1. Introduction

Interest in Veselago’s slab lens [1] with simultaneous negative permittivity and permeability and thus negative index of refraction has significantly increased since the publication of Pendry’s perfect lens [2] and Smith’s experimental confirmation [3] of negative refraction using a composite metamaterial composed of wires and split ring resonators at microwave frequencies. Materials with a negative (effective) refractive index associated with the wave phase are left-handed since the electric field, the magnetic field and the wave vector form a left-handed triplet [1]. It has been found that dielectric photonic crystals (PhCs) can also have negative refraction [4–6]. Photonic crystals are important enabling structures for the control of light, due to their rich dispersion characteristics [7], and have found many applications [8]. A dielectric PhC slab with negative refraction can also realize imaging, and has attracted intensive research recently, both in theory and experiments [9–16, 18–26]. Studies of the first-band square-lattice PhCs near the M point of the first Brillouin zone show that imaging is not associated with negative effective refractive index, instead, the “photonic mass” is negative [9]; it does not obey the rules of geometrical optics or ray model [10]; it is restricted [11], and the channeling or self-collimating effect mainly transports the evanescent waves [10, 12] due to the square-like equi-frequency surface; negative refraction occurs at large angles of incidence due to the phase modulation of the wave front [12]; an excessively thick slab cannot form a quality image [10] although a thicker slab is expected to form images for more distant objects [9]. In addition, there are some reports on the imaging of the second-band hexagonal lattice PhCs near the Γ point [6]. It has been shown that because the equi-frequency surface is circular, a negative effective index of refraction can be defined [6]; imaging is unrestricted and the image can be found by ray tracing (geometrical optics) [11]. Furthermore, the effective index of refraction calculated for finite-sized PhCs is smaller than that obtained from infinite PhCs [13]; the wave vector of the main component of the Bloch mode lies outside the first Brillouin zone [14]; there are certain conditions for one single beam propagation [15], and the negative refraction is not associated with the “rightness” of the wave [16].

Many previous studies of PhC imaging have been restricted to the near field or to the cases where both the object distance and the image distance are smaller than the thickness of the slab. This is mainly because geometrical optics predicts that the sum of the object distance L from the object to the slab and the image distance L’ from the slab to the image is equal to the slab thickness d for a homogenous material or metamaterial with a refractive index n of -1 [2]. Mathematically, the relation is expressed as L + L'= d, which is referred as Veselago relation for convenience. Therefore, no object at a greater distance than d can produce an image, and no image can be produced further than d [17] since the former leads to a virtual image and the latter requires a virtual object. However, even when the equi-frequency contours of a photonic crystal are circular, dielectric photonic crystals differ from left-handed materials with n = -1 in several important aspects. For example, the periods of the PhCs are on the same order as the operating wavelength, therefore scattering and diffraction inside the slab is strong; the impedance of the PhC is not matched with the homogenous material surrounding to it, thus strong reflection is expected; the thickness of the PhC slab using dielectric materials is on the order of the wavelength whereas for left-handed materials the thickness is typically much smaller than the wavelength due to high material absorption. In addition, even if only “one beam” is formed inside the PhC slab, the phase front has a “wiggly” profile [15] and thus differs from the conventional cylindrical or plane waves. The wiggly phase front indicates that there is no internal focus inside the slab even when the Veselago relation is valid. Most importantly, a recent study shows that photonic crystals are inherently right-handed [24] and thus one cannot explain PhC imaging using theories of left-handed materials.

This paper shows that for PhC slab imaging of the second band if the object distance is larger than the slab thickness, a real image can also be formed. We first obtain the object-image relation for object distance smaller than and equal to the slab thickness, which agrees with the prediction from Veselago relation, then extend the object distance to values larger than the thickness. We will show that in this regime the image distance changes little, thus it is referred to as “saturated” imaging [27]. However, we find that in some cases approximately the same degree of sub-wavelength resolution is obtained as for near-field imaging. By analyzing the angular frequency spectrum of the image intensity we will show that in this regime the image contains higher spatial frequencies which correspond to evanescent waves but which originate from within the photonic crystal due to scattering and diffraction [8, 24] and surface resonance [18], rather than directly from the object. Some other issues, such as effects of apertures and terminations on the image distance and potential applications of the saturated imaging regime are discussed.

2. Near-field imaging and saturated imaging

We investigate a two-dimensional (2D) hexagonal lattice with air-holes in a dielectric with a refractive index of 3.6, and r/a=0.4 where r is the air-hole radius and a is the lattice constant [6]. The cylindrical air-holes are infinitely long in the third dimension. This photonic crystal exhibits negative refraction at the second band, and is designed for E-polarization where the electric field is parallel to the axis of the air-holes. According to Fig. 8 in [6], the effective index of refraction is around -1 for a normalized frequency of 0.312. However, different authors have found different values for this normalized frequency. For example, values of 0.3[11], 0.305 [12] and 0.31 [14] have been used. The origin of the inconsistency is in differing implementations of the Plane Wave Expansion (PWE) method which is used to calculate the band structures of the photonic crystal. We calculated the effective index of refraction for the second band from its band structure (similar to Fig. 8 [6]). The dispersion of the effective refractive index is shown in Fig. 1. The normalized frequency Ω is close to 0.3 for the effective refractive index n_eff of -1, which is used in this paper. Thus, the lattice constant can be calculated as a = Ωλ where λ is the operating wavelength. When working at wavelength of λ=1.55 μm, it has a lattice constant of a=0.468 μm and air-hole radius of r=0.4a= 0.1872 μm.

The simulations were performed using the finite-difference time-domain (FDTD) method. Since FDTD simulation is quite computationally intensive, a 10 μm by 10 μm square with 0.5 μm thick perfectly-matched layers (PML) is used for the simulation space with a spatial discrete increment of 10 nm. The total number of simulation time steps is chosen to be 12,500 or more at the Courant time step size. A point source is used as an object to be imaged.

 

Fig. 1. Calculated effective refractive index for the hexagonal PhC (holes) with r/a=0.4 and refractive index of 3.6

Download Full Size | PDF

 

Fig. 2. The 3-row PhC flat slab

Download Full Size | PDF

Since excessively thick PhC slabs cannot form a good image [10], and thin slab imaging may find application in nanophotonic devices, we only investigate relatively thin slabs. We examine imaging using slabs composed of three to ten rows of holes. First, a very thin slab with three rows of holes, shown in Fig. 2, is investigated. The thickness is d=2.53a, which is about 0.76λ, and the full aperture is 8.5d. For near-field imaging, an object distance of L=0.1d is used where L is measured from the object to the left surface of the slab (see Fig. 2). Fig. 3(a) shows the instantaneous electric field after 12,500 time steps, in which the slab is shown in between the two vertical lines. It can be seen that, on the right-hand side of the flat slab, there is a focus “point” at a distance L’=0.983d where L’ is measured from the right surface of the slab to the image (see Fig. 2). This is confirmed by extracting the phase information, as shown in Fig. 3(b). From Fig. 3 it can be seen that on the image side, the wavefronts converge to a point coincident with the peak intensity of the electric field, and then diverge as expected. We can therefore identify the point of peak intensity as an image point. The irregularity of the field Fig. 3(a) and the phase Fig. 3(b) in the image side is mainly originated from the modulation of the waves inside the PhC slab, since the waves inside the PhC slab are Bloch waves [8]. In contrast to conventional cylindrical waves Bloch waves in general do not possess regular wave shapes with a simple analytical expression.

Next, we examine the case for which the object distance is equal to the slab thickness. For a homogenous left-handed material that obeys geometrical optics, it is predicted that the image should be located at the right-hand interface [2] (i.e. L’=0). However, the dielectric PhC slab behaves differently. The image is displaced from the right-hand interface of the slab with an imaging distance of 0.16d, as shown in Fig. 4a. From the phase distribution in Fig. 4(b) we can also see that the waves first converge to a “point” that lies in the air region to the right of photonic crystal slab. In this case the object is not in the near-field range. However, the field and phase patterns are almost the same as those for near-field imaging. Increasing the object distance further still, to a value of 2d, results in an image that is still slightly separated from the right-hand surface of the PhC slab, at a distance of 0.07d. Obviously the PhC slab imaging behavior observed here departs from Veselago relation given previously.

 

Fig. 3. (a) The instantaneous electric field after 12,500 time steps at the Courant time step size for an object distance of 0.1d, for the 3-row PhC slab.

Download Full Size | PDF

 

Fig. 3. (b) The phase distribution for the field of Fig. 3(a)

Download Full Size | PDF

 

Fig. 4. (a) The instantaneous electric field after 12,500 time steps at the Courant time step size for an object distance of 1.0d

Download Full Size | PDF

 

Fig. 4. (b) The phase distribution for the field of Fig. 4(a)

Download Full Size | PDF

One may attribute such behavior to the thinness of the slab and the proximity of the object when compared to the wavelength. However, the same behavior is also observed with relatively thick slabs. For example, for a 10-row slab with a thickness of d=2.59λ, when the object is at 1.0d, the image distance is 0.10d. If the object is far away, for example at 1.5d, the image distance of 0.028d In this case, the object distance is about 3.89λ, which is far beyond the “near-field” range. It can be seen that the image distance changes little when object distance increases up to 1.3λ from 1.0d to 1.5d. This imaging behavior is different from unrestricted imaging [10, 11] which predicts a linear relation. It also differs from the restricted imaging [8] in which restricted imaging refers to the near-field and is observed for the first band of photonic crystal. Thus we refer such behavior as “saturated” imaging [27], meaning that further increases in object distance do not change the object distance accordingly. It is observed when object distances are larger than the slab thickness (i.e. usually not in the near-field) and when the second photonic crystal band is used.

To systematically investigate the relation between the object and image distances for PhC slab imaging, we performed a series of simulations for object distances from 0.01d to 2.00d for different numbers of rows (from 3 to 10). To remove the effect of the different thicknesses, Fig. 5 graphs the relation between normalized object and image distances for 3-, 6-, 7- and 10-row slabs. The Veselago relation becomes L/d+L’/d=l. From Fig. 5 it can be seen that the object-image relation is almost “linear” for object distances smaller than the slab thickness, although there are some variations, for all the cases. A closer examination shows that from 0.0d to about 0.5d of the object distance, the simulated results for PhC slab imaging agree with Veselago relation quite well. In this case, the object and image locations form conjugate pairs. A small increase in object distance causes the image to move approximately the same distance in the opposite direction. This illustrates the unrestricted imaging property of the second band [11]. For object distances between 0.5d to 1.0d, imaging is also unrestricted, but deviates further from the Veselago relation. This is the approximate range of the region of linear object-image relation. Beyond this range, it is evident that the image is saturated as its location changes very little despite large changes in the object distance. Here we should stress that the linear object-image relation does not imply that a ray model can be applied in all aspects of analyzing the PhC imaging properties. For example, we do not observe an internal focus point with the field distributions. One reason is that inside the slab the wavefronts are irregular [15].

It is important to note that the PhC slab itself does not have focusing power. In other words, the PhC slab cannot focus a time-harmonic plane wave into a “point”. Therefore, if the object is too far away from the slab, it cannot be imaged. This is one reason that we do not investigate the cases where the object distances are too large.

 

Fig. 5. The relation between normalized object and image distances for slabs with 3 rows, 6 rows, 7 rows and 10 rows of holes.

Download Full Size | PDF

 

Fig. 6. The transversal distribution of the electric field intensity at the image locations for different object distances of 3-row slab.

Download Full Size | PDF

Note that the above simulations use perfectly-matched layers (PML) to truncate the simulation space. The PML absorbs waves that impinge on it with a reflection coefficient around -60 ~ -80 dB. However, unlike the case of homogenous materials where a PML provides an analogue for infinite space, the PML does not simulate an infinite photonic crystal structure. This is because theoretically the field at any one point inside the photonic crystal is a function of the total scattering and diffraction of the whole PhC, and the amplitude distribution inside a PhC may take a long time to reach steady-state due to scattering and diffraction from the holes [18]. Therefore using a PML to truncate the photonic crystal eliminates any contributions from the remainder of the PhC beyond the PML. In order to assess the validity of the PML we performed three additional types of simulation. For the first of these, we increased the aperture (also using PML) and at the same time proportionally increased the total number of simulation time steps. For example, using an aperture of 20μm in place of the previous aperture of 10μm of the 3-row slab, the image distances only decreases 0.03 μm for an object distance 0.1d, and the image distances decrease at most 0.01μm for object distance larger than 1.0d. In both cases the results tend to be closer to the theoretical line in Fig. 5. Thus a larger aperture nearly does not change the image distance. This result indicates that PhC slab imaging of n_eff=-1 at the second band is also all-angled [18]. This is different from the case where the effective refractive index is -0.8 and the operating wavelength is 1.50 μm, in which our simulations show that the image distance strongly depends on the size of the aperture: the larger the aperture, the larger the image distance.

For the second type of simulation the PhC slab was terminated using an air interface with an aperture of 10 μm. The air is truncated by the PML layers, which is close to reality since the PhC slab has finite aperture in practical devices. The distance between the air and the PML is 1 μm. The image distance changes little: the maximum change is 0.05~0.08 μm, and again brings the results closer to the theoretical prediction. On average the change in image distance as a result of this change in boundary conditions is less than one computational cell, 0.01μm.

For the third type of simulation a much larger number of time steps 200,000 were used in order to allow for more convergence of the multiple scattering and diffraction processes. The change of the image distance is at most 0.03μm. In summary, the results from the three types of simulation support the validity of the results obtained using the initial parameters.

4. Discussion

For left-handed materials, the refractive index is associated with the phase velocity and the wave vector. Under these circumstances the rules of geometrical optics hold and the Veselago relation is valid (the straight line in Fig. 5). If the object distance is larger than the slab thickness, the slab cannot form a real image. However, the imaging properties of a PhC slab differ from those of a left-handed material because the photonic crystals are not left-handed and the effective negative refractive index is not associated with the phase velocity [24].

For PhC imaging, when the object is located at a distance smaller than the slab thickness, a real image can be formed. The imaging relation is approximately linear from Fig. 5. But in the saturated imaging regime, the relationship is highly non-linear. In this case, even when the object distance is larger than the slab thickness, the PhC slab can still form a real image. This is due to the complicated scattering and diffraction interactions of the Bloch waves inside the PhC slab. As shown in [12], rays with large angles dominate the negative refraction. When the object distance is large, rays with large angles are reduced and thus the negative refraction is weak. The result is to cause the focus to shift to the right.

The linear part of the object-image relation shown in Fig. 5 can be used to estimate the effective index of refraction of the finite-sized PhC slab. When the refractive index deviates slightly from -1, the object-distance relation can be revised as L + L'= d / |n_eff| . From the simulation data, the best fit value for the effective index is found to be -0.810, -0.893, -0.926 and -0.917 for the 3-row, 6-row, 7-row and 10-row slabs, respectively. The smaller values of effective index may be understood to result from insufficient wave interaction inside the slabs due to finite-thickness. Here we have not taken into account of the ambiguity in determining the exact thickness of the slab [29]. In the case of vertically-air terminated slabs, the extracted effective index is -0.897, -0.949 and -0.935 and -0.918 for the 3-row, 6-row, 7-row and 10-row slabs, respectively. These values are closer to -1 than those obtained from direct PML truncation, because now the reflections arising at the vertical end faces contribute more scattering and diffraction. The actual values of the effective refractive index may lie in between the two data sets since in the first case there are no wave reflections from the end-faces and in the second case there are stronger reflections than infinitely-long slab. The object-image relationship can be used as a practical tool to obtain the effective refractive index of finite-sized photonic crystals.

Note that the effective index in Fig. 1 is extracted from the band structure, which is calculated by using the PWE method for an infinite PhC. The imaging results obtained here are for PhC slabs with finite sizes. The effective index extracted in this way should not be expected to be the same as that of an infinite photonic crystal. Nonetheless, the extracted effective index is quite close to the theoretical value of -1. We should stress here that this effective index of refraction is not associated to the phase velocity, as discussed in [24].

In imaging, resolution is a very important image quality parameter. There are several different criteria to evaluate image quality [30]. However, the Full-Width-Half-Maximum (FWHM) of the image intensity is simple to obtain and many authors have used it to measure the transversal resolution. Therefore we also use the FWHM. Fig. 6 and Fig. 7 graph the normalized transversal intensity distribution across the image for some object distances for the 3-row slab and the 10-row slab. The FWHM of the main lobe is about 0.43λ for the 3-row slab, which is better than the conventional diffraction limit and is smaller than the wavelength. Since side lobe levels increase with increasing object distance larger object distances become less interesting. For the 10-row slab, The FWHM is about 0.55λ, wider than that for 3-row slab, but the side lobes are lower, as can be seen from Fig. 7.

It can be seen from Fig. 6 and Fig. 7 that the FWHM of the main lobe has almost no change in the linear regime. However, as the object distance increases further and approaches the saturated imaging regime, the side lobes become larger and larger. Consequently, the resolution of the saturated images is reduced mainly because of the side lobes (see Fig. 6 and Fig. 7). Since the simulation uses PML to absorb the waves at the edge of the slab edge, the effects of aperture diffraction can be neglected. Where do these side lobes come from?

To explore this issue, the field intensity distribution for object distances of 0.1d and 1.6d is shown in Fig. 8 and Fig. 9 for the 3-row slab. We can see that for both cases, there are secondary lobes (smaller intensity peaks) besides the images. However, for near-field imaging (Fig. 8), those secondary lobes do not lie in the image plane and thus have little effect on the image resolution. As the object distance increases to the saturated imaging region or the far field (Fig. 9), the image distance changes little, but the secondary lobes move into the image plane and increase the overall FWHM. For the 10-row slab with an object distance 1.5d, the intensity distribution is shown in Fig. 10. The side lobes are also evident and their transversal distances to the main lobe are small compared to the 3-row slab. Thus as the object distance increases further, the side lobes will be completely mixed with the main lobe, leading to reduction of the resolution as can be seen in Fig. 7.

 

Fig. 7. The transversal distribution of the electric field intensity at the image locations for different object distances of 10-row slab.

Download Full Size | PDF

 

Fig. 8. Intensity distribution of 3-row slab for an object distance of 0.1d

Download Full Size | PDF

 

Fig. 9. Intensity distribution of 3-row slab for an object distance of 1.6d

Download Full Size | PDF

 

Fig. 10. Intensity distribution of 10-row slab for an object distance of 1.5d

Download Full Size | PDF

Note that those secondary lobes are not the same as the side lobes of the conventional Airy pattern [33]which accompany the image in conventional optics due to diffraction. For PhC imaging, these side lobes are interference peaks arising from the multiple scattering events inside the photonic crystal and represent the leakage of energy from the main image focus.

To obtain a better understanding of the mechanism of the image formation process, we analyze the angular frequency spectrum of the image intensity by performing a Fourier transform. The spectral curves for the 3-row PhC slab are shown in Fig. 11 for different object distances where k is the wavenumber and the largest wavenumber of a propagating wave is k ₀ =2π/λ. It can be seen that there are indeed higher-spatial frequencies present than those corresponding to propagating waves. The waves with larger wave numbers than the propagating waves contribute to the sub-wavelength resolution. In particular, there is a secondary peak, centered around k = 1.44k ₀. This is due to the evanescent waves [8, 24] and the surface resonance mode of the PhC slab [18]. The surface resonance depends on the surface termination, but depends less on the object distance. From Fig. 11 it can be seen that for an object distance of 1.0d, the spectra have only a slight decrease compared to near-field imaging. However, for the saturated images, the spectra drop at higher spatial frequencies corresponding to the propagating waves, partly because of the finite slab aperture. The largest wavenumbers for the propagating waves are 0.94 k ₀ and 0.90 k ₀ for object distances 1.6d and 2.0d. These two wave numbers are the cutoff points if conventional optics is used. Since evanescent waves are induced by scattering and diffraction [8,24] and the surface resonance modes [18], they are also present in the saturated images giving rise the secondary peaks in Fig. 10. Similar behaviors are also found for other slabs with more rows of holes.

We can also investigate the wave amplitude. Although the amplitude at the input surface of the slab decreases as the object distance increases, the corresponding variation in the image amplitude is much less significant. Fig. 12 shows the wave amplitude on the central symmetric axis y=0 for the 3-row slab imaging as a function of axial distance x, where the amplitudes are normalized to the image maximum for an object distance of 0.01d (the solid line) for the 3-row slab. Other slabs have similar behaviors. It can be seen that for a larger range of object distance from 0.01d to 1.6d, the image amplitudes change little compared to the incident amplitudes which are roughly inversely proportional to the square-root of the distance. This once again implies that the PhC slab restores the amplitude of the object to some extent by capturing evanescent waves that would otherwise be lost, rather than simply “amplifying” the incident amplitude. Compared to left-handed materials however, such restoration is far less significant. This is probably because there is scattering and interface reflection losses, and the slab has a larger thickness than the left-handed material and thus the resulting standing waves inside the slab have limited amplitudes. This is not the same as the left-handed materials with n=-1 where theoretically the slab can amplify the amplitude enough to restore the object amplitude to its original value [2, 16]. Similar behaviors are also found for other slabs with more rows of holes.

 

Fig. 11. The angular frequency spectra of the image intensities shown in Fig. 6 for the 3-row slab

Download Full Size | PDF

 

Fig. 12. Longitudinal amplitudes along y=0 for different object distances of the 3-row slab

Download Full Size | PDF

From Figs. 3 and 4 and Figs. 8 and 9 it can be seen that the reflection is strong. To reduce reflection and improve light coupling, additional termination with partial holes [28], complex basis [29] and tapered transition [30] may be used. Another approach is to adopt mode matching structures [34] which behave in a similar way to interference coatings used in free-space optics. With large coupling efficiency, the resolution is also improved, and can reach about 0.2λ [30].

The image location where the intensity reaches its peak depends on the relative contributions of propagating and evanescent waves [18]. In the linear imaging regime discussed in this paper, the contributions of the evanescent waves to the image intensity are not so significant that the image location is shifted from that predicted by Veselago relation. If the evanescent waves are strong enough, Veselago relation will no longer be a valid predictor of the image position. This behavior is also different from that of the left-handed materials. However, in the saturated regime, the evanescent waves contribute more to the image intensity, thus the image is very close to the right surface of PhC slab.

We can also consider possible applications of these structures. While the saturated imaging region may not be used to identify individual objects in an imaging configuration, it could be used in defocused imaging or to increase the depth of field. For example, in optical data storage, the air gap of 20nm~50nm between the optical disk and the objective lens must be precisely maintained. Deformations in the disk cause the object distance to change dynamically. Thus a conventional recording and readout optical system must re-focus mechanically using a servo system. However, in the saturated imaging region for PhC slab, it may not be necessary to refocus. For example, in the case of the 7-row slab, when the object distance changes from 1.0d to 1.5d, corresponding to about 1.4λ, the change of the image distance is less than 0.1λ, and the resolution is almost the same as that of the unrestricted images. For example, for the next generation Blu-ray recording system which has a working wavelength of 405nm, a change in object distance of 50nm would result in a change of a few of nanometers in the position of the saturated image. Therefore a servo system to refocus the optics may not be required. Previous studies have discussed the application of left-handed materials and PhC imaging in optical recording [36, 37] by taking the advantage of its sub-wavelength resolution. Here we propose that sub-wavelength resolution and saturated imaging could also be used in optical recording to ease the strict requirements on the maintenance of the air-gap. In addition, since the PhC slab thickness is on the order of the wavelength, it could also be applied in microscale optical sensing systems for such as molecular and proteinomic analysis.

5. Conclusion

In this paper, we have quantitatively investigated the relation between the object and image distances for sub-wavelength resolution imaging using a hexagonal air-hole dielectric PhC with negative refraction at the second band. In particular, PhC slab imaging has been extended beyond the near-field regime with object distances larger than the slab thickness. It has shown that the relation between the object distance and image distance is almost linear for object distances smaller than the PhC slab thickness even in the case of a thin PhC slab. This relation can be used to extract the values of the effective index for the finite-sized PhC slab and the results are close to those obtained from the plane wave expansion method. When the object distance is larger than the slab thickness, saturated imaging occurs. We have analyzed the saturated imaging regime of the photonic crystal and have shown that with proper design such an imaging method has a large tolerance for the changes in object distance without compromising the sub-wavelength resolution performance and while maintaining an almost constant image distance. The FWHM resolution is found to be almost the same as for near-field imaging up to the point where the secondary lobes enter the image plane. For an object distance up to two times the slab thickness for a 3-row slab, the resolution is about 0.43λ. For the 10-row slab, the resolution is about 0.55λ for object distances up to 1.5 times the slab thickness. The analysis of the image angular frequency spectrum reveals that the evanescent waves are also present in the saturated imaging regime and contribute to the sub-wavelength resolution.