Simplified Modal Method Research Articles

Design of highly efficient transmission gratings with deep etched triangular grooves

The design of highly efficient fused silica transmission gratings with deep-etched triangular-shaped grooves in the -1st order diffraction is realized by the rigorous coupled wave analysis (RCWA). The antireflective effect of a subwavelength triangular-groove grating with gradient effective refractive index results in the higher diffraction efficiency (>99.9%). The performance of the presented gratings is clearly better than traditional rectangular and blazed ones. The gratings are designed under Littrow mounting at a wavelength of 1064nm to be used in high-power laser systems. A detailed fabrication tolerance, covering not only the errors in height but also the errors in the lateral dimension, is demonstrated. The physical process of the diffraction characteristics for such a triangular-groove grating can be well explained by the simplified modal method based on two-beam interference of the first two propagating modes excited by the incident wave. Based on the fact that the transmittance derived from the modal method is in good agreement with that calculated by the RCWA, the simplified modal method can be effectively utilized as an easily designed tool of the triangular-shaped gratings.

Finite element analysis of the dynamic behavior of a laminated windscreen with frequency dependent viscoelastic core

This paper presents numerical and experimental validation of results obtained by a shell finite element, which has been developed for modeling of the dynamic behavior of sandwich multilayered structures with a viscoelastic core. The proposed shell finite element is very easy to implement in existing finite element solvers, since it uses only the displacements as degrees of freedom at external faces and at inter-layer interfaces. The displacement field is linearly interpolated in the thickness direction of each layer, and analytical integration is made in the thickness direction in order to avoid meshing of each sandwich layer by solid elements. Only the two dimensional mid-surface of reference is meshed, facilitating the mesh generation task. A simplified modal approach using a real modal basis is also proposed to efficiently calculate the dynamic response of the sandwich structure. The proposed method reduces the memory size and computing time and takes into account the frequency-dependence of the polymer core mechanical properties. Results obtained by the proposed element in conjunction with the simplified modal method have been numerically and experimentally validated by comparison to results obtained by commercial software codes (MSC/NASTRAN and ESI/RAYON-VTM), and to measurements done on automobile windscreens.

Design of a high-efficiency seven-port beam splitter using a dual duty cycle grating structure

In this paper, we propose a compact seven-port beam splitter which is constructed using only a single-layer high-density grating with a dual duty cycle structure. The properties of this grating are investigated by a simplified modal method. The diffraction efficiency can be achieved around 10% more than conventional Dammann gratings while the uniformity can still be maintained at less than 1%. The effect of deviations from the design parameters on the performance of the grating is also presented.

Unified design of sinusoidal-groove fused-silica grating

A general design rule of deep-etched subwavelength sinusoidal-groove fused-silica grating as a highly efficient polarization-independent or polarization-selective device is studied based on the simplified modal method, which shows that the device structure depends little on the incident wavelength, but mainly on the ratio of groove depth to incident wavelength and the ratio of wavelength to grating period. These two ratios could be used as the design guidelines for wavelength-independent structure from deep ultraviolet to far infrared. The optimized grating profile with a different function as a polarizing beam splitter, a polarization-independent two-port beam splitter, or a polarization-independent grating with high efficiency of -1st order is obtained at a wavelength of 1064 nm, and verified by using the rigorous coupled-wave analysis. The performance of the sinusoidal grating is better than a conventional rectangular one, which could be useful for practical applications.

Design and fabrication of a polarization-independent wideband transmission fused-silica grating

A fused-silica polarization-independent wideband transmission grating used in the -1st order (Littrow mounting) for chirped-pulse-amplification, high-power lasers is designed and manufactured. An approximate grating profile can be obtained by using the simplified modal method with consideration for the corresponding accumulated phase difference of two excited propagating grating modes. An exact grating profile is optimized by using the rigorous coupled-wave analysis. With the optimized profile parameters, the gratings can theoretically exhibit diffraction efficiencies of greater than 97% at a wavelength of 800 nm for both of TE- and TM-polarized waves. Diffraction efficiencies of greater than 92% can be obtained in a 100 nm bandwidth (from 750 to 850 nm) for both TE- and TM-polarized waves. Holographic recording technology and inductively coupled plasma etching are used to manufacture the fused-silica grating. Experimental results agree well with the theoretical values.

Polarization-independent triangular-groove fused-silica gratings with high efficiency at a wavelength of 1550 nm

We describe polarization-independent triangular-groove fused-silica gratings illuminated by incident lights in the C + L bands as (de)multiplexers for dense wavelength division multiplexing (DWDM) application. The physical mechanisms of the grating can be shown clearly by using the simplified modal method with consideration of the corresponding accumulated average phase difference of two excited propagating grating modes, which illustrates that the grating structure depends mainly on the ratio of the average effective indices difference to the incident wavelength. Exact grating profile is optimized by using the rigorous coupled-wave analysis (RCWA). With the optimized grating parameters, the grating exhibits diffraction efficiencies of greater than 90% under TE- and TM-polarized incident lights for 101 nm spectral bandwidths (1500–1601 nm) and it can reach an efficiency of more than 99% for both polarizations at a wavelength of 1550 nm. Without loss of metal absorption, coating of dielectric film layers, the designed triangular-groove fused-silica grating should be of great interest for DWDM application.

Unified design of wavelength-independent deep-etched fused-silica gratings

We present a unified design of wavelength-independent deep-etched fused-silica gratings as polarizing beam splitters and polarization-independent two-port beam splitters by using the simplified modal method. By defining unified grating parameters as the ratio of incident wavelength to grating period and the ratio of groove depth to grating period, unified grating structures are found to be approximately wavelength-independent, which is based on the modal view of the accumulated phase difference of two excited propagating grating modes. Diffraction efficiencies given by the rigorous coupled-wave analysis (RCWA) verified this unified design at the wavelength of 1064 nm. It should be noted that this unified design of wavelength-independent gratings is an analytic result, which is impossible to be derived with the well-known numerical RCWA. Modal method is powerful and presents a clear physical picture for us to obtain this unified design. Therefore, this unified design can be used as a general guideline for designing deep-etched fused-silica gratings as beam splitters for practical applications from ultraviolet to near infrared.

Deep-etched sinusoidal polarizing beam splitter grating

A sinusoidal-shaped fused-silica grating as a highly efficient polarizing beam splitter (PBS) is investigated based on the simplified modal method. The grating structure depends mainly on the ratio of groove depth to grating period and the ratio of incident wavelength to grating period. These ratios can be used as a guideline for the grating design at different wavelengths. A sinusoidal-groove PBS grating is designed at a wavelength of 1310 nm under Littrow mounting, and the transmitted TM and TE polarized waves are mainly diffracted into the zeroth order and the -1st order, respectively. The grating profile is optimized by using rigorous coupled-wave analysis. The designed PBS grating is highly efficient (>95.98%) over the O-band wavelength range (1260-1360 nm) for both TE and TM polarizations. The sinusoidal grating can exhibit higher diffraction efficiency, larger extinction ratio, and less reflection loss than the rectangular-groove PBS grating. By applying wet etching technology on the rectangular grating, which was manufactured by holographic recording and inductively coupled plasma etching technology, the sinusoidal grating can be approximately fabricated. Experimental results are in agreement with theoretical values.

Design and fabrication of a polarization-independent two-port beam splitter

We design and manufacture a fused-silica polarization-independent two-port beam splitter grating. The physical mechanism of this deeply etched grating can be shown clearly by using the simplified modal method with consideration of corresponding accumulated phase difference of two excited propagating grating modes, which illustrates that the binary-phase fused-silica grating structure depends little on the incident wavelength, but mainly on the ratio of groove depth to grating period and the ratio of incident wavelength to grating period. These analytic results would also be very helpful for wavelength bandwidth analysis. The exact grating profile is optimized by using the rigorous coupled-wave analysis. Holographic recording technology and inductively coupled plasma etching are used to manufacture the fused-silica grating. Experimental results agree well with the theoretical values.

Polarizing beam splitter of two-layer dielectric rectangular transmission gratings in Littrow mounting

We theoretically investigated the design of polarizing beam splitters (PBS) of two-layer dielectric rectangular transmission gratings in Littrow mounting. The PBS grating is aimed to separate TE and TM polarized waves into two different transmitted orders, i.e. the 0th and −1st orders. The polarization-dependent diffraction process is analyzed by using a simplified modal method. The mode reflection and transmission at the interface of the two layers are illustrated. Design equations are given based on the average mode indices of the even and odd modes inside the grating region, which are verified by using rigorous coupled wave analysis. Moreover, with simulated annealing algorithm, two kinds of two-layer PBS gratings are optimized for operation over the C-band (1520–1570 nm). Compared with the single-layer transmission PBS gratings, such two-layer scheme exhibits a great flexibility of design. The PBSs of two-layer dielectric transmission gratings should be useful for practical applications.

Dual-function beam splitter of a subwavelength fused-silica grating

We present the design and fabrication of a novel dual-function subwavelength fused-silica grating that can be used as a polarization-selective beam splitter. For TM polarization, the grating can be used as a two-port beam splitter at a wavelength of 1550 nm with a total diffraction efficiency of 98%. For TE polarization, the grating can function as a high-efficiency grating, and the diffraction efficiency of the -1st order is 95% under Littrow mounting. This dual-function grating design is based on a simplified modal method. By using the rigorous coupled-wave analysis, the optimum grating parameters can be determined. Holographic recording technology and inductively coupled plasma etching are used to manufacture the fused-silica grating. Experimental results are in agreement with the theoretical values.

Modal analysis of deep-etched low-contrast two-port beam splitter grating

Modal analysis of a deep-etched low-contrast two-port beam splitter grating under Littrow mounting is presented. The guideline for the design of a subwavelength transmission fused-silica phase grating as high-efficiency grating, polarizing beam splitter (PBS), and two-port beam splitter, is summarized. As an example, a polarization-independent two-port beam splitter grating is designed at wavelength of 1064 nm. We firstly analyzed the physical essence of the grating by the simplified modal method. The guideline for the grating design and the approximate grating parameters are obtained. Then using the rigorous coupled-wave analysis (RCWA) with parameters varying around the approximate ones, optimum grating parameters can be determined. With the design guideline, the time for the rigorous calculation of the grating profile parameters can be reduced significantly.