## Abstract

This chapter presents the recent progress on structure-induced ultratransparency in both one- and two-dimensional photonic crystals (PhCs). Ultratransparent PhCs not only have the omnidirectional impedance matching with the background medium, but also have the ability of forming aberration-free virtual images. In certain frequency regimes, such ultratransparent PhCs are the most transparent solid materials on earth. The ultratransparency effect has many applications such as perfectly transparent lens, transformation optics (TO) devices, microwave transparent devices, solar cell packaging, etc. Here, we demonstrate that the ultratransparent PhCs with “shifted” elliptical equal frequency contour (EFC) not only provide a low-loss and feasible platform for transformation optics devices at optical frequencies, but also enable new degrees of freedoms for phase manipulation beyond the local medium framework. In addition, microwave transparent devices can be realized by using such ultratransparent PhCs.

### Keywords

- ultratransparency
- photonic crystals
- impedance matching
- spatial dispersion
- transformation optics

## 1. Introduction

Photonic crystals (PhCs), as periodic arrangement of dielectrics, affect the motion of photons and electromagnetic (EM) waves in much the same way that semiconductor crystals affect the propagation of electrons, providing a new mechanism to control and manipulate the flow of light at wavelength scale [1, 2, 3, 4, 5]. The key property of the PhCs is the photonic band gap induced from the periodic modulation of photons and EM waves, which can block wave propagation in certain or all directions. As photonic band-gap materials, PhCs play vital roles in light confinement and optical manipulation, promising many important applications, such as omnidirectional reflectors [6, 7], waveguides [8, 9], fibers [10], high-Q nanocavities and laser [11, 12], and angular filters [13].

However, little attention has been paid to the effect of enhancing transparency. Although there are pass bands in PhCs, they are usually reflective and, therefore, not transparent enough. As we know, transparent media are the foundation of almost all optical instruments, such as optical lens. However, perfect transparency has never been realized in natural transparent solid materials such as glass because of the impedance mismatch with free space. On the other hand, in the past decades, artificial EM materials like metamaterials [14, 15, 16, 17, 18, 19, 20] have been proposed to realize unusual EM properties beyond natural materials. However, most of the researches were focused on the realization of abnormal refractive behaviors such as negative refraction. Transparency over a large range of incident angle or a large frequency spectrum is theoretically possible, but the experimental realization is very difficult as complex and unusual parameters are required.

The photonic band-gap effect is actually induced by the periodic modulation of the reflections on the surfaces of dielectrics. That is, the periodic modulation strengthens the reflections on the surfaces of dielectrics to form a complete band gap at particular frequencies. Then, a natural question is: Is it possible to rearrange the periodic modulation of the reflections on the surfaces of dielectrics to make them cancel each other for all incident angles, so that omnidirectional impedance matching and omnidirectional perfect transmission can be realized?

In this chapter, we show the opposite effect of the band-gap effect in PhCs, i.e., the structure-induced ultratransparency effect [21, 22]. Ultratransparent PhCs not only have the omnidirectional impedance matching with the background medium, but also have the ability of forming aberration-free virtual images. The equal frequency contours (EFCs) of such ultratransparent PhCs are designed to be elliptical and “shifted” in the k-space and thus contain strong spatial dispersions and provide more possibilities for omnidirectional impedance matching. Interestingly, the combination of perfect transparency and elliptical EFCs satisfies the essential requirement of ideal transformation optics (TO) devices [23, 24, 25, 26, 27]. Therefore, such ultratransparent PhCs not only provide a low-loss and feasible platform for TO devices at optical frequencies, but also enable new freedom for phase manipulation beyond the local medium framework. Moreover, such ultratransparent PhCs have shown enormous potential applications in the designs of microwave transparent wall, nonreflection lens, omnidirectional polarizer, and so on.

## 2. Ultratransparency effect: the opposite of the band-gap effect

### 2.1. Definition of ultratransparency: omnidirectional impedance matching and aberration-free virtual image

It is well known that the band-gap effect induced by the periodic distribution of dielectrics can block the propagation of EM waves in certain or all directions, as illustrated in Figure 1(a). The forbiddance of wave propagation is the result of lacking propagation modes within the PhCs, which can be seen from the EFCs in Figure 1(a). The circle in the left denotes the EFC of free space, and there is no dispersion of the PhCs within the band gap. Interestingly, we would like to rearrange the periodic array of dielectrics to obtain the omnidirectional impedance matching effect, which allows near 100% transmission of light at all incident angles, as illustrated in Figure 1(b).

Moreover, we expect such media to have the ability of forming aberration-free virtual images, which is absent in normal transparent media like glass. By using ray optics, it can be easily shown that transmitted rays from a point source behind a dielectric slab would form a “blurred” area of virtual image rather than a point image. For demonstration, we placed a point source on the left side of a glass slab (with a thickness of

Actually, such aberrations of the virtual image originate from the mismatch between the EFCs of free space and dielectrics, i.e., their EFCs do not have the same height in the transverse direction (i.e., the

where

Supposing that the rearrangement of the periodic array of dielectrics not only makes the whole structure impedance matched to free space for all incident angles, but also creates the unique EFC described by Eq. (1); thus, both omnidirectional 100% perfect transmission and aberration-free virtual imaging are enabled simultaneously. Apparently, such a level of transparency is superior to that of normal transparent media like dielectrics and is thus hereby denoted as ultratransparency.

### 2.2. Ultratransparency based on local and nonlocal media

According to Fresnel equations, reflection of light on the surface of dielectrics is inevitable, except at a single-incident angle referred to as the Brewster angle under transverse magnetic (TM) polarization, as demonstrated in Figure 3(a). Here, we extend the impedance matching from one particular angle (i.e., Brewster angle) to all incident angles in a nonlocal or spatial dispersive medium, whose effective permittivity

To begin with, we assume that the nonlocal medium exhibits an irregular-shaped EFC shown in Figure 3(b). An incident wave of transverse electric (TE) polarization with electric fields polarized in the

which determines the relationship between the components of wave vectors

In this case, the wave impedance of free space can be derived as

If Eq. (3) can be satisfied for all

An obvious local medium solution of Eq. (3) is that

Eq. (4) shows that when EM parameters possess spatial dispersions, there exist infinite solutions of

## 3. Structure-induced ultratransparency in two-dimensional PhCs

### 3.1. Nonlocal effective medium theory

For the design of ultratransparent PhCs, we first propose a nonlocal effective medium theory for the homogenization of PhCs. Here, we consider uniform plane wave incidence. In this situation, the validity of the nonlocal effective medium theory lies in the satisfaction of the following four premises: (1) single-mode approximation [28], i.e., only one eigen-mode is excited; (2) the amplitudes of fields at the incident boundary are almost constant; (3) the phases of fields at the incident boundary obey the trigonometric functions; and (4) the electric and magnetic fields are in phase at the incident boundary. Although these premises are seemingly stringent, it turns out that most eigen-modes of the first few bands (e.g., monopolar and dipolar bands) can indeed satisfy these requirements (Figure 4).

With the assumption of the abovementioned premises, eigen-fields of TE polarization at the boundary

where

On the other hand, the PhC satisfying previous premises generally can be described as a uniform medium with effective relative permittivity

Compared with Eqs. (5) and (6), the effective parameters can be derived as,

With Eq. (7), the effective parameters of the PhC can be obtained by analyzing the eigen-fields with Bloch wave vector

### 3.2. Two-dimensional ultratransparent PhCs

PhCs contain strong spatial dispersions and thus provide the perfect candidate for realization of ultratransparency effect. Here, we demonstrate a type of PhCs composed of a rectangular array of dielectric rods in free space, with the unit cell shown in Figure 5(a). Under TE polarization, the band structure is presented in Figure 5(b), and the EFC of the third band is plotted in the reduced first Brillouin zone in Figure 5(c). The working frequency is chosen as

Figure 5(d) shows the impedance difference of the PhC and free space of the third band, i.e.,

Moreover, in Figure 6(a), we present the effective parameters

Now, by substituting the condition

The choice of

For further verification, the transmittance through such a PhC slab consisting of

### 3.3. Microwave experimental verification

In fact, for transparency in a relatively smaller range of incident angles, the design process is much easier and the effect can exist in much simpler structures. In the following, we demonstrate a simple ultratransparent PhC, which is verified by proof-of-principle microwave experiments. The PhC consists of rectangular alumina (

In addition, the impedance difference between the PhC and the free space is calculated by using Eq. (5), as shown in Figure 7(c). Clearly, the impedance difference is very small on the EFC of

Moreover, the effective parameters obtained from Eq. (7) are presented in Figure 8(a) by solid lines with symbols, showing

In Figure 8(a), the dashed lines denote

Furthermore, we calculate the transmittance through the PhC slab with

Next, we show microwave experimental results to verify the above theory. A 23 × 5 array of such a PhC is assembled in the

The measured electric fields for 0°, 30°, and 45° incident angles are displayed in Figure 9(b), (c), and (d), respectively. Clearly, the reflection is barely noticeable, indicating impedance matching for all these incident angles. In Figure 10, the measured transmittance (triangular dots) coincides with simulation results (solid lines) quite well, both showing great enhancement compared with that through an alumina slab with the same thickness (dashed lines). Although the ultratransparency effect is hereby only verified at the microwave frequency regime, the principle can be extended to optical frequency regime by using PhCs composed of silicon or other dielectrics.

## 4. Structure-induced ultratransparency in one-dimensional PhCs

In the above, we have shown the structure-induced ultratransparency in two-dimensional PhCs. In the following, we demonstrate the structure-induced ultratransparency in one-dimensional PhCs [22].

The one-dimensional ultratransparent PhC we studied is composed of two dielectric materials A and B stacked along the

Figure 12(a) presents the band structure of the PhC, whose unit cell is constructed in a symmetric form, i.e., ABA structure, as shown by the inset in Figure 12(a). The relative permittivity and filling ratio of the material A (B) are 2 (6) and 0.6 (0.4), respectively. The dashed line denotes the normalized frequency

In Figure 12(b) and (d), the EFCs at the frequency

For further verification, the transmittance through the PhC slab composed of

Moreover, in Figure 14(a) and (b), the transmittance through a PhC slab (

Although a wide-angle (

## 5. Applications

### 5.1. For transformation optics

In the above, we have demonstrated the ultratransparency in both one- and two-dimensional PhCs. In the following, we show some applications of such ultratransparent PhCs. It is interesting to note that the omnidirectional perfect transparency and elliptical EFCs of the ultratransparent media are essential for ideal TO devices. The theory of TO [23, 24, 25, 26, 27] promises many novel and interesting applications, such as invisibility cloaks [23, 25, 31, 32], concentrators [33], illusion optics devices [34, 35, 36], and simulations of cosmic phenomena [37, 38]. Generally, the TO devices are realized by using metamaterials [14, 15, 16, 17, 18, 19, 20], which require complicated designs of electric and magnetic resonances, hindering the realization and applications in practice. In fact, most of the previous TO experiments were realized by using the so-called reduced parameters, which maintain the refractive behavior, but sacrifice the impedance matching as well as the perfect transparency [25, 39, 40, 41, 42, 43, 44, 45]. Moreover, at optical frequencies, the inherent loss in metallic components of metamaterials makes the realization of perfect transparency as well as the ideal nonreflecting TO devices extremely difficult [46, 47], if not impossible. Interestingly, we find that the ultratransparent PhCs provide a low-loss and feasible platform for TO devices at optical frequencies.

To begin with, we consider a TO medium obtained by stretching the coordinate along the

where

Considering Eq. (12), the dispersion of the TO medium, i.e.,

which has the similar form as that of Eq. (1). The EFC of the TO medium is an ellipse having the same height as the EFC of air in the

The only difference of the EFCs between the TO medium and the ultratransparent medium is that there may exist a “shift” of

For demonstration, we show a specific example in Figure 16. The ultratransparent PhC is one-dimensional and composed of components I and II. The unit cell is constructed in a symmetric way with a lattice constant of

Moreover, simulations of wave propagation through the TO medium slab with a thickness of

Next, we show an example of TO device by using one-dimensional ultratransparent PhCs. The design process is shown in Figure 17(a), in which the original shell of a concentrator [33] is discretized into four layers and each layer is further replaced by a corresponding ultratransparent PhC. Figure 17(b) shows the parameters of the discretized layers of TO media and the ideal profile. The corresponding four types of ultratransparent PhCs are of the same lattice constant **-**space.

The detailed parameters of the PhCs are presented in Figure 18. The insets present the illustrations of unit cells, relative permittivities, and thicknesses of each component of the four different PhCs. Moreover, the transmittance through PhC slabs with 10 unit cells is plotted as the function of incident angles, as shown by the solid lines in Figure 18(**a–d**). During the calculation, the background media are chosen as the discretized TO media (Figure 17(b)) with the parameters

Moreover, numerical simulations are performed to demonstrate the functionality of the concentrator. Figure 19(a) and (b) corresponds to the concentrator composed of the original discretized TO media and the ultratransparent PhCs, respectively. It is seen that under an incident beam of Gaussian wave from the lower left, both concentrators exhibit good concentration effects in the core areas and induce almost no scattering of waves. Interestingly, the waves inside the core areas exhibit a distinct phase difference of

Therefore, we have demonstrated that ultratransparent media can work as the TO media to realize TO devices. Such ultratransparent media not only provide a low-loss and feasible platform for TO devices at optical frequencies, but also enable new freedom for phase manipulation beyond the local medium framework.

### 5.2. For microwave transparency

In the microwave regime, the ultratransparent media are also very useful and may have many applications in the design of radome, transparent wall, and so on. Here, we show an example of microwave transparent wall which allows the WiFi and 4G signals to pass through freely, and thus may find applications in architectural designs.

The microwave transparent wall is composed of one-dimensional ultratransparent PhCs with ABA unit cells. Materials A and B are chosen as polypropylene (

## 6. Conclusions and outlook

In this chapter, we introduced the recent results of the structure-induced ultratransparency effect in both one- and two-dimensional PhCs, which allow near 100% transmission of light for all incident angles and create aberration-free virtual images. The ultratransparency effect is well explained by nonlocal effective medium theory for PhCs and verified by both simulations and proof-of-principle microwave experiments. The design principle lies in systematic tuning of the microstructures of the PhCs based on the retrieved nonlocal effective parameters.

With the ultratransparent media, many applications can be expected such as the perfectly transparent optical lens, ideal TO devices, microwave transparent devices, and solar cell packaging. Interestingly, the ultratransparent media with “shifted” elliptical EFC not only provides a low-loss and feasible platform for TO devices at optical frequencies, but also enables new degrees of freedoms for phase manipulation beyond the local medium framework. In addition, microwave transparent walls allowing the WiFi and 4G signals to pass through freely can also be realized.

Although the ultratransparency effect is mainly demonstrated for TE polarization here, the principle is general and can be extended to TM polarization, or even both polarizations. Polarization-independent ultratransparency has wide and important applications. On the other hand, polarization-dependent ultratransparent media could also have some special applications. For instance, if the PhC is ultratransparent for TE polarization, while the working frequency falls in an omnidirectional band gap for TM polarization, such a PhC would work as an omnidirectional polarizer.

The concept and theory of ultratransparency give a guideline for pursuing solid materials with the ultimate transparency, i.e., broadband, omnidirectional, and polarization-insensitive total transparency. In the future, ultratransparent solid materials may be optimized to exhibit an unprecedented level of transparency and find vital applications in various fields.

## Acknowledgments

This chapter is supported by National Natural Science Foundation of China (No. 11374224, 11574226, 11704271), Natural Science Foundation of Jiangsu Province (No. BK20170326), Natural Science Foundation for Colleges and Universities in Jiangsu Province of China (No. 17KJB140019), Jiangsu Planned Projects for Postdoctoral Research Funds (1701181B) and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).