We analyze focusing of electromagnetic waves inside a photonic crystal slab by means of finite-difference time-domain simulations. At the frequency of the source, the photonic crystal behaves as an effective medium with an effective index of refraction of -1. Despite of the strong Bloch modulation of the field inside the slab, the presence of a well-definite internal focus is evident. The dimensions of the internal focus are similar to those of the external focus. The effect of the frequency of the wave on the focusing is also discussed.
©2005 Optical Society of America
A flat slab of an effective medium with a refractive index of −1 can focus electromagnetic waves . This kind of lens has two foci: one inside and the other outside the slab. It has also been proposed that such a negative-index lens (NIL) can partly overcome the diffraction limit and achieve subwavelength focusing since evanescent waves are restored inside the negative-index medium . Thus, such a “superlens” is able to focus some of the Fourier components of a source that do not propagate in a radiative way. In contrast, a classical lens only restores the phase of the propagating waves so subwavelength information cannot be recovered at the image plane. In general, a NIL is constructed from artificially arranged composites. One choice is to arrange a periodic lattice of split-ring resonators, which has an effective negative permeability, and metallic wires, which have an effective negative permittivity [3–5]. A NIL can also be implemented by means of a planar slab consisting of a grid of printed metallic strips over a ground plane, loaded with series capacitors and shunt inductors . A third choice is the use of photonic crystals (PhC), which has the added values of low absorption losses and straightforward scalability up to visible and infrared wavelengths .
In a PhC the permittivity ε is always locally positive whereas the permeability is µ=1. Electromagnetic propagation inside a PhC takes place in the form of Bloch waves governed by the dispersion diagram that relates the frequency and the wave vector for each electromagnetic mode. Notomi  showed that PhCs can behave as dielectric materials with an effective index of refraction n eff in certain spectral regions where the equifrequency contours (EFCs) become rounded. If the EFC shrinks with increasing frequency then the group velocity points inwards and a phenomenon of negative refraction can be expected at the interface between the PhC and air (or a dielectric medium)  and a PhC can behave as a flat NIL.
In general, negative refraction (and its resulting focusing effects) in PhCs occurs under two conditions : (i) at frequencies in which the propagation is forbidden along one of the main symmetry directions (partial gap) and allowed in the other one; (ii) at frequencies in which the EFCs of the PhC have a rounded shape whose radius diminishes with increasing frequency so the wave vector and the group velocity are antiparallel. The first case is common in the lowest valence band near the band gap  whereas the second case takes place at higher photonic bands . It should be mentioned that the mechanism giving rise to negative refraction is well different in both cases. Negative refraction at the first band occurs due to the anisotropy of the medium but the effective index is positive. Moreover, recent studies by Z. Ye and co-workers [12–13] suggest that a collimation or self-guiding effect instead of a true refraction is the phenomenon that takes place, since the radiation inside the PhCs tends to bend away from the forbidden direction. For example, Snell’s law is not met at the air-PhC interface in this case and a lens working under these conditions  do not follow the rules of geometric optics . For example, the focus predicted to occur inside the NIL when n eff=-1 does not occur in this case since the PhC slab does not behave as an effective medium [12–13,15]. In addition, the external focus only occurs when the source is in the vicinity of the slab (near-field focusing). In this paper, we will focus on the case (ii) for which the EFCs turn rounded so neff can be defined and a PhC slab can behave as a true lens . It should be mentioned that the existence of rounded EFCs that shrink with increasing frequency is not a sufficient condition to construct a PhC NIL. In addition, the slab termination has to be periodic and the mode inside the PhC must be symmetry matched to the plane wave propagating through free-space .
Recently, several authors have reported theoretical and experimental studies related to PhC NIL [16,18–22]. However, the existence of an internal focus, which is a clear evidence of the NIL following the geometric optics rules, has not received much attention. In some of these works, focusing inside the PhC is not clearly observed because the slab thickness is smaller than the wavelength. Throughout this paper we analyze the focusing inside a PhC slab with n eff=-1 and demonstrate that it behaves as the NIL proposed by Veselago more than 35 years ago.
2. Description of the scenario: a PhC slab that behaves as a NIL
We consider a two-dimensional hexagonal lattice of circular air holes (with lattice constant a and radius r=0.4a) on a dielectric background with ε=12.96. We assume that the material has negligible absorption losses, as it would be the case of Si or GaAs at optical frequencies (λ=1550 nm, λ being the free-space wavelength). The lattice extends over the x-z plane and is infinite in the y dimension. Only transverse magnetic modes with the electric field along the y direction are considered. We use a plane wave expansion method to calculate the band diagram of this PhC as shown in Fig. 1(a). Frequency is expressed in normalized units of a/λ. We will focus our attention on the second band. For frequencies over 0.26 the EFCs become rounded (effective refractionlike medium ) and their radius diminishes when the frequency increases (antiparallelism between the wave vector and the group velocity). The effective index n eff is depicted in Fig. 1(b) for the two main symmetry directions. Only propagation along ΓM is interesting because the Bloch mode along ΓK is uncoupled and cannot be excited efficiently [11,17]. If the frequency is 0.306 then n eff=-1, which is the first step towards the NIL. If n eff≠-1 the position of the internal and internal foci will be different depending on the angle of incidence which results in a longitudinal spreading of the focus .
Figure 2 is intended to describe the scenario used to test the behavior of our PhC slab as a NIL by means of finite-difference time-domain (FDTD) simulations . Perfectly-matched layer conditions  are imposed at the boundaries to avoid undesired reflections. The PhC slab is oriented so that the incidence (z direction) is along ΓM. Both interfaces are cut along ΓK. The slab thickness is L. The slab is symmetric with respect to both x and z axes. An optical source is placed L/2 below the slab (z source=-L). Several sets of electric field monitors (represented by dots) are placed along the dotted lines A, B, C, D, E, and F. The objective is to obtain the field (and intensity) profiles to evaluate the lensing effect. Figure 3 shows the FDTD electric field distribution obtained at different time steps of the FDTD simulation. Time is expressed in ct/a units, c being speed of light in vacuum and t absolute time. Both slab terminations are identical (the slab is symmetric with respect to the x axis) and are chosen to ensure high transmission efficiency to sharpen the focusing . To this end, (L=11.5√3-0.4)a=19.52a, which is much larger than the corresponding wavelength. Our FDTD simulations have proven that this kind of termination (also used in other works [19,22]) produces a strong and sharpen external focus. However, we have not optimized the termination to maximize the lens resolution since our aim is to make a qualitative description of the internal focusing. The source has a Gaussian profile that at its origin has a transverse full-width half-maximum (FWHM) of 0.4λ. We have chosen this kind of source instead of a point source as in the focusing analysis of section 3 in order to better observe the shrinking and focusing of the beam inside and outside the NIL. The frequency is 0.306 in order to have n eff=-1. The physical structure of the PhC slab has been removed in the snapshots to better appreciate the electromagnetic propagation inside the PhC. First we see in Fig. 3(a) the input Gaussian wave before impinging the PhC slab. At ct/a=20 the wave enters the slab [Fig. 3(b)] and a Bloch wave is excited. The reflection at the input interface is very low, although a perfect matching cannot be achieved, in contrast with the case of NILs based on left-handed metamaterials. After this, the beam commences to shrink and a negative curvature appears. Internal focusing is observed in Fig. 3(c) approximately at the center of the slab. It should be highlighted that such an internal focus does not occur when the focusing is at a frequency in the valence band . At ct/a=110 we observe that the beam has reached the output interface and exits the PhC slab [see Fig. 3(d)]. However, we do not observe a beam broadening after the internal focus: this is because oblique rays need more time to reach the output interface as they travel a larger distance and still do not contribute to the beam at the time step shown in Fig. 3(d). The internal broadening for positive z is clearly observed in Fig. 3(e) at a time step of ct/a=140, and also the formation of the external focus is evident: at the exit of the slab the electric field has a negative curvature and the radius of the phase fronts gets smaller. In Fig. 3(f) the external focus is totally formed. The beam propagation follows the geometric optics rules as predicted by Veselago . The external focus is not perfect: this is mainly due to the fact that evanescent waves are not completely restored and the power coupling between the plane wave outside the slab and the Bloch mode inside the NIL depends on the angle of incidence (the slab does not behave completely as an isotropic medium), being poor for high incident angles . It should also be mentioned that internal focusing can be clearly observed because reflectance at the interfaces is low due to the choice of the surface termination. If the reflectance was higher, the presence of multiple internal reflected waves could mask the internal focus at a steady-state regime.
3. Analysis of the external focus
First, we make a brief evaluation of the external focus placed at z=L (assuming n eff=-1). To this end, we use the electric field monitors placed along lines A (longitudinal) and B (transversal). In this case, a point source with normalized power is allocated at z=-L. Figure 4 shows the time evolution of the external focus in both the longitudinal (a) and the transversal (b) dimensions for the case of L=19.52a. Blue and red colors stand for minima (zero) and maxima of light intensity respectively. Transversal patterns are represented only for x > 0 due to the symmetry of the NIL. It can be observed that after a transient period, in which the leading edge of the monochromatic wave exits the PhC slab, the focus is formed and it keeps on in a steady state. To analyze more in detail the external focus, Fig. 5 shows the transversal (a) and the longitudinal (b) light intensity obtained as time average of the square electric field at the image side. The position of the intensity maximum along z (z=19.5a≈L) is in good agreement with the laws of refraction assuming n eff=-1, which does not occur when the focusing is in the valence band, for which the focus only arises near the PhC slab output (near-field focusing). The FWHM in each case is also shown in Fig. 5. We find that subwavelength focusing is only observed for the transversal direction. This result agrees with previous works in which it was shown that the external focus was wider in the longitudinal than in the transversal dimension [2–22], which can be related to slight deviations of the effective index from −1 for different angles of incidence (it should be noted that the EFC is not really a circle but a strongly rounded hexagon ). We repeated the simulation but using a thinner PhC slab (L=(5.5√3-0.4)a=9.12a) with the same terminations. The result was a stronger focusing with an intensity maximum of 0.94 a. u. and an FWHM of 1.22λ for the longitudinal direction and 0.42λ for the transversal direction. It can be noted that although the focus is stronger for the thinner slab, the resolution is almost the same and is only slightly improved. It has to be noted that our FDTD simulations have not revealed the excitation of surface modes which are known to be related to the excitation of evanescent waves. However, subwavelength resolution by a PhC lens has been demonstrated even without the presence of surface modes . It should also be mentioned that, in contrast to the case of PhC NILs that include metallic elements , in our case we can assume that no absorption losses are present and the total power transmission is quite high. Anyway, the presence of losses would result in a large decrease of the focus strength but the NIL resolution will be only slightly decreased .
4. Analysis of the internal focus
Now we evaluate focusing inside the PhC slab. It should be emphasized that the propagation inside the PhC is well different from that in free-space: inside the PhC one or several Bloch modes will be excited so a strong periodic modulation of the electromagnetic field is expected. We now use the field monitors located along lines C, D, E and F. For the sake of comparison, the same results have been obtained when the source is a wide Gaussian beam with normalized power and initial waist 20λ, which resembles a plane wave. First we consider propagation along z. Figure 6(a) shows the longitudinal profile of the field intensity along line C. The blue solid curve is the average intensity for the point source whereas the dashed black curve represents the same magnitude for the case of pseudo-plane wave. It can be easily observed that the profile is well different from that depicted in Fig. 4: the field is strongly modulated and there is a series of peaks (corresponding to the dielectric background) and valleys (corresponding to the holes) along the line C. Specially, a strong minimum of intensity is observed at z=0. Despite of the modulation, the focusing is evident: for the plane wave case, the intensity maxima keep almost constant along the slab; for the case of the point source the intensity maxima are larger near the center of the slab (z=0). Figure 6(b) shows the time-space diagram of the intensity along the C line. The formation of the Bloch-modulated focus is clear.
In addition, it deserves to be noted that the intensity is not symmetric with respect to z=0: maxima are slightly larger for positive values of z. Moreover, a second focus seems to appear around z=6a, which is also observed in the external focus (see Fig. 5(a)). This can be explained by considering the amplification of evanescent waves along the slab, which would explain the subwavelength focusing observed in Fig. 5(b) . Since the distance between the source and the slab is high (much larger than the wavelength) it can be expected that evanescent waves arrive strongly attenuated at the PhC input interface so their amplification will be low. However, the asymmetry of the intensity can also be explained if it is considered that the EFC has not a totally rounded form and the collimation properties occurring at lower frequencies due to the hexagonlike shape of the EFCs is also slightly present at the frequency used in the simulations. This is, some rays that impinge the input interface at x 1 are refracted and, since n eff is slightly lower than −1 for incident angles >0°, they exit the PhC slab at x 2<0 that verifies |x 2|<|x 1|. Results shown below in Fig. 8 contribute to this explanation. We think that both phenomena (growing of evanescent waves and residual collimating behavior) contribute to the observed effect.
The transversal intensity along the E line is shown in Fig. 7(a). As in Fig. 6(a), blue solid line stands for the intensity when the field is generated by a point source whereas the dashed black line is for the case of a unit-power plane-like wave impinging the NIL. The result is similar to that depicted in Fig. 6(a): when the field is generated by a point source the intensity tends to be concentrated around the center of the slab despite of the modulation (the minimum in x=0 corresponds to the central hole of the slab). Figure 7(b) shows the time-space diagram of the intensity along the E line. Now we compare the transversal intensity at the slab center (line E) with the intensity near the slab interfaces (lines D and F). The result is shown in Fig. 8: the intensity is represented along the line E (blue curve), D (black curve) and F (red curve). It is clear that the negative refraction produces the focusing around the center of the slab. The intensity profile near the input (line F) and the output (line D) interfaces is similar, although a larger intensity near the output interface in the region close to x=0 is observed, in agreement with the results shown in Fig. 6(a). This is, the input transversal intensity is rearranged as the field propagates through the NIL so the power concentration is higher near x=0 as the wave reaches the output interface. As explained before, both the growing of evanescent waves and the residual collimating behavior of the PhC are probably the origin of this phenomenon.
In the studied PhC slab the center of the NIL coincides with a hole (see Fig. 2) so the field intensity at this point displays a minimum. It would be interesting to analyze the internal focusing in the case the PhC slab is configured in a way that there is dielectric at the center of the slab. To this end, we displace the optical source as well as all the field monitors a distance a/2 rightwards. The obtained longitudinal and transversal intensity profiles are shown as black curves in Fig. 9(a) and Fig. 9(b) respectively. For the sake of comparison, the intensity profiles obtained previously (hole in the center of the slab) are also shown with blue curves. It is interesting to note that in this case an intensity maximum occurs at the center of the slab whereas intensity minima are observed corresponding to the presence of holes. Despite of the well different intensity profiles inside the slab for each slab configuration, it must be noted that the external focus is exactly the same for both of them. We measured the same longitudinal and transversal profiles represented in Fig. 5 for the slab with dielectric at its center, which confirms that the PhC slab behaves as an isotropic NIL. We also include the intensity profile of the external foci as red dashed curves. The longitudinal external focus in Fig. 9(a) has been represented inverted with respect to z=L/2. It can be observed the resemblance of the internal and external focus if the Bloch-modulation is not taken into account. The intensity level is higher in the internal focus, which would mean that approximately half the intensity is lost when the wave travels from the NIL to the external medium. However, taking into account that inside the PhC there is a series of intensity maxima (dielectric) and minima (holes), we can state that the mean intensity is similar for both the internal and external focus, which agrees well with the low reflection observed at the air-PhC interfaces. From Fig. 9 it can also be deduced that the values of FWHM are similar for both the internal and external foci. We must also note that the presence of the second maximum in the longitudinal external focus at z=13.5a in Fig. 5(a) is also observed in the internal focus so it can be concluded that the second interface produces a high-quality image of the filed inside the slab and the first interface is the main origin of distortion of the final image. This is consistent with the fact that evanescent waves are strongly attenuated before entering the NIL and can only be restored in part, so the loss of information takes part mainly in this region.
Let us make a brief discussion on the influence of the frequency of the source on the behavior of the NIL. When the frequency is changes, n eff varies as in Fig. 1(b). Assuming that the NIL under study follows the geometric optics rules, it can be stated if the frequency is chosen so that the index is different from −1, not all the rays converge in the same point both inside and outside the slab and the position of the internal and external foci is modified . In addition, a longitudinal enlargement of the foci is expected. For example, if the frequency increases (decreases), |n eff| decreases (increases) so the internal focus position (proportional to |n eff|) comes to a lower (larger) value of z and the external focus position (proportional to 1/|n eff|) is displaced to larger (smaller) values of z. We measured the characteristics of the internal and external foci for the slab with L=19.52a at frequencies a/λ=0.295, 0.306 and 0.32. The corresponding field intensity along the z axis is represented in Fig. 10, in which it can be observed that the internal and the external foci are displaced as expected for each frequency. The broadening of the external focus is more evident for a/λ=0.32. We think that in the case of a/λ=0.296 the external focus is not broadened because of the proximity of the PhC surface, which gives rise to a near-field focusing. Anyway, we can conclude that both the internal and external foci follow the rules of geometric optics when the frequency of the source is varied.
It should be noted that in this work we have analyzed a two-dimensional PhC slab. Such a structure allows focusing only on the x-z plane since the lens is invariant along the y axis. This is, the source is actually a line source, not a point source. To achieve point focusing from a true point source, a three-dimensional PhC slab is required. In this case, we should work in a frequency region where the EFCs are spheres (instead of circles as in the two-dimensional case) whose radii decrease with increasing frequency. This has been recently demonstrated to be possible using a bicontinuous PhC of a bcc lattice, in which subwavelength resolution was shown to occur along both transversal directions . The same analysis here applied may also be used in such a three-dimensional structure, although the field sampling may be realized both along the y and the z axes.
In this paper we have analyzed the focusing of electromagnetic waves inside a negative-index PhC slab. This structure behaves as a negative index flat lens that follows the simple rules of geometric optics. We have chosen a slab with thickness much larger than the wavelength to appreciate wave propagation inside the lens. Inside the PhC slab, a focus is clearly observed but it is strongly modulated due to the Bloch-wave propagation. The longitudinal and transversal dimensions of the internal focus are on the order of the dimensions of the external focus, so it can be deduced that the loss of subwavelength details occurs during the wave propagation from the source to the slab. By varying the frequency of the source we have demonstrated that the position of the focus is modified as expected from the refraction laws. The intensity increase along the longitudinal direction inside the negative index lens is thought to be originated from a slight growing of evanescent waves and a residual collimating behavior of the PhC since the equifrequency contour is not really a circle but a strongly-rounded hexagon. The extension to a three-dimensionally point focusing has also been discussed.
This work has been partially funded by the Spanish Ministry of Science and Technology under grant TIC2002-01553.
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