## Abstract

Numerical studies on a graded negative index lens made by a slab of graded photonic crystal (PC) are reported. The graded negative index lens is capable of focusing plane waves and can also be made highly insensitive to frequency, enabling broadband negative index imaging. We provide a simple model for the graded PC lens and predict its superior focusing properties such as low chromatic aberrations and broadband operation. Those properties were also confirmed and analyzed by the finite-difference time-domain simulations. We believe the negative index graded PCs will expand the utility of PC lenses and enable new applications in optoelectronic systems.

© 2008 Optical Society of America

## 1. Introduction

Negative index metamaterial attracted great interest and quickly became the subject of extensive worldwide research thanks to the many novel optical phenomena it can enable [1]. One of the most exciting applications of negative index metamaterials is the possibility of imaging with sub-wavelength resolution, which is often called superlensing [2,3]. Negative refraction and negative index imaging are also possible in photonic crystals (PCs) which are periodic dielectric structures. A PC can behave like a material with a negative refractive index within some spectral regions due to its photonic band structure. This feature was exploited for designing and fabricating PC superlenses [4–7]. Although sub-wavelength imaging is of great scientific and technological importance, the negative index lens, whether made of metamaterials or PCs, does not possess a focal length and cannot provide image magnification. It can image a point-like source in the near field but not a distant object generating parallel rays of incident light. To overcome these deficiencies, negative index lenses with curved surfaces such as the planoconcave lenses made of PCs and metamaterials composed of split-ring resonators have been proposed and ultrashort focal length was demonstrated [8,9]. Useful imaging functionalities such as image inversion and magnification can also be achieved by prism structures with flat surfaces, made of negative index metamaterials or PCs [10].

Alternatively, a slab of optical material with graded index profile can be used to focus plane waves and provide imaging with magnification. Recently, flat lenses made by graded negative index metamaterials and their focusing properties have been studied, giving rise to new prospects for developing negative index imaging systems [11–13]. In this paper, we present graded negative index lenses implemented by negative index PC structures. It can be created by gradually modifying the lattice periodicity or the filling factor of a uniform PC structure. Graded PCs have been used to enhance the control of light propagation and produce mirage and superbending effect [14,15]. There were also studies on the collimation and focusing properties of graded PCs as well as their possible applications as a coupler for PC waveguides [16–18]. We examined a graded PC with a graded negative effective index and demonstrated its focusing of plane waves over a broad frequency range. Moreover, the back focal length of the lens is insensitive to the frequency despite the strong dispersion of the PCs. This observation is in sharp contrast to a uniform negative index PC lens which exhibits severe chromatic aberrations [19]. Therefore, the negative index graded PC provides a way to design a negative index imaging system with broad bandwidth and may find applications in future displays and optoelectronic systems [20].

## 2. Negative index graded PC lens

In order for a thin slab of graded negative index material to act as a convex lens and focus plane waves, its refractive index profile in the transverse direction has to vary in such a way that the modulus of the index is smallest at the center and increases towards the edges [21]. This is in contrast to a regular graded index lens with positive refractive index whose index profile has a maximum on the optical axis located at the center. However, the focusing properties such as the focal length of a graded negative index lens are complementary to that of the corresponding graded positive index lens as they are both determined by the Eikonal equation. Therefore, within the paraxial approximation, a thin slab of graded negative index lens with a parabolic index profile can produce a diffraction-limited focal point and the focal length is determined by the index gradient. For a graded index lens with a finite thickness and relatively high numerical aperture (NA), the desired index profile should be able to modify the incident plane wave and generate a wave front close to that of a cylindrical wave at the back surface of the lens. Such a lens can thus produce a diffraction-limited focal point with minimal spherical aberrations.

We construct a planar graded negative index lens with a two dimensional (2D) PC by varying the PC parameters so that its effective refractive index changes along the transverse direction of the lens. A PC composed of triangular lattice of air-holes in a dielectric medium with refractive index of 3.46 was used as it behaves like an effective negative index medium within the spectral region of the second photonic band when excited with transverse magnetic (TM) waves [22]. The group velocity of the light propagation inside the PC, which is perpendicular to the equifrequency surfaces (EFSs) from its definition on *v*̄_{g}=∇_{k⃗}
*ω*(*k*⃗), points towards the center of the EFSs due to the negative slope of the second photonic band, leading to negative refraction at the interface between a positive index medium and the PC [23]. The EFSs become more circular in shape near the top of the second band and thus we can define an effective refractive index with a negative value for the PC. Figure 1(a) shows the photonic band structures of PCs with different air-hole sizes. The solid and dashed curves correspond to PCs in which the radii of air-hole are 0.25*a* and 0.40*a*, respectively, where *a* is the lattice periodicity. At a given frequency, those PCs possess different effective phase indices as the horizontal line at a frequency of 0.26 (c/*a*) intersects their photonic bands at different points. The effective phase index is proportional to the distance from Γ point to the intersecting point, which is equal to the magnitude of the wave vector of the Bloch mode. The distances along the Γ-M and Γ-K direction are approximately the same, indicating an isotropic effective index for the PC. As shown by the solid and dashed arrows in Fig. 1(a), the PC with larger air-holes has a larger wave vector magnitude and thus a larger refractive index modulus.

Therefore, a graded negative index lens can be obtained by increasing the air-hole radius from the center towards the edges along the transverse direction. Figure 1(b) shows the schematic of such a graded index lens implemented with a triangular PC. The optical axis is along the Γ-M direction of the lattice while the transverse direction of the lens is along the Γ- K direction. The distance between nearest neighbor air-holes, *a*, is kept constant throughout the structure. The thickness of the lens along the optical axis is 8.7*a* and there are 25 layers of air-holes along the transverse direction of the lens. The air-hole radius is graded from 0.25*a* on the optical axis to 0.40*a* at the edges. The variation of the air-hole which determines the effective refractive index profile of the lens has to be properly designed in order to achieve the desired focusing properties. We first studied a graded PC lens with a parabolic variation of the air-hole size so that the ratio of air-hole radius to *a* follows a function, *r*(*y*)/*a*=0.25+1.042×10^{-3}·*y*
^{2} where r(y) is the air-hole radius and y is the transverse coordinate in unit of *a*.

When a plane wave is incident on a graded index PC lens, the wave front changes its curvature and becomes convex when exiting the lens. This curved wave front consequently forces the electromagnetic (EM) wave to focus into a focal point at certain distance from the lens. The variation of the wave front is caused by the graded effective index of PC. That is, an off-axis ray will experience a larger phase change than the on-axis ray due to the larger modulus of effective index near the edge of the PC lens. Since the PC has a negative effective index, the phase of the off-axis ray is ahead of the on-axis ray, resulting a convex wave front and a focal point behind the lens. We calculated the photonic band structures for PCs with various air-hole radii and obtained their effective indices. For the parabolic variation of airhole size along the transverse coordinate, we found the spatial index distribution and calculated the phase profile of the EM wave at a plane right after the PC lens by multiplying the effective index with the thickness of the lens. This simplified analysis applies strictly to a thin slab; otherwise the wave front will not be uniform within the lens. Though simplified, this analysis illustrates the mechanism of focusing by the graded PC lens and can be used to guide the PC lens design.

The solid curve in Fig. 2(a) shows the phase profile of waves along the transverse plane after the PC lens. The x axis is the transverse coordinate in unit of *a* and the y axis is the phase profile in unit of 2π. The phase of on-axis light was used as the reference and set to be zero. It has a convex curvature as expected and resembles a parabolic function. The dashed curve is the phase profile of a cylindrical wave obtained on a plane located at distance R away from its origin and it follows the form of *ϕ*2*π*/*λ*
_{0}·(*y*
^{2}+*R*
^{2})^{1/2}. We choose the parameter R=15*a* so that it best fits the phase profile at the back interface of the PC lens. This indicates the back focal length of this graded PC lens is about 15*a* which provides a simple way to estimate and design the focusing properties of this type of graded index lens. The deviations between the actual phase profile and cylindrical wave phase profile show the existence of spherical aberrations.

The phase profile generated by the graded PC lens can be engineered by controlling the variation of the air-holes. By properly designing the graded PC, we can improve the focusing properties of the lens and achieve reduced spherical aberrations. In Fig. 2(b) we show the phase profile after three different graded PC lenses. These PCs all have their radii of the air-holes varying between 0.25*a* and 0.40*a* but follow different functions. The parabolic function is shown again in the figure by the solid red curve. The purple curve represents a graded PC with a linearly graded air-holes, and the blue curve corresponds to a graded PC whose air-hole variation follows the function, *r*(*y*)/*a*=0.25+4.626×10^{-3}·*y*
^{1.4}. The phase profiles shown by the three solid curves have different curvatures, indicative of different back focal lengths for the three graded PCs. The linearly graded PC, which exhibits the largest curvature in its phase profile, has the shortest back focal length while the parabolic graded PC possesses the longest back focal length. Note the blue curve best fits the phase profile of the cylindrical wave shown by the dashed curve. Therefore, this graded PC lens will have the smallest spherical aberrations.

In order to visualize the focusing properties of the graded index PC lens and quantitatively characterize its performance, we used finite-difference time-domain (FDTD) method to study the propagation of EM waves through the lens [24]. The computational domain contained a graded PC slab sandwiched between two air regions and it was terminated with the perfectly matched layer boundary condition in order to avoid unphysical reflections from the computational boundaries [25]. The size of the grid cells are *Δ*
*x*=*Δy*=*a*/30 and the time increments are *Δt*=Δ*x*/2*c*, where *c* is the speed of light in vacuum. We placed a continuous incident wave source with a spatial width of 26*a* in the air region on the left side of the PC slab and recorded the electric field and its intensity distribution after the EM fields reached the steady state. The width of the source was slightly larger than the transverse size of the graded PC so that the lens was fully illuminated. Figure 3(a) shows the focusing effects of three graded PC lenses by plotting the intensity distributions. The variation of the air-holes of these graded PCs corresponds to those shown in Fig. 2(b). As we move from linear to parabolic grading, the focal point moves away from the lenses, as expected from their phase profiles obtained by the simple ray model. The back focal lengths measured by finding the locations with maximum field intensity along the optical axis are 9.7*a*, 10.7*a* and 12.5*a* for the three graded PC lenses. We also measured the intensity distribution at the back focal plane along the transverse coordinate for the three graded PC lenses, shown by Fig. 3(b). They are normalized so that the maximum is 1 on the optical axis. The graded PC lens with air-hole variation corresponding to the function on *r*(*y*)/*a*=0.25+4.626×10^{-3}·*y*
^{1.4} forms a focal point having a transverse spot size of 4.6*a* as shown by the solid blue curve. Its focusing property is superior to that of the linear and parabolic graded PCs, which have focal point spot sizes of 5.0*a* and 5.6*a*, respectively. This is consistent with the analysis based on the phase front because the small focal point can be attributed to the reduced spherical aberrations.

The numerical simulations by FDTD show that the properties of the graded PC lens such as the back focal length do not exactly match the predictions from the ray model using the phase front calculation. This is not surprising as we applied a simplified model and neglected several facts of a real lens system. First, the graded PC lens has a finite thickness and allows the wave front to deform within it. This makes the lens behave differently from the ideal model although the thin lens remains a good approximation for the system. Additionally, PCs with different air-hole sizes have different transmission and reflection properties at their interfaces with air. Therefore, the graded PC lens has a non-uniform transfer function in magnitude along its transverse direction while the ray model assumes no spatial variation in field magnitude. Finally, due to the reflection at both surfaces of the lens, the wave eventually coming out of the lens may have experienced multiple reflections and accumulated phase distortions that are not considered in the simple ray model. However, despite the simplification and approximation in the model we used, it provides a good qualitative description of the graded PC lens and can predict other interesting focusing properties as discussed below.

## 3. Broadband focusing

PCs often possess complex dispersive properties due to their photonic band structures except for long wavelength limit where they behave like a homogeneous dielectric medium. Therefore, the effective refractive index of a PC generally has strong frequency dependence. In the triangular PC, the negative effective index decreases in modulus with increasing frequency. This effect is known to create severe chromatic aberrations for a negative index imaging system made by a uniform negative index PC [19]. This is a natural consequence of the strongly dispersive nature of the negative index PC. However, in a graded PC lens, the grading creates additional chromatic aberrations which may cancel the aberrations caused by the frequency dependence of effective index. To illustrate this point, let us consider uniform PCs with various air-hole sizes. They all exhibit similar dispersion because we are operating in the same second photonic band in all cases. As a result, the profile of the effective refractive index along the graded PC remains the same even as the frequency is varied. Note that the absolute values of the effective index of the individual PCs with various air-hole sizes are still strongly dependent on the operating frequency but the grading of the effective index profile and consequently the resultant transverse phase profile can be made to be insensitive to the frequency. As shown in Fig. 4(a), the phase profile after the graded PC lens with air-hole variation following the function *r*(*y*)/*a*=0.25+4.626×10^{-3}·*y*
^{1.4} exhibits only minor changes as the operating frequency is increased from 0.22 to 0.32 (c/*a*). Therefore, such a graded PC lens will have a small variation in its back focal length over a broad frequency range.

Using FDTD techniques, we studied the focusing properties of this graded PC lens at multiple frequencies. The electric field intensity distributions along the optical axis for different frequencies were obtained simultaneously by using the Fourier transform of a broadband excitation. Figure 4(b) shows the on axis field intensity at frequencies of 0.24, 0.26, 0.28, 0.30 and 0.32 (c/*a*). Note the good focusing by the graded negative index lens, shown by the intensity peaks of different curves in Fig. 4(b), is preserved over a broad bandwidth, about 30% of the center frequency. The transverse size of the focal points shows only a small change of 1.4*a*. The back focal length of the lens increases in response to the increase in frequency. This can be attributed to the slight distortion of the phase profile at different frequencies. The change of the focal length for this graded PC lens is 12.6*a* over a frequency range of 0.08 (c/*a*).

The chromatic aberration of a graded PC lens can be further reduced by carefully adjusting its grading parameters. For instance, in addition to grading the air-hole sizes, we can also gradually vary the spacing between neighboring air-holes along the transverse direction of the lens. Varying the lattice period of a PC will modify its photonic band structure and therefore the effective phase index. Furthermore, the phase profile created at the back surface of a lattice grading PC lens has a reduced curvature when the frequency is increased, which is opposite to the frequency dependence of an air hole grading PC lens as shown in Fig 4(a). As a result, a structure with dual grading on both the air-hole size and lattice periodicity can provide focusing with superior frequency insensitivity. Taking the graded PC lens presented in Fig. 4, we linearly decrease the spacing between air-holes by a step of 0.015*a* per lattice period along its transverse direction starting from the optical axis and examined its focusing properties as a function of frequency. As shown in Fig. 5(a), the electric field intensity distributions by FDTD simulations demonstrate well defined focal points at frequencies of 0.265, 0.275, 0.285 and 0.295 (c/*a*). The location of the focal points hardly changes within this frequency region and the numerically measured back focal length varies less than 1.0*a*. For comparison, Fig. 5(b) shows the focusing of a uniform negative index PC with air-hole radius equal to 0.33*a* and the same thickness as the graded PC lens. It exhibits the focusing effect over a narrower bandwidth (about 10% of the center frequency) than the graded PC lens and has much larger chromatic aberration. The focal length varies by 7.3*a* over the same frequency range which is more than seven times larger than that of the graded PC lens.

In order to quantitatively study the chromatic aberrations of the graded PC lenses, we plotted their back focal lengths as a function of frequency, as shown in Fig. 6. The red solid curve represents a graded PC lens discussed in Fig. 4 and its back focal length forms a moderately increasing function with the frequency within a region between 0.22 and 0.32 (c/*a*). The dashed black line is a linear fit to this curve and the slope of the fitting curve can be used to characterize the chromatic aberration. For this graded PC lens, the slope is 1.3×10^{2} (*a*
^{2}/c). The purple solid curve shows the back focal length of the uniform negative index PC lens described in Fig. 5(b) within a narrower frequency range between 0.26 and 0.295 (c/*a*). The slope of the corresponding curve fit is 2.4×10^{2} (a^{2}/c), indicating much more severe chromatic aberration. The blue solid curve and its linear fitting represent the graded PC lens with dual grading discussed in Fig 5(a). The slope of this graded PC lens is reduced to 0.4×10^{2} (*a*
^{2}/c). This is three times smaller than the original graded PC lens and shows a six-fold improvement compared to the uniform negative index PC lens. In the meantime, the available frequency bandwidth of this graded PC lens is about 40%, four times greater than the uniform negative index PC lens.

## 4. Conclusion

In summary, we studied a graded negative index lens made by a slab of graded PC. By analyzing the photonic band structures of PCs with various air-hole sizes, we created a simple model for a graded PC lens and investigated its focusing properties. Numerical simulations using FDTD methods were also performed to quantitatively analyze the focusing behavior of various graded PC lenses over a range of frequencies. It was found that a graded negative index PC lens can not only add new functionalities such as focusing of parallel incident light, but also reduces the chromatic aberration and improve the bandwidth of the negative index imaging system. The broadband negative index imaging with superior aberration characteristics are expected to significantly expand the utility of PC lens and spawn exciting new applications in, for example, integrated micro-systems such as lab-on-a-chip.

## Acknowledgments

This work was supported in part by the National Science Foundation Grant No. BES-0608934 and by the U.S. Army Research Office under MURI Contract 50432-PH-MUR.

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