Refractive Index Discontinuity Research Articles

Modeling and experimental investigation of bubble entrapment for flow over topography during immersion lithography

In immersion lithography, the air gap that currently exists between the last lens element of the exposure system and the wafer is filled with a liquid that more closely matches the refractive index of the lens. There is a possibility that air bubbles, which represent a refractive index discontinuity, may be present in the liquid within the active exposure region and cause errors in imaging. One potential source of bubble generation is related to the flow of liquid over previously patterned features, or topography, during scanning or filling. This microscale entrainment mechanism is investigated experimentally and analyzed using computational fluid dynamics (CFD) modeling. The contact angle is a critical parameter that governs the behavior of the contact line and therefore the entrainment of air due to topography; the same topography on a hydrophobic surface is more likely to trap air than on a hydrophilic one. The contact angle can be a strong function of the flow velocity; a hydrophilic surface can exhibit hydrophobic behavior when the velocity of the free surface becomes large. Therefore, the contact angle was experimentally measured under static and dynamic conditions for a number of different surfaces, including resist-coated wafers. Finally, the flow of liquid across 500-nm-deep, straight-sidewall spaces of varying width was examined using both experimental visualization and CFD modeling. No air entrainment was observed or predicted over the velocity and contact angle conditions that are relevant to immersion lithography. The sharp-edged features studied here represent an extreme topography relative to the smoother features that are expected on a planarized wafer; therefore, it is not likely that the microscale entrainment of bubbles due to flow over wafer-level topography will be a serious problem in immersion lithography systems.

Comparison of vector finite-difference techniques for modal analysis

We compare a vector finite-difference method that correctly applies the boundary conditions at both horizontal and vertical dielectric interfaces (but not at corners or slanted interfaces) to algorithms that only approximately satisfy these boundary conditions. We find, rather unexpectedly, that for strongly guiding waveguides the boundary conditions imposed at the refractive-index discontinuities typically affect the calculated field distributions less than the procedure employed to assign the refractive index at a computational grid point. In fact, locally averaging the refractive index around each grid point transforms the precision of the most straightforward finite-difference models to that of far more sophisticated techniques. Further, H- and E-field formalisms exhibit identical accuracy.

Reconstruction of refractive index discontinuities from truncated phase-contrast tomography projections

Propagation-based phase-contrast tomography is a noninterferometric imaging technique that can reconstruct the complex refractive index distribution of an object. To accomplish such a reconstruction using existing methods, however, there must be no truncation of the phase-contrast projection data in the detector planes at each tomographic view angle. We demonstrate that local tomography reconstruction theory can be applied for reconstruction of refractive index discontinuities in two-dimensional objects from truncated phase-contrast projections acquired in the near-Fresnel zone. Numerical results are presented to corroborate our theoretical assertions.

Design of a phase-shifted long-period grating using the partial-etching technique

In this paper, a newly designed reconfigurable phase-shifted long period grating using the partial etching technique is presented. In the locally etched long-period grating, there are discontinuities in the effective refractive index of the cladding modes. The phase shift in the long-period grating is generated due to the effective-refractive-index change of the cladding modes in the locally etched long-period grating. The phase shift is proportional to the partial etching depth and etching length. Thus, by controlling the etching depth and the etching area, we can tune the transmission spectrum of a long period grating after it has been written and create an arbitrary spectral response of the grating. Using this characteristic, we design a reconfigurable phase-shifted long-period grating. The phase-shifted long-period grating introduced in this paper will be used in a reconfigurable gain-flattening filter for optical amplifiers, such as the erbium-doped fiber amplifier (EDFA). We believe that this post-processing partial-etching technique can eliminate the complicated fabrication process in order to make a complex grating mask for a gain-flattening filter, thus simplifying the fabrication process and reducing manufacturing cost. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 18–21, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20710

An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition

An accurate and efficient solution method using spectral collocation method with domain decomposition is proposed for computing optical waveguides with discontinuous refractive index profiles. The use of domain decomposition divides the usual single domain into a few subdomains at the interfaces of discontinuous refractive index profiles. Each subdomain can be expanded by a suitable set of orthogonal basis functions and patched at these interfaces by matching the physical boundary conditions. In addition, a new technique incorporating the effective index method and the Wentzel-Kramers-Brillouin method for the a-priori determination of the scaling factor in Hermite-Gauss or Laguerre-Gauss basis functions is introduced to considerably save computational time without experimenting with it. This method shares the same desirable property of the spectral collocation method of providing a fast and accurate solution but avoids the oscillatory solutions and improves the poor convergence problem of the simple spectral collocation method with single domain where regions of discontinuous refractive index profiles exist. Applications to several two- and three-dimensional waveguide structures having exact or accurate approximate solutions are given to test the accuracy and efficiency; all the results are found to be in excellent agreement.

Surface magneto-optic solitons

Solitons, localized at the interface between a linear magneto-optic half-space and a nonlinear optical medium, are investigated with the aid of an approximate model validated by direct numerical simulations. The interface is located in a planar waveguide, and is characterized not only by the magneto-optic properties but also by a linear refractive index discontinuity. The analysis is based upon a global envelope equation and uses waveguide and magneto-optic parameters that are averaged over the whole waveguide structure. A variational analysis shows that, even though surface solitons can be localized in deeply stable stationary states, they can also be generated in parts of the parameter space that facilitate movement between stable and unstable regions. This leads to suggestions for isolator and multi-switch operation.

Opposite-parity orthonormal function expansion for efficient full-vectorial modeling of holey optical fibers.

An improved full-vectorial method exploiting the opposite-parity property of eigenmodes based on orthonormal-functions expansion is proposed to solve the wave equation for holey optical fibers. By use of the parity property of eigenmodes in symmetric structures, the number of orthonormal function integrals involved in the calculation is reduced, and the computation efficiency is greatly enhanced. The coupling between the two transverse field components is considered, and both dominant and minor field components can be calculated for the accurate modeling of fiber modes. This method is useful for efficiently modeling holey fibers, especially those with large air holes, in which the coupling effect that is due to refractive-index discontinuities is strong.

Analysis of rib waveguides by a fourth‐order accurate finite‐difference beam‐propagation method

AbstractWhen a step index waveguide is analyzed by the finite‐difference method, the treatment of the discontinuous refractive index profile is a problem. In this paper, while the boundary condition of the field is taken into account, the improved finite‐difference equation in which the truncation error is reduced to the fourth order is combined with the alternate direction implicit solution method, formulating a three‐dimensional beam propagation method with high computational efficiency. First, by means of the imaginary axis method, an eigenmode analysis of the rib waveguide is carried out so that the effectiveness and validity of the method are proven. Next, a beam propagation analysis is carried out for the rib waveguide taper. By taking into account the discontinuity location of the refractive index, the quantization error is found to be reduced. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 85(12): 20–29, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.10061

Interaction of whispering-gallery electromagnetic waves with acoustic waves in tapered quartz rods

The propagation of whispering-gallery waves in a dynamic cavity formed by a tapered quartz rod and the plane interface between two regions of the rod with different refractive indices moving along the axis of the rod is analysed. It is shown that the limiting frequency shift of light in such a cavity is determined by its Q factor and the attainable refractive index discontinuity. The possibility of using acoustic waves for obtaining a dynamic cavity is considered.

SOMARE‐99: Observations of tropospheric scattering layers using multiple‐frequency range imaging

Results from an experimental implementation of multiple‐frequency range imaging (RIM) are presented. The technique exploits the benefits of frequency diversity to improve range resolution of atmospheric radar systems. The theory has been described in the literature, and simulations have proven its usefulness. Nevertheless, experimental results have been extremely limited. Over a 5‐day period in May 1999 we conducted experiments using RIM on the sounding system (SOUSY) radar in northern Germany to observe the layered structure in the troposphere. The experiment is referred to as the SOUSY Multifrequency Atmospheric Radar Experiment 1999 (SOMARE‐99). Estimates of range brightness produced by the RIM analysis provide insight about the layered structure of the atmosphere. The RIM results show distinct similarities to previous in situ measurements, which have shown sharp refractive index discontinuities to exist throughout the troposphere and stratosphere. Examples from selected time periods show layers modulated by possible short‐period gravity waves or advection of periodic structures as well as other layers with apparent downward motion possibly caused by the progression of a warm front or large‐scale subsidence.

Organic multilayers as distributed Bragg reflectors

We report on a distributed Bragg reflector (DBR) operating in the IR spectral region, based on a fluorocarbon polymer (CFx) multilayer. In this organic structure, the periodic variation of the refractive index is obtained by embedding gold clusters in the CFx, resulting in a periodic CFx/CFx(Au) layered structure with a refractive index discontinuity in the infrared spectral region of about Δn∼0.4. We have fabricated a quarter-wavelength stack consisting of 4 CFx/CFx(Au) pairs, with a nominal Au concentration equal to 17%. The DBR shows a reflectance peak of about 70% at 1.7 μm and a full width at half maximum of about 0.4 μm.

First Frequency-Domain Interferometry Observations of Large-Scale Vertical Motion in the Atmosphere

The spatiotemporal distribution of the vertical velocity at synoptic and subsynoptic scales is key to the patterns of weather and climate on earth. On these scales, the vertical velocity is on the order of one to a few centimeters per second, typically about three orders of magnitude smaller than typical horizontal wind velocities. Because of the smallness of large-scale vertical velocities relative to typical horizontal velocities, a direct observation of the large-scale vertical air velocity is extremely difficult. In a case study on observational material obtained during a 68-h experiment using the SOUSY very high frequency (VHF) radar in the Harz Mountains in Germany, the authors present the first intercomparison between three different sources of physical information that can provide large-scale vertical wind velocities: (i) the Doppler shifts observed with a vertically pointing VHF radar ; (ii) the rates of change of the altitudes of refractiveindex discontinuities as identified with frequency-domain interferometry (FDI), which is still a relatively unexplored technique in meteorology; and (iii) the output of a regional numerical weather prediction model (NWPM), which has been set up to model the meteorological situation during the observational period. There are several phenomena that have been known to possibly cause significant biases in mean vertical velocities retrieved from the Doppler shifts measured with vertically pointing clear-air VHF radars: (i) stationary or nonstationary gravity waves with vertical-velocity amplitudes up to the order o f1ms 21; (ii) stationary or horizontally advected tilted refractive-index discontinuities that are aspect sensitive in the VHF regime; and (iii) a correlation between the radar-reflectivity fluctuations and the vertical-velocity fluctuations within a vertically propagating gravity wave. On the basis of an intercomparison between the vertical velocities retrieved from (i) ‘‘standard Doppler’’ VHF radar observations, (ii) VHF FDI observations, and (iii) the NWPM output, the authors present first evidence that, under ideal conditions, VHF FDI can be used to directly monitor large-scale vertical motion.

On the reality of upper mesospheric/lower thermospheric turbulent “eddies”

The structure of atmospheric turbulence and its relationship to the refractive index discontinuities responsible for the scattering of radio waves are discussed. Emphasis is placed more on the nature of the intermittent scatterers observed at upper mesospheric/lower thermospheric altitudes than on volume scattering. The preferred model is of transient patches composed of vortex strings. An overview of the methods used to estimate the rate of dissipation of turbulent energy using MF radar data which emphasizes the advantages of imaging Doppler interferometry is presented, together with a review of pertinent laboratory investigations of coherent structures. An attempt is made to remove the contamination by gravity waves from imaging Doppler interferometry fluctuating velocity data. Emphasis is given to the fact that all radar measurements of turbulent intensity to date are upper estimates, not only because of the presence of gravity wave fluctuations contaminating the data, but also because the turbulence itself is not stationary and homogeneous but intermittent in both space and time. The intermittent nature of the turbulence, in which the refractive index discontinuities responsible for the radar backscatter are associated with decaying, homogeneous turbulent coherent structures, is in contrast to the theory of volume scattering, which assumes the radar pulse volume to be partially filled with stationary, homogeneous turbulence.

Laser emission from photonic dots

Laser emission was observed in photonic semiconductor dots with a discretized optical mode spectrum. The photonic dots with lateral sizes between 1 and 5 μm provide a three-dimensional optical confinement by using in the vertical direction AlAs/GaAs Bragg mirrors and in the lateral directions the refractive index discontinuity at the etched surfaces. In the optically pumped structures, the laser emission takes place on the fundamental mode of the microcavities. External threshold excitation densities of 200 W/cm2, which correspond to a very low internal optical excitation power of 0.15 μW per microcavity post, were measured for microcavity structures with a lateral size of 2.7 μm.

T-ray tomography

We demonstrate tomographic T-ray imaging, using the timing information present in terahertz (THz) pulses in a reflection geometry. THz pulses are reflected from refractive-index discontinuities inside an object, and the time delays of these pulses are used to determine the positions of the discontinuities along the propagation direction. In this fashion a tomographic image can be constructed.

The inverse problem in optics of stratified media with discontinuous parameters

Analytical properties of spectral coefficients of a non-absorbing stratified medium with discontinuous parameters are studied. Explicit formulae establishing one-to-one correspondence between the spectral coefficients' phases and amplitudes are derived. Conditions under which these formulae can be applied are discussed. The uniqueness of the determination of the discontinuous refractive index from the amplitude reflection coefficient is proved under practically sensible conditions.

Multiple-scattering approach to Mie scattering from a sphere of arbitrary size

The classic Mie scattering problem is revisited via an integral-equation formalism. Our approach consists in resumming the multiple-scattering Born series in terms of which the solution can be formally expressed. For small dielectric spheres, the resummation can be done exactly without invoking the full apparatus of multipoles. For a large dielectric sphere with a small refractive-index discontinuity relative to the surrounding medium, the Born series may be approximately resummed to generate correctly a description of the phenomenon of anomalous diffraction. For spheres of arbitrary size, the Born series for each excited electric or magnetic multipole has a complicated structure, but one that is shown to be exactly equivalent to the celebrated Mie results. A significant advantage over the usual differential approach is the implicit inclusion of the electromagnetic boundary conditions at the scattering surface in the integral approach, a feature that may provide valuable insights into scattering from arbitrarily shaped particles.

Generalised impedance boundary method of moments for multilayer graded-index dielectric waveguide structures

A generalisation of the impedance boundary method of moments (IBMOM) for application to multilayer graded-index dielectric waveguides is proposed. The transverse field in each layer is expanded in local basis functions and matched at the interfaces between the layers. Exact impedance boundary conditions are applied at the two interfaces between the guiding region and cladding regions. Comparison with the IBMOM shows that the generalised impedance boundary method of moments (GIBMOM) is more efficient in calculating the propagation constants of structures with discontinuous refractive index distributions. Numerical results for a two-layer graded-index structure and a step-index coupler are presented to illustrate the improvement in the rate of convergence.

Ray splitting and quantum chaos.

Ray splitting is the phenomenon whereby a ray incident on a boundary splits into more than one ray traveling away from the boundary. The most common example of this is the situation, originally considered by Snell in 1621, in which an incident light ray splits into reflected and transmitted rays at a discontinuity in refractive index. This paper seeks to extend techniques and results from quantum chaos to short wavelength problems in which ray splitting surfaces are present. These extensions are tested using a simple model problem for the Schr\"odinger equation in two dimensions with a finite step potential discontinuity. Numerical solutions for the energy spectrum and eigenfunctions in this system are then compared with predictions based on quasiclassical theoretical results suitably extended to include ray splitting. Among the topics treated are the ray orbits for our problem, energy level statistics, scars, trace formulas, the quasiclassical transfer operator technique, and the effect of lateral waves. It is found that these extensions of quantum chaos are very effective for treating problems with ray splitting.

Ray splitting and quantum chaos.

Ray splitting is the phenomenon whereby a ray incident on a boundary splits into more than one ray traveling away from the boundary. The most common example of this is the situation, originally considered by Snell in 1621, in which an incident light ray splits into reflected and transmitted rays at a discontinuity in refractive index. This paper seeks to extend techniques and results from quantum chaos to short wavelength problems in which ray splitting surfaces are present. These extensions are tested using a simple model problem for the Schr\odinger equation in two dimensions with a finite step potential discontinuity. Numerical solutions for the energy spectrum and eigenfunctions in this system are then compared with predictions based on quasiclassical theoretical results suitably extended to include ray splitting. Among the topics treated are the ray orbits for our problem, energy level statistics, scars, trace formulas, the quasiclassical transfer operator technique, and the effect of lateral waves. It is found that these extensions of quantum chaos are very effective for treating problems with ray splitting.