定义：
具有特殊光学性质的纳米结构材料。

光子超材料是人造的包含纳米结构的材料，特殊结构赋予了其特殊的光学性质。它们的结构有至少两种不同的材料得到（通常包含金属和介电材料）。材料的结构通常是周期性的，并且周期小于光波长。因此，其特殊光学性质不是由光子带隙产生的，而是由类似于与普通固体材料中离子或原子的相互作用产生的。

与光子带隙材料不同的是，光子超材料与普通材料一样可看做是均匀光学材料，但是其材料参数与普通材料不同。例如，一些超材料的相对介电常数ε小于0，或者相对磁导率 μ小于0. 也有的材料的 ε不为0。这些性质会产生很特殊的波传播现象，例如负折射或者传播过程中的相位延迟非常小。 

常见的一种类型为电磁超材料，可以在不同频率范围起作用。已经实现了微波范围的这种材料，即波长远大于光学范围，因此制备亚波长材料较容易。光子超材料较难制备，但是自2000至今，在这一领域已经得到了很大的发展。 

包含纳米谐振腔的结构 
光子超材料通常包含一些金属纳米电磁场谐振腔，尽管从其结构的形状可能不明显。早期的方案是从微波领域发展过来的，采用的是开口谐振环（图1）。可看做简化的LC回路，这里非闭口的金属环取代了电感，打开金属环就形成电容器。

谐振腔通常具有矩形结构或（当开口变大时）像马蹄铁。谐振腔的共振发生在中红外区域（波长为几微米），并且其带宽变为只有几百纳米。也可能在不同频率会发生多个共振，通常最低频率的共振是实际中需要用到的。 

图1：光子超材料的一层，金属纳米谐振腔制备在非导电（介电材料）衬底上。当入射光与所示平面垂直时，可以激发共振，产生特殊的传播效应。 
可以采用光刻方法制备金属-介电材料混合物。常用的金属是金。银更适宜于高频应用，但是由于其易被氧化，所以比较难加工。 一些实验只采用以上所示的单层材料。三维结构可以由上面的单层材料在竖直方向上叠在一起得到，周期同样需要小于光波长。 

当光入射到纳米谐振腔时会激发电磁场振荡。当频率接近共振频率时，激发效应尤其强，但是有时更关心的是大于或小于共振频率时发生的光学效应。由于结构的周期小于一半的光波长，因此不存在光子带隙效应，描述光的传播可以采用光子超材料的有效相对介电常数ε和有效相对磁导率 μ。（即使不采用磁性材料，后者也可能不等于1.）另外，还需要引入双各向异性参数 ξ来描述电场强度E与磁场密度B，或者磁场强度H与电位移D之间的耦合。由于开口谐振腔环中电场会诱导产生环路电流，产生磁场，然后磁场也会诱导电流产生电场，这就是耦合过程。改变结构可以使ξ 由于存在对称性而等于0。 

还有一些简单的谐振腔形式，有的只包含一些周期性排列的短线，或者一些其它形状的金属片。这些形式可以应用到更短波长的应用中，可以达到可见光谱中的红色区域。有的超材料结构中整个金属都是通过电连接在一起，而不是如上所述的孤立的纳米谐振腔。

还有些结构由三维模块组成的，例如一些短的螺旋线（螺旋天线），该线是对圆偏振光方向敏感的。这些三维结构可先在聚合物材料中直接激光写入，再结合后续的加入金属部分的过程就可以制备得到。 

超材料特殊的光学性质 
上面已经提到，超材料的光学性质可由介电常数 ε和磁导率μ来描述，有时还需要额外的双各向异性参数 ξ。正常的光学材料μ约等于1，但是超材料中μ的数值完全不同，尽管超材料通常不包含铁磁材料。 

以上的三个参数都与频率有关，并且与光的传播方向和偏振态无关。另外，这些参数不一定为实数（有些结构中，ξ在任何频率下都为0）。一般超材料中的光学损耗都不能忽略（尤其是高频情况），导致ε和μ的虚部很大。 
折射率的大小由下面的方程决定： 

当为实数ε和μ，并且假设 ξ = 0，折射率n是实数，但是可正可负。折射率的正负号是为了区分根据斯涅耳定律，在光学界面处折射光束的方向。 
材料的阻抗为： 

由于阻抗与 μ / ε有关，而折射率与μ • ε有关，两个介质折射率相同不一定具有相同的阻抗。（正常的光学介质中折射率决定了阻抗，μ约为1.）类似的，两个介质阻抗相同也可能具有不同的折射率。

这在光反射中会用到，因为光学界面处反射光的大小与阻抗有关。例如，在真空与μ = ε = −1的介质界面处不会发生反射，尽管折射率有+1变为-1，但是界面两边的阻抗是相同的。 

下面主要讨论具有某些具体关系性质的光子超材料。 

负折射率材料
当ε和μ都小于0（参阅双负超材料），并且ξ为0，可以在真空和超材料界面处看到负折射现象。（双各向异性材料有ξ ≠ 0，负折射材料中也会存在）如图2所示，折射光束并不是位于常规的一侧。由于折射率小于0，改变了斯涅耳定律角度的符号。 

图2：光束从真空射向负折射率材料（右图），介质中的折射光与入射光在发现的同一侧。这与寻常正折射率材料（左图）正好相反。图中没有画出反射光束。 
负折射材料还具有其它不寻常的光学性质。特别是，折射光束的相位波前是朝着界面传播的，即与能流的方向相反（用坡印廷矢量表示）。坡印廷矢量表示为E × H，但是此时B的方向与H相反，因此坡印廷矢量与矢量的方向是相反的E × B。（这里大写字母都是表示的矢量形式。）因此，这一材料也被称为左手材料（LHM）：即需要根据左手定则从E和B得到坡印廷矢量。“左手”会与下面提到的手性现象有关。 

由负折射率材料制作的凸透镜会使光散焦，而不是像传统的凸透镜可以聚焦光束，负折射率凹透镜可以聚焦光束。如果折射率为-1，甚至可以得到超透镜，可到物体的像的分辨率不受衍射的限制。以上性质只有当折射率严格接近-1时才能得到,而需要足够接近的程度在应用中也非常不切实际。可以采用一种改进的方案得到超透镜，其中物体距透镜的距离需要足够小。另外不采用折射率材料也可以实现超透镜。 

如果在负折射率超材料中加入一些物质可以得到很多特别的效应。根据物质在介质中的位置，在材料外面看可以看到其被放大了或者发生了畸变。还可以制作隐身装置（隐身衣），该装置可以隐藏物体，不过只能在某些特定频率范围内实现隐身。 

手性介质
上面提到有些超材料包含金属螺旋纳米线。这些材料具有一定的手性，不同方向的圆偏振光产生的光学效应是不同的。可以设计超材料使其强烈吸收某一方向圆偏振光，而对另一方向的圆偏振光几乎完全透射。这一装置称为圆偏振器。该现象不需要折射率小于0。 

工作于短波长的困难 
当工作在短波长（例如可见光区域）时遇到的第一个困难是需要制备非常精细的结构，其周期要小于光波长。有些光刻技术可以满足制备条件，可以采用更短的波长制备，或者实现亚波长分辨率。而直接激光写入则不能达到要求。这些制备方法的速度受制于每一部分所需的时间。 

更多的限制来自于材料本身的性质。尤其是，几乎没有导体能工作在几百太赫兹的频率时。因此，通常的装置在很高频率时光学损耗很高。

Definition: nanostructured materials with special optical properties

Photonic metamaterials are artificially engineered optical materials containing nanostructures which give them truly remarkable optical properties. These structures are made from at least two different materials (often involving both metals and dielectrics). They are normally periodic, with the period being small compared to the optical wavelength. Therefore, the special optical properties do not arise from photonic bandgaps as for certain photonic crystals, but rather from an interaction which is more similar to that of atoms or ion in a normal solid medium. In contrast to photonic bandgap materials, photonic metamaterials can be described as homogeneous optical materials, much like natural materials, although with partly rather unusual material parameters. For example, some metamaterials have a negative relative permittivity ε and/or a negative relative permeability μ. Others have a near-zero ε value [20, 22]. Such properties lead to very peculiar wave propagation phenomena, such as negative refraction, propagation with very small phase delays, and an extremely large nonlinear index.

There is a more general class of electromagnetic metamaterials, operating in different frequency regimes. Many of such metamaterials have been realized for microwaves, where the wavelengths are much longer than in the optical domain, and thus it is easier to fabricate sub-wavelength structures. Photonic metamaterials are generally more difficult to realize, but substantial progress in the area of nanotechnology has been achieved particularly since the year 2000.

Structures Containing Nano-Resonators

Photonic metamaterials typically contain some kind of metallic nanoscopic electromagnetic resonators, even though it is not always obvious from looking at the shapes of the structures that these act as resonators. An early approach, which has been taken over from previous work in the microwave domain, is based on split-ring resonators (Figure 1). These can be considered as simplified LC circuits, where the inductance has been replaced with a not fully closed metallic ring and the capacitor is formed by the opening in that ring. Such resonators can also have approximately rectangular forms or (if the opening is made larger) like horseshoes. The resonances of such a resonator can be in the mid-infrared domain (with wavelengths of a few microns) when its width is reduced to the order of a few hundred nanometers. Note that multiple resonances with different frequencies may occur, but the lowest-frequency resonance is often most relevant for the application.

Figure 1: One layer of a photonic metamaterial, containing metallic nano-resonators on a non-conducting (dielectric) substrate. Light impinging such a structure in a direction perpendicular to the shown plane can excite resonances and experience unusual propagation effects.

Such metal–dielectric composites can be fabricated, e.g., with lithographic methods. A frequently used metal is gold. Silver is better suited for high frequencies, but is more difficult to process due to its tendency to be oxidized.

Certain experiments are made only with single layers (monolayers) as described above; these are also called photonic metasurfaces, and many interesting optical functions can already be implemented with those. Three-dimensional structures (3D metamaterials) can be made by stacking such arrays in the vertical direction with a period which is again well below the optical wavelength.

When light impinges such nano-resonators, it can excite electromagnetic oscillations. These are particularly strong for frequencies near the resonance frequency, but the most interesting optical effects may occur somewhat above or below the resonance. As the period of the structure is well below half the optical wavelength, there are no photonic bandgap effects, and the effect on light propagation can be described with a (frequency-dependent) effective relative permittivity ε and relative permeability μ of the metamaterial. (The latter can strongly deviate from 1 even if no (ferro)magnetic materials are used.) In addition, one may need a bi-anisotropy parameter ξ, which describes the coupling of electric field strength E and magnetic field density B, or magnetic field strength H and electric displacement field D. Such couplings can occur, e.g., because in a split-ring resonator and electric field can induce a ring current, generating a magnetic field, and a magnetic field can induce a current which leads to the build-up of an electric field. However, ξ can be zero for reasons of symmetry if structures are combined in a suitable way.

Still more simplified forms of resonators may consist only of a periodic arrangement of short wires, or of metallic pieces with other shapes. Such forms can be used at shorter wavelengths, now reaching the red end of the visible optical spectrum. There are also metamaterial structures where the whole metallic structure is electrically connected, in contrast to the isolated nano-resonators as described above. Furthermore, there are structures with genuinely three-dimensional building blocks, such as short helically wound wires (helical antennas), which are sensitive to the direction of circular polarization. Such three-dimensional structures can be fabricated, e.g., with direct laser writing in polymer materials, combined with subsequent processing steps for introducing metallic parts.

Special Optical Properties of Metamaterials

As explained above, the optical properties of metamaterials can be described with the permittivity ε and the permeability μ, and in some cases in addition the bi-anisotropy parameter ξ. Whereas for ordinary optical materials, μ is very close to 1, μ can have totally different values in metamaterials, even though these usually do not contain ferromagnetic materials.

All three parameters are generally frequency-dependent and often also dependent on the propagation direction and polarization state of the light. Also, these parameters are not necessarily real (apart from ξ, which is zero at all frequencies for certain structures). The non-negligible optical losses of typical metamaterials (particularly when operating at high frequencies) lead to substantial imaginary parts of ε and μ.

The magnitude of the refractive index is determined by the following equation:

For purely real ε and μ, and assuming ξ = 0, the refractive index n is real but can be positive or negative. The sign of the refractive index is chosen such that Snell's law correctly describes the direction of the refracted beam at an optical interface.

The electromagnetic impedance of a material is given by

As the impedance depends on μ / ε, whereas the refractive index depends on μ · ε, two media with the same refractive index do not need to have the same impedance, if μ can vary. (For ordinary optical media, the refractive index determines the impedance and vice versa, as μ is very close to 1.) Similarly, two media can have the same impedance while differing in the refractive index. This is relevant, e.g., for the magnitude of reflections from optical interfaces, which depends on the different impedances. For example, there would be no reflection at the interface between vacuum and a medium with μ = ε = −1, although the refractive index changes from +1 to −1: the impedance is the same on both sides of the interface.

Another interesting point is that the impedance can be extremely large for a material with very low ε but a normal value of μ. In that case, the reflectivity of an interface to another (ordinary) optical material can exhibit very strong reflection; it becomes difficult to get any light into such a low-ε material. That can be a reason to search for materials where μ also gets small, avoiding a huge impedance.

The following subsections discuss photonic metamaterials with optical properties which are of particular interest.

Near-zero Epsilon Materials

Materials with very small ε also exhibit a very low refractive index (assuming that μ is not very large). That can have very peculiar consequences:

• The optical wavelength in such material (for a given optical frequency) becomes much longer than in an ordinary material. The oscillation at different positions in the material can occur with nearly the same optical phase.
• Refraction of light coming from such a medium into an ordinary optical medium works such that the propagation direction in the latter is nearly independent of the incidence angle.
• Radiative properties can be substantially modified: spontaneous emission can be suppressed, while laser gain can be high. That could be used for thresholdless lasers.
• The nonlinear index can become very large, because it is proportional to the χ(3) coefficient divided by the refractive index and its real part.

Negative-index Media

If both ε and μ are negative (→ double-negative metamaterial) and ξ is zero, one can show that the phenomenon of negative refraction occurs at interfaces between vacuum and the metamaterial. (For a bi-anisotropic material with ξ ≠ 0, negative refraction can also occur [16].) As illustrated in Figure 2, the refracted beam is not on the usual side. This can be described with a negative refractive index, changing the sign of the angle in Snell's law.

Figure 2: When a beam coming from vacuum hits a negative-index material (right side), the refractive beam inside the medium is on the same side of the surface normal as the incident beam. This is in striking contrast to the situation for “ordinary” positive-index materials (left side). The reflected beams have been omitted in the drawing.

Negative refraction has a number of consequences, leading to additional very unusual optical properties. In particular, the wavefronts of the refracted beam travel toward the interface, i.e., in a direction opposite that of the energy flow (as indicated by the Poynting vector). The direction of the Poynting vector is still given by E × H, but as B is now in the direction opposite to H, the Poynting vector is opposite to the direction of E × B. For that reason, such media are also called left-handed metamaterials (LHM): a left-hand rule instead of the usual right-hand rule delivers the direction of the Poynting vector from E and B. The term “left-handed” is somewhat unfortunate as it may suggest a (non-existing) relation to the phenomenon of chirality, as discussed below.

A convex lens made of a negative-index material could be defocusing rather than focusing, and a concave lens could be focusing. Even a simple plate (with planar and parallel end faces) of a negative-index medium could act as a lens. If the refractive index is −1, this could even lead to a superlens (perfect lens), creating images of objects with a resolution not limited by diffraction. It has been shown, however, that this property would be achieved only if the refractive index is extremely close to −1 [5] – so close in fact that such an application appears utterly unrealistic. Still, superlenses based on a modified scheme are possible [3, 6], where however the distance from the object to the lens must be extremely small, and the effect is thus less intriguing. Note also that there are superlenses based on effects other than negative refraction.

Very peculiar effects would occur if some object could be embedded in a negative-index metamaterial. Depending on its depth in the medium, it may appear strongly magnified and distorted for an observer outside the material. It should even be possible to make cloaking devices (invisibility cloaks), which can hide objects within them against viewing with light from outside, although probably only in a very limited frequency range.

Chiral Media

As mentioned above, there are also metamaterials containing metallic helical nanowires. These have a certain handedness, which can lead to substantially different optical effects for light with different directions of the circular polarization. It is possible to obtain metamaterials which are highly absorbing for one circular polarization direction and have a high transmittance for the other polarization direction. The resulting device can be called a circular polarizer. A negative refractive index is not required for that phenomenon.

Challenges for Operation at Short Wavelengths

A first difficulty with metamaterials for operation at short wavelengths, e.g., in the visible spectral region, is the fabrication of very fine structures such that their period is still well below the optical wavelength (within the medium, not just in vacuum). Some lithographic techniques are just sufficient to fulfill that condition, partly by using shorter wavelengths for the fabrication and partly by achieving a sub-wavelength resolution. Other techniques such as direct laser writing can also be employed. The speed of such fabrication methods may limit the size of the parts made within a reasonable time.

More fundamental limitations arise from the properties of the materials used. In particular, there are no perfect conductors for frequencies of hundreds of terahertz. Therefore, particularly devices for the highest frequencies exhibit relatively high optical losses.

Bibliography