Photonic crystals (PhCs) can be utilized to control the propagation behaviors of light within a frequency band (i.e., conduction band or stop band) for their periodic arrangements of dielectric. Utilizing the effect of band gap one may introduce defects in a photonic crystal to limit or guide the electromagnetic wave with the frequency in stop gaps. Furthermore, based on synthetic periodic dielectric materials photonic crystals can exhibit various anomalous transmission properties, such as negative refraction, self-focusing, zero phase delay or effective-zero-index properties that are determined by the characteristics of their band structures and equal frequency contours (EFC). These extraordinary results contributing to the design of novel PhC devices and the development of PhC application are demonstrated in this chapter.
- photonic crystal
- photonic band gap
- negative refraction
- zero phase delay
- Dirac-like point
Photonic crystals (PhCs) are structures in which the dielectric constant is periodically modulated on a length scale comparable to the desired wavelength of operation [1, 2], and the resultant photonic dispersion may exhibit photonic band gaps (PBGs) and anomalous propagation behaviors which are useful in controlling light behavior according to different theoretical principles. Based on the PBG effect we may introduce a line defect in a photonic crystal to guide the electromagnetic waves with the frequency in stop gap . The line defect is called a photonic crystal waveguide (PCW), which is very compact (with the typical width is around half-wavelength) and allows sharp bends without losses for its all-dielectric structure . Many attempts have been made to fabricate materials with complete photonic band gaps (PBGs) at near-infrared  and visible frequencies , such as semiconductor microfabrication , self-assembly of colloidal particles , electron-beam lithography , multiphoton polymerization , holographic lithography (HL) , and so on. Among them, HL is a very promising technique for the inexpensive fabrication of high-quality two- and three-dimensional (2D and 3D) PhC templates with the unique advantages of one-step recording, the ability to obtain an inverse lattice by using a template, inexpensive volume recording and rapid prototyping. The PhCs formed by HL which usually have irregular “atoms” or columns. Since the PBG property of resultant structure varies with the shape of “atoms” or columns, thus the PBGs and propagation properties for PhCs made by HL will be different from those of regular structures.
Veselago predicted a kind of materials with negative refractive index in 1968  which has attracted lots of attention in recent years. Such materials are generally referred to as left-handed materials (LHMs), double negative materials , or backward-wave media et al. , whose best known characteristic is to refract light in opposite direction. Shelby et al. have accomplished one of the first experiments to demonstrate the negative-index behavior . However, the absorption loss of the metal limits its potential optical applications. Different from LHMs, PhCs made of periodic all-dielectric materials can exhibit an fascinating dispersion such as negative refraction and self-focusing properties which are determined by the properties of their band structures and equal frequency contours (EFC) [16, 17].
In this chapter, the maximal complete relative photonic band gaps of 22.1% and 25.1% are introduced how to be achieved in 2D- and 3D photonic crystals formed by HL method. The symmetry mismatch between the incident wave and the Bloch modes of PhC can be used to guide light efficiently. The unique features of negative refraction, dual-negative refraction, triple refraction phenomena, and special collimation effects of symmetrical positive-negative refraction have been verified by numerical simulations or experiments. Effective measurement method has been proposed to identify the Dirac-like point of finite PhC arrays accurately. A mechanism for generation of efficient zero phase delay of electromagnetic wave propagation based on wavefront modulation is investigated with parallel wavefronts (or phasefronts) extending along the direction of energy flow. The band structures of PBG are calculated by the plane-wave expansion method , and a finite-difference time-domain (FDTD) method  is used to calculate the transmission property of the guided mode. The method of wave-vector diagram is generally used to predict the properties of beam propagation in PhCs.
2. Characteristics of photonic band gap in 2D- and 3D-PhCs
2.1. Photonic band gap in 2D-PhCs
Since the shape and size of lattice columns in 2D case or atoms in 3D case are usually determined by the isointensity surfaces of the interference field, the columns or atoms formed by this way often are of irregular shapes [20, 21]. Consequently, the PBG property of resultant structure is closely related to the fabrication process. Therefore the complete PBG can be obtained and improved by proper designs of the shape and size of lattice cells . Compared with the 3D case, the 2D PhCs are easier to fabricate and study theoretically for the fact that, the wave propagation of 2D PhCs can be investigated separately for two orthogonal polarizations known as TM and TE. They have important practical significance because they offer the possibility of guiding and manipulating light in planar defect circuits , photonic crystal fibers  and controlling polarization through their anisotropic band structures.
A kind of 2D 3-fold PhCs conforming to the hybrid triangular configuration formed by HL was proposed . This kind of hybrid triangular lattice is formed by two sets of triangular lattices, as depicted in Figure 1, with the big dots indicating the triangular lattice sites with lattice constant
The basis vectors of the hybrid triangular lattice may be chosen as and . To produce this 2D hybrid triangular lattice holographically, we may use the following intensity of interference field,
where the first constant has no essential effect, which can be changed by adjusting the light intensity threshold
In addition, another kind of 2D 6-fold hybrid triangular configuration formed holographically is proposed , in which the complete PBGs can be found even with much lower dielectric constant (ε = 3.8). This 2D periodic structure is a hybrid triangular lattice combining two sets of triangular sublattices as depicted in Figure 3, where the red dots denote a triangular sublattice with lattice constant a, while the blue dots denote another set with the lattice constant of which is rotated by an angle of 30° with respect to the former.
For the triangular structure formed holographically the complete PBGs always appear in inverse structures (air columns in dielectric material) with high dielectric constant instead of normal structures . However, the complete PBG may be obtained for normal structure (dielectric column in air background) with lower dielectric contrast (
2.2. Photonic band gap in 3D-PhCs
An important way to make 3D PhCs by HL is the interference of four umbrellalike beams (IFUB) where three ambient beams (A-beams) form the same apex angle
The interference of four noncoplanar plane waves of the same frequency will result in an intensity distribution
With the angle
The plane-wave expansion method  was used to study the PBG properties of structures of this kind and search for the corresponding optimum volume filling ratio
As mentioned above, the fcc lattice can be obtained by the interference of one central beam and three ambient beams symmetrically scattered around the former, but the structure made in this geometry has only quite a narrow PBG of 5.35%. An alternative beam design was proposed to fabricate the fcc lattice with a large complete PBG, but it requires four beams incident from two opposite surfaces of a sample , making it difficult to realize in practice. So a five-beam symmetric umbrella configuration is proposed to make 3D PhCs with large complete PBGs. The proposed recording geometry of five-beam symmetric umbrella configuration is shown in Figure 8, where the central beam (C-beam) is set along the z direction, while the four ambient beams (A-beams) are in the
This geometry can be realized by using a DBS to obtain a zero-order diffracted beam and four symmetric first-order diffracted beams. The polarization of the central beam is circularly polarized and all the four A-beams are linearly polarized. The unit polarization vectors of five beams are
In general, the lattice structures are tetragonal symmetric structures. The continuous increase of apex angle
Full photonic band gaps exist over a very wide range of apex angle with a relatively low refractive index contrast. Figure 10 represents the relative gap sizes of optimized structures for the apex angle range of 50° <
2.3. Photonic crystal waveguide
A waveguide can be created in the PhC slab by introducing a linear defect in the in-plane 2D periodic structure [34, 35]. Since the ability to guide light waves around sharp corners with high efficiency is crucial for photonic integrated circuits, many studies have been carried out concerning waveguide bends through sharp bends in 2D PhC slabs [36, 37]. However, all these works limit the studies to the structures formed by air rods with regular circular cross sections. PBGs for PhCs made by HL may be different from those of regular structures, so will the propagation properties.
When the 2D periodic structure is a triangular Bravais lattice formed by the interference technique of three noncoplanar beams , the structure was filled with a material of high refractive index and then removing the template, an inverse structure can be obtained. When the intensity threshold
Some peaks of transmission may result from the resonance between two bends in the waveguide of ordinary PhCs with circular air holes . The band diagram of the PhC configuration has been calculated with the intensity threshold of
Different from the total internal reflection and photonic crystal fibers (PCF) with full 2D PBGs by introducing line defect, a 2D photonic crystal waveguide (PCW) formed by an air core and two identical semi-infinite layers of left-handed holographic PhC is proposed to confine light in air waveguide. As shown in Figure 13, The EFCs plot and wave-vector diagram of TM2 band in the HL photonic crystal indicate the PhC is left-handed. Considering the symmetry of Bloch modes of this PhC, the incident interface can be placed in ГM or ГK directions. Simulations have demonstrated that the incident beam can readily travel through the PhC slab with the input surface interface normal to ГM direction, but restrained in ГK direction, which may originate from the symmetry mismatch between the external incident wave and the Bloch mode of this PhC structure .
An air waveguide is introduced in the PhC along ГK direction, as shown in Figure 14(a), with the length 50
Figure 15 shows TM field in the waveguide with 3 layers of air width for the cases when frequencies are (a) 0.28, (b) 0.315 and (c) 0.33 respectively, light attenuation can be seen clearly in Figure 15(a) as a result of vertical scattering loss. Since air thickness decreases from 4 layers to 3 layers, the group velocities slow down and the backscattering loss becomes a dominant loss factor. The light with the frequency of 0.315 display a low transmission in the waveguide as shown in Figure 15(b). Figure 15(c) verifies that the incident wave with frequency beyond the minimum critical excitation frequency can be well confined to the air waveguide. Based on the symmetry mismatch between the incident wave and the Bloch modes of the holographic PhC, a holographic PCW without PBGs can also efficiently guide light in a wide frequency region with high transmission efficiency.
3. Anomalous refraction behaviors modulated by PhC bands
Photonic crystals can exhibit an extraordinarily high, nonlinear dispersion such as negative refraction and self-focusing properties that are solely determined by the characteristics of their band structures and equal frequency contours (EFC) [42, 43, 44]. The 2D honeycomb structures formed by single-exposure interference fabrication methods is used to investigate a series of HL PhC structures in order to obtain comprehensive understanding of the anomalous refractive properties in PhCs.
The filling ratio of the HL PhC is determined by the ratio of the total exposure dose. Silicon with ε = 11.56 (i.e., n = 3.4) is used to analyze the dispersion characteristics of the holographic PhCs. Figure 16(a) gives the cross section of the HL PhC sample. The EFCs plot and wave-vector diagram of TM2 band in Figure 16(b) illustrated the EFCs around point Г are convex and shrink with increasing frequency, indicating the PhC is left-handed. Due to the symmetry mismatch between the external plane wave at normal incidence and the Bloch modes of this PhC as mentioned above, the interface between PhC and free space is arranged along the ГK direction.
As shown in Figure 16(a), the surface of dielectric PhC slab with the trigonal flange (cut 0.4
For a continuous point source of
Multi-refraction effects in the 2D triangular PhC have also been found . The special EFC distributions of different bands can be used to predicate the propagating properties of incident electromagnetic wave (EMW). The EFC plot of the second band is shown in Figure 18(a) with almost straight EFC in the ΓK direction at the frequency of 0.26 a/λ. The group velocity
The k-conservation relation is observed in wave-vector diagrams . Traditional EMWs propagate in media with their wavefronts perpendicular to the energy flow direction. Here, the EMW of 0.36
For the normal HL structure with triangular lattice symmetry of
The more complicated refraction behaviors can be excited in the higher band regions based on the intricate undulation of one band or the overlap of different bands. As shown in Figure 21(a), the sixth band has dual parallel EFCs with opposite curvatures by the red rings within the frequency range from 0.44 to 0.47 a/λ within a wide scope of incident angle. As an example, when the working frequency is chosen to be 0.46a/λ, the incident wave at
4. Phenomena of zero phase delay in PhCs and application
Recently, materials with zero or near-zero-
A plane wave can be described as , where the symbols of
The PhC sample of triangular lattice is composed of dielectric rods with
Another approach to realize zero phase delay in PhCs is based on the accidental degeneracy of two dipolar modes and a single monopole mode generates at the Dirac-like point (DLP) with the linear dispersions of Dirac cone at the Brillouin zone center of PhCs . At the DLP shown in Figure 24(a), the PhC can mimic the zero-index medium (ZIM) with the characteristics of uniform field distribution. Linear dispersions near the center point induced by the triple degeneracy display many unique scattering properties, such as conical diffraction , wave shaping and cloaking . The dispersion properties obeying the 1/L scaling law near the DLP in the normal propagation direction have been verified theoretically and experimentally  through the dielectric PhC ribbons with the finite thicknesses, however, it is difficult to distinguish the precise crossing point from the wide extremum range just relying on the transmittance spectrum.
A Gaussian pulse source of TM mode was placed in front of a PhC square-lattice array with the incident angle of 90°, thus three transmission spectrums can be measured outside the other three output interface of the PhC array with one in parallel direction and two in perpendicular upward and downward directions. As long as the size is large enough, the PhC array can be regarded as an infinite PhC with the similar transmittance at the different exit boundaries with the same interface structure. As shown in Figure 24(b), the upward and downward sharp cusps embedded in the parallel and perpendicular extremum transmittance spectra intersect at the DLP frequency, therefore, the DLP can be identified accurately due to the property of uniform field distribution of PhC with effective zero index at DLP.
In summary, some novel kinds of 3- and 2-D lattices formed by holographic lithography have been investigated to find that the complete PBG over wide ranges of system parameters can be achieved by proper design. The beam design for making this structure is also derived. By using the interference of symmetric four umbrellalike beams one can obtain complete PBGs over a very wide range of apex angle, three special lattice structures of fcc, sc and bcc were achieved with different complete PBGs of 5.35%, 10.23% and 21.57%. The similar PBG of 21.3% for the bcc lattice structures, and the biggest PBG of 25.1% for fcc lattice structure were achieved by five umbrellalike beams, which is much larger than the value of 5.35% formed by four umbrellalike beams for the shape reason of PhC lattice cell. The holographic PCW can efficiently guide light in a wide range of frequency around sharp corners and the resonance between the two bends has a close relation with the configuration of PhC which can be regarded as an important factor to improve the transmission property of 2D PCW. Since the absolute refractive index 1 of the air guiding core is larger than the effective index of PhC cladding, the total internal reflections can be achieved approximately with a high transmittance in the PhC waveguide with the effective refraction index near zero. For PhC waveguide when the width of air layer is greater than the critical thickness, the problems of scattering loss can be avoided. The proposed left-handed holographic PCW is more suitable for many applications in the area of photonic integrated circuits, and the idea and method of analysis here open a new freedom for PCW engineering.
Different from the anomalous phenomena of PhCs in the high bands, the PhCs of honeycomb-lattice in the lower band region are more suitable to realize the effect of all-angle left-handed negative refraction. For the straight EFCs distribution in special directions, the regular PhC can be applied to the design of optical collimator. Moreover, multi-refraction has been found in the higher bands. There are two ways to realize multi-refractions, one arises from a single band with intricate undulations and another originates from the overlap of multi-bands. The unique phenomena of dual-negative refraction and positive-negative refraction have been found in the higher bands of PhC. Based on the design flexibility of PhC, the transmission properties of PhC can be engineered with a large freedom. These results are important and useful for the design of PhC device.
An efficient way to achieve EMW propagation with zero phase delay is to modulate the wavefronts paralleling to the direction of energy flow. This method can be extended from 2D to 3D cases or other artificially engineered materials, which opens a new door to obtain perfect zero-phase-delay propagation of EMW and has significant potential in many applications. Furthermore, the research found that the Dirac-like point of PhC can be identified accurately by measuring the transmission spectra through a finite photonic crystal square array.
All these results can be extended to the fabrication of other artificially engineered materials and provide guidelines to the design of new type optical devices.
This work was supported by National Natural Science Foundation of China (11574311, 51532004, 61275014)