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Anomalous behaviour of light reflection in crystals with different homogeneity
Optical Materials 33 (2011) 1258–1261
Contents lists available at ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Anomalous behaviour of light reflection in crystals with different homogeneity Michele Bellingeri a, Francesco Scotognella b,⇑ a b
Dipartimento di Scienze Ambientali, Università di Parma, Parco Area delle Scienze, 33/A 43100 Parma, Italy Dipartimento di Fisica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy
a r t i c l e
i n f o
Article history: Received 4 December 2010 Received in revised form 17 February 2011 Accepted 21 February 2011 Available online 17 March 2011 Keywords: Photonic crystals Shannon–Wiener index Structure-property relationship
a b s t r a c t The light reflection as a function of the sample length has been studied for an ideal two-dimensional photonic crystal and for a two-dimensional photonic structure with smaller homogeneity with respect to the photonic crystal. We have found that, although the number of the scattering elements is constant for the two structures, the behaviour of the light reflection increases linearly with the sample length in the less homogeneous photonic structure, while it is strongly sub-linear in the photonic crystals. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In the last two decades great attention has been devoted to the study of the light transmission in photonic structures. In such structures, for a certain range of energies and certain wave vectors, light is not allowed to propagate through the medium [1–3]. This behaviour is very similar to the one of electrons in a semiconductor, where energy gaps arise owing to the periodic crystal potential at the atomic scale. Photonic structures can possess a periodical modulation of the dielectric constant. These structures are called photonic crystals and they are present in nature or can be fabricated through a wide range of techniques, with the dielectric periodicity in one, two and three dimensions [4–7]. Several efficient mathematical methods can predict the light transmission in photonic crystals [8–11] and these instruments can be useful for different applications, such as the fabrication of distributed feedback lasers [12]. Instead, these calculations become very cumbersome for aperiodic and random structures. Recently, concepts and methods widely used in statistics have been successfully applied to explain light transport phenomena in materials where the local density of scattering elements is position-dependent [13–15]. To efficiently predict the optical properties of such complicated systems, also as a function of the sample length, the implementation of simple and not time consuming methods can be very useful. In this work, we have studied the light reflection as a function of the sample length in a non-trivial engineered two-dimensional photonic structure. This structure is less homogeneous with respect to a perfectly ordered structure, i.e. a 2D photonic crystal ⇑ Corresponding author. E-mail address: [email protected] (F. Scotognella). 0925-3467/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2011.02.026
[16]. We have observed that the less homogeneous structure shows a linear behaviour of the average reflection, over a wide range of wavelengths, as a function of the sample length, while an ordered photonic crystal, with the same number of scattering centres, shows a strong sub-linear behaviour. 2. Outline of the method For this study, we first consider an ideal two-dimensional photonic crystal [3]. This photonic structure is a square lattice of dielectric circular pillars, where the pillars have a diameter d of 75 nm and are made of Titanium dioxide. The lattice constant a of the crystal is 300 nm and the matrix where the pillars are embedded is Silicon dioxide. The refractive indexes of TiO2 and SiO2 are nT = 2.45 and nS = 1.46, respectively. Note that, for such a geometrical setting nTd nS (a–d) is satisfied [3]. We consider 12 12 cell photonic crystal, to have a size of 3.6 3.6 lm (Fig. 1, crystal PC1). In order to analyse the light transmission as a function of the sample length, we have built the structures depicted in Fig. 1, where PC2 is PC1 repeated two times and PC3 is PC1 repeated three times. We have realised structures from PC1 up to PC7, where PC7 is PC1 repeated sevenfold. The other photonic structure we have used for this study is already reported in Ref [16]. Briefly, to design this crystal, we have assigned pillars in cells by a fitness model [16,17]. We have used this model in order to realise a crystal space with skewed clusters size without benchmark distribution. Thus, we have obtained a random crystal in which the clusters size distribution (i.e. pillars for cells distribution) is skewed. The whole structure has the same size 3.6 3.6 lm of PC1 and is depicted in Fig. 2 (R1 diagram). Also for this structure, R2 is the structure R1 repeated two times, R3 is
M. Bellingeri, F. Scotognella / Optical Materials 33 (2011) 1258–1261
1259
It is possible to correlate the distribution of the pillars in the structure to the Shannon–Wiener index [18,19]. The Shannon– Wiener H’ index is a diversity index widely used in statistics and information theory, defined as
H0 ¼
s X
pj log pj
ð1Þ
j¼1
where pj is the proportion of the j-fold species and s is the number of the species. For PC1, i.e. the ideal two-dimensional photonic crystal, the Shannon–Wiener index has a value of 1, corresponding to the maximum of the homogeneity. Instead, for the fitness model crystal R1 the value of the Shannon–Wiener index is 0.7, implying that R1 is less homogeneous than PC1. For what concerns the calculation of the light transmission (and reflection) of the photonic structures through finite element method, we assumed a TM-polarized field and used the scalar equation for the transverse electric field component EZ 2
ð@ 2x þ @ 2y ÞEZ þ n2 k0 EZ ¼ 0
ð2Þ
where n is the refractive index distribution and k0 is the free space wave number [3,20]. As input field, a plane wave with wave vector k directed along the x-axis has been assumed. Scattering boundary conditions in the y direction has been used. 3. Results and discussion
Fig. 1. Ideal two-dimensional photonic crystal (PC1). PC2 is PC1 repeated twice, PC3 is PC1 repeated threefold.
By using a finite element method, we have calculated the transmission spectra for the structures PCn and Rn [3,20]. Then, we have compared the light transmission as a function of the sample length for these two different systems. In Fig. 3 the transmission spectra, in the range 450–1400 nm, for PCn and Rn are displayed. As one
Fig. 2. Photonic structure made by a fitness model (R1). R2 is R1 repeated twice, R3 is R1 repeated threefold.
R1 repeated three times, up to structure R7, where R7 is R1 repeated sevenfold. To simplify, we call the two series of structures PCn and Rn, where n = 1,. . .,7.
Fig. 3. Light transmission in the range 450–1400 nm for structures PCn and Rn.
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M. Bellingeri, F. Scotognella / Optical Materials 33 (2011) 1258–1261
Fig. 4. Average light absorption as a function of the sample length for PCn and Rn.
could expect, for both the systems, the average transmission is decreasing by increasing the length of the sample. For PC1, it is remarkable to note the photonic band gap, that is in good agreement with the Bragg–Snell law, i.e. kBragg ¼ neff K, where kBragg is the centre wavelength of the stop band, neff is the effective refractive index of the lattice and k the spatial period (in this case, k = a = 300 nm). By increasing the sample length, the only variations in the transmission spectra are the depth of the photonic band gap, which tends to T = 0 in all the wavelengths comprised in its width, and the fringes at longer wavelengths. At about 500 nm the second order of the photonic band gap is present. The scenario for the structures Rn is consistently dissimilar. The spectral features in the R1 spectrum are very different with respect to PC1. Structure R1 shows very deep transmission peak at shorter wavelengths, up to 700 nm, while peaks at longer wavelengths are shallow. Instead, increasing the sample length, light transmission becomes very low in all the range of wavelengths. An interesting spectral feature is the peak at about 1320 nm, that is not present in R1 spectrum, but it occurs from R2 to R7. Since we are analysing structures made with transparent materials in the range 450–1400 nm, we can assume that the absorption of the structure is only due to light reflection. In Fig. 4 the average light absorption all over the range 450–1400 nm as a function of the sample length for Rn and PCn is shown. It is clear that the light absorption for structures Rn is linear as a function of the sample length, while for the ideal photonic crystal a strong sub-linear behaviour occurs. Light absorption in fitness model crystals Rn fits perfectly the linear function y ax + b (where a = 19.998 and b = 4.778, R2 = 0.99) meaning a constant amount of light entrapped for each unit of crystal length. For the ideal photonic crystal PCn the best fit is y axb, (a = 16.71 and b = 0.4886); the decreasing shape indicates the decay of light absorption as a function of crystal length. The values of average absorption of the two series of structures are reported in Table 1. Table 1 Values of average absorption as a function of the sample length for the two set of crystals. Crystal length (number of crystals)
Average absorption for PCn
Average absorption for Rn
1 2 3 4 5 6 7
17.2931 23.1270 28.0651 32.9646 36.8647 40.0218 43.4024
24.2661 44.6571 64.7264 85.3384 105.5120 125.4044 143.4876
In the spectral range that we have selected, i.e. 450–1400 nm, the first- and second-order of the photonic band gap of PCn crystal are present. By increasing the length of the crystal, light absorption increases for the spectral components, which fall inside the two gaps, saturating at PC3. From PC3 to PC7 the photonic band gaps do not contribute substantially to an absorption increase, while in the transparent regions we observe a slight increase due to fringes (Fig. 3). Therefore, the average light absorption all over the range 450–1400 nm is sub-linear as a function of the sample length. Conversely, for the structure Rn the absorption features are spread all over the selected spectral range. This is due to the random position of a skewed size distribution of TiO2 pillar clusters, which contributes to light scattering on a large range of wavelengths, similarly to what happens in quasi-crystals. As depicted in Fig. 3, the transmission features do not saturate by increasing from R1 to Rn, so that we observe, in such a spectral range, a linear behaviour of the average light absorption as a function of the sample length. Such a behaviour could be of major interest in applications such as light harvesting, in which enhancement of absorption of incoherent light over a broad spectral range is desired. To confirm the previous in silico experiment, we have performed the calculations of the average light transmission as a function of the sample length for the series of structures R2n. In this new series the sample length has been increased by adding each time a different cell permutation of the crystal R. The bootstrap test output is noteworthy: the permuted crystal displays exactly the behaviour of light penetration shown in the original structure. In other word, a change in the pillar distribution maintaining the global evenness, i.e. maintaining unmodified H’, does not alter the light transmission pattern of the medium. This phenomenon can be ascribed to the fact that the light transmission is affected by the Shannon index of the crystals, i.e. the homogeneity of these crystals, and not from a particular cluster distribution. 4. Conclusion In conclusion, we have studied the light reflection as a function of the sample length in a non-trivial engineered two-dimensional photonic structure. This structure is less homogeneous with respect to a perfectly ordered structure, i.e. a 2D photonic crystal. We have observed that the less homogeneous structure shows a linear behaviour of the average reflection as a function of the sample length, while the photonic crystal, with the same number of scattering centres, shows a strong sub-linear behaviour. Furthermore, we have noticed that this light reflection trend is not affected by the permutation of the scattering elements in the photonic structure, if the homogeneity (quantified by the Shannon–Wiener index) is unaltered. This result may be interesting for the study of the light transmission in real disordered materials for different applications, e.g. light harvesting [21], or in colloidal solutions where aggregation of nano-objects play a significant role in sensing particular analytes [22]. Acknowledgements The authors acknowledge Riccardo Gatti, Prof. Guglielmo Lanzani and Prof. Stefano Longhi for helpful discussions. References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486. [3] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995. [4] L.D. Bonifacio, B.V. Lotsch, D.P. Puzzo, F. Scotognella, G.A. Ozin, Adv. Mater. 21 (2009) 1641.
M. Bellingeri, F. Scotognella / Optical Materials 33 (2011) 1258–1261 [5] F. Scotognella, D.P. Puzzo, A. Monguzzi, D.S. Wiersma, D. Maschke, R. Tubino, G.A. Ozin, Small 5 (2009) 2048. [6] C. Lopez, Adv. Mater. 15 (2003) 1679. [7] J.E.G.J. Wijnhoven, W.L. Vos, Science 281 (1998) 802. [8] W. Axmann, P. Kuchment, J. Comput. Phys. 150 (1999) 468. [9] D.C. Dobson, J. Comput. Phys. 149 (1999) 363. [10] S.G. Johnson, J.D. Joannopoulos, Opt. Express 8 (2001) 173. [11] F. Scheffold, G. Maret, Phys. Rev. Lett. 81 (1998) 5800. [12] G.A. Turnbull, P. Andrew, M.J. Jory, W.L. Barnes, I.D.W. Samuel, Phys. Rev. B 64 (2001) 125122. [13] P. Barthelemy, J. Bertolotti, D.S. Wiersma, Nature 453 (2008) 495. [14] J. Bertolotti, K. Vynck, L. Pattelli, P. Barthelemy, S. Lepri, D.S. Wiersma, Adv. Funct. Mater. 20 (2010) 965.
[15] [16] [17] [18] [19] [20] [21] [22]
1261
J. Bertolotti, K. Vynck, D.S. Wiersma, Phys. Rev. Lett. 105 (2010) 163902. M. Bellingeri, S. Longhi, F. Scotognella, JEOS RP 10041 (2010). R. Albert, A.L. Barabási, Rev. Mod. Phys. 74 (2002) 47. C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379. W. Weaver, C.E. Shannon, The Mathematical Theory of Communication, Univ. of Illinois Press, Urbana, 1949. C.S. Lien, Physics of Optoelectronic Devices, John Wiley & Sons, New York, 1995. V.E. Ferry, M.A. Verschuuren, H.B.T. Li, E. Verhagen, R.J. Walters, R.E.I. Schropp, H.A. Atwater, A. Polman, Opt. Express 18 (2010) A237. L. Cademartiri, F. Scotognella, P.G. O’Brien, B.V. Lotsch, J. Thompson, S. Petrov, N.P. Kherani, G.A. Ozin, Nano Lett. 9 (2009) 1482.
Contents lists available at ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Anomalous behaviour of light reflection in crystals with different homogeneity Michele Bellingeri a, Francesco Scotognella b,⇑ a b
Dipartimento di Scienze Ambientali, Università di Parma, Parco Area delle Scienze, 33/A 43100 Parma, Italy Dipartimento di Fisica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy
a r t i c l e
i n f o
Article history: Received 4 December 2010 Received in revised form 17 February 2011 Accepted 21 February 2011 Available online 17 March 2011 Keywords: Photonic crystals Shannon–Wiener index Structure-property relationship
a b s t r a c t The light reflection as a function of the sample length has been studied for an ideal two-dimensional photonic crystal and for a two-dimensional photonic structure with smaller homogeneity with respect to the photonic crystal. We have found that, although the number of the scattering elements is constant for the two structures, the behaviour of the light reflection increases linearly with the sample length in the less homogeneous photonic structure, while it is strongly sub-linear in the photonic crystals. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction In the last two decades great attention has been devoted to the study of the light transmission in photonic structures. In such structures, for a certain range of energies and certain wave vectors, light is not allowed to propagate through the medium [1–3]. This behaviour is very similar to the one of electrons in a semiconductor, where energy gaps arise owing to the periodic crystal potential at the atomic scale. Photonic structures can possess a periodical modulation of the dielectric constant. These structures are called photonic crystals and they are present in nature or can be fabricated through a wide range of techniques, with the dielectric periodicity in one, two and three dimensions [4–7]. Several efficient mathematical methods can predict the light transmission in photonic crystals [8–11] and these instruments can be useful for different applications, such as the fabrication of distributed feedback lasers [12]. Instead, these calculations become very cumbersome for aperiodic and random structures. Recently, concepts and methods widely used in statistics have been successfully applied to explain light transport phenomena in materials where the local density of scattering elements is position-dependent [13–15]. To efficiently predict the optical properties of such complicated systems, also as a function of the sample length, the implementation of simple and not time consuming methods can be very useful. In this work, we have studied the light reflection as a function of the sample length in a non-trivial engineered two-dimensional photonic structure. This structure is less homogeneous with respect to a perfectly ordered structure, i.e. a 2D photonic crystal ⇑ Corresponding author. E-mail address: [email protected] (F. Scotognella). 0925-3467/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2011.02.026
[16]. We have observed that the less homogeneous structure shows a linear behaviour of the average reflection, over a wide range of wavelengths, as a function of the sample length, while an ordered photonic crystal, with the same number of scattering centres, shows a strong sub-linear behaviour. 2. Outline of the method For this study, we first consider an ideal two-dimensional photonic crystal [3]. This photonic structure is a square lattice of dielectric circular pillars, where the pillars have a diameter d of 75 nm and are made of Titanium dioxide. The lattice constant a of the crystal is 300 nm and the matrix where the pillars are embedded is Silicon dioxide. The refractive indexes of TiO2 and SiO2 are nT = 2.45 and nS = 1.46, respectively. Note that, for such a geometrical setting nTd nS (a–d) is satisfied [3]. We consider 12 12 cell photonic crystal, to have a size of 3.6 3.6 lm (Fig. 1, crystal PC1). In order to analyse the light transmission as a function of the sample length, we have built the structures depicted in Fig. 1, where PC2 is PC1 repeated two times and PC3 is PC1 repeated three times. We have realised structures from PC1 up to PC7, where PC7 is PC1 repeated sevenfold. The other photonic structure we have used for this study is already reported in Ref [16]. Briefly, to design this crystal, we have assigned pillars in cells by a fitness model [16,17]. We have used this model in order to realise a crystal space with skewed clusters size without benchmark distribution. Thus, we have obtained a random crystal in which the clusters size distribution (i.e. pillars for cells distribution) is skewed. The whole structure has the same size 3.6 3.6 lm of PC1 and is depicted in Fig. 2 (R1 diagram). Also for this structure, R2 is the structure R1 repeated two times, R3 is
M. Bellingeri, F. Scotognella / Optical Materials 33 (2011) 1258–1261
1259
It is possible to correlate the distribution of the pillars in the structure to the Shannon–Wiener index [18,19]. The Shannon– Wiener H’ index is a diversity index widely used in statistics and information theory, defined as
H0 ¼
s X
pj log pj
ð1Þ
j¼1
where pj is the proportion of the j-fold species and s is the number of the species. For PC1, i.e. the ideal two-dimensional photonic crystal, the Shannon–Wiener index has a value of 1, corresponding to the maximum of the homogeneity. Instead, for the fitness model crystal R1 the value of the Shannon–Wiener index is 0.7, implying that R1 is less homogeneous than PC1. For what concerns the calculation of the light transmission (and reflection) of the photonic structures through finite element method, we assumed a TM-polarized field and used the scalar equation for the transverse electric field component EZ 2
ð@ 2x þ @ 2y ÞEZ þ n2 k0 EZ ¼ 0
ð2Þ
where n is the refractive index distribution and k0 is the free space wave number [3,20]. As input field, a plane wave with wave vector k directed along the x-axis has been assumed. Scattering boundary conditions in the y direction has been used. 3. Results and discussion
Fig. 1. Ideal two-dimensional photonic crystal (PC1). PC2 is PC1 repeated twice, PC3 is PC1 repeated threefold.
By using a finite element method, we have calculated the transmission spectra for the structures PCn and Rn [3,20]. Then, we have compared the light transmission as a function of the sample length for these two different systems. In Fig. 3 the transmission spectra, in the range 450–1400 nm, for PCn and Rn are displayed. As one
Fig. 2. Photonic structure made by a fitness model (R1). R2 is R1 repeated twice, R3 is R1 repeated threefold.
R1 repeated three times, up to structure R7, where R7 is R1 repeated sevenfold. To simplify, we call the two series of structures PCn and Rn, where n = 1,. . .,7.
Fig. 3. Light transmission in the range 450–1400 nm for structures PCn and Rn.
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Fig. 4. Average light absorption as a function of the sample length for PCn and Rn.
could expect, for both the systems, the average transmission is decreasing by increasing the length of the sample. For PC1, it is remarkable to note the photonic band gap, that is in good agreement with the Bragg–Snell law, i.e. kBragg ¼ neff K, where kBragg is the centre wavelength of the stop band, neff is the effective refractive index of the lattice and k the spatial period (in this case, k = a = 300 nm). By increasing the sample length, the only variations in the transmission spectra are the depth of the photonic band gap, which tends to T = 0 in all the wavelengths comprised in its width, and the fringes at longer wavelengths. At about 500 nm the second order of the photonic band gap is present. The scenario for the structures Rn is consistently dissimilar. The spectral features in the R1 spectrum are very different with respect to PC1. Structure R1 shows very deep transmission peak at shorter wavelengths, up to 700 nm, while peaks at longer wavelengths are shallow. Instead, increasing the sample length, light transmission becomes very low in all the range of wavelengths. An interesting spectral feature is the peak at about 1320 nm, that is not present in R1 spectrum, but it occurs from R2 to R7. Since we are analysing structures made with transparent materials in the range 450–1400 nm, we can assume that the absorption of the structure is only due to light reflection. In Fig. 4 the average light absorption all over the range 450–1400 nm as a function of the sample length for Rn and PCn is shown. It is clear that the light absorption for structures Rn is linear as a function of the sample length, while for the ideal photonic crystal a strong sub-linear behaviour occurs. Light absorption in fitness model crystals Rn fits perfectly the linear function y ax + b (where a = 19.998 and b = 4.778, R2 = 0.99) meaning a constant amount of light entrapped for each unit of crystal length. For the ideal photonic crystal PCn the best fit is y axb, (a = 16.71 and b = 0.4886); the decreasing shape indicates the decay of light absorption as a function of crystal length. The values of average absorption of the two series of structures are reported in Table 1. Table 1 Values of average absorption as a function of the sample length for the two set of crystals. Crystal length (number of crystals)
Average absorption for PCn
Average absorption for Rn
1 2 3 4 5 6 7
17.2931 23.1270 28.0651 32.9646 36.8647 40.0218 43.4024
24.2661 44.6571 64.7264 85.3384 105.5120 125.4044 143.4876
In the spectral range that we have selected, i.e. 450–1400 nm, the first- and second-order of the photonic band gap of PCn crystal are present. By increasing the length of the crystal, light absorption increases for the spectral components, which fall inside the two gaps, saturating at PC3. From PC3 to PC7 the photonic band gaps do not contribute substantially to an absorption increase, while in the transparent regions we observe a slight increase due to fringes (Fig. 3). Therefore, the average light absorption all over the range 450–1400 nm is sub-linear as a function of the sample length. Conversely, for the structure Rn the absorption features are spread all over the selected spectral range. This is due to the random position of a skewed size distribution of TiO2 pillar clusters, which contributes to light scattering on a large range of wavelengths, similarly to what happens in quasi-crystals. As depicted in Fig. 3, the transmission features do not saturate by increasing from R1 to Rn, so that we observe, in such a spectral range, a linear behaviour of the average light absorption as a function of the sample length. Such a behaviour could be of major interest in applications such as light harvesting, in which enhancement of absorption of incoherent light over a broad spectral range is desired. To confirm the previous in silico experiment, we have performed the calculations of the average light transmission as a function of the sample length for the series of structures R2n. In this new series the sample length has been increased by adding each time a different cell permutation of the crystal R. The bootstrap test output is noteworthy: the permuted crystal displays exactly the behaviour of light penetration shown in the original structure. In other word, a change in the pillar distribution maintaining the global evenness, i.e. maintaining unmodified H’, does not alter the light transmission pattern of the medium. This phenomenon can be ascribed to the fact that the light transmission is affected by the Shannon index of the crystals, i.e. the homogeneity of these crystals, and not from a particular cluster distribution. 4. Conclusion In conclusion, we have studied the light reflection as a function of the sample length in a non-trivial engineered two-dimensional photonic structure. This structure is less homogeneous with respect to a perfectly ordered structure, i.e. a 2D photonic crystal. We have observed that the less homogeneous structure shows a linear behaviour of the average reflection as a function of the sample length, while the photonic crystal, with the same number of scattering centres, shows a strong sub-linear behaviour. Furthermore, we have noticed that this light reflection trend is not affected by the permutation of the scattering elements in the photonic structure, if the homogeneity (quantified by the Shannon–Wiener index) is unaltered. This result may be interesting for the study of the light transmission in real disordered materials for different applications, e.g. light harvesting [21], or in colloidal solutions where aggregation of nano-objects play a significant role in sensing particular analytes [22]. Acknowledgements The authors acknowledge Riccardo Gatti, Prof. Guglielmo Lanzani and Prof. Stefano Longhi for helpful discussions. References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486. [3] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995. [4] L.D. Bonifacio, B.V. Lotsch, D.P. Puzzo, F. Scotognella, G.A. Ozin, Adv. Mater. 21 (2009) 1641.
M. Bellingeri, F. Scotognella / Optical Materials 33 (2011) 1258–1261 [5] F. Scotognella, D.P. Puzzo, A. Monguzzi, D.S. Wiersma, D. Maschke, R. Tubino, G.A. Ozin, Small 5 (2009) 2048. [6] C. Lopez, Adv. Mater. 15 (2003) 1679. [7] J.E.G.J. Wijnhoven, W.L. Vos, Science 281 (1998) 802. [8] W. Axmann, P. Kuchment, J. Comput. Phys. 150 (1999) 468. [9] D.C. Dobson, J. Comput. Phys. 149 (1999) 363. [10] S.G. Johnson, J.D. Joannopoulos, Opt. Express 8 (2001) 173. [11] F. Scheffold, G. Maret, Phys. Rev. Lett. 81 (1998) 5800. [12] G.A. Turnbull, P. Andrew, M.J. Jory, W.L. Barnes, I.D.W. Samuel, Phys. Rev. B 64 (2001) 125122. [13] P. Barthelemy, J. Bertolotti, D.S. Wiersma, Nature 453 (2008) 495. [14] J. Bertolotti, K. Vynck, L. Pattelli, P. Barthelemy, S. Lepri, D.S. Wiersma, Adv. Funct. Mater. 20 (2010) 965.
[15] [16] [17] [18] [19] [20] [21] [22]
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J. Bertolotti, K. Vynck, D.S. Wiersma, Phys. Rev. Lett. 105 (2010) 163902. M. Bellingeri, S. Longhi, F. Scotognella, JEOS RP 10041 (2010). R. Albert, A.L. Barabási, Rev. Mod. Phys. 74 (2002) 47. C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379. W. Weaver, C.E. Shannon, The Mathematical Theory of Communication, Univ. of Illinois Press, Urbana, 1949. C.S. Lien, Physics of Optoelectronic Devices, John Wiley & Sons, New York, 1995. V.E. Ferry, M.A. Verschuuren, H.B.T. Li, E. Verhagen, R.J. Walters, R.E.I. Schropp, H.A. Atwater, A. Polman, Opt. Express 18 (2010) A237. L. Cademartiri, F. Scotognella, P.G. O’Brien, B.V. Lotsch, J. Thompson, S. Petrov, N.P. Kherani, G.A. Ozin, Nano Lett. 9 (2009) 1482.