## 1 Introduction to Wave Interaction with Periodic Structures

Compared to their behavior in unstructured, bulk materials, the propagation of photons and phonons in periodic micro- or nano-structures can be dramatically altered enabling the deliberate engineering of the photon or phonon interaction with matter. Although control via periodic structures has been demonstrated in a host of material systems, our review will focus on polymer-based structures and their associated applications. Polymer-based structures have several exclusive advantages over other materials: (1) scalability; (2) wide tunability; (3) ease of functionalization; (4) mechanical flexibility; (5) compatibility with various patterning methods; (6) relatively low cost; (7) light-weight. Moreover, the artificially created non-linear photonic/phononic behavior of the periodic structures can be further diversified and enhanced by introduction of intrinsic non-linear behavior of additives to achieve state-of-the-art multifunctional properties (optical, electrical, mechanical, etc.) (**Figure** **1**).

### 1.1 Current Trends

Research on photonic crystals (PhCs) has been explosive for nearly three decades since the concept of the 3D PhCs was proposed independently for inhibition of the spontaneous emission of atoms by Yablonovitch[1] and for strong localization of photons by John[2] in 1987. The capability to create artificial photonic dispersion and photonic bandgaps (PBGs) has led to the emergence of a major interdisciplinary field of science and technology as is evidenced by the publication rate shown in **Figure** **2**a. Initially, PhC research was predominantly driven by physics; however much of the current research activity occurs in engineering disciplines, surpassing the output of physics. The contribution from biologically focused research is small, but rapidly growing. With regards to application areas, the main research focus has been laser applications due to the initial concepts of Yablonovitch and John, but now sensors and energy applications are fast-growing without any apparent slow-down in their growth rates compared to laser applications (Figure 2b). At this rate, interest in sensor applications is likely to surpass that of laser applications in a few years. Recently, the exotic properties promised by photonic metamaterials have added a new growth area. In the continued development of photonics, polymer structures will play an increasingly critical role, and it is therefore timely for a review of the recent progress in fabrication and applications of photonic polymer structures.

Phononic crystals (PnCs) are periodic structures comprised of materials having different mechanical impedances. These micro- and nano-composites can create artificial acoustic dispersion and phononic bandgaps (PnBGs) similar to PhCs. The annual number of publications on PnCs is relatively smaller than that of PhCs, but since 2000 the number has grown rapidly as can be seen in Figure 2c. Importantly, due to the speed of sound being several orders of magnitude slower than the speed of light as well as the absence of a fundamental speed of sound, phononic structures can be easily designed to have interesting dispersion relations in the deep sub-wavelength regime. This has led to burgeoning research in acoustic metamaterials to control the flow of acoustic waves below the diffraction limit. A number of non-conventional phenomena can arise from phononic metamaterials such as acoustic cloaking,[3-5] negative acoustic refraction,[6-8] and negative modulus.[9] As the engineering of mechanical waves in solid/solid and fluid/solid structures has been demonstrated in the acoustic and ultrasonic regimes, polymer-based phononic metamaterials enable reconfigurable, adaptive, and active functionality.

### 1.2 Basic Ideas of Band Diagrams

We next briefly introduce basic concepts and ideas related to the band structure of a periodic material. The group velocity governing the energy flow of both electromagnetic and elastic waves is defined by , where *ω* and **k** are an angular frequency and a wavevector, respectively, and a plot of *ω*(**k**) vs. **k** is known as the dispersion relation or the (photonic, phononic) band diagram. When the slope of dispersion curve approaches zero, for example, at the edge of a Brillouin zone (BZ), the propagation speed of the waves is significantly slowed or the modes become a standing wave () which can result in the enhanced interaction of the photon/phonon with the material or the prohibition of energy flow. When a PhC or a PnC has a so called *complete* bandgap, no propagation mode exists within a range of frequencies (PBG or PnBG) regardless of directions and polarizations of photons or phonons. For a given propagation direction, the width of a bandgap, represented by a gap-to-midgap ratio, in frequency is generally proportional to the optical parameter contrast, for PBGs, or the elastic parameter contrast, for PnBGs, where and are magnetic permeability and dielectric permittivity, and and are the mass density and bulk elastic modulus of at least two different constituent materials. In a 2D or 3D lattice, since these ratios depend on the propagation direction, a sufficiently large contrast is essential to open a complete PBG or a PnBG to prohibit the propagation of waves in all directions.

A band diagram showing the dispersion relation, *ω*(**k**) up to the boundary of the irreducible BZ is an effective means to view the overall characteristics of *ω*(**k**) of a 2D or 3D photonic or phononic crystal. For simplicity, we consider a 2D PhC with a square symmetry to illustrate a 2D photonic band diagram for in-plane propagation with two different polarizations. Our chosen 2D PhC is made of air cylinders arranged on the square lattice (symmetry *p4mm*) in a polystyrene (PS) matrix with permittivity (**Figure** **3**a1). The photonic band structure for the PhC is plotted against dimensionless frequency and only needs to be considered as a function of the wave vector **k** over one quadrant of the BZ (the irreducible BZ) due to rotational and mirror symmetries in this particular PhC.

For transverse electric (TE)-polarized light, the 1st PBG appears for dimensionless frequency between 0.35∼0.42 (red gap) for propagation along the Γ−X direction while the PBG shifts to 0.45∼0.54 (blue gap) for the Γ−M direction. Thus, there is no complete PBG for TE-polarization; this is also true for transverse magnetic (TM)-polarization. For relatively long wavelengths (), *ω*(**k**) always approaches a linear relationship with a slope of for all PhCs, where and are the speed of light in vacuum and an effective refractive index for a given polarization. An effective index can be used because the PhC acts as a homogeneous medium for these long wavelengths. This does not mean, however, that the PhC is an effectively isotropic medium. Indeed, for normalized frequencies less than 0.3, the TM polarization mode (with the magnetic field oscillating along the *z*-axis) has a slope greater than that of TE polarization mode (with the electric field oscillating along *z*-axis), because the electric field of TE polarization can concentrate more within the high dielectric constant regions than can the TM polarization.

To see the influence at a much higher permittivity value, we replot the PBG diagram for silicon () (Figure 3a2). Since is also strongly increased, all modes are scaled down in frequency; for example, the slope of for long wavelengths () decreases corresponding to the larger value. No TE propagation modes exist for the frequencies within the range 0.22–0.25 (green gap) regardless of propagation direction. Similarly, no TM propagation modes exist over a different frequency range 0.29–0.32 (yellow gap). As the two forbidden frequency ranges do not overlap, this particular square PhC does not have a complete PBG (simultaneous PBG for both polarizations). Note that *ω*(**k**) for all possible wavevectors **k** in the BZ cannot be represented by lines but by curved surfaces as a function of and . For example, the photonic band diagram in Figure 3a2 shows the lines of intersection between the curved surfaces of *ω*(**k**) and the BZ planar boundaries of the BZ as seen in Figure 3a3. Despite this limited representation of *ω*(**k**), the band diagram in Figure 3a2 is still very useful for most cases because it shows propagation modes associated from zero to the maximum wavevectors along the zone boundaries as illustrated by the colored arrows on the wavevector plane in Figure 3a3.

A dielectric material is not only an optical material, but of course is also a mechanical material as well. For example, the previous 2D PhC is also a 2D PnC for elastic waves and exhibits the phononic band diagram shown in Figure 3b. In this case, we assign the *y* axis to the dimensionless phonon frequency given by , where is the speed of transverse waves in the constituent material. In a homogeneous and isotropic solid, elastic waves can propagate through three different polarization modes, two transverse modes and one longitudinal mode, in contrast to only two transverse modes (TE and TM modes) for optical waves. However, in a PnC, these modes will in general be coupled depending on the symmetry of the particular PnC. In this 2D PnC, as the *x*- and *y*-displacements are coupled, elastic waves with displacement in the *xy*-plane (*in-plane* waves) are neither purely transverse nor purely longitudinal. Due to the lack of discrete periodicity along the z-direction, elastic waves with displacements in the z-direction (*out-of-plane* waves) are independent from the in-plane waves. In particular, for very low frequencies (or equivalently ), there are three modes, two in-plane modes and one out-of-plane mode. Similar to the photonic dispersion, the phononic dispersion, *ω*(**k**), is linear for , but in this case, we have three different slopes, representing three different speeds of sound for the PnC (Figure 3b). Although the two in-plane modes are coupled to each other, one mode has higher slope (or faster speed of sound) than the other. The speed of sound for longitudinal waves is faster than that of transverse waves in a homogeneous solid having a density , bulk modulus , and shear modulus (since ). Thus, the higher propagation speed mode is more longitudinal-like (compression-dominant) mode and the other is transverse-like (shear-dominant) mode (see **Table** **1**). In contrast to the incomplete PBG of the PhC, this PnC having realistic mechanical parameters of typical of an epoxy polymer exhibits a complete PnBG in which for a range of frequencies no elastic waves can propagate regardless of the elastic polarizations and 2D propagation directions. Moreover, the of the complete PnBG is even wider than those of the incomplete PBGs due to the larger contrast in the mechanical impedance, for longitudinal waves. Note that the photonic and phononic band diagrams are plotted in two different dimensionless frequency scales, one is scaled by the speed of light (3.0 × 10^{8} m/s) and the other by the speed of transverse sound in this specific background medium (e.g., 1.8 × 10^{3} m/s). Thus, the real frequencies (or energies) of photons and phonons are different by four-to-five orders of magnitude. As a single periodic structure can be both a PhC and a PnC (see Figure 3), one can imagine novel opto-acoustic devices, called phoXonic crystals, enabled by engineering both the photonic and phononic dispersions together.[10]

Material | [g cm^{−3}] | [GPa] | [GPa] | [km s^{−1}] | [km s^{−1}] | ||
---|---|---|---|---|---|---|---|

Polydimethylsiloxane (PDMS) | 1.43 (580 nm) | 0.97 | 1 | 0.003 | 3.0 | 0.8 | 3100 |

Poly(methyl methacrylate) (PMMA) | 1.49 (550 nm) | 1.18 | 6.2 | 2.2 | 2.7 | 1.3 | 8600 |

Polystyrene (PS) | 1.59 +10^{−4}i (500 nm) | 1.05 | 3.6 | 1.5 | 2.3 | 1.2 | 6100 |

Polycarbonate (PC) | 1.59 (550 nm) | 1.20 | 2.4 | 1.9 | 2.8 | 1.3 | 5400 |

Silicon | 3.50 (1.36 μm) | 2.33 | 98 | 65 | 9.0 | 5.4 | 47900 |

Aluminum | 1.01 + 6.6i (550 nm) | 2.70 | 69 | 26 | 6.4 | 3.1 | 43200 |

Steel | 2.76 + 3.8i (633 nm) | 7.80 | 160 | 79 | 5.8 | 3.1 | 111800 |

Diamond | 2.42 (550 nm) | 3.5 | 442 | 478 | 18.4 | 9.2 | 124400 |

water | 1.33 (550 nm) | 1.0 | 2.2 | 10^{−6} | 1.5 | N/A | 4700 |

Air | 1.00 (550 nm) | 10^{−3} | 10^{−4} | N/A | 0.34 | N/A | 1 |

As materials generally exhibit intrinsic dispersion of optical as well as mechanical parameters with frequency (especially polymers), the material parameters should be considered at the particular desired working frequencies. Table 1 shows representative values of each material to emphasize the difference among the materials. The refractive indices of polymers are reasonably constant in the visible and near IR regime; however, they show higher dispersion in the UV and anomalies in mid-IR due to intrinsic molecular group optical resonances. Thus, their refractive index becomes a complex function of frequency with a significantly increased imaginary part (or an extinction coefficient). Compared to the attainable mechanical contrasts, refractive index contrasts of polymers-based PhCs are limited due to low and small range of refractive indices of polymers range from 1.4 to 1.7 with a theoretical lower limit of ∼1.29 for an amorphous fluoropolymer without pores.[11] The effective refractive index can be increased by the addition of nanoscale high-index inorganic materials[12] or decreased by creating nanopores[13] but this often results in higher light scattering depending on the size, refractive index difference, and spatial distribution of the foreign materials.

The mechanical dispersion of a polymer at a given temperature can generally be expressed in terms of dynamic moduli, where the real and imaginary parts are related to in-phase and out-of-phase displacements with respect to the driving force. Polymers show strong dispersion at various temperatures and frequencies due to intrinsic molecular mechanical resonances. Since a modulus of a viscoelastic polymer tends to increase monotonically as a function of frequency while temperature softens the material (which leads to the time-temperature superposition principle), the dispersion of a modulus beyond a rheologically measurable frequency limit ∼ 500 Hz can be predicted from its temperature dependent moduli using the Williams-Landel-Ferry (WLF) empirical model[14, 15] for viscoelastic linear chain polymers.