Rigorous and efficient grating-analysis method made easy for optical engineers

The coordinate-transformation-based differential method of Chandezon et al. [J. Opt. (Paris) 11, 235 (1980); J. Opt. Soc. Am. 72, 839 (1982)] (the C method) is one of the simplest and most versatile methods for modeling surface-relief gratings. However, to date it has been used by only a small number of people, probably because, traditionally, elementary tensor theory is used to formulate the method. We reformulate the C method without using any knowledge of tensor, thus, we hope, making the C method more accessible to optical engineers.     

重叠体光栅相干组束的耦合波分析

目前基于重叠体光栅的相干组束系统还没有完整的理论模型和参数分析,针对这个问题,本文采用耦合波理论,建立了比较完整的基于重叠体光栅的双光束组束模型,给出了各光束在重叠体光栅内传播的解析解.模型表明,基于重叠体光栅的双光束组束系统其输出光强受两入射光的振幅和位相、体光栅间的相移和折射率调制振幅、光栅厚度,失谐量等多个光学参数的影响.通过数值计算,详细分析了各光学参数对组束结果的影响.最后总结了设计高效重叠体光栅相干组束系统时必须遵循的条件,如必须对光栅厚度进行优化、各光栅的折射率调制振幅必须相同,保证失谐量足够小等.     

Antireflection structures with use of multilevel subwavelength zero-order gratings

It is known that the zero-order two-phase-level gratings with a period much smaller and a thickness much larger than the wavelength may have antireflection properties the same as appropriate dielectric layers under normal incidence. On the basis of the rigorous coupled-wave analysis method formulation, it is shown that multilevel unidimensional phase gratings, for both TE and TM polarization, are functionally equivalent to antireflection structures of multilevel dielectric layers, even if the period is close to the wavelength.     

Design of retrodiffraction gratings for polarization-insensitive and polarization-sensitive characteristics by using the Taguchi method

We present the design of retrodiffraction gratings that utilize total internal reflection (TIR) in a lamellar configuration to achieve high performance for both TE and TM polarized light and polarization-sensitive performance for gratings behaving as polarizer filters; the design was based on rigorous coupled wave analysis (RCWA) and the Taguchi method. The components can thus be fabricated from a single dielectric material and do not have to be coated with a metallic or dielectric film layer to enhance the reflectance. The effects of the structural and optical parameters of lamellar gratings were investigated, and the TIR gratings in a lamellar configuration were structurally and optically optimized in terms of the signal-to-noise ratio (S/N) and a statistical analysis of variance (ANOVA) of the refractive index, grating period, filling factor, and grating depth as control factors and the estimated efficiency by RCWA as a noise factor. For more accurate robustness, a two-step optimization process was used for each purpose. For TIR gratings designed to perform similarly for TE and TM incident polarization, the -1st-order efficiencies were estimated to be up to 92.0% and 88.5% for TE and TM polarization, respectively. Also, for the TIR gratings designed to achieve polarization-sensitive performance when behaving as a polarizer filters, the -1st-order diffraction efficiencies for TE and TM polarization were estimated to be up to 95.5% and 2.7%, respectively. From these analysis results, it was confirmed that the Taguchi method shows feasibility for an optimization approach to a technique for designing optical devices.     

Completeness and Modal Expansion Methods in Diffraction Theory

As a consequence of their structure, modal formulations for diffraction gratings may satisfy reciprocity and conservation of energy criteria exactly, independently of the errors incurred when using truncated field expansions. We consider completeness relations, which provide a means of assessing truncation errors in modal formulations. Taking the lamellar diffraction grating as an example, we describe the completeness relations as they apply to infinitely conducting, dielectric and lossy structures.     

Improved method for computing of light-matter interaction in multilayer corrugated structures

An improved method for the calculation of light-matter interaction that appears with the light propagation through multilayer periodically corrugated structures consisting of any dielectric or absorptive media is reported. The method is based on the differential formalism for a system of Maxwell's equations when the boundary conditions are simplified by the introduction of a curvilinear nonorthogonal coordinate system. The solution for electromagnetic fields was written in the form of the superposition of partial plane waves. The obtained method essentially reduces computation time and increases accuracy compared with the Chandezon method. For a numerical demonstration of the proposed method, calculation of long-range surface plasmon polaritons was performed. The presented method can be enhanced for calculations of light propagation through thin absorptive films with various periodic profiles at both film interfaces.     

Operator approach to electromagnetic coupled-wave calculations of lamellar gratings: infrared optical properties of intrinsic silicon gratings

The system of coupled-wave equations for electromagnetic calculations of lamellar gratings is transformed to a new operator-vector form. The numerical procedure is based on truncation of the transformed system and proves to be stable, to be free of ill conditioning, and to preserve the power-conservation requirement for a lossless dielectric with high accuracy. To execute the procedure a very compact MATLAB-based program is developed, and numerical simulations for thick intrinsic silicon gratings are performed. Zero-reflectance phenomena at normal incidence for both TE and TM polarizations are studied. The ratios of the grating dimensions to be wavelengths at which these anomalies occur are found numerically. It is shown that by keeping the period- and slot-width-towavelength ratios constant and by increasing the slot depth one can repeat the anomalies. An antiblazing property at oblique incidence is also considered. The connection with recent directional polarized-emission experiments on intrinsic silicon g atings is discussed.     

Infrared quarter-wave reflection retarders designed with high-spatial-frequency dielectric surface-relief gratings on a gold substrate at oblique incidence

One- and two-dimensional high-spatial-frequency dielectric surface-relief gratings on a Au substrate are used to design a high-reflectance quarter-wave retarder at 70° angle of incidence and 10.6-μm light wavelength. The equivalent homogeneous anisotropic layer model is used. It is shown that equal and high reflectances (>98.5%) for the p and the spolarizations and quarter-wave retardation can be achieved with two-dimensional ZnS surface-relief gratings. Sensitivities to changes of incidence angle, light wavelength, grating filling factor, and grating layer thickness are considered.     

Independent electromagnetic optimization of the two coating thicknesses of a dielectric layer on the facets of an echelle grating in Littrow mount

Abstract

Electromagnetic investigations using the integral equation system method with parametrization (IESMP) show that the two-coating thicknesses of a dielectric layer on the facets of an echelle grating in a Littrow mount have to be independently optimized. While the optimal coating thickness on the blaze facet is the same for maximal efficiency and minimal absorption in both polarizations, this is not the case for the anti-blaze facet. Therefore, it is only possible to optimize the two-coating thicknesses for one of the purposes. On the blaze facet, a simple formula based on thin-film optical considerations describes the optimal thickness very well. Additionally, we found that resonance anomalies can significantly reduce efficiency if the wrong coating thickness is used on the anti-blaze facet. The coating thickness creating the resonance anomaly can be deduced by investigating the poles of the reflection coefficient of a dielectric coated metallic mirror in grazing incidence. This value can be used to optimize the layer for maximal efficiency. Consequently, we are generally able to describe the optimal coating thicknesses for minimal absorption as well as for maximum efficiency in both, TE- and TM-polarization, using only thin-film optical considerations without any further rigorous calculation.     

Diffraction of electromagnetic waves by periodic arrays of rectangular cylinders

Reflection, transmission, and absorption of electromagnetic waves by periodic arrays of conducting or dielectric rectangular cylinders are studied by a finite-difference time-domain technique. Truncated gratings made of lossless and lossy conducting and dielectric elements are considered. Results for surface current density, transmission, and reflection coefficients are calculated and compared with corresponding results in the literature, which are obtained by approximate or rigorous methods applicable only to idealized infinite models. An excellent agreement is observed in all cases, which demonstrates the accuracy and efficacy of our proposed analysis technique. Additionally, this numerical method easily analyzes practical gratings that contain a finite number of elements made of lossless, lossy, or even inhomogeneous materials. The results rapidly approach those for the idealized infinite arrays as the number of elements is increased. The method can also solve nested gratings, stacked gratings, and holographic gratings with little analytical or computational effort.     

Metal Gratings with Dielectric Coating of Variable Thickness within a Period

Abstract

The rigorous theory of diffraction is applied to metallic gratings with dielectric coating. The modulation depth of the profile between coating and air is assumed to be lower than the modulation depth of the profile between metal and coating. We call this levelling. For monochromator mounting together with a special coating thickness levelled single-coated gratings with blaze show a marked increase of the efficiencies for wavelengths greater than the blaze maximum wavelength. Optimal improvements run over two octaves of the wavelength.     

Meetings Calendar

Abstract

A rigorous modal analysis of lamellar gratings, i.e. gratings having rectangular grooves, in conical mountings is presented. It is an extension of the analysis of Botten et al. which considered non-conical mountings. A key step in the extension is a decomposition of the electromagnetic field in the grating region into two orthogonal components. A computer program implementing this extended modal analysis is capable of dealing with plane wave diffraction by dielectric and metallic gratings with deep grooves, at arbitrary angles of incidence, and having arbitrary incident polarizations. Some numerical examples are included.     

Microstructure and dielectric characteristics of high-k tetragonal ZrO2 films with various thicknesses processed by sol-gel method

High-k ZrO2/Si films were fabricated by a sol-gel method and the effects of the thickness of ZrO2 on the phase formation, interface chemical structure, and dielectric performance were studied. The 0.1 M precursor sol was prepared by using Zr acetylacetonate, coated, dried on Si substrates, and finally annealed at 500 degrees C. The thickness of ZrO2 was varied in the range from 7 to 51 nm by repeating the coating and drying sequences. The deposited ZrO2 was amorphous for the sample with a thickness of -7 nm, but tetragonal (t-) phases appeared as the thickness increased. As the thickness increased, the flat-band voltage and hysteresis width in the capacitance-voltage curves increased. The sol-gel deposited ZrO2 dielectrics showed a high k value (-33) due to the formation of the t-phase, while retaining gate leakage current levels of less than -4.0 x 10(-5) A/cm2 at 1 MV/cm.     

Study of the differential theory of lamellar gratings made of highly conducting materials

Differential theory is said to be difficult to apply to surface-relief gratings made of metals with very high conductivity even though the formulation follows Li's Fourier factorization rules. Recently, Popov et al. [J. Opt. Soc. Am. 21, 199 (2004)] pointed out this difficulty and explained that its origin is related to the inversion of Toeplitz matrices constructed by the permittivity distribution inside the groove region. The current paper provides information about the differential theory for highly conducting gratings and considers the numerical instability problems. A stable calculation for lossless gratings is described, based on the extrapolation technique with the assumption of small losses.     

Submicrometer gratings for solar energy applications

Diffractive optical structures for increasing the efficiency of crystalline silicon solar cells are discussed. As a consequence of the indirect band gap, light absorption becomes very ineffective near the band edge. This can be remedied by use of optimized diffraction gratings that lead to light trapping. We present blazed gratings that increase the optically effective cell thickness by approximately a factor of 5. In addition we present a wideband antireflection structure for glass that consists of a diffraction grating with a dielectric overcoat, which leads to an average reflection of less than 0.6% in the wavelength range between 300 and 2100 nm.     

Method for measuring thickness of dielectric films using microdielectric fringe-effect sensors

A method for noninvasive thickness measurements of dielectric films using fringe-effect (FE) sensors is developed and experimentally validated. The fringing electrical field, created by electrodes microfabricated at the film substrate, depends on the film thickness and dielectric permittivity of the film under test (FUT). The unknown film thickness is estimated by matching the theoretical prediction of thickness-dependent sensor admittance with the measured value. In the case of FE sensors with spatially periodic, interdigitated electrode (IDE) configuration, the admittance prediction is simplified, which allows for the real-time measurements of changing thickness. The developed method can be used to continuously measure the changing dielectric permittivity of the FUT material, which makes it possible to determine the thickness of films of changing dielectric properties, caused by chemical or other transformations. The application of the developed method is demonstrated experimentally by measuring the thickness of silicon nitride film deposited in several increments on the quartz substrate of the IDE sensor. In the expected range of sensor sensitivity, the results show an excellent agreement with the independent thickness measurements.     

Densification and electrical properties of solution coated lead zirconate titanate thick films

Ferroelectric Pb(Zr_0.7Ti_0.3)O₃ (PZT(70/30)) powder was prepared by a sol-gel method and PZT thick films were fabricated by the screen-printing method on the alumina substrates. A concentration of a coating solution was 0.5, 1.0 mol/L and the number of coating was repeated from 0-6. The thickness of the all PZT thick films was about 60 μm. All PZT multilayered thick films showed the XRD patterns of typical peroveskite polycrystalline structure. The relative dielectric constant of the 0.5 M coated PZT-6 (6-number of sol coatings) and 1.0 M coated PZT-6 thick films were 540 and 656.2, respectively. The dielectric loss of the 0.5 M coated PZT-6 and 1.0 M coated PZT-6 thick films were 3.2% and 2.7%, respectively.     

Polarization conversion in conical diffraction by metallic and dielectric subwavelength gratings

Subwavelength metallic and dielectric diffraction gratings which rotate the linear polarization of incident light by 90 degrees are examined. Using rigorous diffraction theory in total-internal-reflection configuration, it is shown that full conversion from incident transverse electric field to transverse magnetic zero-order field can be achieved with both dielectric and metallic elements, but dielectric gratings provide higher efficiency and are thus preferable. The fabrication aspects and constraints are discussed in detail and the behavior of the gratings over broad wavelength bands is presented.     

Pseudospectral modal method for conical diffraction of gratings

For lamellar gratings and other layered periodic structures, the modal methods (including both analytic and numerical ones) are often the most efficient, since they avoid the discretization of one spatial variable. The pseudospectral modal method (PSMM) previously developed for in-plane diffraction problems of one-dimensional gratings achieves high accuracy for a small number of discretization points, and it outperforms most other modal methods. In this paper, an extension of the PSMM to conical diffraction problems is presented and implemented. Numerical examples are used to demonstrate the high accuracy and excellent convergence property of this method for both dielectric and metallic gratings.     

Surface plasmon resonance radiation from germanium

It is well known that when certain reflection gratings which are coated with different metal or dielectric-metal layers are illuminated by p-polarized light surface plasmon resonance radiation appears in the diffraction spectra in the form of polarization anomalies (Wood's anomalies). Surface plasmons can be excited in this way if the real part ε₁ of the dielectric constant of the metal layer has a value less than -1. This technique has been extended to semiconductors in the present work by coating a grating with Ge and investigating the diffracted light in the 4–6 eV region, where ε₁ for Ge is less than -1. Experimental results are presented which show that certain peaks in the observed spectra may be due to the expected resonance radiation.