- Email: [email protected]
Thermal radiation in quasiperiodic photonic crystals
ARTICLE IN PRESS Microelectronics Journal 40 (2009) 848–850
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Thermal radiation in quasiperiodic photonic crystals P.W. Mauriz a,, M.S. Vasconcelos a, F.F. de Medeiros b, E.L. Albuquerque b a b
˜o, 65025-000 Sa ˜o Luı´s-MA, Brazil Departamento de Fı´sica, Instituto Federal de Educac- ˜ ao, Cieˆncia e Tecnologia do Maranha Departamento de Fı´sica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil
a r t i c l e in f o
a b s t r a c t
Available online 20 January 2009
In this work we investigated the thermal power spectrum of the electromagnetic radiation through onedimensional stacks of alternating negative and positive refractive index layers, arranged as truncated quasiperiodic photonic structures obeying the Fibonacci (FB), Thue–Morse (TM), and double-period (DP) sequences. The thermal radiation power spectra are determined by means of a theoretical model based on a transfer matrix formalism for both normal and oblique incidence geometries, together with Kirchoff’s second law. We studied the radiation spectra by considering the case where both refractive indices of layers A and B are assumed to be a constant, as well as a more realistic case which takes into account the frequency-dependent electric permittivity and magnetic permeability m to characterize the negative refractive index n in layer B. Crown Copyright & 2008 Published by Elsevier Ltd. All rights reserved.
PACS: 87.15.Aa 81.07.Nb 72.80.Le 73.20.Jc Keywords: Fibonacci Thermal radiation Photonic crystals
Materials with simultaneously negative permittivity and negative permeability m, yielding a negative refractive index n (i.e., pffiffiffiffiffiffi the negative square root n ¼ m had to be chosen), the so-called negative index materials (NIMs) or metamaterials, have recently been extensively studied in several distinct artificial physical settings, inspired by Veselago’s work [1] many years ago. However, Veselago’s work was mainly an academic curiosity for a long time, as real materials with negative and m were not available. This situation changed when theoretical proposals demonstrated that it is possible to obtain both negative and m, within a certain frequency domain, if metallic periodically structured photonic media (microwave materials) are used [2], bringing Veselago’s result into the limelight. Further, the striking demonstration by Pendry [3] that NIM can be used to make perfect lenses with resolution capabilities not limited by the conventional diffraction limit has given an enormous boost to the interest in these materials. Experimentally, they were achieved for structures fabricated by interspersing effective media (thin metal wires or electric resonators) with effective media m (magnetic resonators such as split rings or high dielectric fibers) [4]. Besides, negative refraction phenomena can also occur in other situations, such as in materials with strong dispersion and losses [5]. Until recently, no comprehensive analysis of the influence of the LHM to the thermal radiation distribution appeared in literature. There are few papers dealing with the quantum field description of LHM related phenomena, like the modification of
Corresponding author.
E-mail address: [email protected] (P.W. Mauriz).
spontaneous emission and super-radiance effect [6]. Recently, Maksimovic´ et al. [7] analyzed the modification of thermal radiation when it pass over through a periodic and quasiperiodic (Cantor) multilayer structure incorporating both LHM and conventional materials for both s- and p-polarizations. They showed that thermal radiation spectra are strongly influenced by structures containing LHM in comparison to all-positive refractive index multilayered structures. It is the aim of this work to investigate the behavior of a light beam normally and obliquely incident on a one-dimensional multilayer photonic structure, composed of SiO2 /LHM layers arranged in a quasiperiodical fashion, which follows the Fibonacci (FB), Thue–Morse (TM), and double-period (DP) substitutional sequences (for an up to date review of these quasiperiodic structures see Ref. [8]). The main reason to study these structures is that they are realizable experimentally, so they are not mere academic examples of a quasicrystal. These quasiperiodic structures can be generated by their inflation rules, as follows: A ! AB, B ! A (FB); A ! AB, B ! BA (TM); A ! AB, B ! AA (DP). Here, A (thickness dA ) and B (thickness dB ) are the building blocks modelling the SiO2 and LHM layers, respectively. We use a theoretical model based on Kirchoff’s second law to calculate the emission spectra of the thermal radiation in these multilayered structures, together with a transfer matrix formalism, which simplify enormously the algebra involved in the calculation. Consider the quasiperiodic multilayer structure, medium A, with thickness dA , is fulfilled by SiO2, and is characterized by a pffiffiffiffiffiffiffiffiffiffi positive refractive index nA ¼ A mA and an impedance pffiffiffiffiffiffiffiffiffiffiffiffi ffi Z A ¼ mA =A , both constants. Medium B is a metamaterial with thickness dB , and is characterized by a p negative ffiffiffiffiffiffiffiffiffiffiffiffiffi refractive index pffiffiffiffiffiffiffiffiffiffi nB ¼ B mB and an impedance Z B ¼ mB =B . The multilayer
0026-2692/$ - see front matter Crown Copyright & 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2008.11.060
ARTICLE IN PRESS P.W. Mauriz et al. / Microelectronics Journal 40 (2009) 848–850
structure is grown on an absorbing substrate S, with a constant refractive index nS . The entire structure is embedded in a transparent medium C (considered to be air or vacuum) with a constant refractive index nC . To calculate the spectral properties of the one-dimensional optical quasiperiodic structures, i.e., the Fibonacci, Thue–Morse and doubled-period sequences, we can use a transfer matrix method [9]. This method consists in to relate amplitudes of the electromagnetic fields in a layer with those of the former layer, and so on, by successive applications of Maxwell’s electromagnetic boundary conditions at each interface along the multilayer system. Therefore, the transfer matrix relates the electromagnetic incident field amplitudes (A01C and A02C ) of one side of the multilayer system (at zo0), with the transmitted amplitude AN 1C of the electromagnetic field in the other side, at z4L, L being the size of the multilayer system, by means of the product of the interface matrices M ab (a, b being any A, B, S and C media) and the propagation matrices Mg (g ¼ A, B and S), as follows [10]: ! ! A01C AN 1C M M M M M M ¼ M , (1) BS S SC CA A AB B A02C 0 where 1 M ab ¼ 2 Mg ¼
1 þ Z a =Z b
1 Z a =Z b
1 Z a =Z b
1 þ Z a =Z b
expðikg dg Þ
0
0
expðikg dg Þ
,
(2)
! with kg ¼ ng o=c.
(3)
The above matrices were obtained for the normal incidence case. For the oblique incidence one, we need to replace Z a ! Z a = cos ya for s-polarization or TE mode, and Z a ! Z a cos ya for ppolarization or TM mode in the interface matrices, as well as ng ! ng cos yg for both TE and TM polarizations in the propagation matrices. The transmittance and the reflectance coefficients are simply given by RðoÞ ¼ jM21 =M 11 j2 ;
TðoÞ ¼ j1=M 11 j2 ,
(4)
where M i;j ði; j ¼ 1; 2Þ are the elements of the optical transfer matrix MðoÞ ¼ MCA M A M AB MB M BS M S M SC . If no absorbing material is introduced in the multilayer system, i.e., if the refractive indices are all real (lossless), then RðoÞ þ TðoÞ ¼ 1 by conservation of energy. When we introduce a material with complex refractive index, i.e., the absorption is present, RðoÞ and TðoÞ can be used to define a real absorptance by AðoÞ ¼ 1 RðoÞ TðoÞ, which is again a statement of conservation of energy. However, from Kirchoff’s second law, we know that the ratio of the thermal emittance EðoÞ to the absorptance AðoÞ is a constant, independent of the nature of the material, being the unity when the source is a perfect blackbody [11,12]. Hence, EðoÞ ¼ AðoÞ and therefore EðoÞ ¼ AðoÞ ¼ 1 RðoÞ TðoÞ.
(5)
Once the emittance is obtained, it is multiplied by Planck’s power spectrum rBB ðo; bÞ, b ¼ 1=kB T being the thermal factor (kB is Boltzmann’s constant), to give the power spectrum of the quasiperiodic photonic structure. In this way, the power spectrum rðo; bÞ of the layered structure is given, in terms of its emittance EðoÞ, by
rðo; bÞ ¼ EðoÞrBB ðo; bÞ;
Considering the quasiperiodic multilayer structure in thermal equilibrium, with its surroundings at a given temperature T, we present now the numerical simulations for the spectral emissivity. Fundamentally, we want to know how the characteristic curve of Planck’s blackbody spectrum (at a given temperature T) should be modified when we use a multilayered photonic crystal filter to model our physical system. Let us first assume an ideal case in which both the magnetic permeability and the electric permittivity can be approximated by constants, for the same frequency range of interest. Considering medium A as SiO2 , whose refractive index is nA ¼ 1:45, while medium B is considered to have a negative complex refractive index nB ¼ 1:0 þ i0:01, i.e., it is an absorbing LHM layer which can be stimulated thermally to emit. Moreover, we assume the individual layers as quarter-wave layers, for which the quasiperiodicity is expected to be more effective [14], with the central wavelength l0 ¼ 700 nm. These conditions yield the physical thickness in each layer defined by the following optical relation: n0A dA ¼ n0B dB ¼ l0 =4, n0A
rBB ðo; bÞ ¼
_o3 1 , p2 c2 expðb_o 1Þ
(6)
which is also known as Planck’s law of blackbody radiation [13]. Here, c is the speed of the light in the vacuum and _ is Planck’s constant divided by 2p.
(7) n0B
¼ ReðnA Þ and ¼ ReðnB Þ are the real parts of nA and nB , where respectively. The reversed phase shifts in the two layers are
dA ¼ ðp=2ÞO cosðyA Þ;
!
849
dB ¼ ðp=2ÞO cosðyB Þ,
(8)
where O ¼ o=o0 ¼ l0 =l is the reduced frequency and the angles yA and yB are the incidence angles that the light beam makes with the normal of the layers. This multilayered photonic band-gap stack is grown on an absorbing dielectric substrate characterized also by a complex refractive index nS ¼ 3:0 þ i0:03, whose thickness is given by dS ¼ 100l0 =n0S, n0S being the real part of nS . The thermal radiation spectra for the multilayered sequences, as a function of the reduced frequency O ¼ o=o0, are depicted in Figs. 1a (periodic case), b (ninth generation of the Fibonacci sequence), c (ninth generation of the Thue–Morse sequence) and d (ninth generation of the double-period sequence). In all cases, we have considered the normal incidence yA ¼ yB ¼ 0 and dA ¼ dB . The temperature T is defined in terms of the midgap frequency, o0 ¼ 2pc=l0 , from differentiation of Planck’s law. The solid lines describe the power spectra of the multilayered structures, while the dashed curves represent the power spectra of a perfect blackbody substrate. The dotted curves are the power spectra for a non-ideal absorbing material, with complex refractive index nS ¼ 3 þ i0:03. Observe that Planck’s curves for a perfect blackbody substrate represent the higher limit of the thermal emission at multilayered sequences. In comparison with the curves related to the thermal emission at non-ideal absorbing substrate (graybody spectrum), the stacked thin-film photonic band-gap structures arranged in a quasiperiodic way, strongly alter the power spectrum of this substrate when we introduce these sequences as filters, giving rise the so-called band-gap structures. For the periodic case (Fig. 1a) the thermal radiation spectrum presents a very soft behavior, without modulation, with low values for rðOÞ, when compared with the quasiperiodic cases. We can stand out in the range O ¼ 1:75–2.25, a peak of emission around O ¼ 2:0. Fig. 1b shows the thermal radiation spectrum for the ninth sequence of the Fibonacci quasiperiodic structure for O ¼ o=o0 between 0 and 2.5. The spectrum around the resonance frequency, O ¼ 1:0, has a pronounced profile as compared to the periodic case depicted in Fig. 1a, mainly those lying in the frequency region 0:5oOo1:5. The central and widest peak has a value next to rðOÞ ¼ 0:96, while the observed local minimum emission, next to O ¼ 1:0, corresponds to rðOÞ ¼ 0:33. If we move away from the central peak, the curve adjusts gradually towards the radiation emission profile of the gray body. In Fig. 1c, we show the emission spectrum for the ninth sequence of the quasiperiodic Thue–Morse structure, whose main aspects are the presence of a narrow peak
ARTICLE IN PRESS 850
P.W. Mauriz et al. / Microelectronics Journal 40 (2009) 848–850
Fig. 1. The thermal radiation spectra (solid lines) as a function of the reduced frequency O ¼ o=o0 considering the normal incidence case, in which nA ¼ 1:45 and nB ¼ 1:0 þ i0:01. The photonic multilayer structure is considered to be as follows: (a) a periodic structure; (b) a ninth generation Fibonacci quasiperiodic structure; (c) a ninth generation Thue–Morse quasiperiodic structure; (d) a ninth generation double-period quasiperiodic structure. The normalized characteristic curves of Planck’s thermal radiation are represented by dashed lines (the perfect blackbody), while the spectra for an absorbing material with refractive index nS ¼ 3 þ i0:03 are represented by dotted lines. The temperature is chosen so that the blackbody peak is aligned with the midgap frequency o0 ¼ 2pc=l0.
at the frequency O ¼ 1:0, as well as a secondary peak at O ¼ 2:0. Around O ¼ 1:0, the energy of the electromagnetic wave is strongly absorbing, resembling a light-filter, re-emitting strongly the light with rðOÞ ¼ 0:96. Finally, Fig. 1d shows the spectrum of emission corresponding to the ninth sequence of the doubleperiod quasiperiodic structure. Different from Figs. 1b and c, we notice a window of non-emission around O ¼ 1:0. The reason for that is because its transmission spectrum has a large band gap around O ¼ 1:0, with no counterpart either with the other quasiperiodic sequences studied here or with those obtained for a multilayered system without the presence of a metamaterial [10]. Besides, the spectrum now presents a strong emission for several values of the frequency range, as we move away from the resonance frequency O ¼ 1:0. More important, as in all case considered here, we have in Fig. 1d another constructive interference for the thermal radiation spectra at O ¼ 2:0.
We would like to thank partial financial support from the Brazilian Research Agencies CNPq-Rede Nanobioestruturas, FINEP and FAPEMA.
References [1] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [2] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, J. Phys. Condens. Matter 10 (1998) 4785; J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microwaves Theory Tech. 47 (1999) 2075. [3] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. [4] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [5] G. Dolling, C. Enkrich, M. Wegener, C.M. Soukoulis, S. Linden, Science 312 (2006) 892. [6] J. Ka¨stel, M. Fleischhauer, Phys. Rev. A 71 (2005) 01180; J. Ka¨stel, M. Fleischhauer, Laser Phys. 15 (2005) 1. [7] M. Maksimovic´, Z. Jaksic´, Phys. Lett. A 342 (2005) 497; M. Maksimovic´, Z. Jaksic´, J. Opt. A Pure Appl. Opt. 8 (2006) 355. [8] E.L. Albuquerque, M.G. Cottam, Polaritons in Periodic and Quasiperiodic Structures, Elsevier, Amsterdam, 2004. [9] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376 (2003) 225; M.S. Vasconcelos, E.L. Albuquerque, Phys. Rev. B 59 (1999) 11128. [10] F.F. de Medeiros, E.L. Albuquerque, M.S. Vasconcelos, J. Phys. Condens. Matter 18 (2006) 8737. [11] P. Pigeat, D. Rouzel, B. Weber, Phys. Rev. B 57 (1998) 9293. [12] C.M. Cornelius, J.P. Dowling, Phys. Rev. A 59 (1999) 4736. [13] M. Planck, The Theory of Heat Radiation, Dover, New York, 1959. [14] K. Kono, S. Nakada, Y. Narahara, Y. Ootuka, J. Phys. Soc. Jpn. 60 (1991) 368.
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Thermal radiation in quasiperiodic photonic crystals P.W. Mauriz a,, M.S. Vasconcelos a, F.F. de Medeiros b, E.L. Albuquerque b a b
˜o, 65025-000 Sa ˜o Luı´s-MA, Brazil Departamento de Fı´sica, Instituto Federal de Educac- ˜ ao, Cieˆncia e Tecnologia do Maranha Departamento de Fı´sica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil
a r t i c l e in f o
a b s t r a c t
Available online 20 January 2009
In this work we investigated the thermal power spectrum of the electromagnetic radiation through onedimensional stacks of alternating negative and positive refractive index layers, arranged as truncated quasiperiodic photonic structures obeying the Fibonacci (FB), Thue–Morse (TM), and double-period (DP) sequences. The thermal radiation power spectra are determined by means of a theoretical model based on a transfer matrix formalism for both normal and oblique incidence geometries, together with Kirchoff’s second law. We studied the radiation spectra by considering the case where both refractive indices of layers A and B are assumed to be a constant, as well as a more realistic case which takes into account the frequency-dependent electric permittivity and magnetic permeability m to characterize the negative refractive index n in layer B. Crown Copyright & 2008 Published by Elsevier Ltd. All rights reserved.
PACS: 87.15.Aa 81.07.Nb 72.80.Le 73.20.Jc Keywords: Fibonacci Thermal radiation Photonic crystals
Materials with simultaneously negative permittivity and negative permeability m, yielding a negative refractive index n (i.e., pffiffiffiffiffiffi the negative square root n ¼ m had to be chosen), the so-called negative index materials (NIMs) or metamaterials, have recently been extensively studied in several distinct artificial physical settings, inspired by Veselago’s work [1] many years ago. However, Veselago’s work was mainly an academic curiosity for a long time, as real materials with negative and m were not available. This situation changed when theoretical proposals demonstrated that it is possible to obtain both negative and m, within a certain frequency domain, if metallic periodically structured photonic media (microwave materials) are used [2], bringing Veselago’s result into the limelight. Further, the striking demonstration by Pendry [3] that NIM can be used to make perfect lenses with resolution capabilities not limited by the conventional diffraction limit has given an enormous boost to the interest in these materials. Experimentally, they were achieved for structures fabricated by interspersing effective media (thin metal wires or electric resonators) with effective media m (magnetic resonators such as split rings or high dielectric fibers) [4]. Besides, negative refraction phenomena can also occur in other situations, such as in materials with strong dispersion and losses [5]. Until recently, no comprehensive analysis of the influence of the LHM to the thermal radiation distribution appeared in literature. There are few papers dealing with the quantum field description of LHM related phenomena, like the modification of
Corresponding author.
E-mail address: [email protected] (P.W. Mauriz).
spontaneous emission and super-radiance effect [6]. Recently, Maksimovic´ et al. [7] analyzed the modification of thermal radiation when it pass over through a periodic and quasiperiodic (Cantor) multilayer structure incorporating both LHM and conventional materials for both s- and p-polarizations. They showed that thermal radiation spectra are strongly influenced by structures containing LHM in comparison to all-positive refractive index multilayered structures. It is the aim of this work to investigate the behavior of a light beam normally and obliquely incident on a one-dimensional multilayer photonic structure, composed of SiO2 /LHM layers arranged in a quasiperiodical fashion, which follows the Fibonacci (FB), Thue–Morse (TM), and double-period (DP) substitutional sequences (for an up to date review of these quasiperiodic structures see Ref. [8]). The main reason to study these structures is that they are realizable experimentally, so they are not mere academic examples of a quasicrystal. These quasiperiodic structures can be generated by their inflation rules, as follows: A ! AB, B ! A (FB); A ! AB, B ! BA (TM); A ! AB, B ! AA (DP). Here, A (thickness dA ) and B (thickness dB ) are the building blocks modelling the SiO2 and LHM layers, respectively. We use a theoretical model based on Kirchoff’s second law to calculate the emission spectra of the thermal radiation in these multilayered structures, together with a transfer matrix formalism, which simplify enormously the algebra involved in the calculation. Consider the quasiperiodic multilayer structure, medium A, with thickness dA , is fulfilled by SiO2, and is characterized by a pffiffiffiffiffiffiffiffiffiffi positive refractive index nA ¼ A mA and an impedance pffiffiffiffiffiffiffiffiffiffiffiffi ffi Z A ¼ mA =A , both constants. Medium B is a metamaterial with thickness dB , and is characterized by a p negative ffiffiffiffiffiffiffiffiffiffiffiffiffi refractive index pffiffiffiffiffiffiffiffiffiffi nB ¼ B mB and an impedance Z B ¼ mB =B . The multilayer
0026-2692/$ - see front matter Crown Copyright & 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2008.11.060
ARTICLE IN PRESS P.W. Mauriz et al. / Microelectronics Journal 40 (2009) 848–850
structure is grown on an absorbing substrate S, with a constant refractive index nS . The entire structure is embedded in a transparent medium C (considered to be air or vacuum) with a constant refractive index nC . To calculate the spectral properties of the one-dimensional optical quasiperiodic structures, i.e., the Fibonacci, Thue–Morse and doubled-period sequences, we can use a transfer matrix method [9]. This method consists in to relate amplitudes of the electromagnetic fields in a layer with those of the former layer, and so on, by successive applications of Maxwell’s electromagnetic boundary conditions at each interface along the multilayer system. Therefore, the transfer matrix relates the electromagnetic incident field amplitudes (A01C and A02C ) of one side of the multilayer system (at zo0), with the transmitted amplitude AN 1C of the electromagnetic field in the other side, at z4L, L being the size of the multilayer system, by means of the product of the interface matrices M ab (a, b being any A, B, S and C media) and the propagation matrices Mg (g ¼ A, B and S), as follows [10]: ! ! A01C AN 1C M M M M M M ¼ M , (1) BS S SC CA A AB B A02C 0 where 1 M ab ¼ 2 Mg ¼
1 þ Z a =Z b
1 Z a =Z b
1 Z a =Z b
1 þ Z a =Z b
expðikg dg Þ
0
0
expðikg dg Þ
,
(2)
! with kg ¼ ng o=c.
(3)
The above matrices were obtained for the normal incidence case. For the oblique incidence one, we need to replace Z a ! Z a = cos ya for s-polarization or TE mode, and Z a ! Z a cos ya for ppolarization or TM mode in the interface matrices, as well as ng ! ng cos yg for both TE and TM polarizations in the propagation matrices. The transmittance and the reflectance coefficients are simply given by RðoÞ ¼ jM21 =M 11 j2 ;
TðoÞ ¼ j1=M 11 j2 ,
(4)
where M i;j ði; j ¼ 1; 2Þ are the elements of the optical transfer matrix MðoÞ ¼ MCA M A M AB MB M BS M S M SC . If no absorbing material is introduced in the multilayer system, i.e., if the refractive indices are all real (lossless), then RðoÞ þ TðoÞ ¼ 1 by conservation of energy. When we introduce a material with complex refractive index, i.e., the absorption is present, RðoÞ and TðoÞ can be used to define a real absorptance by AðoÞ ¼ 1 RðoÞ TðoÞ, which is again a statement of conservation of energy. However, from Kirchoff’s second law, we know that the ratio of the thermal emittance EðoÞ to the absorptance AðoÞ is a constant, independent of the nature of the material, being the unity when the source is a perfect blackbody [11,12]. Hence, EðoÞ ¼ AðoÞ and therefore EðoÞ ¼ AðoÞ ¼ 1 RðoÞ TðoÞ.
(5)
Once the emittance is obtained, it is multiplied by Planck’s power spectrum rBB ðo; bÞ, b ¼ 1=kB T being the thermal factor (kB is Boltzmann’s constant), to give the power spectrum of the quasiperiodic photonic structure. In this way, the power spectrum rðo; bÞ of the layered structure is given, in terms of its emittance EðoÞ, by
rðo; bÞ ¼ EðoÞrBB ðo; bÞ;
Considering the quasiperiodic multilayer structure in thermal equilibrium, with its surroundings at a given temperature T, we present now the numerical simulations for the spectral emissivity. Fundamentally, we want to know how the characteristic curve of Planck’s blackbody spectrum (at a given temperature T) should be modified when we use a multilayered photonic crystal filter to model our physical system. Let us first assume an ideal case in which both the magnetic permeability and the electric permittivity can be approximated by constants, for the same frequency range of interest. Considering medium A as SiO2 , whose refractive index is nA ¼ 1:45, while medium B is considered to have a negative complex refractive index nB ¼ 1:0 þ i0:01, i.e., it is an absorbing LHM layer which can be stimulated thermally to emit. Moreover, we assume the individual layers as quarter-wave layers, for which the quasiperiodicity is expected to be more effective [14], with the central wavelength l0 ¼ 700 nm. These conditions yield the physical thickness in each layer defined by the following optical relation: n0A dA ¼ n0B dB ¼ l0 =4, n0A
rBB ðo; bÞ ¼
_o3 1 , p2 c2 expðb_o 1Þ
(6)
which is also known as Planck’s law of blackbody radiation [13]. Here, c is the speed of the light in the vacuum and _ is Planck’s constant divided by 2p.
(7) n0B
¼ ReðnA Þ and ¼ ReðnB Þ are the real parts of nA and nB , where respectively. The reversed phase shifts in the two layers are
dA ¼ ðp=2ÞO cosðyA Þ;
!
849
dB ¼ ðp=2ÞO cosðyB Þ,
(8)
where O ¼ o=o0 ¼ l0 =l is the reduced frequency and the angles yA and yB are the incidence angles that the light beam makes with the normal of the layers. This multilayered photonic band-gap stack is grown on an absorbing dielectric substrate characterized also by a complex refractive index nS ¼ 3:0 þ i0:03, whose thickness is given by dS ¼ 100l0 =n0S, n0S being the real part of nS . The thermal radiation spectra for the multilayered sequences, as a function of the reduced frequency O ¼ o=o0, are depicted in Figs. 1a (periodic case), b (ninth generation of the Fibonacci sequence), c (ninth generation of the Thue–Morse sequence) and d (ninth generation of the double-period sequence). In all cases, we have considered the normal incidence yA ¼ yB ¼ 0 and dA ¼ dB . The temperature T is defined in terms of the midgap frequency, o0 ¼ 2pc=l0 , from differentiation of Planck’s law. The solid lines describe the power spectra of the multilayered structures, while the dashed curves represent the power spectra of a perfect blackbody substrate. The dotted curves are the power spectra for a non-ideal absorbing material, with complex refractive index nS ¼ 3 þ i0:03. Observe that Planck’s curves for a perfect blackbody substrate represent the higher limit of the thermal emission at multilayered sequences. In comparison with the curves related to the thermal emission at non-ideal absorbing substrate (graybody spectrum), the stacked thin-film photonic band-gap structures arranged in a quasiperiodic way, strongly alter the power spectrum of this substrate when we introduce these sequences as filters, giving rise the so-called band-gap structures. For the periodic case (Fig. 1a) the thermal radiation spectrum presents a very soft behavior, without modulation, with low values for rðOÞ, when compared with the quasiperiodic cases. We can stand out in the range O ¼ 1:75–2.25, a peak of emission around O ¼ 2:0. Fig. 1b shows the thermal radiation spectrum for the ninth sequence of the Fibonacci quasiperiodic structure for O ¼ o=o0 between 0 and 2.5. The spectrum around the resonance frequency, O ¼ 1:0, has a pronounced profile as compared to the periodic case depicted in Fig. 1a, mainly those lying in the frequency region 0:5oOo1:5. The central and widest peak has a value next to rðOÞ ¼ 0:96, while the observed local minimum emission, next to O ¼ 1:0, corresponds to rðOÞ ¼ 0:33. If we move away from the central peak, the curve adjusts gradually towards the radiation emission profile of the gray body. In Fig. 1c, we show the emission spectrum for the ninth sequence of the quasiperiodic Thue–Morse structure, whose main aspects are the presence of a narrow peak
ARTICLE IN PRESS 850
P.W. Mauriz et al. / Microelectronics Journal 40 (2009) 848–850
Fig. 1. The thermal radiation spectra (solid lines) as a function of the reduced frequency O ¼ o=o0 considering the normal incidence case, in which nA ¼ 1:45 and nB ¼ 1:0 þ i0:01. The photonic multilayer structure is considered to be as follows: (a) a periodic structure; (b) a ninth generation Fibonacci quasiperiodic structure; (c) a ninth generation Thue–Morse quasiperiodic structure; (d) a ninth generation double-period quasiperiodic structure. The normalized characteristic curves of Planck’s thermal radiation are represented by dashed lines (the perfect blackbody), while the spectra for an absorbing material with refractive index nS ¼ 3 þ i0:03 are represented by dotted lines. The temperature is chosen so that the blackbody peak is aligned with the midgap frequency o0 ¼ 2pc=l0.
at the frequency O ¼ 1:0, as well as a secondary peak at O ¼ 2:0. Around O ¼ 1:0, the energy of the electromagnetic wave is strongly absorbing, resembling a light-filter, re-emitting strongly the light with rðOÞ ¼ 0:96. Finally, Fig. 1d shows the spectrum of emission corresponding to the ninth sequence of the doubleperiod quasiperiodic structure. Different from Figs. 1b and c, we notice a window of non-emission around O ¼ 1:0. The reason for that is because its transmission spectrum has a large band gap around O ¼ 1:0, with no counterpart either with the other quasiperiodic sequences studied here or with those obtained for a multilayered system without the presence of a metamaterial [10]. Besides, the spectrum now presents a strong emission for several values of the frequency range, as we move away from the resonance frequency O ¼ 1:0. More important, as in all case considered here, we have in Fig. 1d another constructive interference for the thermal radiation spectra at O ¼ 2:0.
We would like to thank partial financial support from the Brazilian Research Agencies CNPq-Rede Nanobioestruturas, FINEP and FAPEMA.
References [1] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [2] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, J. Phys. Condens. Matter 10 (1998) 4785; J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microwaves Theory Tech. 47 (1999) 2075. [3] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. [4] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [5] G. Dolling, C. Enkrich, M. Wegener, C.M. Soukoulis, S. Linden, Science 312 (2006) 892. [6] J. Ka¨stel, M. Fleischhauer, Phys. Rev. A 71 (2005) 01180; J. Ka¨stel, M. Fleischhauer, Laser Phys. 15 (2005) 1. [7] M. Maksimovic´, Z. Jaksic´, Phys. Lett. A 342 (2005) 497; M. Maksimovic´, Z. Jaksic´, J. Opt. A Pure Appl. Opt. 8 (2006) 355. [8] E.L. Albuquerque, M.G. Cottam, Polaritons in Periodic and Quasiperiodic Structures, Elsevier, Amsterdam, 2004. [9] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376 (2003) 225; M.S. Vasconcelos, E.L. Albuquerque, Phys. Rev. B 59 (1999) 11128. [10] F.F. de Medeiros, E.L. Albuquerque, M.S. Vasconcelos, J. Phys. Condens. Matter 18 (2006) 8737. [11] P. Pigeat, D. Rouzel, B. Weber, Phys. Rev. B 57 (1998) 9293. [12] C.M. Cornelius, J.P. Dowling, Phys. Rev. A 59 (1999) 4736. [13] M. Planck, The Theory of Heat Radiation, Dover, New York, 1959. [14] K. Kono, S. Nakada, Y. Narahara, Y. Ootuka, J. Phys. Soc. Jpn. 60 (1991) 368.