

High-field transport with hot phonons in degenerate semiconductors

The effect of phonons in non-thermodynamic equilibrium (hot-phonons) on steady-state transport in degenerate bulk and two-dimensional (2D) semiconductor structures is studied. The phonon-drift effects are taken to be negligible, and, the formulation is confined to electrons in a single parabolic band interacting with a - population of polar-optic phonons, restricted to interface modes in the 2D case. Hot phonons in bulk semiconductors lead not only to a decrease in the energy relaxation rate, but, also, to a decrease in the mobility as the carrier concentration n_3D is increased. Hot phonons in 2D systems, on the other hand, cause the energy relaxation rate and mobility to go through a minima as the electron concentration n_2D is increased. The electron drift velocity, contrary to the bulk case, is seen to with n_2D for n_2D > 10¹² cm⁻², and reflects the difference in the nature of electron density of states in the two cases.