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Where, f is the focal length of the focus lens, λ is the wavelength of interference beams and b is the beam aperture on the mask. The area of tight focusing can be calculated by the equation : As a result of the tight focusing, the interference area is limited. 14, in order to form the interference pattern, a lens was used to make the parallel beamlets overlap in the interference area. This technique will be very useful in the manufacturing of the integrated optics devices. Furthermore, using this method a large variety of complex optical structures can be fabricated, such as 3D PCs with periodic defects, quasicrystals and functional metamaterials. Since it is a single optical axis system, the aberrations due to multi-optical-axes can be avoided. In this report, by introducing a diffraction optical element (DOE) and an LC-SLM, large area PCs with periodic defects have been fabricated. It is quite difficult to avoid aberrations due to the unparallel optical axes and to the defects of small lenses. Unfortunately, a lens array is a multi-optical-axis system. To solve this problem, a lens array was used to expand the interference region.

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The uniform area is about 500 μm2 and only one defect could be fabricated in the PC structure. However, as a lens had been used to focus the interference beams, the interference area was quite small and not uniform. This technique is based on multi-beam interference, the pixel of SLM can be very large (up to 1~2 mm), and ten pixels are enough to fabricate a photonic crystal structure with defects. A liquid crystal spatial light modulator (LC-SLM) was used to control the phases of interference beamlets. Recently, we developed a novel method to introduce defects into the PCs using one-step holographic lithography.

Since the smallest pixel of the controlled phase mask is about 1 μm, the period of the fabricated PCs is larger than a few μm by using this technique. This technique is based on multi-beam diffraction with a phase mask, whose pixels should be much smaller than the period of the PCs, and thousands of pixels are required. In 2004, the phase-control holographic lithography was used to introduce defects in a single step. However, this technique requires a costly and complicated second step and an accuracy aligning to the manufacturing process. One way is directly writing by femtosecond laser after the crystal has already been formed.

Since the applications of PCs usually require accurate introduction of defects into the PCs’ structure, developing a method to effectively and accurately incorporate functional defects directly into PCs is necessary. So far holographic lithography has been successful in fabricating large area, high quality PCs templates.

The pattern can be controlled by the beam propagation direction, the number of the interfering beams, the beam intensities, their respective polarizations and relative phases.

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Using HL to make PCs is one of the most promising techniques for mass production of large area, defect free PC structures. These techniques include e-beam lithography for multilayer stacking of woodpile structures, colloidal self-assembly, multi-photon direct laser writing, and holographic lithography (HL) based on multiple beam interference. Several techniques have succeeded to a certain degree in fabricating 3D PCs. Fabrication of 3D PCs was a great challenge over the past decades. We show that the tuning capability is available if the cubic nonlinearity is deployed in the internal springs supporting the resonators.Photonic crystals (PCs), proposed in 1987, have undergone rapid development in recent years.

We then extend the framework to locally resonant periodic structures to explore the implications of this phenomenon on bandgap tunability. We demonstrate that the wavenumber shifts result in an unusual dispersion correction effect, which we term wavenumber-space band clipping. To overcome this limitation, we present a multiple scales framework to analytically capture the wavenumber shift experienced by dispersion relation of nonlinear monatomic chains under harmonic boundary excitations. However, these models are not compatible with harmonic boundary excitations, which represent the conditions encountered in most practical applications. Several models have been proposed to capture the frequency shifts observed when the system is subjected to harmonic initial excitations. This phenomenon can manifest either as a frequency shift or as a wavenumber shift depending on whether the excitation is prescribed as an initial condition or as a boundary condition, respectively. In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation.

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