It was suggested recently that photonic crystals (PC) may exhibit an index of refraction n less than one or even negative [121, 122] which would enable one to design and fabricate novel optical elements with enticing properties . While the first consideration of a negative n by Veselago in 1968 was largely ignored , Pendry's suggestions  started an intense debate about the meaning of a negative n and its practical realization. In this subchapter we will discuss shortly some simulations of the interaction between electromagnetic waves and 2D photonic crystal in the form of porous structure using a modified version of the multiple scattering techniques .
Calculations were undertaken for a 2D photonic crystal consisting of a cubic lattice of 8x32=256 parallel pores embedded in a dielectric matrix with the dielectric constant ?. The radius of pores r=0.49a, where a is the lattice constant. A new approach was used enabling to determine whether n is well defined (at least in a good approximation) or undefined. The approach takes into account that the interface between two (homogeneous) materials with indices of refraction n and -n would act as a perfect lens, i.e. a point source in the medium with index n will result in a symmetric point image in the medium with index - n. According to Veselago , there is also no reflection at such an interface.
The basic idea of the method is the following: add a fictitious optical probe material with a variable refractive index np (the index p stands for "probe") to the PC under investigation, introduce a point light source into the photonic crystal, and calculate the wave propagation for various np. If conditions can be found where an acceptable image of the point source occurs in the probe material under mirror symmetry conditions, simple geometric optics dictates that a) nPC is well defined, and b) nPC=-np. These conditions are quite general and apply to both nPC>0 and nPC. More details are presented elsewhere .
Figure 5.4: Photonic band structure (a) and transmittance spectrum (b) of 2D PC consisting of square lattice of pores with r=0.49a in dielectric matrix with ε=11.4.
Figure 5.4 illustrates the photonic band structure and the calculated transmittance spectrum (TM mode) of the PC described above. The photonic band structure is essentially comparable to that published by J.D. Joannopoulos  and proves the general validity of the approach used for simulations. In order to identify the frequency ranges for which the photonic crystal can be described as an (approximately) optically homogeneous medium with a given, if unconventional index of refraction, the concept of the effective medium according to Lalanne  was used. After identifying those wavelength regions, the nPC value obtained then can be used to calculate the formation of images with PCs. This can be demonstrated for nPC>1 as well as for the particularly interesting case of nPC<0. Simple geometric optics (which should be applicable for a well-defined nPC) predicts that a concave lens then should focus light for the case when the refractive index of the lens is less than that of surrounding material. The simulations show that this is indeed the case.
Ordered parallel pores embedded in a dielectric matrix to form a concave lens as shown in Figure 5.5a indeed focus light in the long wavelength limit associated with n>1. Note that there is only one row of pores in the center of the lens. Calculations were made for a radiation point source situated left from the lens at a distance 100a where there is homogeneous dielectric material with λ=11.4. Figure 5.5b shows the distribution of the light intensity as a function of the distance from the middle of the lens. As one can see from Figures 5.5a and b, the focusing effect proves to be rather strong.
Figure 5.5: Focusing effect of a porous PC concave lens: (a) the electromagnetic field distribution in vacuum for long wavelength limit and (b) the transmittance coefficient along the optical axis of the lens for λ=10a>>a; effective nPC=1.8.
Figure 5.6 illustrates the results of calculations made for a frequency where nPC<0 (a/λ=0.81). The radiation coming from a point source placed in homogeneous dielectric material with λ=11.4 proves to be focused by the PC concave lens although the electric field modulus exhibits a more complicated spatial distribution than in the case of nPC >1 (compare Figures 5.5a and 5.6a). The focusing effect is seen also in Figure 5.6b where the distribution of the light intensity as a function of the distance from the middle of the lens is illustrated.
Figure 5.6: Focusing effect of a porous PC concave lens: (a) the electromagnetic field distribution in vacuum and (b) the transmittance coefficient along the optical axis of the lens for wavelength λ=1/0.81a; effective refractive index nPC <0.
In conclusion, the calculations show that concave lenses made from porous dielectrics focus electromagnetic waves both in the long wavelength limit where n>1 and in the spectral regions characterized by negative refractive index. The obtained results may be used for the purpose of designing and manufacturing novel micro-lenses ready to be integrated in optoelectronic circuits. Note that single crystals of nanopores can be easily introduced into semiconductor materials using electrochemical etching techniques as discussed in the previous chapters .