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NASA NTRS1993

Curved grating fabrication techniques for concentric-circle grating, surface-emitting semiconductor lasers

We describe the fabrication and operational characteristics of a novel, surface-emitting semiconductor laser that makes use of a concentric-circle grating to both define its resonant cavity and to provide surface emission. A properly fabricated circular grating causes the laser to operate in radially inward- and outward-going circular waves in the waveguide, thus, introducing the circular symmetry needed for the laser to emit a beam with a circular cross-section. The basic circular-grating-resonator concept can be implemented in any materials system; an AlGaAs/GaAs graded-index, separate confinement heterostructure (GRINSCH), single-quantum-well (SQW) semiconductor laser, grown by molecular beam epitaxy (MBE), was used for the experiments discussed here. Each concentric-circle grating was fabricated on the surface of the AlGaAs/GaAs semiconductor laser. The circular pattern was first defined by electron-beam (e-beam) lithography in a layer of polymethylmethacrylate (PMMA) and subsequently etched into the semiconductor surface using chemically-assisted (chlorine) ion-beam etching (CAIBE). We consider issues that affect the fabrication and quality of the gratings. These issues include grating design requirements, data representation of the grating pattern, and e-beam scan method. We provide examples of how these techniques can be implemented and their impact on the resulting laser performance. A comparison is made of the results obtained using two fundamentally different electron-beam writing systems. Circular gratings with period lambda = 0.25 microns and overall diameters ranging from 80 microns to 500 microns were fabricated. We also report our successful demonstration of an optically pumped, concentric-circle grating, semiconductor laser that emits a beam with a far-field divergence angle that is less than one degree. The emission spectrum is quite narrow (less than 0.1 nm) and is centered at wavelength lambda = 0.8175 microns.

Jordan, Rebecca H.↗

NASA NTRS1975

Calculation of arbitrary-order diffraction efficiencies of thick gratings with arbitrary grating shape

A method for calculating arbitrary-order diffraction efficiencies of thick, lossless transmission gratings with arbitrary periodic grating shapes has been developed. A Fourier-series representation of the grating is employed, along with a coupled-mode theory of diffraction. For illustration, numerical values of the diffraction efficiencies at the first three Bragg angles are calculated for sinusoidal, square-wave, triangular, and saw-tooth gratings. Numerical results for the same grating shapes with the same parameters are also calculated for comparison, by extending Burckhardt's numerical method for analyzing thick sinusoidal gratings. The comparison shows that the coupled-mode theory provides results with relative computational ease and results that are in agreement with calculations obtained by extending the more-rigorous Burckhardt theory to nonsinusoidal grating shapes and to higher-order Bragg angles.

Su, S. F.↗

NASA NTRS1991

The fabrication of toroidal and coma-corrected toroidal diffraction gratings from spherical master gratings using elastically-deformable substrates - A progress report

A technique has been developed which permits toroidal, and coma-corrected toroidal, diffraction gratings to be replicated from spherical master gratings by the use of elastically-deformable substrates. Toroidal gratings correct for astigmatism and, thus, make it possible to construct stigmatic spectrometers that employ a single reflective diffraction grating. These spectrometers are particularly useful for the extreme-ultraviolet (EUV) wavelength range, where reflection coefficients are low, since the single optical surface provides for dispersion, focusing, and astigmatism correction. The fabrication procedures for the pure toroidal, and coma-corrected toroidal, gratings are described, and initial test results are presented. The use of the toroidal gratings in a high-resolution sounding-rocket EUV spectroheliometer, and in both the coronal diagnostics spectrometer and the ultraviolet coronagraph spectrometer on the ESA/NASA solar and heliospheric observatory mission, is described briefly, and the use of this technique for the fabrication of a coma-corrected toroidal grating for the prime Rowland spectrograph of the FUSE/Lyman mission is briefly discussed.

Huber, Martin C. E.↗

NASA NTRS2001

Reflection Grating Array Associated with the Reflection Grating Spectrometer Developed by the Space Research Organization of the Netherlands for the X-ray Multi-Mirror Mission (XMM)

The University of California, Berkeley (UCB) served as the Principal Investigator institution for the United States participation in the development of the Reflection Grating Spectrometer (RGS) which included the design, development, fabrication, and testing of the Reflection Grating Assembly (RGA). UCB was assisted in this role by the Lawrence Livermore National Laboratory and Columbia University who provided the primary facilities, materials, services and personnel necessary to complete the development. UC Berkeley's Dr. Steven Kahn provided the technical and scientific oversight for the design. development and testing of the RGA units by monitoring the performance of the units at various stages in their development. Dr. Kahn was also the primary contact with the Space Research Organization of the Netherlands (SRON) and represented the RGA development at all SRON and European Space Agency (ESA) reviews of the RGA status. In accordance with the contract, the team designed and developed novel optical technology to meet the unique requirements of the RGS. The ESA XMM-Newton Mission carries two identical Reflection Grating Spectrometers (RGS) behind two of its three nested sets of Wolter I type mirrors. The instrument allows high-resolution measurements in the soft X-ray range (6 to 38 angstroms or 2.1 to 0.3 keV) with a maximum effective area of about 140 sq cm at 15 angstroms. Its design is optimized for the detection of the K-shell transitions of carbon, nitrogen, oxygen, neon, magnesium, and silicon. as well as the L shell transitions of iron. The RGA itself consists of two units. A structure for each unit was designed to hold up to 220 gratings. In its final configuration, one unit holds 182 gratings and the second hold 181 gratings.

Kahn, Steven M.↗

DOE OSTI2025

Attosecond Transient Grating Spectroscopy with Near-Infrared Grating Pulses and an Extreme Ultraviolet Diffracted Probe

Transient grating spectroscopy has become a mainstay among metal and semiconductor characterization techniques. Here, we extend the technique toward the shortest achievable time scales by using tabletop high-harmonic generation of attosecond extreme ultraviolet (XUV) pulses that diffract from transient gratings generated with sub-5 fs near-infrared (NIR) pulses. We demonstrate the power of attosecond transient grating spectroscopy (ATGS) by investigating the ultrafast photoexcited dynamics in an Sb semimetal thin film. ATGS provides an element-specific, background-free signal unfettered by spectral congestion, in contrast to transient absorption spectroscopy. With ATGS measurements in Sb polycrystalline thin films, we observe the generation of coherent phonons and investigate the lattice and carrier dynamics. Among the latter processes, we extract carrier thermalization, hot carrier cooling, and electron-hole recombination, which are on the order of 20 fs, 50 fs, and 2 ps time scales, respectively. Furthermore, the simultaneous collection of transient absorption and transient grating data allows us to extract the total complex dielectric constant in the sample dynamics with a single measurement, including the real-valued refractive index, from which we are also able to investigate carrier-phonon interactions and longer-lived phonon dynamics. The outlined experimental technique expands the capabilities of transient grating spectroscopy and attosecond spectroscopies by providing a wealth of information concerning carrier and lattice dynamics with an element-selective technique at the shortest achievable time scales.

Quintero-Bermudez, Rafael↗

NASA NTRS1974

Optical grating evaluator - A device for detailed measurement of diffraction grating efficiencies in the vacuum ultraviolet

A device for detailed measurement of diffraction grating efficiencies and over-all performance in the VUV has been designed and constructed at the Naval Research Laboratory. The system employs semiautomated mechanisms to scan the face of the grating with a narrow monochromatic beam, and an efficiency map of the grating surface is produced on a strip chart recorder. Grating efficiency in the various diffracted orders and intensity of light scattered between orders may also be measured. A unique feature is the ability to determine the angle and effectiveness of grating blaze and variations in blaze under different conditions of illumination.

Michels, D. J.↗

NASA NTRS1998

Stratified Diffractive Optic Approach for Creating High Efficiency Gratings

Gratings with high efficiency in a single diffracted order can be realized with both volume holographic and diffractive optical elements. However, each method has limitations that restrict the applications in which they can be used. For example, high efficiency volume holographic gratings require an appropriate combination of thickness and permittivity modulation throughout the bulk of the material. Possible combinations of those two characteristics are limited by properties of currently available materials, thus restricting the range of applications for volume holographic gratings. Efficiency of a diffractive optic grating is dependent on its approximation of an ideal analog profile using discrete features. The size of constituent features and, consequently, the number that can be used within a required grating period restricts the applications in which diffractive optic gratings can be used. These limitations imply that there are applications which cannot be addressed by either technology. In this paper we propose to address a number of applications in this category with a new method of creating high efficiency gratings which we call stratified diffractive optic gratings. In this approach diffractive optic techniques are used to create an optical structure that emulates volume grating behavior. To illustrate the stratified diffractive optic grating concept we consider a specific application, a scanner for a space-based coherent wind lidar, with requirements that would be difficult to meet by either volume holographic or diffractive optic methods. The lidar instrument design specifies a transmissive scanner element with the input beam normally incident and the exiting beam deflected at a fixed angle from the optical axis. The element will be rotated about the optical axis to produce a conical scan pattern. The wavelength of the incident beam is 2.06 microns and the required deflection angle is 30 degrees, implying a grating period of approximately 4 microns. Creating a high efficiency volume grating with these parameters would require a grating thickness that cannot be attained with current photosensitive materials. For a diffractive optic grating, the number of binary steps necessary to produce high efficiency combined with the grating period requires feature sizes and alignment tolerances that are also unattainable with current techniques. Rotation of the grating and integration into a space-based lidar system impose the additional requirements that it be insensitive to polarization orientation, that its mass be minimized and that it be able to withstand launch and space environments.

Chambers, Diana M.↗

NASA NTRS2008

Shaping Diffraction-Grating Grooves to Optimize Efficiency

A method of shaping diffraction-grating grooves to optimize the spectral efficiency, spectral range, and image quality of a spectral imaging instrument is under development. The method is based on the use of an advanced design algorithm to determine the possibly complex shape of grooves needed to obtain a desired efficiency-versus-wavelength response (see figure). Then electron- beam fabrication techniques are used to realize the required groove shape. The method could be used, for example, to make the spectral efficiency of the grating in a given wavelength range proportional to the inverse of the spectral efficiency of a photodetector array so that the overall spectral efficiency of the combination of the grating and the photodetector array would be flat. The method has thus far been applied to one-dimensional gratings only, but in principle, it is also applicable to two-dimensional gratings. The algorithm involves calculations in the spatial-frequency domain. The spatial-frequency spectrum of a grating is represented as a diffraction-order spectral-peak-width function multiplied by an efficiency function for a single grating groove. This representation affords computational efficiency and accuracy by making it possible to consider only the response from one grating groove (one period of the grating), instead of from the whole grating area, in determining the response from the entire grating. This combination of efficiency and accuracy is crucial for future extensions of the algorithm to two-dimensional designs and to designs in which polarization must also be taken into account. The algorithm begins with the definition of target values of relative efficiency that represent the desired spectral response of the grating in certain spectral frequencies calculated from the diffraction order and wavelength. The grating period is divided into a number of cells - typically, 100. The phase contribution from each cell is determined from the phase of the incident electromagnetic wave and the height of the grating surface in the cell. The total contribution from all cells to each target value is then calculated. Then a method known to specialists as the optimum-rotation-angle method is used to adjust the height of each cell so that the total response from all cells is optimized. The computation is iterative and continues until the desired response is obtained. In the event that the desired response is unphysical, the algorithm nevertheless strives to generate a grating-grove profile for which the response approximates the desired one as closely as possible.

Backlund, John↗

NASA NTRS2003

[Technology Development for X-Ray Reflection for the Constellation-X Reflection Grating Spectrometer (RGS)]

This Grant covers MIT support for the technology development of x-ray reflection gratings for the Constellation-X Reflection Grating Spectrometer (RGS). Since the start of the Grant MIT has extended its previously-developed patterning and super-smooth, blazed grating fabrication technology to ten-times smaller grating periods and ten-times larger blaze angles to demonstrate feasibility and performance in the off-plane grating geometry. In the past year we successfully developed several nanoimprint grating replication methods that achieved very high fidelity replication of master silicon gratings. Grating geometry on the nano and macro scales were faithfully replicated, demonstrating the viability of the process for manufacturing the thousands of gratings required for the RGS. We also successfully developed an improved metrology truss for holding test grating substrates during metrology. The flatness goal of grating substrates is under 500 nm. In the past, grating holders would cause non-repeatable distortion of >> 500 nm to the substrates due to friction and gravity sag. The new holder has a repeatability of under 50 nm which is adequate for the proposed RGS grating substrates.

Schattenburg, Mark L.↗

NASA NTRS2009

Fabricating Radial Groove Gratings Using Projection Photolithography

Projection photolithography has been used as a fabrication method for radial grove gratings. Use of photolithographic method for diffraction grating fabrication represents the most significant breakthrough in grating technology in the last 60 years, since the introduction of holographic written gratings. Unlike traditional methods utilized for grating fabrication, this method has the advantage of producing complex diffractive groove contours that can be designed at pixel-by-pixel level, with pixel size currently at the level of 45 45 nm. Typical placement accuracy of the grating pixels is 10 nm over 30 nm. It is far superior to holographic, mechanically ruled or direct e-beam written gratings and results in high spatial coherence and low spectral cross-talk. Due to the smooth surface produced by reactive ion etch, such gratings have a low level of randomly scattered light. Also, due to high fidelity and good surface roughness, this method is ideally suited for fabrication of radial groove gratings. The projection mask is created using a laser writer. A single crystal silicon wafer is coated with photoresist, and then the projection mask, with its layer of photoresist, is exposed for patterning in a stepper or scanner. To develop the photoresist, the fabricator either removes the exposed areas (positive resist) of the unexposed areas (negative resist). Next, the patterned and developed photoresist silicon substrate is subjected to reactive ion etching. After this step, the substrate is cleaned. The projection mask is fabricated according to electronic design files that may be generated in GDS file format using any suitable CAD (computer-aided design) or other software program. Radial groove gratings in off-axis grazing angle of incidence mount are of special interest for x-ray spectroscopy, as they allow achieving higher spectral resolution for the same grating area and have lower alignment tolerances than traditional in-plane grating scheme. This is especially critical for NASA Constellation- X project that will utilize hundreds of gratings all of which need to be precisely aligned for x-ray observation of space.

Iazikov, Dmitri↗

NASA NTRS2010

Focusing Diffraction Grating Element with Aberration Control

Diffraction gratings are optical components with regular patterns of grooves, which angularly disperse incoming light by wavelength in a single plane, called dispersion plane. Traditional gratings on flat substrates do not perform wavefront transformation in the plane perpendicular to the dispersion plane. The device proposed here exhibits regular diffraction grating behavior, dispersing light. In addition, it performs wavelength transformation (focusing or defocusing) of diffracted light in a direction perpendicular to the dispersion plane (called sagittal plane). The device is composed of a diffraction grating with the grooves in the form of equidistant arcs. It may be formed by defining a single arc or an arc approximation, then translating it along a certain direction by a distance equal to a multiple of a fixed distance ("grating period") to obtain other groove positions. Such groove layout is nearly impossible to obtain using traditional ruling methods, such as mechanical ruling or holographic scribing, but is trivial for lithographically scribed gratings. Lithographic scribing is the newly developed method first commercially introduced by LightSmyth Technologies, which produces gratings with the highest performance and arbitrary groove shape/spacing for advanced aberration control. Unlike other types of focusing gratings, the grating is formed on a flat substrate. In a plane perpendicular to the substrate and parallel to the translation direction, the period of the grating and, therefore, the projection of its k-vector onto the plane is the same for any location on the grating surface. In that plane, no waveform transformation by the grating k-vector occurs, except of simple redirection.

Iazikov, Dmitri↗

NASA NTRS2004

Reannealed Fiber Bragg Gratings Demonstrated High Repeatability in Temperature Measurements

Fiber Bragg gratings (FBGs) are formed by periodic variations of the refractive index of an optical fiber. These periodic variations allow an FBG to act as an embedded optical filter, passing the majority of light propagating through a fiber while reflecting back a narrow band of the incident light. The peak reflected wavelength of the FBG is known as the Bragg wavelength. Since the period and width of the refractive index variation in the fiber determines the wavelengths that are transmitted and reflected by the grating, any force acting on the fiber that alters the physical structure of the grating will change the wavelengths that are transmitted and reflected by it. Both thermal and mechanical forces acting on the grating will alter its physical characteristics, allowing the FBG sensor to detect both the temperature variations and the physical stresses and strains placed upon it. This ability to sense multiple physical forces makes the FBG a versatile sensor. To assess the feasibility of using Bragg gratings as temperature sensors for propulsion applications, researchers at the NASA Glenn Research Center evaluated the performance of Bragg gratings at elevated temperatures for up to 300 C. For these purposes, commercially available polyimide-coated high-temperature gratings were used that were annealed by the manufacturer to 300 C. To assure the most thermally stable gratings at the operating temperatures, we reannealed the gratings to 400 C at a very slow rate for 12 to 24 hr until their reflected optical powers were stabilized. The reannealed gratings were then subjected to periodic thermal cycling from room temperature to 300 C, and their peak reflected wavelengths were monitored. The setup shown is used for reannealing and thermal cycling the FBGs. Signals from the photodetectors and the spectrum analyzer were fed into a computer equipped with LabVIEW software. The software synchronously monitored the oven/furnace temperature and the optical spectrum analyzer as well as processed the data. Experimental results presented in the following graph show typical wavelength versus temperature dependence of a reannealed FBG through six thermal cycles (80 hr). The average standard deviation of the temperature-to-wavelength relationship ranged from 1.86 to 2.92 C over the six thermal cycles each grating was subjected to. This is an error of less than 1.0 percent of full scale throughout the entire evaluation temperature range from ambient to 300 C.

Adamovsky, Grigory↗

NASA NTRS1989

Efficiency measurements of reflection gratings in the 100-300-A band

Zero-order and first-order efficiencies of plane diffraction grazings over the 100-300-A band have been measured using the NRL beamline facility at the National Synchrotron Light Source. Measurements were taken at grating angles of 5, 10, and 15 deg, and the incident radiation was polarized primarily in the transverse magnetic mode. Four (three laminar, one blazed) of the gratings were ion etched in high quality quartz blanks, overcoated with 200 A of gold. A fifth (blazed) grating was ruled in gold. The line densities are 1000, 2000, and 3600 grooves/mm for those with laminar profiles. Both blazed gratings have a blaze angle of 8 deg and line densities of 3600 grooves/mm. Measured first-order efficiencies up to 15 percent were obtained for the laminar gratings and up to 8 percent was obtained for the ion-etched blazed grating. Much lower efficiencies (not greater than 2 percent) were measured for the ruled grating. A computer program was written to calculate grating efficiencies using a full electromagnetic model. Measured efficiencies of the ion-etched gratings agree well with predicted values, but the measured first-order efficiency of the ruled grating is much lower than predicted.

Meekins, John F.↗

NASA NTRS1995

System Construction for the Measurement of Bragg Grating Characteristics in Optical Fibers

Bragg gratings are used to measure strain in optical fibers. To measure strain they are sometimes used as a smart structure. They must be characterized after they are written to determine their spectral response. This paper deals with the test setup to characterize Bragg grating spectral responses.Bragg gratings are a photo-induced phenomena in optical fibers. The gratings can be used to measure strain by measuring the shift in wavelength. They placed the fibers into a smart structure to measure the stress and strain produced on support columns placed in bridges. As the cable is subjected to strain the grating causes a shift to a longer wavelength if the fiber is stretched and a shift to a shorter wavelength shift if the fiber is compacted. Our applications involve using the fibers to measure stress and strain on airborne systems. There are many ways to write Bragg gratings into optical fibers. Our focus is on side writing the grating. Our capabilities are limited in the production rate of the gratings. The Bragg grating is written into a fiber and becomes a permanent fixture. We are writing the grating to be centered at 1300 nm because that is the standard phase mask wavelength.

West, Douglas P.↗

NASA NTRS2000

All-Optical Micro Motors Based on Moving Gratings in Photosensitive Media

An all-optical micro motor with a rotor driven by a traveling wave of surface deformation of a stator being in contact with the rotor is being studied. Instead of an ultrasonic wave produced by an electrically driven piezoelectric actuator as in ultrasonic motors, the wave is a result of a photo induced surface deformation of a photosensitive material produced by a traveling holographic grating. Two phase modulated coherent optical beams generate the grating. Several types of photosensitive materials are studied such as photorefractive crystals, photosensitive piezoelectric ceramics, and side-chain liquid crystalline polyesters. In order to be considered as a possible candidate for micro motors, the material should exhibit surface deformation produced by moving grating of the order of 10 micron. Deformations produced by static holographic gratings are studied in photorefractive crystals of LiNbO3 using high vertical resolution surface profilometer Dektak 3 and surface interferometer WYKO. An experimental set-up with moving grating has been developed. The set-up uses a two-beam interferometry configuration with one beam being reflected by a thin mirror mounted on a loud speaker. A ramp voltage signal generator drives the speaker. Changing voltage, polarity, and frequency of the signal can easily generate vibrating gratings or moving gratings in both directions. A vibrating grating has been applied to a photorefractive crystal of BSO controlled by an external electric field of the order of 104 V/cm. We have additionally studied effects of moving grating interaction with light absorbing fluids such as solutions of 2,9,16,23-Tetrakis(phenylthio)-29H, 31 Hphthalocyanine in chlorobenzene in capillary tubes. The purpose of using a liquid is to show that the moving gratings can force a liquid to shift. The interaction of a single low power focused laser beam at 633 nm with such fluid produced an intensive circular motion, which also might be applied to all-optical micro motors or actuators.

Curley, M.↗

NASA NTRS2003

Binary-Phase Fourier Gratings for Nonuniform Array Generation

We describe a design method for a binary-phase Fourier grating that generates an array of spots with nonuniform, user-defined intensities symmetric about the zeroth order. Like the Dammann fanout grating approach, the binary-phase Fourier grating uses only two phase levels in its grating surface profile to generate the final spot array. Unlike the Dammann fanout grating approach, this method allows for the generation of nonuniform, user-defined intensities within the final fanout pattern. Restrictions governing the specification and realization of the array's individual spot intensities are discussed. Design methods used to realize the grating employ both simulated annealing and nonlinear optimization approaches to locate optimal solutions to the grating design problem. The end-use application driving this development operates in the near- to mid-infrared spectrum - allowing for higher resolution in grating specification and fabrication with respect to wavelength than may be available in visible spectrum applications. Fabrication of a grating generating a user-defined nine spot pattern is accomplished in GaAs for the near-infrared. Characterization of the grating is provided through the measurement of individual spot intensities, array uniformity, and overall efficiency. Final measurements are compared to calculated values with a discussion of the results.

Keys, Andrew S.↗

NASA NTRS2005

[Development of X-ray Reflection Grating Technology for the Constellation-X Mission]

This Grant supports MIT technology development of x-ray reflection gratings for the Constellation-X Reflection Grating Spectrometer (RGS). Since the start of the Grant MIT has extended its previously-developed patterning and super-smooth, blazed grating fabrication technology to ten-times smaller grating periods and ten-times larger blaze angles to demonstrate feasibility and performance in the off-plane grating geometry. In the past year we have focused our efforts on extending our Nanoruler grating fabrication tool to enable it to perform variable-period scanning-beam interference lithography (VP-SBIL). This new capability required extensive optical and mechanical improvements to the system. The design phase of this work is largely completed and key components are now on order and assembly has begun. Over the next several months the new VP-SBIL Nanoruler system will be completed and testing begun. We have also demonstrated a new technique for patterning gratings using the Nanoruler called Doppler mode, which will be important for patterning the radial groove gratings for the RGS using the new VP-SBIL system. Flat and thin grating substrates will be critical for the RGS. In the last year we demonstrated a new technique for flattening thin substrates using magneto-rheologic fluid polishing (MRF) and achieved 2 arcsecond flatness with a 0.5 mm-thick substrate-a world's record. This meets the Con X requirement for grating substrate flatness.

Schattenburg, Mark L.↗

NASA NTRS2000

Cross-fiber Bragg grating transducer

A transducer has been invented that uses specially-oriented gratings in waveguide a manner that allows the simultaneous measurement of physical phenomena (such as shear force, strain and temperature) in a single sensing element. The invention has a highly sensitive, linear response and also has directional sensitivity with regard to strain. The transducer has a waveguide with a longitudinal axis as well as two Bragg gratings. The transducer has a first Bragg grating associated with the waveguide that has an angular orientation .theta..sub.a relative to a perpendicular to the longitudinal axis such that 0.degree.<.theta..sub.a <.theta..sub.max. The second Bragg grating is associated with the waveguide in such a way that the angular orientation .theta..sub.b of the grating relative to a perpendicular to the longitudinal axis is (360.degree.-.theta..sub.max)<.theta..sub.b <360.degree.. The first Bragg grating can have a periodicity .LAMBDA..sub.a and the second Bragg grating can have a periodicity .LAMBDA..sub.b such that the periodicity .LAMBDA..sub.a of the first Bragg grating does not equal the periodicity .LAMBDA..sub.b of the second Bragg grating. The angle of the gratings can be such that .theta..sub.a =360.degree.-.theta..sub.b. The waveguide can assume a variety of configurations, including an optical fiber, a rectangular waveguide and a planar waveguide. The waveguide can be fabricated of a variety of materials, including silica and polymer material.

Albin, Sacharia↗