Nonlinear dispersive wave problems

The nonlinear partial differential equations for dispersive waves have special solutions representing uniform wavetrains. An expansion procedure is developed for slowly varying wavetrains, in which full nonlinearity is retained but in which the scale of the nonuniformity introduces a small parameter. The first order results agree with the results that Whitham obtained by averaging methods. The perturbation method provides a detailed description and deeper understanding, as well as a consistent development to higher approximations. This method for treating partial differential equations is analogous to the "multiple time scale" methods for ordinary differential equations in nonlinear vibration theory. It may also be regarded as a generalization of geometrical optics to nonlinear problems.

To apply the expansion method to the classical water wave problem, it is crucial to find an appropriate variational principle. It was found in the present investigation that a Lagrangian function equal to the pressure yields the full set of equations of motion for the problem. After this result is derived, the Lagrangian is compared with the more usual expression formed from kinetic minus potential energy. The water wave problem is then examined by means of the expansion procedure.

Propagation of the fast magnetosonic wave in a tokamak plasma

The propagation of the fast magnetosonic wave in a tokamak plasma has been investigated at low power, between 10 and 300 watts, as a prelude to future heating experiments.

The attention of the experiments has been focused on the understanding of the coupling between a loop antenna and a plasma-filled cavity. Special emphasis has been given to the measurement of the complex loading impedance of the plasma. The importance of this measurement is that once the complex loading impedance of the plasma is known, a matching network can be designed so that the r.f. generator impedance can be matched to one of the cavity modes, thus delivering maximum power to the plasma. For future heating experiments it will be essential to be able to match the generator impedance to a cavity mode in order to couple the r.f. energy efficiently to the plasma.

As a consequence of the complex impedance measurements, it was discovered that the designs of the transmitting antenna and the impedance matching network are both crucial. The losses in the antenna and the matching network must be kept below the plasma loading in order to be able to detect the complex plasma loading impedance. This is even more important in future heating experiments, because the fundamental basis for efficient heating before any other consideration is to deliver more energy into the plasma than is dissipated in the antenna system.

The characteristics of the magnetosonic cavity modes are confirmed by three different methods. First, the cavity modes are observed as voltage maxima at the output of a six-turn receiving probe. Second, they also appear as maxima in the input resistance of the transmitting antenna. Finally, when the real and imaginary parts of the measured complex input impedance of the antenna are plotted in the complex impedance plane, the resulting curves are approximately circles, indicating a resonance phenomenon.

The observed plasma loading resistances at the various cavity modes are as high as 3 to 4 times the basic antenna resistance (~ .4 Ω). The estimated cavity Q’s were between 400 and 700. This means that efficient energy coupling into the tokamak and low losses in the antenna system are possible.

Generation of millimeter-wave in optical pulse carrier by using an apodized Moire fiber grating

A novel scheme is proposed to transform a Gaussian pulse to a millimeter-wave frequency modulation pulse by using an apodized Moire fiber Bragg grating in radio-over-fiber system. The relation between the input and output pulses is analyzed theoretically by Fourier transformation method and the requirements for the proposed fiber grating are presented. An apodized Moire fiber Bragg grating is designed and its characteristics are studied. It is shown that the proposed device is feasible, and the new scheme is believed to be an effective solution for the generation of millimeter-wave sub-carrier in future radio-over-fiber systems. (c) 2006 Elsevier B.V. All rights reserved.

Highly sensitive wave-front sensor with a non-zero-order phase plate

We propose a novel highly sensitive wave front detection method for a quick check of a flat wave front by taking advantage of a non-zero-order pi phase plate that yields a non-zero-order diffraction pattern. When a light beam with a flat wave front illuminates a phase plate, the zero-order intensity is zero. When there is a slight distortion of the wave front, the zero-order intensity increases. The ratio of first-order intensity to that of zero-order intensity is used as the criterion with which to judge whether the wave front under test is flat, eliminating the influence of background light. Experimental results demonstrate that this method is efficient, robust, and cost-effective and should be highly interesting for a quick check of a flat wave front of a large-aperture laser beam and adaptive optical systems. (c) 2005 Optical Society of America.

Quantitative measurement of domain inversion in RuO2 : LiNbO3 crystal by digital holographic interferometry

We quantitatively study the domain inversion in a RuO2:LiNbO3 crystal wafer by the digital holographic interferometry. The crystal wafer is placed into one arm of a Mach-Zehnder-type interferometer to record a series of holograms. Making use of the angular spectrum backward propagation algorithm, we reconstruct the optical wave field in the crystal plane. The extracted phase difference from the reconstructed optical wave field is a well linear function of the applied external voltage. We deduce that the linear electro-optic coefficient of the detected RuO2:LiNbO3 crystal sample is 9.1x10(-12) m/V. An unexpected phase contrast at the antiparallel domain wall is observed and the influence of the applied external voltage on it is studied in detail. Also the built-in internal field is quantitatively measured as 0.72 kV/mm. (c) 2006 American Institute of Physics.

Wavefront conversion by local volume holograms between cylindrical and plane waves with reconstruction at a different readout wavelength

Based on the two-dimensional coupled-wave theory, the wavefront conversion between cylindrical and plane waves by local volume holograms recorded at 632.8 nm and reconstructed at 800 nm is investigated. The proposed model can realize the 90 degrees holographic readout at a different readout wavelength. The analytical integral solutions for the amplitudes of the space harmonics of the field inside the transmission geometry are presented. The values of the off-Bragg parameter at the reconstructed process and the diffracted beam's amplitude distribution are analysed. In addition, the dependences of diffraction efficiency on the focal length of the recording cylindrical wave and on the geometrical dimensions of the grating are discussed. Furthermore, the focusing properties of this photorefractive holographic cylindrical lens are analysed.

Time-dependent monoenergetic neutron transport in two adjacent semi-infinite media

An exact solution to the monoenergetic Boltzmann equation is obtained for the case of a plane isotropic burst of neutrons introduced at the interface separating two adjacent, dissimilar, semi-infinite media. The method of solution used is to remove the time dependence by a Laplace transformation, solve the transformed equation by the normal mode expansion method, and then invert to recover the time dependence.

The general result is expressed as a sum of definite, multiple integrals, one of which contains the uncollided wave of neutrons originating at the source plane. It is possible to obtain a simplified form for the solution at the interface, and certain numerical calculations are made there.

The interface flux in two adjacent moderators is calculated and plotted as a function of time for several moderator materials. For each case it is found that the flux decay curve has an asymptotic slope given accurately by diffusion theory. Furthermore, the interface current is observed to change directions when the scattering and absorption cross sections of the two moderator materials are related in a certain manner. More specifically, the reflection process in two adjacent moderators appears to depend initially on the scattering properties and for long times on the absorption properties of the media.

This analysis contains both the single infinite and semi-infinite medium problems as special cases. The results in these two special cases provide a check on the accuracy of the general solution since they agree with solutions of these problems obtained by separate analyses.

Wavefront measurement of double-clad pulsed ytterbium fiber laser

Because of high efficiencies, compact structure, and excellent heat dissipation, high-power fiber lasers are extremely useful for applications such as cutting, welding, precision drilling, trimming, sensing, optical transmitter, material processing, micromachining, and so on. However, the wavefront of the double clad fiber laser doped with ytterbium is still unknown. In this paper, wavefront of a fiber laser is measured and the traditional Hartmann-shack wavefront sensing method is adopted. We measured a double clad fiber laser doped with ytterbium which produces pulse wave output at infrared wavelength. The wavefront shape and contour are reconstructed and the result shows that wavefront is slightly focused and not an ideal plane wavefront. Wavefront measurement of fiber laser will be useful to improving the lasers' performance and developing the coherent technique for its applications.

Wave induced oscillations in harbors of arbitrary shape

Theoretical and experimental studies were conducted to investigate the wave induced oscillations in an arbitrary shaped harbor with constant depth which is connected to the open-sea.

A theory termed the “arbitrary shaped harbor” theory is developed. The solution of the Helmholtz equation, ∇²f + k²f = 0, is formulated as an integral equation; an approximate method is employed to solve the integral equation by converting it to a matrix equation. The final solution is obtained by equating, at the harbor entrance, the wave amplitude and its normal derivative obtained from the solutions for the regions outside and inside the harbor.

Two special theories called the circular harbor theory and the rectangular harbor theory are also developed. The coordinates inside a circular and a rectangular harbor are separable; therefore, the solution for the region inside these harbors is obtained by the method of separation of variables. For the solution in the open-sea region, the same method is used as that employed for the arbitrary shaped harbor theory. The final solution is also obtained by a matching procedure similar to that used for the arbitrary shaped harbor theory. These two special theories provide a useful analytical check on the arbitrary shaped harbor theory.

Experiments were conducted to verify the theories in a wave basin 15 ft wide by 31 ft long with an effective system of wave energy dissipators mounted along the boundary to simulate the open-sea condition.

Four harbors were investigated theoretically and experimentally: circular harbors with a 10° opening and a 60° opening, a rectangular harbor, and a model of the East and West Basins of Long Beach Harbor located in Long Beach, California.

Theoretical solutions for these four harbors using the arbitrary shaped harbor theory were obtained. In addition, the theoretical solutions for the circular harbors and the rectangular harbor using the two special theories were also obtained. In each case, the theories have proven to agree well with the experimental data.

It is found that: (1) the resonant frequencies for a specific harbor are predicted correctly by the theory, although the amplification factors at resonance are somewhat larger than those found experimentally,(2) for the circular harbors, as the width of the harbor entrance increases, the amplification at resonance decreases, but the wave number bandwidth at resonance increases, (3) each peak in the curve of entrance velocity vs incident wave period corresponds to a distinct mode of resonant oscillation inside the harbor, thus the velocity at the harbor entrance appears to be a good indicator for resonance in harbors of complicated shape, (4) the results show that the present theory can be applied with confidence to prototype harbors with relatively uniform depth and reflective interior boundaries.

A novel method to determine the FOCAL energy range

Determination of the energy range is an important precondition of focus calibration using alignment procedure (FOCAL) test. A new method to determine the energy range of FOCAL off-lined is presented in this paper. Independent of the lithographic tool, the method is time-saving and effective. The influences of some process factors, e.g. resist thickness, post exposure bake (PEB) temperature, PEB time and development time, on the energy range of FOCAL are analyzed.

平面交叉型微透镜阵列的制作及成像特性研究

介绍了采用光刻离子交换工艺制作平面交叉型微透镜阵列的方法。利用积分形式的光线方程式讨论了平面交叉型微透镜的近轴光学特性，研究了微透镜的光线轨迹方程式和一些重要的近轴成像特性，利用ABCD定理得到了平面交叉型微透镜像距、焦距、像高、横向放大率和主平面位置的数学表达式，焦距的理论计算结果和实验数据吻合得很好。

Optical generaton of millimeter-wave pulses using a fiber Bragg grating in a fiber-optics system

A scheme is proposed to transform an optical pulse into a millimeter-wave frequency modulation pulse by using a weak fiber Bragg grating (FBG) in a fiber-optics system. The Fourier transformation method is used to obtain the required spectrum response function of the FBG for the Gaussian pulse, soliton pulse, and Lorenz shape pulse. On the condition of the first-order Born approximation of the weak fiber grating, the relation of the refractive index distribution and the spectrum response function of the FBG satisfies the Fourier transformation, and the corresponding refractive index distribution forms are obtained for single-frequency modulation and linear-frequency modulation millimeter-wave pulse generation. The performances of the designed fiber gratings are also studied by a numerical simulation method for a supershort pulse transmission. (c) 2007 Optical Society of America.

Large plane deformations of thin elastic sheets of neo-hookean material

Large plane deformations of thin elastic sheets of neo-Hookean material are considered and a method of successive substitutions is developed to solve problems within the two-dimensional theory of finite plane stress. The first approximation is determined by linear boundary value problems on two harmonic functions, and it is approached asymptotically at very large extensions in the plane of the sheet. The second and higher approximations are obtained by solving Poisson equations. The method requires modification when the membrane has a traction-free edge.

Several problems are treated involving infinite sheets under uniform biaxial stretching at infinity. First approximations are obtained when a circular or elliptic inclusion is present and when the sheet has a circular or elliptic hole, including the limiting cases of a line inclusion and a straight crack or slit. Good agreement with exact solutions is found for circularly symmetric deformations. Other examples discuss the stretching of a short wide strip, the deformation near a boundary corner which is traction-free, and the application of a concentrated load to a boundary point.

Spin wave resonance in ferromagnetic films

The objective of this investigation has been a theoretical and experimental understanding of ferromagnetic resonance phenomena in ferromagnetic thin films, and a consequent understanding of several important physical properties of these films. Significant results have been obtained by ferromagnetic resonance, hysteresis, torque magnetometer, He ion backscattering, and X-ray fluorescence measurements for nickel-iron alloy films.

Taking into account all relevant magnetic fields, including the applied, demagnetizing, effective anisotropy and exchange fields, the spin wave resonance condition applicable to the thin film geometry is presented. On the basis of the simple exchange interaction model it is concluded that the normal resonance modes of an ideal film are expected to be unpinned. The possibility of nonideality near the surface of a real film was considered by means of surface anisotropy field, inhomogeneity in demagnetizing field and inhomogeneity of magnetization models. Numerical results obtained for reasonable parameters in all cases show that they negligibly perturb the resonance fields and the higher order mode shapes from those of the unpinned modes of ideal films for thicknesses greater than 1000 Å. On the other hand for films thinner than 1000 Å the resonance field deviations can be significant even though the modes are very nearly unpinned. A previously unnoticed but important feature of all three models is that the interpretation of the first resonance mode as the uniform mode of an ideal film allows an accurate measurement of the average effective demagnetizing field over the film volume. Furthermore, it is demonstrated that it is possible to choose parameters which give indistinguishable predictions for all three models, making it difficult to uniquely ascertain the source of spin pinning in real films from resonance measurements alone.

Spin wave resonance measurements of 81% Ni-19% Fe coevaporated films 30 to 9000 Å thick, at frequencies from 1 to 8 GHz, at room temperature, and with the static magnetic field parallel and perpendicular to the film plane have been performed. A self-consistent analysis of the results for films thicker than 1000 Å, in which multiple excitations can be observed, shows for the first time that a unique value of exchange constant A can only be obtained by the use of unpinned mode assignments. This evidence and the resonance behavior of films thinner than 1000 Å strongly imply that the magnetization at the surfaces of permalloy films is very weakly pinned. However, resonance measurements alone cannot determine whether this pinning is due to a surface anisotropy, an inhomogeneous demagnetizing field or an inhomogeneous magnetization. The above analysis yields a value of 4πM=10,100 Oe and A = (1.03 ± .05) x 10^-6 erg/cm for this alloy. The ability to obtain a unique value of A suggests that spin wave resonance can be used to accurately characterize the exchange interaction in a ferromagnet.

In an effort to resolve the ambiguity of the source of pinning of the magnetization, a correlation of the ratio of magnetic moment and X-ray film thickness with the value of effective demagnetizing field 4πNM as determined from resonance, for films 45 to 300 Å has been performed. The remarkable agreement of both quantities and a comparison with the predictions of five distinct models, strongly imply that the thickness dependence of both quantities is related to a thickness dependent average saturation magnetization, which is far below 10,100 Oe for very thin films. However, a series of complementary experiments shows that this large decrease of average saturation magnetization cannot be simply explained by either oxidation or interdiffusion processes. It can only be satisfactorily explained by an intrinsic decrease of the average saturation magnetization for very thin films, an effect which cannot be justified by any simple physical considerations.

Recognizing that this decrease of average saturation magnetization could be due to an oxidation process, a correlation of resonance measurements, He ion backscattering, X-ray fluorescence and torque magnetometer measurements, for films 40 to 3500 Å thick has been performed. On basis of these measurements it is unambiguously established that the oxide layer on the surface of purposefully oxidized 81% Ni-19% Fe evaporated films is predominantly Fe-oxide, and that in the oxidation process Fe atoms are removed from the bulk of the film to depths of thousands of angstroms. Extrapolation of results for pure Fe films indicates that the oxide is most likely α-Fe₂O₃. These conclusions are in agreement with results from old metallurgical studies of high temperature oxidation of bulk Fe and Ni-Fe alloys. However, X-ray fluorescence results for films oxidized at room temperature, show that although the preferential oxidation of Fe also takes place in these films, the extent of this process is by far too small to explain the large variation of their average saturation magnetization with film thickness.

Corrections to the paraxial approximation of an arbitrary free-propagation beam

A relatively simple transform from an arbitrary solution of the paraxial wave equation to the corresponding exact solution of the Helmholtz wave equation is derived in the condition that the evanescent waves are ignored and is used to study the corrections to the paraxial approximation of an arbitrary free-propagation beam. Specifically, the general lowest-order correction field is given in a very simple form and is proved to be exactly consistent with the perturbation method developed by Lax et nl. [Phys. Rev. A 11, 1365 (1975)]. Some special examples, such as the lowest-order correction to the paraxial approximation of a fundamental Gaussian beam whose waist plane has a parallel shin from the z = 0 plane, are presented. (C) 1998 Optical Society of America.