一个不等式

命题若0<λ≤2,则对所有正实数a、b,有aa λb bb λa≤21 λ(1)证明(1)式等价于11 λba 11 λab≤21 λ(2)令x=ab,y=ba,则(2)式等价于:x>0,y>0,xy=1,有11 λx 11 λy≤21 λ(3)(3)式等价于11 λx 11 λy 2(1 λx)(1 λy)≤41 λ2(1 λx)(1 λy)≤41 λ-11 λx-11 λy2(1 λ)(1     

Bifurcated overtones of one-way localized Fabry-Pérot resonances in parity-time symmetric optical lattices

Since the first observation of parity-time(PT) symmetry in optics, varied interesting phenomena have been discovered in both theories and experiments, such as PT phase transition and unidirectional invisibility, which turns PT-symmetric optics into a hotspot in research. Here, we report on the one-way localized Fabry-Pérot(FP) resonance, where a welldesigned PT optical resonator may operate at exceptional points with bidirectional transparency but unidirectional field localization. Overtones of such one-way localized FP resonance can be classified into a blue shifted branch and a red shifted branch. Therefore, the fundamental resonant frequency is not the lowest one. We find that the spatial field distributions of the overtones at the same absolute order are almost the same, even though their frequencies are quite different.     

Optical total reflection and transmission with mode control in a dielectric subwavelength nanorod chain

We report a polarization dependent reflection phenomenon beyond the normal incidence with a subwavelength nanorod chain.Light waves of the transverse electric mode will be totally reflected while those of the transverse magnetic mode will transmit through.The total reflection or transmission phenomenon can be seen as a constructive or destructive interference of the incident field with the transverse mode field of the chain.With the low-loss feature and the ultra-compact characteristic,this structure may find applications in photonic circuits.     

也谈Minc-Sathre不等式的上下界

H．Minc—L．Sathre在1964年证明了下面的不等式：对于一切自然数n，有     

High efficiency grating couplers based on shared process with CMOS MOSFETs

Grating couplers are widely investigated as coupling interfaces between silicon-on-insulator waveguides and optical fibers.In this work,a high-efficiency and complementary metal-oxide-semiconductor(CMOS) process compatible grating coupler is proposed.The poly-Si layer used as a gate in the CMOS metal-oxide-semiconductor field effect transistor(MOSFET) is combined with a normal fully etched grating coupler,which greatly enhances its coupling efficiency.With optimal structure parameters,a coupling efficiency can reach as high as ～ 70% at a wavelength of 1550 nm as indicated by simulation.From the angle of fabrication,all masks and etching steps are shared between MOSFETs and grating couplers,thereby making the high performance grating couplers easily integrated with CMOS circuits.Fabrication errors such as alignment shift are also simulated,showing that the device is quite tolerant in fabrication.     

Self-Collimation in Planar Photonic Crystals Fabricated by CMOS Technology

We report a self-collimating demonstration in planar photonic crystals （PhCs） fabricated in silicon-on-insulator （SOI） wafers using 0.18 μm silicon complimentary metal oxide semiconductor （CMOS） techniques. The emphasis was on demonstrating the self-collimation effect by using the standard CMOS equipment and process development of an optical test chip using a high-volume manufacturing facility. The PhCs are designed on the 230-nm-top-Si layer using a square lattice of air holes 280 nm in diameter. The lattice constant of the PhCs is 380 nm. The experimentally obtained wavelengths for self-collimation are in excellent agreement with theory.     

一个不等式的加强

贵刊2004年10月号问题1519为:ma、wa分别表示△ABC的a边上的中线和角平分线长,求证mawa≥2b bcc.(1)该不等式可加强为:mawa≥b22 bcc2.(2)事实上,由恒等式m2a=w2a 21(b-c)2-41b ac2(b-c)2,w2a=bc(b c( b a )(c)b2 c-a)得m2a≥w2a 21(b-c)2-21b ac2(b-c)2=w2a 21(b-c)2·(b c (b     

从威森波克不等式的证明谈起

从威森波克不等式的证明谈起武爱民（河南鹤壁四矿中学４５８０１０）威氏不等式：ａ２＋ｂ２＋ｃ２４３△（其中ａ，ｂ，ｃ和△分别为△ＡＢＣ的边和面积）．目前人们已发现了它的十多种证法，而且被加强为ａ２＋ｂ２＋ｃ２４３△＋（ａ－ｂ）２＋（ｂ－ｃ）２＋（ｃ...