- Email: [email protected]
Negative index photonic crystal lenses based on carbon nanotube arrays
Available online at www.sciencedirect.com
Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505 www.elsevier.com/locate/photonics
Negative index photonic crystal lenses based on carbon nanotube arrays Haider Butt a, Qing Dai a, Timothy D. Wilkinson a,*, Gehan A.J. Amaratunga a,b a
b
Department of Engineering, University of Cambridge, Cambridge CB3 0FA, UK Sri Lanka Institute of Nanotechnology (SLINTEC), Lot 14, Zone A, EPZ, Biyagama, Sri Lanka Received 20 February 2012; received in revised form 26 March 2012; accepted 8 April 2012 Available online 16 April 2012
Abstract We report a novel utilization of periodic arrays of carbon nanotubes in the realization of diffractive photonic crystal lenses. Carbon nanotube arrays with nanoscale dimensions (lattice constant 400 nm and tube radius 50 nm) displayed a negative refractive index in the optical regime where the wavelength is of the order of array spacing. A detailed computational analysis of band gaps and optical transmission through the nanotubes based planar, convex and concave shaped lenses was performed. Due to the negative-index these lenses behaved in an opposite fashion compared to their conventional counter parts. A plano-concave lens was established and numerically tested, displaying ultra-small focal length of 1.5 mm (2.3 l) and a near diffraction-limited spot size of 400 nm (0.61 l). # 2012 Elsevier B.V. All rights reserved. Keywords: Multiwalled carbon nanotube arrays; Photonic crystals; Negative index; Negative refraction; Photonic lenses
1. Introduction Multiwalled carbon nanotubes (MWCNTs) first reported in [1] are very promising materials and have been the focus of enormous research in the past decade. MWCNTs are structurally similar to the concentric arrays of cylindrical tubes made out of single graphite sheets [2]. They are mostly metallic and are able to carry high current densities which paves way towards their extensive utilization in electrical applications, like field emission displays, rectifier electrodes, solar cells and optical antenna arrays [3]. Myriad optical applications of MWCNTs have also been reported exploiting their * Corresponding author. Tel.: +44 1223 748353; fax: +44 1223 748348. E-mail address: [email protected] (T.D. Wilkinson). 1569-4410/$ – see front matter # 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.photonics.2012.04.003
interesting optical properties [4]. Individual MWCNTs display a frequency dependent dielectric function, which is anisotropic in nature and matches very closely with that of bulk graphite [5,6]. However, the highly dense periodic arrays of MWCNTs may display an artificial dielectric function, with a lower effective plasma frequency in a few hundreds of terahertz, acting as metamaterials [7,8]. It was demonstrated by Pendry et al. [9] that the metallic thin wire arrays, when excited by an electric field parallel to the wires, display a low-density plasma like the electromagnetic response with a reduced pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi plasma wavelength l p ¼ a 2plnða=rÞ, where a is the lattice constant of the 2D wire array, r is the radius of the wires. The arrays display an artificial negative dielectric constant for wavelength regime higher than the plasmon wavelength lp, causing reflection. As
500
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
demonstrated in our previous work [7], the CNTs arrays act as plasmonic high pass filters by allowing only the transmission of frequencies modes with wavelengths lower than lp. Below the plasmonic wavelength, where the wavelength is of the order of lattice constant a, diffraction dominates and the arrays display effects like photonic band gaps and negative refractive index which are usually affiliated with photonic crystals. The analysis of photonic band gaps and visible diffraction from MWCNT arrays has been previously reported by Kempa et al. [10,11]. However, in this manuscript we report the utilization of two-dimensional (2D) periodic arrays of MWCNTs for achieving negative index photonic crystal lenses, with ultra small focal lengths and near diffraction-limit spot sizes. To the best of our knowledge we for the first time report the detailed investigation of wave propagation through the CNT arrays and negative index lensing effects. The arrays with nanoscale dimensions (lattice constants of 400 nm) displayed diffraction based negative index lensing effects in the optical regime. Plasmonic filtering effects were observed for higher wavelength. 2. Negative refraction in CNT array based photonic crystals Negative index lenses can be constructed by using negative dielectric materials like thin metallic slabs [12] which acts as a metamaterials or by using photonic crystals which present a periodicity in their dielectric constant [13]. In the case of photonic crystals, the size and periodicity of the scattering elements (nanotubes) are on the order of wavelength of incident light, causing Bragg diffraction. The diffractive phenomena in photonic crystals can lead to the excitation of waves for which phase and group velocities are reversed in the same manner as in negative index metamaterials. Thus, under the right conditions, negative refraction can be observed in photonic crystals [14]. While following the same principle we report computational analysis of negative diffraction effects in periodic carbon nanotube arrays. We first performed band structure calculations by using plane wave expansion (PWE) method [15], in order to find out the potential frequency ranges where the CNT arrays exhibit a negative refractive index. The analysis was performed for transverse electric (TE) mode of light polarized parallel to the CNTs. Due to the computational constraints a dispersionless model was used without the absorption characteristics of the nanotubes. Only the real part Calculated band structure results for an array with lattice constants of 400 nm and tube radius
Fig. 1. Calculated TE band structure for a square lattice carbon nanotube array with a = 400 nm and r = 50 nm.
of 50 nm are presented in Fig. 1. A negative band slope was observed at two frequency ranges around 420 nm (a/l 0.95) in band 3 and 670 nm (a/l 0.6) in band 2. It has been reported that photonic crystal exhibiting negative band slope demonstrate a backwards electromagnetic wave propagation [16,17]. Therefore, an effective negative refractive index can be defined at these frequencies, meaning negative refraction could occur in through the carbon nanotube array. To further study the negative refraction effects, we performed a finite element method (FEM) analysis of optical transmission through the carbon nanotube array. For the sake of simplicity we considered a 2D geometry of square lattice CNT array (Fig. 2) and the electromagnetic field was assumed to be invariant along the axis of the nanotubes (z axis). The dielectric constant of the multiwalled carbon nanotubes was obtained using the Drude–Lorentz model reported in [4] and was incorporated into the model as a frequency dependent function. The calculations were performed for the light polarized parallel to the CNTs (TE) and ranging from 400 nm to 1000 nm. First we simulated the propagation of an oblique incident plane wave (with a 308 angle of incidence) across a CNT array, as shown in Fig. 2(a). Negative refraction was observed across the array near the wavelengths of 470 nm and 675 nm. Transmission spectrum calculation across the CNT array also revealed the highest transmission intensity near these wavelength regimes (Fig. 2(b)). The simulation was also repeated for a point source of light, as shown in Fig. 2(c). The CNT array exhibited a negative effective index and reconstructed the image of the point source on the opposite side. The array acts as a negative index planer lens [16]. The transmission spectrum was calculated near the image of the point
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
501
Fig. 2. (a) Simulated TE propagation of a 30 degrees oblique incident plane wave across a CNT array and the (b) transmission spectrum calculated at the marked cross. (c) Transmission of a point source placed on the left of the array. (d) The transmission spectrum of the point source.
source, as shown in Fig. 2(d). High transmission intensities were observed at the negative index frequency regimes predicted by the PWE method. The highest intensity was observed at 675 nm. The results from both simulations were in good agreement with the PWE band structure calculations. The simulation results show that the CNT array exhibit the negative refection in the optical regime. The operating frequencies can be tailored by changing the array dimensions and geometry. 3. CNT based negative index convex and concave lenses Next we designed and modeled the curved surface lenses based on arrays of multiwalled carbon nanotubes. The refractive transmission of light across the nanotube array based lenses was computed. Both the convex and
concave shaped CNT lenses were modeled and strong negative index responses were observed. The microlenses acted in a reverse fashion compared to their glass based counterparts. As shown in Fig. 3 the convex shaped lens based on high density CNTs acted as a diverging lens towards the incident plane waves. The diverging nature was wavelength dependent and confirmed the negative index behavior. The computed transmission spectrum and wave propagation across the lens showed that the frequencies belonging to the negative index range were diverted to different angels. Most parts of the convex lens was composed of the regular square lattice of CNTs with 400 nm lattice constant (a). Therefore, the lens also offered a plasmonic band gap towards larger wavelength ranges, as observed from the spectrum in Fig. 3(b). This infinitely extending plasmonic band gap (starting from 750 nm), usually found in metals, is also reported in thin
502
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
Fig. 3. (a) Modeled geometry of a CNT based convex lens (with a = 400 nm). (b) Calculated transmission spectrum though the lens. The insets show the wave propagation of 470, 540, 700 and 1100 nm waves through the lens. The transmission spectrum was calculated at the crosses shown in the insets.
Fig. 4. (a) Modeled geometry of a CNT based concave lens. (b–e) The wave propagation of 480, 585, 620, and 670 nm waves across the lens. (f) Calculated transmission spectrum though the lens.
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
metal wire based metamaterials [7,9]. All the electromagnetic waves with wavelengths larger than the plasma wavelength are reflected by the structure and it acts like a high pass filter. As shown is Fig. 3(b), the 1100 nm wave is completely reflected from the lens. Some closer CNT spacings were also used near the edges of the convex lens. This might have introduced some further negative index regimes and lead to the additional transmission peaks observed, such as near 540 nm. The FEM analysis of a concave shaped CNT microlens was also performed. Due to the negative index it behaved like a converging lens and demonstrated a strong wavelength dependant focusing. The transmission spectrum in Fig. 4 shows, the lens demonstrated a negative index response towards light ranging near 480 nm, 585 nm and 620 nm. The highest intensity spot observed at 620 nm. The calculated lens focal length at 620 nm wavelength was around 1500 nm. The lens structure being much thinner than the convex lens, lacked periodicity and did not display any band gaps toward the incident light with larger wavelength. The ultra small focal length and strong negative index response displayed by the concave lens is of interest here as it can be effectively utilized to overcome the diffraction limit for observing objects placed at subwavelength separations.
503
we performed a numerical experiment. The performance of the lens was checked, using two point sources placed at its focal point with a subwavelength separation of just 50 nm (l/13). Rather than using the analyzed concave lens we established a planoconcave lens by transacting half of the nanotubes from it. Negative-index plano-concave lenses perform better than concave lenses as they possess a larger numerical aperture, display less spherical aberration and have shorter focal lengths [18]. Our analysis showed that the plano-concave lens performed strong optical focusing as show in Fig. 5. Highest intensity focus spots were observed for 540 and 660 nm wavelengths. As shown by the x-directed transmission intensity profiles in Fig. 3(d), the lens displayed ultra small focal length of 1.5 mm (2.3 l) for the wavelength of 660 nm. The y-directed intensity profile at the focal plane illustrates that the focused beam has a near diffraction-limited spot size of 400 nm (0.61 l) at full width at half maximum. The focal length and spot size displayed by the lens were smaller compared to the previously reported photonic crystal lenses exhibiting focal lengths of 12 mm (8 l) and a diffraction-limited spot size of 1.05 mm (0.68 l) [18]. 5. Sub-diffraction limited resolution
4. Subwavelength focusing using a CNT planoconcave lens In order to check our hypothesis regarding the subwavelength focusing of CNT based concave lenses
To test the resolving power of the lens, two point sources with 50 nm (l/13) spacing were placed at the focal plane (1.5 mm from the lens) as shown in Fig. 6. The wave propagation and transmission intensity
Fig. 5. Modeled geometry of a plano-concave lens. (b) Calculated transmission spectrum though the lens, with highest focusing at 540 and 660 nm. The insets show the wave propagation of 540 and 660 nm waves. (c) X-directed transmission intensity profiles across the model showing the focal length. (d) Y-directed transmission intensity profiles showing the widths of the subwavelength focus spots.
504
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
Fig. 6. (a) FEM model of two point sources (with 50 nm separation). (b) The model geometry with the point sources placed at the focal length of the plano-concave lens. TE propagation of a 660 nm wave emitted by the point sources was simulated (c) through free space and (d) across the CNT lens. (e) The near-field intensity profile across the y-directed distance without the CNT lens. (f) The same with the lens in place, two distinct peaks are observed.
profiles were computed at the wavelength of 660 nm, both with and without the negative index plano-concave lens. As shown in Fig. 6(c) and (e), in the absence of the CNT lens the near field produced by the two point sources corresponded to that of a single point source.
However, with the lens in place two distinct peaks are observed in the near field clearly distinguishing the two point sources (Fig. 6(d)). The reported plano-concave microlens clearly demonstrates negative-index behavior and resolves two subwavelength objects, over-coming
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
the diffraction-limit. While operating in the optical regime the lens displays a superior ultra-small focal length and diffraction-limited spot size. With the high level of design flexibility of CNT arrays, the lens geometry can be further optimized and easily scaled to operate in any frequency region. Such lenses could find application in high resolution microscopes, telescopes and imaging systems. 6. Conclusion In conclusion we have demonstrated photonic crystal lenses based on arrays of metallic multiwalled carbon nanotubes, which display negative refractive index in the optical regime. Due to negative index, both the convex and concave lenses behave in opposite fashion compared to their glass based counterparts. The modelled plano-concave lens displayed ultra-small focal lengths of the order 1.5 mm (2.3 l) for the wavelength of 660 nm and a near diffraction-limit spot size of 400 nm (0.61 l). Acknowledgment This work was partly funded under the NokiaCambridge Strategic Partnership in Nanoscience and Nanotechnology (Energy Programme). The authors also thank Ranjith R., Jeremy Baumberg, and Petros Farah for the fruitful discussions. References [1] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (1991) 56–58. [2] R.H. Baughman, A.A. Zakhidov, W.A. de Heer, Carbon nanotubes – the route toward applications, Science 297 (2002) 787– 792, August 2. [3] K. Jensen, J. Weldon, H. Garcia, A. Zettl, Nanotube radio, Nano Letters 7 (2007) 3508–3511. [4] E. Lidorikis, A.C. Ferrari, Photonics with multiwall carbon nanotube arrays, ACS Nano 3 (2009) 1238–1248.
505
[5] M.F. Lin, F.L. Shyu, R.B. Chen, Optical properties of wellaligned multiwalled carbon nanotube bundles, Physical Review B 61 (2000) 14114. [6] B.D. Aleksandra, E.H. Li, Optical properties of graphite, Journal of Applied Physics 85 (1999) 7404–7410. [7] H. Butt, Q. Dai, P. Farah, T. Butler, T.D. Wilkinson, J.J. Baumberg, G.A.J. Amaratunga, Metamaterial high pass filter based on periodic wire arrays of multiwalled carbon nanotubes, Applied Physics Letters 97 (2010) 163102–163103. [8] H. Butt, Q. Dai, R. Rajesekharan, T.D. Wilkinson, G.A.J. Amaratunga, Plasmonic band gaps and waveguide effects in carbon nanotube arrays based metamaterials, ACS Nano 5 (2011) 9138–9143. [9] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, Low frequency plasmons in thin-wire structures, Journal of Physics: Condensed Matter 10 (1998) 4785–4809. [10] K. Kempa, B. Kimball, J. Rybczynski, Z.P. Huang, P.F. Wu, D. Steeves, M. Sennett, M. Giersig, D.V.G.L.N. Rao, D.L. Carnahan, D.Z. Wang, J.Y. Lao, W.Z. Li, Z.F. Ren, Photonic crystals based on periodic arrays of aligned carbon nanotubes, Nano Letters 3 (2002) 13–18. [11] J. Rybczynski, K. Kempa, Y. Wang, Z.F. Ren, J.B. Carlson, B.R. Kimball, G. Benham, Visible light diffraction studies on periodically aligned arrays of carbon nanotubes: experimental and theoretical comparison, Applied Physics Letters 88 (2006) 203122. [12] J.B. Pendry, Negative refraction makes a perfect lens, Physical Review Letters 85 (2000) 3966. [13] T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, B. Gralak, Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability, Physical Review Letters 97 (2006) 073905. [14] D.R. Smith, J.B. Pendry, M.C.K. Wiltshire, Metamaterials and negative refractive index, Science 305 (2004) 788–792, August 6. [15] I.A. Sukhoivanov, I.V. Guryev, Photonic Crystals – physics and practical modelling, vol. 152, Springer, 2009. [16] S. Haxha, F. AbdelMalek, A novel design of photonic crystal lens based on negative refractive index, Progress in Electromagnetics Research Symposium 4 (2008) 296–300. [17] S. Foteinopoulou, C.M. Soukoulis, Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects, Physical Review B 72 (2005) 165112. [18] B.D.F. Casse, W.T. Lu, Y.J. Huang, S. Sridhar, Nano-optical microlens with ultrashort focal length using negative refraction, Applied Physics Letters 93 (2008) 053111–53113.
Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505 www.elsevier.com/locate/photonics
Negative index photonic crystal lenses based on carbon nanotube arrays Haider Butt a, Qing Dai a, Timothy D. Wilkinson a,*, Gehan A.J. Amaratunga a,b a
b
Department of Engineering, University of Cambridge, Cambridge CB3 0FA, UK Sri Lanka Institute of Nanotechnology (SLINTEC), Lot 14, Zone A, EPZ, Biyagama, Sri Lanka Received 20 February 2012; received in revised form 26 March 2012; accepted 8 April 2012 Available online 16 April 2012
Abstract We report a novel utilization of periodic arrays of carbon nanotubes in the realization of diffractive photonic crystal lenses. Carbon nanotube arrays with nanoscale dimensions (lattice constant 400 nm and tube radius 50 nm) displayed a negative refractive index in the optical regime where the wavelength is of the order of array spacing. A detailed computational analysis of band gaps and optical transmission through the nanotubes based planar, convex and concave shaped lenses was performed. Due to the negative-index these lenses behaved in an opposite fashion compared to their conventional counter parts. A plano-concave lens was established and numerically tested, displaying ultra-small focal length of 1.5 mm (2.3 l) and a near diffraction-limited spot size of 400 nm (0.61 l). # 2012 Elsevier B.V. All rights reserved. Keywords: Multiwalled carbon nanotube arrays; Photonic crystals; Negative index; Negative refraction; Photonic lenses
1. Introduction Multiwalled carbon nanotubes (MWCNTs) first reported in [1] are very promising materials and have been the focus of enormous research in the past decade. MWCNTs are structurally similar to the concentric arrays of cylindrical tubes made out of single graphite sheets [2]. They are mostly metallic and are able to carry high current densities which paves way towards their extensive utilization in electrical applications, like field emission displays, rectifier electrodes, solar cells and optical antenna arrays [3]. Myriad optical applications of MWCNTs have also been reported exploiting their * Corresponding author. Tel.: +44 1223 748353; fax: +44 1223 748348. E-mail address: [email protected] (T.D. Wilkinson). 1569-4410/$ – see front matter # 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.photonics.2012.04.003
interesting optical properties [4]. Individual MWCNTs display a frequency dependent dielectric function, which is anisotropic in nature and matches very closely with that of bulk graphite [5,6]. However, the highly dense periodic arrays of MWCNTs may display an artificial dielectric function, with a lower effective plasma frequency in a few hundreds of terahertz, acting as metamaterials [7,8]. It was demonstrated by Pendry et al. [9] that the metallic thin wire arrays, when excited by an electric field parallel to the wires, display a low-density plasma like the electromagnetic response with a reduced pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi plasma wavelength l p ¼ a 2plnða=rÞ, where a is the lattice constant of the 2D wire array, r is the radius of the wires. The arrays display an artificial negative dielectric constant for wavelength regime higher than the plasmon wavelength lp, causing reflection. As
500
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
demonstrated in our previous work [7], the CNTs arrays act as plasmonic high pass filters by allowing only the transmission of frequencies modes with wavelengths lower than lp. Below the plasmonic wavelength, where the wavelength is of the order of lattice constant a, diffraction dominates and the arrays display effects like photonic band gaps and negative refractive index which are usually affiliated with photonic crystals. The analysis of photonic band gaps and visible diffraction from MWCNT arrays has been previously reported by Kempa et al. [10,11]. However, in this manuscript we report the utilization of two-dimensional (2D) periodic arrays of MWCNTs for achieving negative index photonic crystal lenses, with ultra small focal lengths and near diffraction-limit spot sizes. To the best of our knowledge we for the first time report the detailed investigation of wave propagation through the CNT arrays and negative index lensing effects. The arrays with nanoscale dimensions (lattice constants of 400 nm) displayed diffraction based negative index lensing effects in the optical regime. Plasmonic filtering effects were observed for higher wavelength. 2. Negative refraction in CNT array based photonic crystals Negative index lenses can be constructed by using negative dielectric materials like thin metallic slabs [12] which acts as a metamaterials or by using photonic crystals which present a periodicity in their dielectric constant [13]. In the case of photonic crystals, the size and periodicity of the scattering elements (nanotubes) are on the order of wavelength of incident light, causing Bragg diffraction. The diffractive phenomena in photonic crystals can lead to the excitation of waves for which phase and group velocities are reversed in the same manner as in negative index metamaterials. Thus, under the right conditions, negative refraction can be observed in photonic crystals [14]. While following the same principle we report computational analysis of negative diffraction effects in periodic carbon nanotube arrays. We first performed band structure calculations by using plane wave expansion (PWE) method [15], in order to find out the potential frequency ranges where the CNT arrays exhibit a negative refractive index. The analysis was performed for transverse electric (TE) mode of light polarized parallel to the CNTs. Due to the computational constraints a dispersionless model was used without the absorption characteristics of the nanotubes. Only the real part Calculated band structure results for an array with lattice constants of 400 nm and tube radius
Fig. 1. Calculated TE band structure for a square lattice carbon nanotube array with a = 400 nm and r = 50 nm.
of 50 nm are presented in Fig. 1. A negative band slope was observed at two frequency ranges around 420 nm (a/l 0.95) in band 3 and 670 nm (a/l 0.6) in band 2. It has been reported that photonic crystal exhibiting negative band slope demonstrate a backwards electromagnetic wave propagation [16,17]. Therefore, an effective negative refractive index can be defined at these frequencies, meaning negative refraction could occur in through the carbon nanotube array. To further study the negative refraction effects, we performed a finite element method (FEM) analysis of optical transmission through the carbon nanotube array. For the sake of simplicity we considered a 2D geometry of square lattice CNT array (Fig. 2) and the electromagnetic field was assumed to be invariant along the axis of the nanotubes (z axis). The dielectric constant of the multiwalled carbon nanotubes was obtained using the Drude–Lorentz model reported in [4] and was incorporated into the model as a frequency dependent function. The calculations were performed for the light polarized parallel to the CNTs (TE) and ranging from 400 nm to 1000 nm. First we simulated the propagation of an oblique incident plane wave (with a 308 angle of incidence) across a CNT array, as shown in Fig. 2(a). Negative refraction was observed across the array near the wavelengths of 470 nm and 675 nm. Transmission spectrum calculation across the CNT array also revealed the highest transmission intensity near these wavelength regimes (Fig. 2(b)). The simulation was also repeated for a point source of light, as shown in Fig. 2(c). The CNT array exhibited a negative effective index and reconstructed the image of the point source on the opposite side. The array acts as a negative index planer lens [16]. The transmission spectrum was calculated near the image of the point
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
501
Fig. 2. (a) Simulated TE propagation of a 30 degrees oblique incident plane wave across a CNT array and the (b) transmission spectrum calculated at the marked cross. (c) Transmission of a point source placed on the left of the array. (d) The transmission spectrum of the point source.
source, as shown in Fig. 2(d). High transmission intensities were observed at the negative index frequency regimes predicted by the PWE method. The highest intensity was observed at 675 nm. The results from both simulations were in good agreement with the PWE band structure calculations. The simulation results show that the CNT array exhibit the negative refection in the optical regime. The operating frequencies can be tailored by changing the array dimensions and geometry. 3. CNT based negative index convex and concave lenses Next we designed and modeled the curved surface lenses based on arrays of multiwalled carbon nanotubes. The refractive transmission of light across the nanotube array based lenses was computed. Both the convex and
concave shaped CNT lenses were modeled and strong negative index responses were observed. The microlenses acted in a reverse fashion compared to their glass based counterparts. As shown in Fig. 3 the convex shaped lens based on high density CNTs acted as a diverging lens towards the incident plane waves. The diverging nature was wavelength dependent and confirmed the negative index behavior. The computed transmission spectrum and wave propagation across the lens showed that the frequencies belonging to the negative index range were diverted to different angels. Most parts of the convex lens was composed of the regular square lattice of CNTs with 400 nm lattice constant (a). Therefore, the lens also offered a plasmonic band gap towards larger wavelength ranges, as observed from the spectrum in Fig. 3(b). This infinitely extending plasmonic band gap (starting from 750 nm), usually found in metals, is also reported in thin
502
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
Fig. 3. (a) Modeled geometry of a CNT based convex lens (with a = 400 nm). (b) Calculated transmission spectrum though the lens. The insets show the wave propagation of 470, 540, 700 and 1100 nm waves through the lens. The transmission spectrum was calculated at the crosses shown in the insets.
Fig. 4. (a) Modeled geometry of a CNT based concave lens. (b–e) The wave propagation of 480, 585, 620, and 670 nm waves across the lens. (f) Calculated transmission spectrum though the lens.
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
metal wire based metamaterials [7,9]. All the electromagnetic waves with wavelengths larger than the plasma wavelength are reflected by the structure and it acts like a high pass filter. As shown is Fig. 3(b), the 1100 nm wave is completely reflected from the lens. Some closer CNT spacings were also used near the edges of the convex lens. This might have introduced some further negative index regimes and lead to the additional transmission peaks observed, such as near 540 nm. The FEM analysis of a concave shaped CNT microlens was also performed. Due to the negative index it behaved like a converging lens and demonstrated a strong wavelength dependant focusing. The transmission spectrum in Fig. 4 shows, the lens demonstrated a negative index response towards light ranging near 480 nm, 585 nm and 620 nm. The highest intensity spot observed at 620 nm. The calculated lens focal length at 620 nm wavelength was around 1500 nm. The lens structure being much thinner than the convex lens, lacked periodicity and did not display any band gaps toward the incident light with larger wavelength. The ultra small focal length and strong negative index response displayed by the concave lens is of interest here as it can be effectively utilized to overcome the diffraction limit for observing objects placed at subwavelength separations.
503
we performed a numerical experiment. The performance of the lens was checked, using two point sources placed at its focal point with a subwavelength separation of just 50 nm (l/13). Rather than using the analyzed concave lens we established a planoconcave lens by transacting half of the nanotubes from it. Negative-index plano-concave lenses perform better than concave lenses as they possess a larger numerical aperture, display less spherical aberration and have shorter focal lengths [18]. Our analysis showed that the plano-concave lens performed strong optical focusing as show in Fig. 5. Highest intensity focus spots were observed for 540 and 660 nm wavelengths. As shown by the x-directed transmission intensity profiles in Fig. 3(d), the lens displayed ultra small focal length of 1.5 mm (2.3 l) for the wavelength of 660 nm. The y-directed intensity profile at the focal plane illustrates that the focused beam has a near diffraction-limited spot size of 400 nm (0.61 l) at full width at half maximum. The focal length and spot size displayed by the lens were smaller compared to the previously reported photonic crystal lenses exhibiting focal lengths of 12 mm (8 l) and a diffraction-limited spot size of 1.05 mm (0.68 l) [18]. 5. Sub-diffraction limited resolution
4. Subwavelength focusing using a CNT planoconcave lens In order to check our hypothesis regarding the subwavelength focusing of CNT based concave lenses
To test the resolving power of the lens, two point sources with 50 nm (l/13) spacing were placed at the focal plane (1.5 mm from the lens) as shown in Fig. 6. The wave propagation and transmission intensity
Fig. 5. Modeled geometry of a plano-concave lens. (b) Calculated transmission spectrum though the lens, with highest focusing at 540 and 660 nm. The insets show the wave propagation of 540 and 660 nm waves. (c) X-directed transmission intensity profiles across the model showing the focal length. (d) Y-directed transmission intensity profiles showing the widths of the subwavelength focus spots.
504
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
Fig. 6. (a) FEM model of two point sources (with 50 nm separation). (b) The model geometry with the point sources placed at the focal length of the plano-concave lens. TE propagation of a 660 nm wave emitted by the point sources was simulated (c) through free space and (d) across the CNT lens. (e) The near-field intensity profile across the y-directed distance without the CNT lens. (f) The same with the lens in place, two distinct peaks are observed.
profiles were computed at the wavelength of 660 nm, both with and without the negative index plano-concave lens. As shown in Fig. 6(c) and (e), in the absence of the CNT lens the near field produced by the two point sources corresponded to that of a single point source.
However, with the lens in place two distinct peaks are observed in the near field clearly distinguishing the two point sources (Fig. 6(d)). The reported plano-concave microlens clearly demonstrates negative-index behavior and resolves two subwavelength objects, over-coming
H. Butt et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 499–505
the diffraction-limit. While operating in the optical regime the lens displays a superior ultra-small focal length and diffraction-limited spot size. With the high level of design flexibility of CNT arrays, the lens geometry can be further optimized and easily scaled to operate in any frequency region. Such lenses could find application in high resolution microscopes, telescopes and imaging systems. 6. Conclusion In conclusion we have demonstrated photonic crystal lenses based on arrays of metallic multiwalled carbon nanotubes, which display negative refractive index in the optical regime. Due to negative index, both the convex and concave lenses behave in opposite fashion compared to their glass based counterparts. The modelled plano-concave lens displayed ultra-small focal lengths of the order 1.5 mm (2.3 l) for the wavelength of 660 nm and a near diffraction-limit spot size of 400 nm (0.61 l). Acknowledgment This work was partly funded under the NokiaCambridge Strategic Partnership in Nanoscience and Nanotechnology (Energy Programme). The authors also thank Ranjith R., Jeremy Baumberg, and Petros Farah for the fruitful discussions. References [1] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (1991) 56–58. [2] R.H. Baughman, A.A. Zakhidov, W.A. de Heer, Carbon nanotubes – the route toward applications, Science 297 (2002) 787– 792, August 2. [3] K. Jensen, J. Weldon, H. Garcia, A. Zettl, Nanotube radio, Nano Letters 7 (2007) 3508–3511. [4] E. Lidorikis, A.C. Ferrari, Photonics with multiwall carbon nanotube arrays, ACS Nano 3 (2009) 1238–1248.
505
[5] M.F. Lin, F.L. Shyu, R.B. Chen, Optical properties of wellaligned multiwalled carbon nanotube bundles, Physical Review B 61 (2000) 14114. [6] B.D. Aleksandra, E.H. Li, Optical properties of graphite, Journal of Applied Physics 85 (1999) 7404–7410. [7] H. Butt, Q. Dai, P. Farah, T. Butler, T.D. Wilkinson, J.J. Baumberg, G.A.J. Amaratunga, Metamaterial high pass filter based on periodic wire arrays of multiwalled carbon nanotubes, Applied Physics Letters 97 (2010) 163102–163103. [8] H. Butt, Q. Dai, R. Rajesekharan, T.D. Wilkinson, G.A.J. Amaratunga, Plasmonic band gaps and waveguide effects in carbon nanotube arrays based metamaterials, ACS Nano 5 (2011) 9138–9143. [9] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, Low frequency plasmons in thin-wire structures, Journal of Physics: Condensed Matter 10 (1998) 4785–4809. [10] K. Kempa, B. Kimball, J. Rybczynski, Z.P. Huang, P.F. Wu, D. Steeves, M. Sennett, M. Giersig, D.V.G.L.N. Rao, D.L. Carnahan, D.Z. Wang, J.Y. Lao, W.Z. Li, Z.F. Ren, Photonic crystals based on periodic arrays of aligned carbon nanotubes, Nano Letters 3 (2002) 13–18. [11] J. Rybczynski, K. Kempa, Y. Wang, Z.F. Ren, J.B. Carlson, B.R. Kimball, G. Benham, Visible light diffraction studies on periodically aligned arrays of carbon nanotubes: experimental and theoretical comparison, Applied Physics Letters 88 (2006) 203122. [12] J.B. Pendry, Negative refraction makes a perfect lens, Physical Review Letters 85 (2000) 3966. [13] T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, B. Gralak, Photonic crystal lens: from negative refraction and negative index to negative permittivity and permeability, Physical Review Letters 97 (2006) 073905. [14] D.R. Smith, J.B. Pendry, M.C.K. Wiltshire, Metamaterials and negative refractive index, Science 305 (2004) 788–792, August 6. [15] I.A. Sukhoivanov, I.V. Guryev, Photonic Crystals – physics and practical modelling, vol. 152, Springer, 2009. [16] S. Haxha, F. AbdelMalek, A novel design of photonic crystal lens based on negative refractive index, Progress in Electromagnetics Research Symposium 4 (2008) 296–300. [17] S. Foteinopoulou, C.M. Soukoulis, Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects, Physical Review B 72 (2005) 165112. [18] B.D.F. Casse, W.T. Lu, Y.J. Huang, S. Sridhar, Nano-optical microlens with ultrashort focal length using negative refraction, Applied Physics Letters 93 (2008) 053111–53113.