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Optimized focusing properties of photonic crystal slabs
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Optics Communications 281 (2008) 3571–3577 www.elsevier.com/locate/optcom
Optimized focusing properties of photonic crystal slabs Nathalie Fabre, Xavier Melique, Didier Lippens, Olivier Vanbesien * Institut d’Electronique, de Microe´lectronique et de Nanotechnologie (IEMN), UMR CNRS 8520, Universite´ des Sciences et Technologies de Lille, Avenue Poincare´, BP 60069, 59655 Villeneuve d’Ascq Cedex, France Received 2 October 2007; received in revised form 18 February 2008; accepted 19 February 2008
Abstract We propose an optimization procedure for focusing operation in finite two dimensional photonic crystal slabs. The device consists of a triangular lattice air holes etched in a semiconductor matrix at a nanometer scale to operate at 1.55 lm. To reach simultaneously an effective refractive index equal to 1 along with a very high transmission coefficient whatever optical wave incidence, the parameters as the lattice period and/or filling factor are precisely adjusted depending on the slab thickness. The method relies on Fabry–Perot resonances engineering in the air/crystal/air cavity constituting the lens. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Since the tremendous development of metamaterials along with the concept of negative refraction a few years ago, photonic crystals (PC’s) have appeared to be suitable candidates to present such abnormal propagation properties in the optical domain with the targeted application of a ‘‘superlens” [1–5]. Indeed, most of the proposed lefthanded materials proposed so far are partially constituted of metallic inclusions [6–8]. Their ability to preserve reasonable losses for optics remains an open question, even if preliminary results of a negative refraction index using a metallo-dielectric material in the visible have been obtained [9]. In this field, dielectric photonic crystals have shown promise when specific isotropic propagation regimes of the photonic structure, induced by the crystal periodicity, are exploited. More precisely, frequency domains can be identified where group and phase velocities are antiparallel. Such a property is true for all propagation directions being equivalent to the definition of a negative refractive index
*
Corresponding author. Tel.: +33 32 019 7876; fax: +33 32 019 7892. E-mail address: [email protected] (O. Vanbesien).
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.02.038
[10]. Alternative proposals are also based on anisotropy to promote negative and/or ultra refraction [11,12]. The requirements for a ‘‘superlens” were clearly identified: a negative refractive index equal to 1 is needed if the optical signal originates from air (n = 1) and a perfect surface impedance matching is also required (z = 1) to avoid any reflection at input [13]. Moreover, for subwavelength resolution based on the evanescent wave amplification effect, it is necessary that the effective permeability and permittivity of the focusing medium to be both equal to 1. These two conditions arep obviously compatible with ffiffiffiffiffiffiffi the definitions n2 ¼ el and z ¼ l=e. If the first parameter n can be easily identified for dielectric photonic crystals, it appears more tricky to define clearly the other ones compared to metallic microstructures aimed to operate in microwaves or up to the infrared region for which the four parameters can be extracted both from simulation (parameter extraction methods, as Fresnel inversion for example). Under certain conditions also in the lower part of the spectrum, these four parameters can be also recovered accurately from transmission and reflection measurements. One important requisite is the subwavelength patterning of the metamaterial, condition only partially fulfilled for photonic crystals. Indeed, the negative refraction regime is obtained in two dimensional photonic crystals for a semiconductor patterning with a periodicity
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N. Fabre et al. / Optics Communications 281 (2008) 3571–3577
close to k0/3 k0/4 (where k0 is the wavelength in air), leading to complex diffraction and refraction effects at the interface air/material under optical excitation. Nevertheless, as long as propagation is monomode at a given wavelength (the two first bands of the PC), parameter extraction remains meaningful [14–16]. In this context, the main question is: Is a ‘‘flat lens” or a ‘‘superlens” with enhanced performances compared to classical approaches possible using two dimensional photonic crystals for integrated nanophotonics? If the debate is still open for the ‘‘superlens”, we will show in this paper that efficient flat lenses can be designed. To the best of our knowledge, most of published papers up to now have focused on the necessity of an optimized isotropy of the band structure and on the optimization of the interface air/crystal to minimize reflection [17–19]. In the following, we will demonstrate that there is also a complex link between the desired operating wavelength and the lens geometry. Above the exact definition of equivalent parameters for the lens, we show that the lens behaves as a cavity with its own resonance which fixes the focusing properties. The basic photonic crystal and our investigation procedure will be presented in Section 2. This will be applied to a specific six row flat lens will presented in Section 3 and an optimization method will be proposed in Section 4. 2. Numerical procedure Our approach is based on a photonic crystal constituted of air holes etched in a semiconductor heterostructure [20,21]. Calculations are performed throughout the paper using the commercial codes BandSolve (for band structures) and Fullwave (FDTD – lens experiment) by RSoft. For the two dimensional simulation, an effective index of 3.32 is assumed corresponding to an InP/GaInAsP/InP based double heterojunction. The desired operating wavelength is 1.55 lm. To obtain a negative refractive index equal to 1 at this wavelength in second band of the crystal, we use a triangular lattice of air holes to promote isotropy with a periodicity a = 476 nm and a hole diameter d = 350 nm (filling factor: 38%). Fig. 1 gives the corresponding TM band structure (E field parallel to the holes) along with the light line. The second band stands from a/ k = 0.24 to 0.36 (wavelength range 1.25–1.98 lm). As shown in Fig. 1b this second band is almost isotropic (circular iso-frequency plots around C point) up to high wave vector values. Let us remain at this point that working with n = 1 (crossing between the band and the light line) is a necessary condition to obtain focusing of an incident point source with a slab but is not sufficient to obtain the ‘‘superlens”. Indeed, surface impedance matching between air and the crystal is required to minimize reflection at input. To illustrate this, Fig. 2a and b show a virtual prototyping of such a lens fed by an optical tapered waveguide ended by a small (k/10) diffracting hole to mimic a point source. For the left hand side of Fig. 2 the crystal is CM oriented (injection perpendicular to the triangle sides)
Fig. 1. (a) Two dimensional band structure for the triangular lattice with period a = 476 nm and air hole diameter d = 350 nm (filling factor of 0.38) – TM mode (E field parallel to the holes) – The index value of the patterned semiconductor is 3.32; (b) 2D iso-frequency plot for the second band where negative refraction index can be defined (red line or circle labelled 0.307: n = 1 circle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Lens experiment using a triangular photonic crystal slab: CM oriented (left side) and CK oriented (right side).
whereas, CK orientation (injection parallel to the triangle sides) is used for the right side of Fig. 2. Even if the dispersion curve is isotropic for this frequency and an effective
N. Fabre et al. / Optics Communications 281 (2008) 3571–3577
refraction index can be deduced, the lens behavior is strongly dependent on incidence. The focusing is clearly apparent in the first case, but has almost disappeared for the second. Moreover, the signal intensity behind the lens is strongly attenuated since reflection at input is huge. In Fig. 2a, a clear focus is present issued from the lens and optical rays can be superimposed as in classical geometrical optics. The extension along propagation axis of the focus point is very small which indicates a refractive index value very close to 1 in this case. Using Gauss approximation of small angles and the distance ratios between focus points inside and outside the lens leads to an estimated value of 0.95. At the opposite, the width of the focus point behind the lens is larger than the incident electric field pattern revealing the fact that subwavelength resolution is not reached, and consequently that effective permittivity and permeability are not close to 1. Also, reflection is important in input and needs to be optimized. Such behavior has already been observed and studied in the literature [22,23] and will be analyzed more deeply in paragraph three. As a first indication, let us mention that the lens thickness is different in CM and CK directions since an equivalent number of air hole rows has been assumed in both cases but distance two successive rows is different (equal to pffiffiffiffiffiffiffibetween ffi a 3=2 in Fig. 2a and equal to a in Fig. 2b). In terms of cavities, the two lenses possess distinct Fabry–Perot modes. Even if n = 1 in both cases, we face a quasi-matched case in terms of impedance with low reflection and high transmission in Fig. 2a but the impedance mismatch prevails in Fig. 2b with opposite performances. This first all-angle interpretation needs now to be refined. Indeed, at this point, it appears clearly that the lens properties depend strongly on the incidence angle of the optical wave and thus on the first interface. More than this first interface, we will see that the thickness and the number of crystal periods used in the main propagation direction are also of prime importance. To analyze all these effects, the waveguide is suppressed at input and we will use a simple Gaussian pulse excitation. Its width is chosen so that transmission and reflection spectra on both sides of the lens can be calculated by fast Fourier transform over the second pass
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band of the photonic crystal and around 1.55 lm, the targeted wavelength of operation. For the study, the incident angle between the pulse direction of propagation and the normal to the lens will be varied. Finally, special attention will be paid to the position and size of the input and output ports of our FDTD simulation in order to recover the totality of the incident optical signal so that the sum of transmitted power and reflected power is equal to the incident power carried by the pulse. For sake of simplicity, we will focus our attention in the following on a six row lens to evidence the main limitations and performances of such lenses. 3. The six row flat lens Fig. 3 gives the transmission spectra obtained for a six row lens for incidence angles varying from 0° to 30° as a function of 1/k (1.55 lm corresponds to 0.645 lm1). As mentioned in Ref. [14], special care has to be devoted to the interface cut to promote the pffiffiffiffiffiffiffi ffi focusing properties. The row width is defined as a 3=2 with the air hole located at the center. This ensures a perfect symmetry along the direction perpendicular to the triangle sides if an odd number of rows are used. However it has been checked that our optimization procedure is valid whatever the number of rows, odd or even. The fact that air holes are staggered does not affect focusing properties. Under normal incidence (here labeled 0°), five transmission peaks are clearly visible, each very close to unity. In between, the transmission decreases rapidly and can be less than 40% even if the wavelength stands within the second band. This impedance mismatch between air and the lens, responsible for a quite high reflection coefficient is not only induced by the first interface as often claimed but the result of the cavity formed by the crystal embedded with ‘‘air”. The number of peaks is equal to the number of air hole rows minus one within the propagation direction. This effect is very well known for any heterogeneous periodical medium [24,25]. If air hole rows are considered as a specific medium with n1 index and the semiconductor in between as a medium n2 index, this effect corresponds to the Fabry–Perot resonances induced by the n1/n2 alternate layers. It can
Fig. 3. Transmission through a six row flat lens under different incidence angles (0°, 2°, 7°, 15°, 30°).
N. Fabre et al. / Optics Communications 281 (2008) 3571–3577
a Wavelength shift (nm)
be mentioned that the low levels of transmission obtained between peaks evolve very weakly with the number of rows even if the number of peaks increases. Conversely, their position within the band depends of the number of rows. This means that if an operating wavelength is targeted, care has to be taken that the thickness of the slab allows the presence of a maximum in transmission at this wavelength. Here, the structure is designed to give a maximum under normal incidence at 1.55 lm for that given number of rows. This first precaution is not enough to ensure good performances for the lens. Indeed, to image a point source even in near field we face a circular wavefront at the first interface with an extension which is function of the source–lens separation. This means a multi-incidence beam interacting with the lens. Obviously, the matching (unitary transmission) is not preserved for any incidence, as exemplified by the additional spectra reported in Fig. 3. This effect was already present in Fig. 2 for the two extreme incidence angles (0° – CM and 30° – CK). But this effect has to be considered whatever the incidence angle value. As this angle increases, one can note a shift to higher wavelengths of the respective transmission peaks along with a decrease in amplitude. This figure shows clearly the sensitivity to incidence angle is more pronounced as k decreases experiencing deeper the interspace between hole rows. These evolutions are summarized in Fig. 4 where wavelength shifts are reported versus incidence angle. For the higher wavelength, the peak position is insensitive (less than 5 nm) to reasonable (in the sense of geometrical optics Gauss approximation) incidence angles, whereas for the shorter wavelength the peak almost disappears for an incidence of 30°. For the intermediate peaks, the shifts are respectively 13, 32 and 55 nm for peaks 2, 3 and 4 between 0° and 30°. Looking at Fig. 4b, it is clear that peaks 2 and 3 leads to high transmission levels for all incidence angles lower than 30°, whereas peaks 1 and 4 decrease faster as soon as angles higher than 10° are under concern. This appears as a strong restriction for good focusing properties. Moreover, the ripple between the peaks has to be correctly evaluated for a specific operating wavelength. As a matter of example, even if the maximum of transmission remains higher than 80%, one can note that the wavelength associated to the maximum under normal incidence induces a transmission drop to less than 50% at this same wavelength for the incidence of 30°. Different theories exist to optimize and decrease such ripples, notably in ‘‘filtering” theory. In general, it consists in considering the structure under study as a series of cascaded cells and in modifying the characteristic properties of each cell (but preserving lens symmetry about its middle in the propagation direction) to reach flat band conditions. Such approaches have also been applied to semiconductor superlattices in nanoelectronics. Concerning photonic crystal slabs, a first attempt has been presented recently with the proposal of a shape modification of the first row of air holes, replacing circles by ovoid forms, to minimize input reflection [2,17]. The associated results are very promising in terms of matching but more has to be done to reach good focusing
50 45 40 35 30 25 20 15 10 5 0
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Peak 3 Peak 2
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Fig. 4. (a) Wavelength shifts versus incidence of maximum transmission peaks for a six row photonic crystal based slab in negative refraction regime; (b) maximum transmission values associated.
performances in terms of associated negative refractive index. We believe that this hole shape engineering should not be limited to the first row but the slab has to be considered entirely to reach such flat band conditions so that it could work efficiently for various incidence angles. As shown in this last work, such engineering is particularly tricky in terms of fabrication due to the precision needed. An alternate way is to incorporate some extra material by atomic layer deposition, as proposed in [26], in order to modify locally the structural parameters. The best results can be obtained at the two particular wavelengths labeled k1 and k2 in Fig. 3 for which high transmission levels (higher than 80%) can be obtained whatever the angle of incidence but with the detriment of a negative index shift around 1. These two values k1 and k2 are equal to 1.71 lm and 1.58 lm. By considering the iso-frequency plots of Fig. 1b, we obtain, respectively, negative refractive index of 1.52 for k1 and 1.08 for k2. However, neither n = 1 nor k = 1.55 lm requirements are fulfilled. Or, as mentioned by several authors, optimal performances for the focusing required n = 1 if fed by an optical wave in air. 4. Discussion In the following, an original optimization procedure will be followed which does not need any shape engineering.
N. Fabre et al. / Optics Communications 281 (2008) 3571–3577
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Fig. 5. Transmission spectra for a 12 row photonic crystal slab for various incidence angles and geometry (a) a = 476 nm, d = 350 nm; (b) a = 476 nm, d = 366 nm; (c) a = 469 nm, d = 350 nm; (d) a = 469 nm, d = 332 nm.
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N. Fabre et al. / Optics Communications 281 (2008) 3571–3577
Our goal is to achieve simultaneously a maximum of transmission and n = 1 by design. Here, the search for maximal transmission does not mean surface impedance matching since for the second band of such a photonic crystal the impedance is close to the dielectric impedance value that is to say about 0.3 rather than the expected 1. The intrinsic nature of high transmission is the geometrical dependence on the number of rows of the lens. Concerning the photonic crystal itself, two parameters can be adjusted, the lattice period and the filling factor (hole diameter). The first dimension to fix is the lens thickness desired. Then as a function of a typical lattice period around 450–500 nm and a filling factor about 30–50% chosen to obtain n = 1 at 1.55 lm to work in second band, the number of rows can be deduced. For this given number of rows, transmission spectra as function of wave incidence can be calculated (as the one of Fig. 3 for six rows or the one of Fig. 5a for 12 rows). Here for clarity, incidences up to 7° are plotted but the study remains strictly valid up to 15°. If ‘‘lucky”, coincidence between a middle-band transmission peak (or high values whatever incidence) and n = 1 at 1.55 lm can be directly obtained, then the lens is quasi-optimized. If not, two routes can be followed as illustrated in Fig. 5b and c depending on prerequisites. As shown in Fig. 5a for a lattice period of 476 nm and hole diameter of 350 nm, transmission is minimum (less than 3 dB) at 1.55 lm where n = 1. If we decide to choose the sixth transmission peak, high values of transmission (higher than 1 dB up to 15° incidence angle) can be obtained at k = 1.522 lm. Here the idea is to shift the n = 1 value to this wavelength by changing the filling factor. Using band calculation procedure, this goal can be reached for a hole diameter of 366 nm instead of 350 nm with the same period. As shown in Fig. 5b, the new transmission spectra shows that the maximum peak coincides with the n = 1 wavelength. It can be mentioned here that the filling factor change induces also slight changes in the
transmission spectra, notably as various incidence angles are explored. But these changes are of second order, and if needed a refinement in filling factor can be operated to reach more accuracy. This first route shows that it is possible to reach a high transmission level along with n = 1, but the operating wavelength is dependent on the transmission peak chosen and on the lens number of rows. If k = 1.55 lm, or any other wavelength, is a strong prerequisite, one have to add at least one step in the optimization procedure. Once again we can start from Fig. 5a. Let us modify now, for a fixed hole diameter, the lattice period in order to shift one transmission peak to 1.55 lm for that given number of rows. This is illustrated in Fig. 5c with a lattice period of 469 nm where the seventh peak coincides now with the targeted wavelength 1.55 lm. Looking at the band structure, now n = 1 corresponds to k = 1.52 lm. As a final step, the filling factor can now be modulated to shift the n = 1 value back to 1.55 lm. This is obtained with a hole diameter of 332 nm. The procedure can be repeated when small additional adjustments are required. At the end the lens is optimized with n = 1 and high transmission levels at 1.55 lm for a given lens thickness, which was the pursued goal (Fig. 5d). Fig. 6 gives a three dimensional field map of the lens operation for the optimized structure. Reflection appears minimized owing to the optimization of the transmission which reaches unity and thus improves the impedance matching. A clear focus spot appears also behind the lens. The source–lens distance is 2.23 lm for a 4.87 lm thick lens. The spot position at 9.70 lm confirms the value of n very close to 1. It has a full width at half maximum (FWHM) of about 1 lm, which corresponds to 2k/3. Strictly speaking, we are far from a superlens operating regime. Let us mention that such a possibility of an ideal lens with PC’s structured at a scale comparable to the wavelength is still questionable [13,27]. In the lower part of the spectrum, a resolution of k/5 with a photonic crystal
Fig. 6. 3D map of the electric field for the optimized lens (a = 469 nm; d = 332 nm) operating at 1.55 lm.
N. Fabre et al. / Optics Communications 281 (2008) 3571–3577
has been deduced from experiments [28]. However, if subwavelength resolution is of major concern, it has to be mentioned that such a flat lens has no optical axis, that is to say that the image is invariant by any translation parallel to the lens interface. This property only afforded by negative refraction justifies solely such lens optimization. 5. Concluding remarks We have shown in this paper that the full concept of a two dimensional photonic crystal ‘‘superlens”, that is to say simultaneously n = 1 and z = 1, or in other terms e = 1 and l = 1 whatever the incidence angle, is a tricky task. However, very efficient optimization procedures can be developed to build efficient lenses at 1.55 lm. Our method is based on the exploitation of Fabry–Perot transmission peaks which appear due to the finite thickness of such photonic crystal slabs and of the second isotropic pass-band wavelength domain of the lattice which allow us to reach, with a high degree of isotropy, negative index values very close to 1. In short, the fundamental parameters are the lens thickness, the targeted wavelength region and the material system used as the dielectric matrix in which holes are etched for example. The other parameters such as lattice period, filling factor and number of rows are in general deduced from a joint calculation of band structure and transmission spectra for various incidences (by means of a FDTD software). It is believed that such optimization methods to be of great importance to improve the performances of future integrated photonic nanodevices based on photonic crystal slabs. Acknowledgments This work was partly supported by the research program METAPHORE (AC Nanosciences et Nanotechnologies of the French ministe`re de la Recherche et des nouvelles technologies). N. Fabre would like to thank De´le´gation Ge´ne´rale pour l’Armement for its fellowship.
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References [1] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. [2] T. Matsumoto, K. Eom, T. Baba, Opt. Lett. 40 (2001) 5920. [3] E. Schonbrun, T. Yamashita, W. Park, C.J. Summers, Phys. Rev. B 73 (2006) 195117. [4] A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, S. Anand, Phys. Rev. Lett. 93 (2004) 073902. [5] C. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 68 (2003) 045115. [6] A. Grbic, G.V. Eleftheriades, Phys. Rev. Lett. 92 (2004) 117403. [7] K. Aydin, I. Bulu, E. Ozbay, Appl. Phys. Lett. 90 (2007) 205402. [8] N. Fang, H. Lee, C. Sun, X. Zhang, Science 308 (2005) 534. [9] G. Dolling, M. Wegener, C.M. Soukoulis, S. Linden, Opt. Lett. 32 (2007) 53. [10] C. Luo, S.G. Jonhson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. Lett. B 62 (2002) 201104. [11] M. Notomi, Phys. Rev. Lett. B 62 (2000) 10696. [12] E. Centeno, D. Cassagne, Opt. Lett. 30 (2005) 2278. [13] D. Maystre, S. Enoch, J. Opt. Soc. Am. A21 (2004) 122. [14] T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, B. Gralak, Phys. Rev. Lett. 97 (2006) 073905. [15] W.B. Weir, Proc. IEEE 62 (1974) 33. [16] D.R. Smith, S. Schultz, P. Markos, C.M. Soukoulis, Phys. Rev. B 65 (2002) 195104. [17] T. Baba, D. Ohsaki, Jpn. J. Appl. Phys. 40 (2001) 5920. [18] A. Martinez, J. Marti, Phys. Rev. B 71 (2005) 235115. [19] B. Zhang, M.Y. Li, The Eur. J. Phys. D, doi:10.1140/epjd/e200700242-2. [20] S. Fasquel, X. Me´lique, O. Vanbe´sien, D. Lippens, Opt. Commun. 246 (2005) 91. [21] N. Fabre, S. Fasquel, C. Legrand, X. Me´lique, M. Muller, M. Francßois, O. Vanbe´sien, D. Lippens, Opto-Electron. Rev. 14 (2006) 225. [22] A. Martinez, H. Miguez, A. Griol, J. Marti, Phys. Rev. B 69 (2004) 165119. [23] Z. Ruan, M. Qiu, S. Xiao, S. He, L. Thylen, Phys. Rev. B 71 (2005) 045111. [24] O. Vanbe´sien, H. Leroux, D. Lippens, Solid State Electron. 35 (1992) 665. [25] J. Carbonell, O. Vanbe´sien, D. Lippens, Superlattices Microstruct. 22 (1997) 597. [26] E. Graugnard, D.P. Gaillot, S.N. Nudham, C.W. Neff, T. Yamashita, C.J. Summers, Appl. Phys. Lett. 89 (2006) 181108. [27] A.L. Efros, C.Y. Li, Solid State Phenom. 121–123 (2007) 1065. [28] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, C.M. Soukoulis, Phys. Rev. Lett. 91 (2003) 207401.
Optics Communications 281 (2008) 3571–3577 www.elsevier.com/locate/optcom
Optimized focusing properties of photonic crystal slabs Nathalie Fabre, Xavier Melique, Didier Lippens, Olivier Vanbesien * Institut d’Electronique, de Microe´lectronique et de Nanotechnologie (IEMN), UMR CNRS 8520, Universite´ des Sciences et Technologies de Lille, Avenue Poincare´, BP 60069, 59655 Villeneuve d’Ascq Cedex, France Received 2 October 2007; received in revised form 18 February 2008; accepted 19 February 2008
Abstract We propose an optimization procedure for focusing operation in finite two dimensional photonic crystal slabs. The device consists of a triangular lattice air holes etched in a semiconductor matrix at a nanometer scale to operate at 1.55 lm. To reach simultaneously an effective refractive index equal to 1 along with a very high transmission coefficient whatever optical wave incidence, the parameters as the lattice period and/or filling factor are precisely adjusted depending on the slab thickness. The method relies on Fabry–Perot resonances engineering in the air/crystal/air cavity constituting the lens. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Since the tremendous development of metamaterials along with the concept of negative refraction a few years ago, photonic crystals (PC’s) have appeared to be suitable candidates to present such abnormal propagation properties in the optical domain with the targeted application of a ‘‘superlens” [1–5]. Indeed, most of the proposed lefthanded materials proposed so far are partially constituted of metallic inclusions [6–8]. Their ability to preserve reasonable losses for optics remains an open question, even if preliminary results of a negative refraction index using a metallo-dielectric material in the visible have been obtained [9]. In this field, dielectric photonic crystals have shown promise when specific isotropic propagation regimes of the photonic structure, induced by the crystal periodicity, are exploited. More precisely, frequency domains can be identified where group and phase velocities are antiparallel. Such a property is true for all propagation directions being equivalent to the definition of a negative refractive index
*
Corresponding author. Tel.: +33 32 019 7876; fax: +33 32 019 7892. E-mail address: [email protected] (O. Vanbesien).
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.02.038
[10]. Alternative proposals are also based on anisotropy to promote negative and/or ultra refraction [11,12]. The requirements for a ‘‘superlens” were clearly identified: a negative refractive index equal to 1 is needed if the optical signal originates from air (n = 1) and a perfect surface impedance matching is also required (z = 1) to avoid any reflection at input [13]. Moreover, for subwavelength resolution based on the evanescent wave amplification effect, it is necessary that the effective permeability and permittivity of the focusing medium to be both equal to 1. These two conditions arep obviously compatible with ffiffiffiffiffiffiffi the definitions n2 ¼ el and z ¼ l=e. If the first parameter n can be easily identified for dielectric photonic crystals, it appears more tricky to define clearly the other ones compared to metallic microstructures aimed to operate in microwaves or up to the infrared region for which the four parameters can be extracted both from simulation (parameter extraction methods, as Fresnel inversion for example). Under certain conditions also in the lower part of the spectrum, these four parameters can be also recovered accurately from transmission and reflection measurements. One important requisite is the subwavelength patterning of the metamaterial, condition only partially fulfilled for photonic crystals. Indeed, the negative refraction regime is obtained in two dimensional photonic crystals for a semiconductor patterning with a periodicity
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N. Fabre et al. / Optics Communications 281 (2008) 3571–3577
close to k0/3 k0/4 (where k0 is the wavelength in air), leading to complex diffraction and refraction effects at the interface air/material under optical excitation. Nevertheless, as long as propagation is monomode at a given wavelength (the two first bands of the PC), parameter extraction remains meaningful [14–16]. In this context, the main question is: Is a ‘‘flat lens” or a ‘‘superlens” with enhanced performances compared to classical approaches possible using two dimensional photonic crystals for integrated nanophotonics? If the debate is still open for the ‘‘superlens”, we will show in this paper that efficient flat lenses can be designed. To the best of our knowledge, most of published papers up to now have focused on the necessity of an optimized isotropy of the band structure and on the optimization of the interface air/crystal to minimize reflection [17–19]. In the following, we will demonstrate that there is also a complex link between the desired operating wavelength and the lens geometry. Above the exact definition of equivalent parameters for the lens, we show that the lens behaves as a cavity with its own resonance which fixes the focusing properties. The basic photonic crystal and our investigation procedure will be presented in Section 2. This will be applied to a specific six row flat lens will presented in Section 3 and an optimization method will be proposed in Section 4. 2. Numerical procedure Our approach is based on a photonic crystal constituted of air holes etched in a semiconductor heterostructure [20,21]. Calculations are performed throughout the paper using the commercial codes BandSolve (for band structures) and Fullwave (FDTD – lens experiment) by RSoft. For the two dimensional simulation, an effective index of 3.32 is assumed corresponding to an InP/GaInAsP/InP based double heterojunction. The desired operating wavelength is 1.55 lm. To obtain a negative refractive index equal to 1 at this wavelength in second band of the crystal, we use a triangular lattice of air holes to promote isotropy with a periodicity a = 476 nm and a hole diameter d = 350 nm (filling factor: 38%). Fig. 1 gives the corresponding TM band structure (E field parallel to the holes) along with the light line. The second band stands from a/ k = 0.24 to 0.36 (wavelength range 1.25–1.98 lm). As shown in Fig. 1b this second band is almost isotropic (circular iso-frequency plots around C point) up to high wave vector values. Let us remain at this point that working with n = 1 (crossing between the band and the light line) is a necessary condition to obtain focusing of an incident point source with a slab but is not sufficient to obtain the ‘‘superlens”. Indeed, surface impedance matching between air and the crystal is required to minimize reflection at input. To illustrate this, Fig. 2a and b show a virtual prototyping of such a lens fed by an optical tapered waveguide ended by a small (k/10) diffracting hole to mimic a point source. For the left hand side of Fig. 2 the crystal is CM oriented (injection perpendicular to the triangle sides)
Fig. 1. (a) Two dimensional band structure for the triangular lattice with period a = 476 nm and air hole diameter d = 350 nm (filling factor of 0.38) – TM mode (E field parallel to the holes) – The index value of the patterned semiconductor is 3.32; (b) 2D iso-frequency plot for the second band where negative refraction index can be defined (red line or circle labelled 0.307: n = 1 circle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Lens experiment using a triangular photonic crystal slab: CM oriented (left side) and CK oriented (right side).
whereas, CK orientation (injection parallel to the triangle sides) is used for the right side of Fig. 2. Even if the dispersion curve is isotropic for this frequency and an effective
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refraction index can be deduced, the lens behavior is strongly dependent on incidence. The focusing is clearly apparent in the first case, but has almost disappeared for the second. Moreover, the signal intensity behind the lens is strongly attenuated since reflection at input is huge. In Fig. 2a, a clear focus is present issued from the lens and optical rays can be superimposed as in classical geometrical optics. The extension along propagation axis of the focus point is very small which indicates a refractive index value very close to 1 in this case. Using Gauss approximation of small angles and the distance ratios between focus points inside and outside the lens leads to an estimated value of 0.95. At the opposite, the width of the focus point behind the lens is larger than the incident electric field pattern revealing the fact that subwavelength resolution is not reached, and consequently that effective permittivity and permeability are not close to 1. Also, reflection is important in input and needs to be optimized. Such behavior has already been observed and studied in the literature [22,23] and will be analyzed more deeply in paragraph three. As a first indication, let us mention that the lens thickness is different in CM and CK directions since an equivalent number of air hole rows has been assumed in both cases but distance two successive rows is different (equal to pffiffiffiffiffiffiffibetween ffi a 3=2 in Fig. 2a and equal to a in Fig. 2b). In terms of cavities, the two lenses possess distinct Fabry–Perot modes. Even if n = 1 in both cases, we face a quasi-matched case in terms of impedance with low reflection and high transmission in Fig. 2a but the impedance mismatch prevails in Fig. 2b with opposite performances. This first all-angle interpretation needs now to be refined. Indeed, at this point, it appears clearly that the lens properties depend strongly on the incidence angle of the optical wave and thus on the first interface. More than this first interface, we will see that the thickness and the number of crystal periods used in the main propagation direction are also of prime importance. To analyze all these effects, the waveguide is suppressed at input and we will use a simple Gaussian pulse excitation. Its width is chosen so that transmission and reflection spectra on both sides of the lens can be calculated by fast Fourier transform over the second pass
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band of the photonic crystal and around 1.55 lm, the targeted wavelength of operation. For the study, the incident angle between the pulse direction of propagation and the normal to the lens will be varied. Finally, special attention will be paid to the position and size of the input and output ports of our FDTD simulation in order to recover the totality of the incident optical signal so that the sum of transmitted power and reflected power is equal to the incident power carried by the pulse. For sake of simplicity, we will focus our attention in the following on a six row lens to evidence the main limitations and performances of such lenses. 3. The six row flat lens Fig. 3 gives the transmission spectra obtained for a six row lens for incidence angles varying from 0° to 30° as a function of 1/k (1.55 lm corresponds to 0.645 lm1). As mentioned in Ref. [14], special care has to be devoted to the interface cut to promote the pffiffiffiffiffiffiffi ffi focusing properties. The row width is defined as a 3=2 with the air hole located at the center. This ensures a perfect symmetry along the direction perpendicular to the triangle sides if an odd number of rows are used. However it has been checked that our optimization procedure is valid whatever the number of rows, odd or even. The fact that air holes are staggered does not affect focusing properties. Under normal incidence (here labeled 0°), five transmission peaks are clearly visible, each very close to unity. In between, the transmission decreases rapidly and can be less than 40% even if the wavelength stands within the second band. This impedance mismatch between air and the lens, responsible for a quite high reflection coefficient is not only induced by the first interface as often claimed but the result of the cavity formed by the crystal embedded with ‘‘air”. The number of peaks is equal to the number of air hole rows minus one within the propagation direction. This effect is very well known for any heterogeneous periodical medium [24,25]. If air hole rows are considered as a specific medium with n1 index and the semiconductor in between as a medium n2 index, this effect corresponds to the Fabry–Perot resonances induced by the n1/n2 alternate layers. It can
Fig. 3. Transmission through a six row flat lens under different incidence angles (0°, 2°, 7°, 15°, 30°).
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a Wavelength shift (nm)
be mentioned that the low levels of transmission obtained between peaks evolve very weakly with the number of rows even if the number of peaks increases. Conversely, their position within the band depends of the number of rows. This means that if an operating wavelength is targeted, care has to be taken that the thickness of the slab allows the presence of a maximum in transmission at this wavelength. Here, the structure is designed to give a maximum under normal incidence at 1.55 lm for that given number of rows. This first precaution is not enough to ensure good performances for the lens. Indeed, to image a point source even in near field we face a circular wavefront at the first interface with an extension which is function of the source–lens separation. This means a multi-incidence beam interacting with the lens. Obviously, the matching (unitary transmission) is not preserved for any incidence, as exemplified by the additional spectra reported in Fig. 3. This effect was already present in Fig. 2 for the two extreme incidence angles (0° – CM and 30° – CK). But this effect has to be considered whatever the incidence angle value. As this angle increases, one can note a shift to higher wavelengths of the respective transmission peaks along with a decrease in amplitude. This figure shows clearly the sensitivity to incidence angle is more pronounced as k decreases experiencing deeper the interspace between hole rows. These evolutions are summarized in Fig. 4 where wavelength shifts are reported versus incidence angle. For the higher wavelength, the peak position is insensitive (less than 5 nm) to reasonable (in the sense of geometrical optics Gauss approximation) incidence angles, whereas for the shorter wavelength the peak almost disappears for an incidence of 30°. For the intermediate peaks, the shifts are respectively 13, 32 and 55 nm for peaks 2, 3 and 4 between 0° and 30°. Looking at Fig. 4b, it is clear that peaks 2 and 3 leads to high transmission levels for all incidence angles lower than 30°, whereas peaks 1 and 4 decrease faster as soon as angles higher than 10° are under concern. This appears as a strong restriction for good focusing properties. Moreover, the ripple between the peaks has to be correctly evaluated for a specific operating wavelength. As a matter of example, even if the maximum of transmission remains higher than 80%, one can note that the wavelength associated to the maximum under normal incidence induces a transmission drop to less than 50% at this same wavelength for the incidence of 30°. Different theories exist to optimize and decrease such ripples, notably in ‘‘filtering” theory. In general, it consists in considering the structure under study as a series of cascaded cells and in modifying the characteristic properties of each cell (but preserving lens symmetry about its middle in the propagation direction) to reach flat band conditions. Such approaches have also been applied to semiconductor superlattices in nanoelectronics. Concerning photonic crystal slabs, a first attempt has been presented recently with the proposal of a shape modification of the first row of air holes, replacing circles by ovoid forms, to minimize input reflection [2,17]. The associated results are very promising in terms of matching but more has to be done to reach good focusing
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Peak 3 Peak 2
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Fig. 4. (a) Wavelength shifts versus incidence of maximum transmission peaks for a six row photonic crystal based slab in negative refraction regime; (b) maximum transmission values associated.
performances in terms of associated negative refractive index. We believe that this hole shape engineering should not be limited to the first row but the slab has to be considered entirely to reach such flat band conditions so that it could work efficiently for various incidence angles. As shown in this last work, such engineering is particularly tricky in terms of fabrication due to the precision needed. An alternate way is to incorporate some extra material by atomic layer deposition, as proposed in [26], in order to modify locally the structural parameters. The best results can be obtained at the two particular wavelengths labeled k1 and k2 in Fig. 3 for which high transmission levels (higher than 80%) can be obtained whatever the angle of incidence but with the detriment of a negative index shift around 1. These two values k1 and k2 are equal to 1.71 lm and 1.58 lm. By considering the iso-frequency plots of Fig. 1b, we obtain, respectively, negative refractive index of 1.52 for k1 and 1.08 for k2. However, neither n = 1 nor k = 1.55 lm requirements are fulfilled. Or, as mentioned by several authors, optimal performances for the focusing required n = 1 if fed by an optical wave in air. 4. Discussion In the following, an original optimization procedure will be followed which does not need any shape engineering.
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Fig. 5. Transmission spectra for a 12 row photonic crystal slab for various incidence angles and geometry (a) a = 476 nm, d = 350 nm; (b) a = 476 nm, d = 366 nm; (c) a = 469 nm, d = 350 nm; (d) a = 469 nm, d = 332 nm.
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Our goal is to achieve simultaneously a maximum of transmission and n = 1 by design. Here, the search for maximal transmission does not mean surface impedance matching since for the second band of such a photonic crystal the impedance is close to the dielectric impedance value that is to say about 0.3 rather than the expected 1. The intrinsic nature of high transmission is the geometrical dependence on the number of rows of the lens. Concerning the photonic crystal itself, two parameters can be adjusted, the lattice period and the filling factor (hole diameter). The first dimension to fix is the lens thickness desired. Then as a function of a typical lattice period around 450–500 nm and a filling factor about 30–50% chosen to obtain n = 1 at 1.55 lm to work in second band, the number of rows can be deduced. For this given number of rows, transmission spectra as function of wave incidence can be calculated (as the one of Fig. 3 for six rows or the one of Fig. 5a for 12 rows). Here for clarity, incidences up to 7° are plotted but the study remains strictly valid up to 15°. If ‘‘lucky”, coincidence between a middle-band transmission peak (or high values whatever incidence) and n = 1 at 1.55 lm can be directly obtained, then the lens is quasi-optimized. If not, two routes can be followed as illustrated in Fig. 5b and c depending on prerequisites. As shown in Fig. 5a for a lattice period of 476 nm and hole diameter of 350 nm, transmission is minimum (less than 3 dB) at 1.55 lm where n = 1. If we decide to choose the sixth transmission peak, high values of transmission (higher than 1 dB up to 15° incidence angle) can be obtained at k = 1.522 lm. Here the idea is to shift the n = 1 value to this wavelength by changing the filling factor. Using band calculation procedure, this goal can be reached for a hole diameter of 366 nm instead of 350 nm with the same period. As shown in Fig. 5b, the new transmission spectra shows that the maximum peak coincides with the n = 1 wavelength. It can be mentioned here that the filling factor change induces also slight changes in the
transmission spectra, notably as various incidence angles are explored. But these changes are of second order, and if needed a refinement in filling factor can be operated to reach more accuracy. This first route shows that it is possible to reach a high transmission level along with n = 1, but the operating wavelength is dependent on the transmission peak chosen and on the lens number of rows. If k = 1.55 lm, or any other wavelength, is a strong prerequisite, one have to add at least one step in the optimization procedure. Once again we can start from Fig. 5a. Let us modify now, for a fixed hole diameter, the lattice period in order to shift one transmission peak to 1.55 lm for that given number of rows. This is illustrated in Fig. 5c with a lattice period of 469 nm where the seventh peak coincides now with the targeted wavelength 1.55 lm. Looking at the band structure, now n = 1 corresponds to k = 1.52 lm. As a final step, the filling factor can now be modulated to shift the n = 1 value back to 1.55 lm. This is obtained with a hole diameter of 332 nm. The procedure can be repeated when small additional adjustments are required. At the end the lens is optimized with n = 1 and high transmission levels at 1.55 lm for a given lens thickness, which was the pursued goal (Fig. 5d). Fig. 6 gives a three dimensional field map of the lens operation for the optimized structure. Reflection appears minimized owing to the optimization of the transmission which reaches unity and thus improves the impedance matching. A clear focus spot appears also behind the lens. The source–lens distance is 2.23 lm for a 4.87 lm thick lens. The spot position at 9.70 lm confirms the value of n very close to 1. It has a full width at half maximum (FWHM) of about 1 lm, which corresponds to 2k/3. Strictly speaking, we are far from a superlens operating regime. Let us mention that such a possibility of an ideal lens with PC’s structured at a scale comparable to the wavelength is still questionable [13,27]. In the lower part of the spectrum, a resolution of k/5 with a photonic crystal
Fig. 6. 3D map of the electric field for the optimized lens (a = 469 nm; d = 332 nm) operating at 1.55 lm.
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has been deduced from experiments [28]. However, if subwavelength resolution is of major concern, it has to be mentioned that such a flat lens has no optical axis, that is to say that the image is invariant by any translation parallel to the lens interface. This property only afforded by negative refraction justifies solely such lens optimization. 5. Concluding remarks We have shown in this paper that the full concept of a two dimensional photonic crystal ‘‘superlens”, that is to say simultaneously n = 1 and z = 1, or in other terms e = 1 and l = 1 whatever the incidence angle, is a tricky task. However, very efficient optimization procedures can be developed to build efficient lenses at 1.55 lm. Our method is based on the exploitation of Fabry–Perot transmission peaks which appear due to the finite thickness of such photonic crystal slabs and of the second isotropic pass-band wavelength domain of the lattice which allow us to reach, with a high degree of isotropy, negative index values very close to 1. In short, the fundamental parameters are the lens thickness, the targeted wavelength region and the material system used as the dielectric matrix in which holes are etched for example. The other parameters such as lattice period, filling factor and number of rows are in general deduced from a joint calculation of band structure and transmission spectra for various incidences (by means of a FDTD software). It is believed that such optimization methods to be of great importance to improve the performances of future integrated photonic nanodevices based on photonic crystal slabs. Acknowledgments This work was partly supported by the research program METAPHORE (AC Nanosciences et Nanotechnologies of the French ministe`re de la Recherche et des nouvelles technologies). N. Fabre would like to thank De´le´gation Ge´ne´rale pour l’Armement for its fellowship.
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