1. Introduction

As a special type of metamaterials, the periodic structure and graded refractive index-based photonic crystals (PCs) have attracted numerous researchers to explore due to their possessed optical allowed band and prohibited band. Just like semiconductor materials inherented with periodic barriers, the photonic crystals also exhibit a band gap [1]. The photonic crystals can modulate the transmission of incident light by means of tailoring the periodic structures as well as corresponding optical band. Thus photonic crystals are regard as a special type of semiconductor in optics. Recently, a lot of applications applied on the basis of PCs are reported such as beam splitter [2–4], waveguide bends [5, 6], photonic crystal fibers (PCF) [7], channel-drop filters [8], and photonic crystal LED [9–11]. By appropriately modulating the parameters of lattice constant of the PCs, the radius of air hole, and the filling factor, the graded index PCs can be obtained accordingly [12–14].

The photonic crystals designed with the high index contrast dielectric materials govern some of the remarkable properties of the PCs such as self-collimation [15–17] and superprism [18, 19]. But the imperfect PCs attracted more researchers to study and reveal their new performance and functions. The pure periodic PCs can be structured by introducing spatial perturbations in terms of the point or line type-based defects [20–24]. By the approach of changing scale of substrate or refractive index in some points of the PCs, the point defects can be achieved. Similarly, the line defects can be obtained in the same way. Introducing defects can break the forbidden band of the PCs. Thus it can produce many special properties such as enhancement of the spontaneous emission, transmission, and confinement of beam propagation. The point defects are usually designed to constitute microcavity of the photonic crystal [25]. The line defects can confine the propagation of incident light which propagates along some specific paths. Even 100% transmission efficiency can be derived for the PCs guided with 90° propagation route. Therefore, the line defects can be used for superbending in practical applications [26].

Different shapes of the defects can generate different results. There has been an interest in study of the influence of the shape of air holes on focusing properties. It was shown that the lenses with rectangular air holes present a strong tuning ability than the other shapes of the air holes [27, 28]. It can possess large amplitude and small FWHM. In this paper, we introduce the special defects into the PCs for the purpose of studying the influence of the defects on ultra-fine nanofocusing of the PCs. In literature, the graded index PCs designed with the light beam incidents from the broadside area of the PCs lenses are commonly reported (i.e., the beam incident from left side of the PCs in direction of perpendicular to the air holes). Specially, we change the incident direction of the light here, i.e., the incident direction is parallel to axis of the air holes something like photonic crystal fibers. Then we etch a periodic array of graded-radius circular air holes and introduce single rectangular air hole at center of the PCs. To our surprise, extremely small FWHM as fine as λ/75 can be obtained. The ultra-fine nanofocusing spot can locate at the subsurface and surface of the PC lenses. The rectangle air hole perforated with periodic array of the circular air holes can enable the PC lenses focusing far beyond the diffraction limit. With significant advantage of finely nanofocusing the incident beams, our designed structure may find extensive applications in high resolution fluorescent imaging/analysis, sub-surface nanometrology, and biosensing with ultra-high resolution.

2. Design of PC lenses

We construct a two-dimensional (2D) hexagonal lattice PCs which consist of a dielectric medium with a periodic array of graded radius air holes. The substrate material is silicon (Si). The PCs compose of 6 × 12 air holes array distributed in x- and y-directions respectively. The graded radius of air holes is set ranging from 0.0018 μm to 0.06 μm, and distributed from the centre to outside along the y-direction. The radius of next row of the air holes is two times of the previous holes. The lattice constant is set to be 0.17126λ. A rectangular air hole acting as single defect here is set at the center of the PCs lenses. Schematic diagram of the PC lenses is shown in Figs. 1(a) and 1(b).

 

Fig. 1 (a) Schematic diagram for the graded index PCs lenses. (b) Schematic of the PCs lenses perforated with graded-radius air-holes and single rectangular air-hole. The original coordinate point (0,0,0) is set at the center of the structure.

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It is known that several-dozen nanometers wide slit can be used for confining and guiding light waves, similarly to a waveguide [29]. This effect may be theroretically described as follows.

Let us consider a thin conducting sheet mounted across a rectangular waveguide with dimensions a and b. Consider further, a narrow resonant slit with the width b1 and the length a1 which is cut in this sheet parallel to its wide side. If the following relationship holds [30]:

The authors of [31] shown that efficient light focusing at the diffraction limit with higher transmission can be obtained with microstructures (microslit in screen) much easier to fabricate than nano ones. The motivation of current research was paper by Almeida V. R [29]. In this paper, the authors showed the possibility of wave localization in waveguide with slit.

To verify our design, we employ a commercial professional software (FDTD Solution developed by Lumerical Solution Inc.) to calculate the focusing properties of the PC lenses. The whole structure is illuminated by means of setting a vertically polarized plane wave (E_y) with its propagation direction along the optical z-axis. And the incident wavelength is 1.5 μm. A plane wave incidents from backside of the lenses (parallel to axis of the circular air holes). The boundary condition is set as perfect match layer (PML). In the numerial calculation, we need a mesh size at least of 2-4 grid per width of the rectangular air hole.

3. Results and analyses

As illustrated in Fig. 1(a), dimension of the rectangular air hole is 0.5 μm × 0.02 μm × 0.8 μm in x-, y-, and z-directions respectively. The defect can alter the focusing performance of the PC lenses. The power of focusing behavior can be quantified by means of investigating conversion ratio of the focusing spot size FWHM. Figure 2 shows the intensity profile of the lenses with different structures. And Fig. 3 shows the intensity profile in the x-z plane of the lens with the perforated defect. As can be seen, intensity distribution of the lenses constructedwith the perforated defects has large peak amplitude accompanied with weak sidelobes as well as an extremely sharp and ultra-fine spot size. Amazingly, the corresponding FWHM is approximately 0.02 μm, and approaximately equals to λ/75. The distance from the focal point to surface is only 0.01 μm. It means the focal spot locates at the sub-surface region of the lenses. The FWHM is so small that other similar devices are too difficult to possess such a narrow spot size. For example, the FWHM of the structure given by A. O. Cakmak, “High Efficiency of Graded Index Photonic Crystals as an Input Coupler” is 0.4λ [32]. To our knowledge, it is the first time to find such a tiny nanofocusing spot realized by the PC lenses. Further study reveals that the lenses without the defects do not produce such focusing effect anymore. And the PC lenses perforted with single rectangular air hole possess the ability of finely focusing, but some of them are not strong enough, the focal length is short also. The defect sits at inner of the PCs lenses (the distance is 0.25 μm from the focal point to surface of the lenses). It can be seen that the lenses perforated with a periodic array of circular air holes with graded radius and single rectangular air hole have the ability to break through the diffraction limit with the FWHM much smaller than λ/2. It may attribute to the negative refraction of the PC lenses which are mainly determined by second band. Width of the second band for the case of rectangular air hole is wider than the other shapes in the same condition [27, 28]. This is why the lenses with a rectangular hole can derive such a high amplitude and extremely small FWHM. The influences of width and length of the rectangular air hole on focusing performance are further studied, as shown in Fig. 4 and Fig. 5. As can be seen, the length of rectangular air hole varies from 0.3 μm to 0.8 μm also can produce such an extremely small FWHM. But the PC lenses designed with the length ranging from 0.3 μm and 0.8 μm do not have the desired results. The sidelobes are strong and the peak amplitude is not high enough. With changing of width of the rectangular air hole, FWHM is nearly unvaried, and thus the influence of the width can be ignored here. But the width variation cannot be too large to overlap the rectangular hole. In addition, one more surprising issue is that an ultra-fine and sharp nanofocusing (FWHM = λ/75) appearing at top surface of the PC lenses is observed, as shown in Fig. 6.

 

Fig. 2 Intensity distributions along the transverse direction. The red line is the graded PCs, the blue line is the PCs only have single rectangular air-hole and the black line is the graded PCs with a rectangular air-hole with the ultra-fine focusing occurred at sub-surface of the PC lenses.

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Fig. 3 Intensity distributions in the x-z plane of the structure perforated with defects.

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Fig. 4 Intensity distributions along the transverse direction with the change of width (d) of the rectangular air-hole. The inset figure is zoom in of the central peak intensity of the curve.

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Fig. 5 Intensity distributions along the transverse direction with the change of length (l) of the rectangular air-hole. The inset figure is zoom in of the central peak intensity of the curve.

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Fig. 6 Intensity distributions along the transverse direction for the width of the rectangular air-hole 0.01 μm. The ultra-sharp focusing occurred at top surface of the PCs lenses.

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Our calculation demonstrates that the substrate materials strongly influence upon the focusing performance, as shown in Fig. 7. As can be seen, besides Si, the focusing profermance of the lenses are not ideal for the other materials focusing with low peak intensity profile and strong sidelobes. But the curve of InP and InAs are similar to that of theSi. The refractive index of InP and InAs are close to Si, which can lead to the similar phenomenon accordingly. Our further study reveal that the same ultra-fine focusing phenomenon can be observed as long as the refractive index of the substrate material is set ranging from 3.0 to 3.5. The influence of wavelength on focusing is calculated also, as shown in Fig. 8. It can be seen that this structure is unsuitable for the regime of visible and near ultraviolet. No any focal points can be found and the incident light beam is scattered in the PCs lenses. Theoretically, this construction is applicable to near infrared region as well.

 

Fig. 7 Intensity distributions along the transverse direction with different substrate materials. The inset figure zoom in of the central peak intensity of the curve.

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Fig. 8 Intensity distribution along the propagation direction with different incident wavelength. The position of the lens is y = 0.

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From literature review we know that one of the methods to decrease the optical beam width is the use of localization of SPP on periodical structure [33]. Sharp localization of optical wave may be observed in the medium with anisotropy properties, for example, consisting of dielectric and metallic layers [34] or array of metallic rods in dielectric [35]. In [35], the localization of optical EM was about FWHM = λ/9, and in [34], the sharp focusing was about FWHM = λ/268 in transverse direction. But realization of such structure with periodic layers of metal (ε = −12.9) and dielectric (ε = 13.9) was about 13.2 nm. Apparently, this medium is difficult for practical realization and it is important that local field maximum is inside of medium.

The PCs are widely used for light focusing. For example, in [36] the focal spot was about 0.26λ. And in [37], the one-dimensional calculation of the PCs showed that the focal spot has FWHM = 0.164λ.

4. Conclusion

In conclusion, we designed a novel hexagonal lattice PCs lenses perforated with single rectangular defect. By means of varying the lens parameters and changing the length and width of the rectangular air hole, the influence of the defect on focusing performance is investigated through our numerical calculation. A FDTD algorithm is employed here for the computational calculation. Our calculation results verify the reliability of our design. The graded index PCs lenses can focus the incident plane wave to a narrow area in both subsurface and surface region. The lenses perforated with a rectangular air hole and graded-radius circular air hole array have the ability to genrate an ultra-fine nanofocusing with the beam spot size (FWHM = λ/75) far beyond the diffraction limit. Possessing with such a significant advantage of the focusing performance, the designed PCs lenses may find extensive applications in fluorescent imaging/analysis, sub-surface nanometrology, biosensing with ultra-high resolution, and optical tweezers.