Focal shift and focal switch of partially polarized Gaussian Schell-model beams passing through a system with the aperture and spherically aberrated lens separated

Based on the beam coherent-polarization (BCP) matrix approach and propagation law of partially coherent beams, the focal shift and focal switch of partially polarized Gaussian Schell-model (PGSM) beams passing through a system with the aperture and spherically aberrated lens separated is studied in detail. Our main attention is focused on the effect of spherical aberration and partial coherence on the focal shift and focal switch of PGSM beams. It is shown that for polarizer-free case there is no focal switch of PGSM beams, the focal shift of PGSM beams is closely related with spherical aberration coefficient C₄, auto-coherence length σ_a, truncation parameter δ and relative position s/f between the aperture and lens in general, and is independent of the cross-coherence length σ_c. After inserting a polarizer the focal switch can take place. Numerical calculation results are given to illustrate how the spherical aberration and partial coherence affect the focal shift and focal switch of PGSM beams.     

光谱开关与多色光场的奇点光学效应

推导出多色高斯光束和高斯-谢尔模型光束通过杨氏实验双缝传输的谱强度公式.对完全空间相干光和部分空间相干光照明杨氏实验装置出现的光谱开关作了详细研究,并判断其是否属于奇点光学效应. 结果表明：多色场奇点光学效应的判据应当是光谱开关出现时的谱强度极小值S_min=0,而不是总光强极小值I_min=0. 当用多色高斯-谢尔模型光束照明杨氏实验装置时,在近场和远场产生的光谱开关都不属于奇点光学效应. 当用多色高斯光束照明杨氏实验装置时,只有远场产 关键词： 光谱开关 奇点光学效应 完全相干和部分相干 多色光场     

Spectral shifts and spectral switches of spatially and spectrally partially coherent pulsed beams in Young’s interference experiment

Taking the Gaussian Schell-model pulsed (GSMP) beam as a typical example of spatially and spectrally partially coherent pulsed beams, an analytical expression for the far-field spectrum of diffracted GSMP beams in Young’s interference experiment is derived, and used to study the spectral shifts and spectral switches of GSMP beams in the far field. Numerical calculation results are given to illustrate the dependence of spectral shifts and spectral switches on the obscuration ratio ε, temporal coherent length T_c, spatial correlation parameter β and diffraction angle α. It is shown that the critical angle α_c of the first- and second-order spectral switches moves away from the z axis with increasing T_c, but α_c of the third-order spectral switch moves towards the z axis with increasing T_c. The spectral transition height Δ decreases and spectral minimum S_min increases as T_c increases. α_c decreases, Δ increases and S_min decreases as β increases. The potential application of spectral switches of spatially and spectrally partially coherent pulsed beams in information encoding and transmission is proposed.     

Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture

The focal shift and focal switch of Bessel–Gaussian (B–G) beams passing through a lens system with or without aperture is studied in detail. For the unapertured case, the necessary condition for the focal switch and the expression for the amplitude of the focal switch are derived. It is shown that if the truncation parameter δδ_c or beam parameter ββ_c (δ_c, β_c are the corresponding critical values), there exits only one axial intensity maximum, and the focusing without focal shift can be achieved for u/f=1 (u is the separation between the aperture and lens, f is the focal length of the lens); if δ>δ_c and β>β_c, there exist two axial intensity maxima, and the focal switch can take place at the turning point u/f=1 for both apertured and unapertured cases. The dependence of the amplitude of focal switch and normalized axial intensity minimum on the truncation parameter, beam parameter and Fresnel number is also studied.     

Focal switch of cosine-Gaussian beams

The focal switch of cosine-Gaussian (CsG) beams passing through a system with the aperture and lens separated is studied analytically and numerically. It is shown that the focal switch of CsG beams can appear not only for the apertured case, but also for the unapertured case. The necessary condition for the focal switch is that truncation parameter α > α_c and the beam parameter β > β_c, α_c, β_c being the corresponding critical values. There exists a maximum of the relative transition height Δz _sw as α varies, and Δz _sw increases with increasing β and decreasing N _w. The normalized axial intensity minimum I _min / I _max decreases with an increase of α and β, and I _min / I _max remains unchanged as N _w varies.     

Effect of quartic-phase aberrations on the focal switch of Hermite–Gaussian beams

The focal switch of Hermite–Gaussian beams diffracted at an aperture and subsequently focused by a spherically aberrated lens is studied. Our main attention is focused on the effect of quartic-phase aberrations on the behavior of the focal switch. It is shown that quartic-phase aberrations affect the relative focal shift Δz_f, turning position s_1,t, and relative transition height Δz_sw. Apart from a critical maximum truncation parameter α_c,max, there is a critical minimum truncation parameter α_c,min. Within the region α_c,min<α<α_c,max the focal switch can take place, but quartic-phase aberrations give rise to a decrease of α_c,max-α_c,min in comparison with the aberration-free case.     

Influence of the spectral power distribution of a LED on the illuminance responsivity of a photometer

The measurement accuracy in the photometric quantities measured through photometer head is determined by the value of the spectral mismatch correction factor (c(S_t,S_s)), which is defined as a function of spectral power distribution of light sources, besides illuminance responsivity of the photometer head used. This factor is more important when photometric quantities of the light-emitting diode (LED) style optical sources, which radiate within relatively narrow spectral bands as compared with that of other optical sources, are being measured. Variations of the illuminance responsivities of various V(λ)-adopted photometer heads are discussed. High-power-colored LEDs, manufactured by Lumileds Lighting Co., were used as light sources and their relative spectral power distributions (RSPDs) were measured using a spectrometer-based optical setup. Dependences of the c(S_t,S_s) factors of three types of photometer heads (f₁′=1.4%, f₁′=0.8% and f₁′=0.5%) with wavelength and influences of the factors on the illuminance responsivities of photometer heads are presented.     

Spectral shifts and spectral switches of partially coherent vector beams propagating through an astigmatic aperture lens

By using the polarization matrix and propagation equation of the cross-spectral density function matrix, the spectral shifts and spectral switches of partially coherent vector beams propagating through an astigmatic aperture lens are studied, where the effect of lens astigmatism and beam cross-correlation on the spectral switches is stressed. It is shown that the critical position, spectral transition height and spectral minimum at spectral switches depend on the astigmatism. The beam cross-correlation plays a role in spectral switches under the condition that a polarizer is inserted. The spectral switch vanishes when the astigmatic coefficient is larger than a certain value, or the cross-correlation length is less than a certain value.     

Spectral changes of diffracted polychromatic Gaussian beams in the far field

The spectral behavior of polychromatic spatially fully coherent Gaussian beams diffracted at an aperture in the far field is studied based on the propagation law of the cross-spectral density function. Detailed numerical calculation results are presented, and illustrated. It is shown that the spectral changes take place in the vicinity of zeros of the Airy pattern, and the spectral modifier depends on the truncation parameter δ and diffraction angle α. The critical diffraction angle α_c, at which the spectrum is split into two lines with equal height i.e., the spectral switch appears, varies with truncation parameter δ. The spectral switch vanishes if δ becomes large enough. A detailed comparison with the previous publications is given, showing somewhat extension made in our work.     

Propagation of spectral Stokes singularities of stochastic electromagnetic beams through atmospheric turbulence

The propagation of spectral Stokes singularities (vortices) of stochastic electromagnetic vortex beams through atmospheric turbulence is studied, where the electromagnetic Gaussian Schell-model (GSM) vortex beam is taken as an illustrative example. It is shown that the spectral Stokes vortices S ₁₂ (C-points), S ₂₃ and S ₃₁ introduced to describe the polarization singularities of stochastic electromagnetic beams appear in turbulence. The motion, creation, annihilation and polarization changes of S ₁₂, S ₂₃ and S ₃₁ vortices, as well as the handedness inversion of S ₁₂ vortices may appear as the propagation distance or one beam parameter varies. In the process the topological relationship holds true. In comparison with the free-space propagation, the variation of the refractive index structure constant C_n²C_{n}^{2} in atmospheric turbulence results in similar effects as above. The dependence of S ₁₂, S ₂₃ and S ₃₁ vortices on the propagation distance and beam and turbulence parameters are illustrated by numerical examples.     

Burning velocity correlation of methane/air turbulent premixed flames at high pressure and high temperature

Turbulent burning velocities for methane/air mixtures at pressures ranging from atmospheric pressure up to 1.0 MPa and mixture temperatures of 300 and 573 K were measured, which covers the typical operating conditions of premixed-type gas-turbine combustors. A bunsen-type flame stabilized in a high-pressure chamber was used, and OH-PLIF visualization was performed with the pressure and mixture temperature being kept constant. In addition to a burner with an outlet diameter of 20 mm for the high-pressure experiments, a large-scale burner with an outlet diameter of 60 mm was used at atmospheric pressure to extend the turbulence Reynolds number based on the Taylor microscale, R_λ, as a common parameter to compare the pressure and temperature effects. It was confirmed that R_λ over 100 could be attained and that u′/S_L could be extended even at atmospheric pressure. Based on the contours of the mean progress variable c = 0.1 determined using OH-PLIF images, turbulent burning velocity was measured. S_T/S_L was also found to be greatly affected by pressure for preheated mixtures at 573 K. The bending tendency of the S_T/S_L curves with u′/S_L was seen regardless of pressure and mixture temperature and the R_λ region where the bending occurs corresponded well to the region where the smallest scale of flame wrinkling measured as a fractal inner-cutoff approaches the characteristic flame instability scale and becomes almost constant. A power law of S_T/S_L with (P/P₀)(u′/S_L) was clearly seen when S_T was determined using c = 0.1 contours, and the exponent was close to 0.4, indicating agreement with the previous results using the mean flame cone method and the significant pressure effects on turbulent burning velocity.     

Far-field properties and beam quality of vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation

The far-field properties and beam quality of vectorial nonparaxial Hermite–Laguerre–Gaussian (HLG) beams are studied in detail, where, instead of the second-order-moments-based M² factor, the extended power in the bucket (PIB) and βparameter are used to characterize the beam quality in the far field and the intensity in the formulae is replaced by the z component of the time-averaged Poynting vector S_z. It is found that the S_z PIB and βparameter of vectorial nonparaxial HLG beams depend on the mode indices n, m, αparameter and waist-width-to-wavelength ratio w₀/λ and the PIB and βparameter are additionally dependent on the bucket's size taken.     

Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence

The closed-form expression for the angular spread of Gaussian Schell-model (GSM) array beams propagating through atmospheric turbulence is derived. It is shown that the angular spread θ _sp of GSM array beams for the superposition of the cross-spectral density function is smaller than of those for the superposition of the intensity. However, the θ _sp of GSM array beams for the superposition of the intensity is less sensitive to turbulence than that for the superposition of the cross-spectral density function. For the superposition of the cross-spectral density function, θ _sp of GSM array beams with smaller coherence length σ ₀, smaller waist width w ₀, smaller beam number N, and larger separation distance x _d are less affected by turbulence than of those with larger σ ₀,w ₀,N, and smaller x _d ; while, for the superposition of the intensity, the effect of turbulence on θ _sp is independent of N and x _d . In addition, the angular spread is nearly the same for the two types of superposition when σ ₀ or w ₀ is small enough, or x _d is large enough. On the other hand, it is found that there exist equivalent GSM array beams for the two types of superposition which may have the same directionality as the corresponding fully coherent Gaussian beam in free space and also in turbulence.     

Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue

Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in L^p spaces of scattering solutions e^−itHP_cu, where P_c is the orthogonal projection onto the continuous spectral subspace of L²(R³) for H. Under suitable decay assumptions on V(x) it is shown that they satisfy the so-called L^p-L^q estimates ||e^−itHP_cu||_p≤(4π|t|)^{−3(1/2−1/p)}||u||_q for all 1≤q≤2≤p≤∞ with 1/p+1/q=1 if H has no threshold resonance and eigenvalue; and for all 3/2<q≤2≤p<3 if otherwise.     

Decay of Bc (3S) → B D

The decay constant for the vector state of 3S-level in the heavy (

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c)-quarkonium is evaluated in the framework of sum rules for the mesonic currents. A scaling relation for the constants of vector quarkonia with different quark contents is derived. The numerical estime gives Γ (B^*+_c(3S) → B⁺ D⁰) = 90 ± 35 MeV.     

Strong atomic displacements around nitrogen in tantalum determined by energy-dispersive x-ray diffraction

The nitrogen induced attenuation of Bragg reflections (static Debye-Waller Factor) of tantalum single crystals containing nitrogen has been measured by energy-dispersive x-ray diffraction. N has been dissolved in Ta up to a concentration ofc=3.3% N/Ta. From a detailed study of the concentration andK dependence (K=scattering vector) of the static Debye-Waller Factor exponent 2L we conclude:i) 2L increases linearly with the nitrogen content forc1.8% N/Ta,ii) The displacements of the atomic shells due to N on the octahedral sites areu ₁=0.45 (5) Å,u ₂=(–)0.11 (2) Å.u ₃=0.25 (2) Å,iii) The thermal vibrations of those atoms close to N are changed, due to the strong static displacements that they experience. The unusual strong and extended displacements around N in Ta are in contradiction to calculations where two-force models have been used. We propose arguments that displacements are responsible for the specific site chosen by hydrogen atoms when they are trapped by nitrogen in Ta.     

The AΠu state of : Electric properties, fine and hyperfine coupling constants, and magnetic moments (g-factors). A theoretical study

Several properties are calculated for A²Π_u of —the majority for the first time—including electric and magnetic moments, and fine/hyperfine structure (fs/hfs) parameters. The new results are compared with our previous ones for X² and B² of [P.J. Bruna, F. Grein, J. Mol. Spectrosc. 227 (2004) 67–80]. The electric quadrupole Θ and hexadecapole Φ moments, polarizability α, and hfs constants a, b, c, d, eQq₀, eQq₂ are evaluated at the density functional theory (DFT) level [B3LYP/aug-cc-pVQZ]. The fs constants (spin–orbit coupling A_Π, Λ-doubling p, q, spin-rotation γ_Π), and magnetic moments (g-factors) are obtained via 2nd-order sum-over-states expansions, using wavefunctions and matrix elements obtained with a multireference configuration interaction (MRDCI) method, and the Breit–Pauli Hamiltonian. At equilibrium, 2nd-order properties of A²Π_u are dominated by its coupling with B². For the A state, two independent components are reported for traceless tensor properties (multipoles Θ and Φ; hfs parameters c/d and q₀/q₂) and three for traced properties (polarizability α and g-factors), i.e., one more component than for axially symmetric Σ states. The currently available experimental data on — limited to A_Π, p, and q—are well reproduced by our theoretical results.     

Some exact solutions to the sine-Gordon equations

We present some new exact (solitary and traveling) solutions to the sine-Gordon equationsc ²u_xx–u_tt = sinu andcu _xt+ u_tt = sinu, and the dissipative sine-Gordon equationc ²u_xx–u_tt–u_t=₁ sinu+ ₂ sin(2u). Here,, ₁, and ₂ are the coupling constants, real numbers;c is theintrinsic speed of the propagating wave determined by the system; and is the damping factor. The physical implication is briefly discussed.     

Constraints on hadronic spectral functions from continuous families of finite energy sum rules

Hadronic τ decay data is used to study the reliability of various finite energy sum rules (FESR's). For intermediate scales (s₀2–3 GeV²), those FESR's with weights s^k are found to have significant errors, whereas those with weights having a zero at the juncture (s=s₀) of the cut and circular part of the contour work very well. It is also shown that a combination of two such sum rules allows rather strong constraints to be placed on the hadronic spectral function without sacrificing the excellent agreement between OPE and hadronic representations. The method then is applied to the strangeness S=−1 and pseudoscalar isovector channels, where it is shown to provide novel constraints on those ansatze for the unmeasured continuum parts of the spectral functions employed in existing extractions of m_u+m_d, m_s. The continuum ansätz in the latter channel, in particular, is shown to produce a rather poor match to the corresponding OPE representation, hence re-opening the question of the value of the light quark mass combination m_u+m_d.     

Anisotropic optical properties of single-crystal GdBa2Cu3O7−δ

The optical spectrum of reduced-T _c GdBa₂Cu₃O_7– has been measured for polarizations parallel and perpendicular to theab plane. The sample was an oxygen-deficient single crystal with a large face containing thec axis. The polarized reflectance from this face was measured from 20–300 K in the spectral region from 30–3000 cm^–1, with 300 K data to 30000 cm^–1. Kramers-Kronig analysis was used to determine the spectral dependence of theab and thec components of the dielectric tensor. The optical properties are strongly anisotropic. Theab-plane response resembles that of other reduced-T _c materials whereas thec axis, in contrast, shows only the presence of several phonons. There is a complete absence of charge carrier response alongc aboveand belowT _c. This observation allows us to set an upper limit to the free-carrier spectral weight for transport perpendicular to the CuO₂ planes.Permanent address: Institute of Physics, CSAV, Prague, Czechoslovakia