All fourteen Bravais lattices can be formed by interference of four noncoplanar beams

Photolithographic realization of target nanostructures in 3D space by inverse design of phase modulation

The mass production of precise three-dimensional (3D) nanopatterns has long been the ultimate goal of fabrication technology. While interference lithography and proximity-field nanopatterning (PnP) may provide partial solutions, their setup complexity and limited range of realizable structures, respectively, remain the main problems. Here, we tackle these challenges by applying an inverse design to the PnP process. Our inverse design platform based on the adjoint method can efficiently find optimal phase masks for diverse target lattices and motifs. We fabricate a 2D rectangular array of nanochannels, which has not been reported for conventional PnP with normally incident light, as a proof of concept. With further demonstration of material conversion, our work provides versatile platforms for nanomaterial fabrication.

What are the traveling waves composing the Hermite-Gauss beams that make them structured wavefields?

To the best of our knowledge, at the present time there is no answer to the fundamental question stated in the title that provides a complete and satisfactory physical description of the structured nature of Hermite-Gauss beams. The purpose of this manuscript is to provide proper answers supported by a rigorous mathematical-physics framework that is physically consistent with the observed propagation of these beams under different circumstances. In the process we identify that the paraxial approximation introduces spurious effects in the solutions that are unphysical. By removing them and using the property of self-healing, that is characteristic to structured beams, we demonstrate that Hermite-Gaussian beams are constituted by the superposition of four traveling waves.

Dynamic photonic crystal in a colloidal quantum-dot solution: formation, structure analysis, and dimensionality switching

We demonstrated, for the first time, to the best of our knowledge, a simple method to create three-dimensional (3D) dynamic photonic crystal (PhC) with controllable lattice symmetry through the interference of four non-coplanar laser beams in a non-linear optical medium [colloidal solution of CdSe/ZnS quantum dots (QDs)]. 3D dynamic PhC was formed due to the periodically changing refraction and absorption of resonantly excited excitons in the colloidal solution of QDs. The formation of dynamic PhC was confirmed by the observed self-diffraction of the laser beams on the dynamic structure which they have created. Tuning of the PhC dimensionality to the two-dimensional (2D) and one-dimensional (1D) was done through the reduction of the number of interfering beams to three and two, respectively, and by controlling the polarization of interacting beams. Physical processes responsible for the observed self-action effects that arise in CdSe/ZnS QDs are discussed in detail.

Continuous roll-to-roll patterning of three-dimensional periodic nanostructures

In this work, we introduce a roll-to-roll system that can continuously print three-dimensional (3D) periodic nanostructures over large areas. This approach is based on Langmuir-Blodgett assembly of colloidal nanospheres, which diffract normal incident light to create a complex intensity pattern for near-field nanolithography. The geometry of the 3D nanostructure is defined by the Talbot effect and can be precisely designed by tuning the ratio of the nanosphere diameter to the exposure wavelength. Using this system, we have demonstrated patterning of 3D photonic crystals with a 500 nm period on a 50 × 200 mm² flexible substrate, with a system throughput of 3 mm/s. The patterning yield is quantitatively analyzed by an automated electron beam inspection method, demonstrating long-term repeatability of an up to 88% yield over a 4-month period. The inspection method can also be employed to examine pattern uniformity, achieving an average yield of up to 78.6% over full substrate areas. The proposed patterning method is highly versatile and scalable as a nanomanufacturing platform and can find application in nanophotonics, nanoarchitected materials, and multifunctional nanostructures.

Periodic particle arrangements using standing acoustic waves

We determine crystal-like materials that can be fabricated by using a standing acoustic wave to arrange small particles in a non-viscous liquid resin, which is cured afterwards to keep the particles in the desired locations. For identical spherical particles with the same physical properties and small compared to the wavelength, the locations where the particles are trapped correspond to the minima of an acoustic radiation potential which describes the net forces that a particle is subject to. We show that the global minima of spatially periodic acoustic radiation potentials can be predicted by the eigenspace of a small real symmetric matrix corresponding to its smallest eigenvalue. We relate symmetries of this eigenspace to particle arrangements composed of points, lines or planes. Since waves are used to generate the particle arrangements, the arrangement's periodicity is limited to certain Bravais lattice classes that we enumerate in two and three dimensions.

Metasurface-generated complex 3-dimensional optical fields for interference lithography

Fast submicrometer-scale 3D printing techniques are of interest for various applications ranging from photonics and electronics to tissue engineering. Interference lithography is a versatile 3D printing method with the ability to generate complicated nanoscale structures. Its application, however, has been hindered by either the complicated setups in multibeam lithography that cause sensitivity and impede scalability or the limited level of control over the fabricated structure achievable with mask-assisted processes. Here, we show that metasurface masks can generate complex volumetric intensity distributions with submicrometer scales for fast and scalable 3D printing. These results push the limits of optical devices in controlling the light intensity distribution and significantly increase the realm of possibilities for 3D printing.

Fast, large-scale, and robust 3-dimensional (3D) fabrication techniques for patterning a variety of structures with submicrometer resolution are important in many areas of science and technology such as photonics, electronics, and mechanics with a wide range of applications from tissue engineering to nanoarchitected materials. From several promising 3D manufacturing techniques for realizing different classes of structures suitable for various applications, interference lithography with diffractive masks stands out for its potential to fabricate complex structures at fast speeds. However, the interference lithography masks demonstrated generally suffer from limitations in terms of the patterns that can be generated. To overcome some of these limitations, here we propose the metasurface-mask–assisted 3D nanofabrication which provides great freedom in patterning various periodic structures. To showcase the versatility of this platform, we design metasurface masks that generate exotic periodic lattices like gyroid, rotated cubic, and diamond structures. As a proof of concept, we experimentally demonstrate a diffractive element that can generate the diamond lattice.

Recent progress in near-field nanolithography using light interactions with colloidal particles: from nanospheres to three-dimensional nanostructures

The advance of nanotechnology is firmly rooted in the development of cost-effective, versatile, and easily accessible nanofabrication techniques. The ability to pattern complex two-dimensional and three-dimensional nanostructured materials are particularly desirable, since they can have novel physical properties that are not found in bulk materials. This review article will report recent progress in utilizing self-assembly of colloidal particles for nanolithography. In these techniques, the near-field interactions of light and colloids are the sole mechanisms employed to generate the intensity distributions for patterning. Based on both 'bottom-up' self-assembly and 'top-down' lithography approaches, these processes are highly versatile and can take advantage of a number of optical effects, allowing the complex 3D nanostructures to be patterned using single exposures. There are several key advantages including low equipment cost, facile structure design, and patterning scalability, which will be discussed in detail. We will outline the underlying optical effects, review the geometries that can be fabricated, discuss key limitations, and highlight potential applications in nanophotonics, optoelectronic devices, and nanoarchitectured materials.

Flexible Holographic Fabrication of 3D Photonic Crystal Templates with Polarization Control through a 3D Printed Reflective Optical Element

In this paper, we have systematically studied the holographic fabrication of three-dimensional (3D) structures using a single 3D printed reflective optical element (ROE), taking advantage of the ease of design and 3D printing of the ROE. The reflective surface was setup at non-Brewster angles to reflect both s- and p-polarized beams for the interference. The wide selection of reflective surface materials and interference angles allow control of the ratio of s- and p-polarizations, and intensity ratio of side-beam to central beam for interference lithography. Photonic bandgap simulations have also indicated that both s and p-polarized waves are sometimes needed in the reflected side beams for maximum photonic bandgap size and certain filling fractions of dielectric inside the photonic crystals. The flexibility of single ROE and single exposure based holographic fabrication of 3D structures was demonstrated with reflective surfaces of ROEs at non-Brewster angles, highlighting the capability of the ROE technique of producing umbrella configurations of side beams with arbitrary angles and polarizations and paving the way for the rapid throughput of various photonic crystal templates.

Multiple beam interference lithography: A tool for rapid fabrication of plasmonic arrays of arbitrary shaped nanomotifs

A novel method enabling rapid fabrication of 2D periodic arrays of plasmonic nanoparticles across large areas is presented. This method is based on the interference of multiple coherent beams originating from diffraction of large-diameter collimated beam on a transmission phase mask. Mutual orientation of the interfering beams is determined by parameters of the used phase mask. Herein, parameters of the phase mask (periods and modulation depth) are selected to yield an interference pattern with high contrast and narrow well-separated maxima. Finally, multiple beam interference lithography (MBIL)-based fabrication of periodic plasmonic arrays with selected nanomotifs including discs, disc dimers, rods and bowtie antennas is demonstrated.

Laser Scanning Holographic Lithography for Flexible 3D Fabrication of Multi-Scale Integrated Nano-structures and Optical Biosensors

Three-dimensional (3D) periodic nanostructures underpin a promising research direction on the frontiers of nanoscience and technology to generate advanced materials for exploiting novel photonic crystal (PC) and nanofluidic functionalities. However, formation of uniform and defect-free 3D periodic structures over large areas that can further integrate into multifunctional devices has remained a major challenge. Here, we introduce a laser scanning holographic method for 3D exposure in thick photoresist that combines the unique advantages of large area 3D holographic interference lithography (HIL) with the flexible patterning of laser direct writing to form both micro- and nano-structures in a single exposure step. Phase mask interference patterns accumulated over multiple overlapping scans are shown to stitch seamlessly and form uniform 3D nanostructure with beam size scaled to small 200 μm diameter. In this way, laser scanning is presented as a facile means to embed 3D PC structure within microfluidic channels for integration into an optofluidic lab-on-chip, demonstrating a new laser HIL writing approach for creating multi-scale integrated microsystems.

Invited Article: Progress in coherent lithography using table-top extreme ultraviolet lasers

Compact (table top) lasers emitting at wavelengths below 50 nm had expanded the spectrum of applications in the extreme ultraviolet (EUV). Among them, the high-flux, highly coherent laser sources enabled lithographic approaches with distinctive characteristics. In this review, we will describe the implementation of a compact EUV lithography system capable of printing features with sub-50 nm resolution using Talbot imaging. This compact system is capable of producing consistent defect-free samples in a reliable and effective manner. Examples of different patterns and structures fabricated with this method will be presented.

Polarization control in flexible interference lithography for nano-patterning of different photonic structures with optimized contrast

Half-wave plates were introduced into an interference-lithography scheme consisting of three fibers that were arranged into a rectangular triangle. Such a flexible and compact geometry allows convenient tuning of the polarizations of both the UV laser source and each branch arm. This not only enables optimization of the contrast of the produced photonic structures with expected square lattices, but also multiplies the nano-patterning functions of a fixed design of fiber-based interference lithography. The patterns of the photonic structures can be thus tuned simply by rotating a half-wave plate.

Flexible method based on four-beam interference lithography for fabrication of large areas of perfectly periodic plasmonic arrays

A novel nanofabrication technique based on 4-beam interference lithography is presented that enables the preparation of large macroscopic areas (>50 mm2) of perfectly periodic and defect-free two-dimensional plasmonic arrays of nanoparticles as small as 100 nm. The technique is based on a special interferometer, composed of two mirrors and a sample with photoresist that together form a right-angled corner reflector. In such an interferometer, the incoming expanded laser beam is split into four interfering beams that yield an interference pattern with rectangular symmetry. The interferometer allows setting the periods of the array from about 220 nm to 1500 nm in both directions independently through the rotation of the corner-reflector assembly around horizontal and vertical axes perpendicular to the direction of the incident beam. Using a theoretical model, the implementation of the four-beam interference lithography is discussed in terms of the optimum contrast as well as attainable periods of the array. Several examples of plasmonic arrays (on either glass or polymer substrate layers) fabricated by this technique are presented.

Custom-modified three-dimensional periodic microstructures by pattern-integrated interference lithography

By combining interference lithography and projection photolithography concurrently, pattern-integrated interference lithography (PIIL) enables the wafer-scale, rapid, and single-exposure fabrication of multidimensional periodic microstructures that integrate arbitrary functional elements. To date, two-dimensional PIIL has been simulated and experimentally demonstrated. In this paper, we report new simulated results of PIIL exposures for various custom-modified three-dimensional (3D) periodic structures. These results were generated using custom PIIL comprehensive vector modeling. Simulations include mask-integrated and mask-shaped 3D periodic arrangements as well as microcavities on top of or fully embedded within 3D periodic structures. These results indicate PIIL is a viable method for making versatile 3D periodic microstructures.

Performance simulation of 2D photonic-crystal devices fabricated by pattern-integrated interference lithography

Pattern-integrated interference lithography (PIIL) has recently been proposed as a rapid, single-step, and wafer-scale fabrication technique for custom-modified one-, two- and three-dimensional periodic structures. Among these structures, photonic-crystal devices have significant potential applications. In this work, we simulate the fabrication of two-dimensional photonic-crystal devices by PIIL using a rigorous vector modeling and realistic photolithographic conditions. We also model the etched patterns in silicon and evaluate the photonic-crystal motif-area and motif-displacement errors. We further calculate the device intensity transmission spectra and show that the performance of PIIL-produced devices are comparable to, and in some cases are superior to, that of their idealized equivalents.

Nanoarchitectures with controllable anisotropic features in structures and properties from simple and robust holographic lithography

Anisotropic nanostructures with precise orientations or sharp corners display unique properties that may be useful in a variety of applications; however, precise control over the anisotropy of geometric features, using a simple and reproducible large-area fabrication technique, remains a challenge. Here, we report the fabrication of highly uniform polymeric and metallic nanostructure arrays prepared using prism holographic lithography (HL) in such a way that the isotropy that can be readily and continuously tuned. The prism position on the sample stage was laterally translated to vary the relative intensities of the four split beams, thereby tuning the isotropy of the resulting polymer nanostructures through the following shapes: circular nanoholes, elliptical nanoholes, and zigzag-shaped nanoarrays. Corresponding large-area, defect-free anisotropic metallic nanostructures could then be fabricated using an HL-featured porous polymer structure as a milling mask. Removal of the polymer mask left zigzag-shaped metallic nanostructure arrays in which nanogaps separated adjacent sharp edges. These structures displayed two distinct optical properties, depending on the direction along which the excitation beam was polarized (longitudinal and transverse modes) incident on the array. Furthermore, bidirectional anisotropic wetting was observed on the anisotropic polymer nanowall array surface.

Pattern-integrated interference [Invited]

Pattern-integrated interference (PII) is described as a logical progression starting from the primary precursors of interference and holography. PII produces, in a single-exposure step, a periodic interference pattern with preselected periods absent. These blocked periods, for example, can form the nonperiodic functional elements of a photonic-crystal device or the circuit elements in a periodic-layout-design semiconductor chip. Various possible system configurations for PII are presented and compared. Example PII-produced intensity patterns for a photonic-crystal microresonator filter and an optical switch are simulated and discussed.

Synthesis of spatially variant lattices

It is often desired to functionally grade and/or spatially vary a periodic structure like a photonic crystal or metamaterial, yet no general method for doing this has been offered in the literature. A straightforward procedure is described here that allows many properties of the lattice to be spatially varied at the same time while producing a final lattice that is still smooth and continuous. Properties include unit cell orientation, lattice spacing, fill fraction, and more. This adds many degrees of freedom to a design such as spatially varying the orientation to exploit directional phenomena. The method is not a coordinate transformation technique so it can more easily produce complicated and arbitrary spatial variance. To demonstrate, the algorithm is used to synthesize a spatially variant self-collimating photonic crystal to flow a Gaussian beam around a 90° bend. The performance of the structure was confirmed through simulation and it showed virtually no scattering around the bend that would have arisen if the lattice had defects or discontinuities.

Pattern-integrated interference lithography: single-exposure fabrication of photonic-crystal structures

Multibeam interference represents an approach for producing one-, two-, and three-dimensional periodic optical-intensity distributions with submicrometer features and periodicities. Accordingly, interference lithography (IL) has been used in a wide variety of applications, typically requiring additional lithographic steps to modify the periodic interference pattern and create integrated functional elements. In the present work, pattern-integrated interference lithography (PIIL) is introduced. PIIL is the integration of superposed pattern imaging with IL. Then a pattern-integrated interference exposure system (PIIES) is presented that implements PIIL by incorporating a projection imaging capability in a novel three-beam interference configuration. The purpose of this system is to fabricate, in a single-exposure step, a two-dimensional periodic photonic-crystal lattice with nonperiodic functional elements integrated into the periodic pattern. The design of the basic system is presented along with a model that simulates the resulting optical-intensity distribution at the system sample plane where the three beams simultaneously interfere and integrate a superposed image of the projected mask pattern. Appropriate performance metrics are defined in order to quantify the characteristics of the resulting photonic-crystal structure. These intensity and lattice-vector metrics differ markedly from the metrics used to evaluate traditional photolithographic imaging systems. Simulation and experimental results are presented that demonstrate the fabrication of example photonic-crystal structures in a single-exposure step. Example well-defined photonic-crystal structures exhibiting favorable intensity and lattice-vector metrics demonstrate the potential of PIIL for fabricating dense integrated optical circuits.

X-ray computed tomography of holographically fabricated three-dimensional photonic crystals

X-ray computed tomography is used to reconstruct the 3D structure of a polymeric photonic crystal. The reconstructed structure is compared to the structure predicted by a model. This analysis provides means to better understand deformations that occur during holographic fabrication of photonic crystals.

Pattern-integrated interference lithography instrumentation

Multi-beam interference (MBI) provides the ability to form a wide range of sub-micron periodic optical-intensity distributions with applications to a variety of areas, including photonic crystals (PCs), nanoelectronics, biomedical structures, optical trapping, metamaterials, and numerous subwavelength structures. Recently, pattern-integrated interference lithography (PIIL) was presented as a new lithographic method that integrates superposed pattern imaging with interference lithography in a single-exposure step. In the present work, the basic design and systematic implementation of a pattern-integrated interference exposure system (PIIES) is presented to realize PIIL by incorporating a projection imaging capability in a novel three-beam interference configuration. A fundamental optimization methodology is presented to model the system and predict MBI-patterning performance. To demonstrate the PIIL method, a prototype PIIES experimental configuration is presented, including detailed alignment techniques and experimental procedures. Examples of well-defined PC structures, fabricated with a PIIES prototype, are presented to demonstrate the potential of PIIL for fabricating dense integrated optical circuits, as well as numerous other subwavelength structures.

Three-beam interference lithography methodology

Three-beam interference lithography represents a technology capable of producing two-dimensional periodic structures for applications such as micro- and nanoelectronics, photonic crystal devices, metamaterial devices, biomedical structures, and subwavelength optical elements. In the present work, a systematic methodology for implementing optimized three-beam interference lithography is presented. To demonstrate this methodology, specific design and alignment parameters, along with the range of experimentally feasible lattice constants, are quantified for both hexagonal and square periodic lattice patterns. Using this information, example photonic crystal rodlike structures and holelike structures are fabricated by appropriately controlling the recording wavevector configuration along with the individual beam amplitudes and polarizations, and by changing between positive- or negative-type photoresists.

Sensor for monitoring the vibration of a laser beam based on holographic polymer dispersed liquid crystal films

A continuous multiple exposure diffraction grating (CMEDG) is fabricated holographically on polymer dispersed liquid crystal (PDLC) films using two-beam interference with multiple exposures. The grating is fabricated by exposing a PDLC film to 18 repeated exposure/non-exposure cycles with an angular step of ~10°/10° while it revolves a circle on a rotation stage. The structure of the sample thus formed is analyzed using a scanning electron microscope (SEM) and shows arc-ripples around the center. From the diffraction patterns of the formed grating obtained using a normally incident laser beam, some or all of the 18 recorded arc beams can be reconstructed, as determined by the probing location. Thus, it can be applied for use as a beam-vibration sensor for a laser.

Three-dimensional optically induced reconfigurable photorefractive nonlinear photonic lattices

We experimentally investigate the formation of reconfigurable three-dimensional (3D) nonlinear photonic lattices in an externally biased cerium doped strontium barium niobate photorefractive crystal by a spatial light modulator-assisted versatile simplified single step optical induction approach. The analysis of the generated 3D nonlinear photonic lattices by plane wave guiding, momentum space spectroscopy, and far field diffraction pattern imaging is presented, which points to the embedded potential of these 3D structures as reconfigurable platform to investigate advanced nonlinear light-matter interaction in periodic structures.

Conditions for primitive-lattice-vector-direction equal contrasts in four-beam-interference lithography

Four distinct conditions for primitive-lattice-vector-direction equal contrasts in four-beam interference are introduced and described. By maximizing the absolute contrast subject to an equal contrast condition, lithographically useful interference patterns are found. Each condition is described in terms of the corresponding constraints on the plane wave wave vectors, polarizations, and intensities. The resulting locations of global intensity maxima, minima, and saddle points are presented. Subordinate conditions for unity absolute contrast are also developed. Three lattices are treated for each condition: simple cubic, face-centered cubic, and body-centered cubic.

Contrast in four-beam-interference lithography

Specific configurations of four linearly polarized, monochromatic plane waves have previously been shown to be capable of producing interference patterns exhibiting the symmetries inherent in all 14 Bravais lattices. We present (1) the range of possible absolute contrasts, (2) the conditions for unity absolute contrast, and (3) the types of interference patterns possible for configurations of four beams interference that satisfy the uniform contrast condition. Results are presented for three Bravais lattice structures: Base- and face-centered cubic and simple cubic.

Holographic design and band gap evolution of photonic crystals formed with five-beam symmetric umbrella configuration

We propose a holographic design of five-beam symmetric umbrella configuration, where there are a central beam and four ambient beams symmetrically scattered around the central one with the same apex angle, for fabrication of three-dimensional photonic crystals with tetragonal or cubic symmetries, and systematically analyzed the band gap properties of resultant photonic crystals when the apex angle is continuously increased. Our calculations reveal that large complete photonic band gaps exist in a wide range of apex angle for a relatively low refractive index contrast. Specifically, the face-centered cubic structure with a relative band gap of 25.1% for epsilon = 11.9 can be obtained with this recording geometry conveniently where all the beams are incident from the same half-space. These results will provide us with more understanding of this important recording geometry and give guidelines to its use in experiments.

Structuring by multi-beam interference using symmetric pyramids

A method for producing optical structures using rotationally symmetric pyramids is proposed. Two-dimensional structures can be achieved using acute prisms. They form by multi-beam interference of plane waves that impinge from directions distributed symmetrically around the axis of rotational symmetry. Flat-topped pyramids provide an additional beam along the axis thus generating three-dimensional structures. Experimental results are consistent with the results of numerical simulations. The advantages of the method are simplicity of operation, low cost, ease of integration, good stability, and high transmittance. Possible applications are the fabrication of photonic micro-structures such as photonic crystals or array waveguides as well as multi-beam optical tweezers.

Conditions for self-collimation in three-dimensional photonic crystals

We introduce the theoretical criterion for achieving three-dimensional self-collimation of light in a photonic crystal. Based on this criterion, we numerically demonstrate a body-center-cubic structure that supports wide-angle self-collimation and is directly compatible with the recently developed holographic fabrication technique. We further show that both bends and beam splitters can be introduced into this structure by the use of interfaces.

Comprehensive modeling of near-field nano-patterning

Near-field nano-patterning greatly simplifies holographic lithography, but deformations in formed structures are potentially severe. A fast and efficient comprehensive model was developed to predict geometry more rigorously. Numerical results show simple intensity-threshold methods do not accurately predict shape or optical behavior. By modeling sources with partial coherence, unpolarized light, and an angular spectrum, it is shown that standard UV lamps can be used to form 3D structures.

Creation of line defects in holographic photonic crystals by a double-exposure thresholding method

Recording of periodic variations of amplitude and phase by the interference of coherent laser beams in a hologram offers a natural means for creating one-, two-, and three-dimensional photonic crystals. For device applications such as waveguides in optical communications, one usually needs to create defects in photonic crystals. We present an analysis and an experimental demonstration of a double-exposure method for creating photonic crystals with line defects. The idea is based on the principle of superposition of holographic grating patterns of different spatial periods while the recording medium is held stationary and on the application of a threshold to the recording medium. We use the same symmetrical optical architecture to achieve nondefective and defective holographic photonic crystals. The technique may be extended to the creation of defects based on functional synthesis by means of Fourier series, by use of light sources of other wavelengths with an appropriate high-contrast recording material.

Excitation strategies for optical lattice microscopy

Recently, new classes of optical lattices were identified, permitting the creation of arbitrarily large two- and three-dimensional arrays of tightly confined excitation maxima of controllable periodicity and polarization from the superposition of a finite set of plane waves. Here, experimental methods for the generation of such lattices are considered theoretically in light of their potential applications, including high resolution dynamic live cell imaging, photonic crystal fabrication, and quantum simulation and quantum computation using ultracold atoms.

Fully three-dimensional modeling of the fabrication and behavior of photonic crystals formed by holographic lithography

A comprehensive and fully three-dimensional model of holographic lithography is used to predict more rigorously the geometry and transmission spectra of photonic crystals formed in Epon SU-8 photoresist. It is the first effort known to the authors to incorporate physics of exposure, postexposure baking, and developing into three-dimensional models of photonic crystals. Optical absorption, reflections, standing waves, refraction, beam coherence, acid diffusion, resist shrinkage, and developing effects combine to distort lattices from their ideal geometry. These are completely neglected by intensity-threshold methods used throughout the literature to predict lattices. Numerical simulations compare remarkably well with experimental results for a face-centered-cube (FCC) photonic crystal. Absorption is shown to produce chirped lattices with broadened bandgaps. Reflections are shown to significantly alter lattice geometry and reduce image contrast. Through simulation, a diamond lattice is formed by multiple exposures, and a hybrid trigonal-FCC lattice is formed that exhibits properties of both component lattices.

Holographic creation of photonic crystals

We describe the creation of general photonic crystals by means of holography with an experimental demonstration. The recordings of periodic variations of amplitude and phase by the interference of coherent laser beams offer a natural means for the creation of one- two- or three-dimensional photonic crystals. Based on the principle of the interference of four noncoplanar beams, we present a comparative analysis of two different approaches for creating photonic crystals and use numerical simulated lattice structures to illustrate the differences between these two approaches. We then use a specific symmetrical optical architecture and select the proper approach to create holographic photonic crystals. The advantages and constraints of this holographic method are discussed.

Three-Dimensional Photonic Crystals

We review our work on two complementary and compatible techniques, namely direct laser writing and holographic lithography which are suitable for fabricating three-dimensional Photonic Crystal templates for the visible and near-infrared. The structures are characterized by electron micrographs and by optical spectroscopy, revealing their high optical quality.

Arrangements of four beams for any Bravais lattice

A single geometric model based on a new concept of a reciprocal primitive pyramid (RPP) in reciprocal space is proposed for investigation of relationships between any three-dimensional (3D) lattice and arrangements of four beams (AFBs) that produce the lattice. A ternary linear equation set, described for the one-to-one correspondence between a RPP and AFB, can readily reveal all AFBs for the same lattice (AFBSLs). Quantitative AFBs for bcc and fcc real lattices are illustrated to show that various AFBSLs can modulate the properties of a photonic bandgap (PBG) both by tuning the lattice constant and by changing the lattice-point shape. This fact may yield the appropriate AFB for a complete 3D PBG with the desired center wavelength. The nonuniqueness of AFBSLs can provide abundant choices for persons who plan interference experiments, especially for holographic fabrication of 3D photonic crystals (PCs).

Triply periodic bicontinuous structures through interference lithography: a level-set approach

Interference lithography holds the promise of fabricating large-area, defect-free photonic structures on the sub-micrometer scale both rapidly and cheaply. There is a need for a procedure to establish a connection between the structures that are formed and the parameters of the interfering beams. There is also a need to produce self-supporting three-dimensional bicontinuous structures. A generic technique correlating parameters of the interfering beams with the symmetry elements present in the resultant structures by a level-set approach is developed. A particular space group is ensured by equating terms of the intensity equation to a representative level surface of the desired space group. Single- and multiple-exposure techniques are discussed. The beam parameters for certain cubic bicontinuous structures relevant to photonic crystals, viz.,the diamond(D), the simple cubic (P), and the chiral gyroid (G) are derived by utilizing either linear or elliptically polarized light.

Tunable face-centered-cubic photonic crystal formed in holographic polymer dispersed liquid crystals

We report on the fabrication and electro-optic measurements of face-centered-cubic (fcc) lattices in holographic polymer dispersed liquid-crystal materials. Four linearly polarized coherent plane waves were interfered to generate a fcc optical lattice that was subsequently and indefinitely recorded as an arrayed pattern of nanometer-sized liquid-crystal droplets (approximately 50 nm) at lattice nodes within a polymer matrix. Observed transmission spectra and Kossel diffraction curves are consistent with fcc crystal structure. A completely reversible 2% wavelength shift of the (+/- 111) stop band was observed on application of an electric field.

Interference of four umbrellalike beams by a diffractive beam splitter for fabrication of two-dimensional square and trigonal lattices

A simple optical interference method for fabricating two-dimensional square and trigonal lattices is demonstrated. A general formula for the interference contrast formed by two arbitrary polarized elliptical waves is deduced, the relation between wave vectors of incident light and the resultant pattern is analyzed, and polarization optimization of all beams to ensure uniform contrast is given.

Polarization optimization in the interference of four umbrellalike symmetric beams for making three-dimensional periodic microstructures

A systematic and comprehensive analysis of the interference of four umbrellalike beams (lFUB) is provided based on the reciprocal space theory. The concept of pattern contrast is extended to the case of the IFUB, and it is indicated that a uniform contrast for all the interference terms can be obtained by properly choosing the beam ratio and the polarization of each beam. Different polarization combinations, including linear light and linear light, circular light and circular light, and linear light and circular light, have been discussed for the purpose of maximum uniform contrast. It is shown that the use of circular light may generally improve the uniform contrast. This study may lay a theoretical foundation for holographic fabrication of three-dimensional (3D) periodic microstructures, such as simple cubic, body-centered cubic, face-centered cubic, or trigonal lattice.