- Email: [email protected]
Critical coupling in a Fabry–Pérot cavity with metamaterial mirrors
Optics Communications 283 (2010) 4764–4769
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Critical coupling in a Fabry–Pérot cavity with metamaterial mirrors Subimal Deb ⁎, S. Dutta Gupta School of Physics, University of Hyderabad, Hyderabad 500046, India
a r t i c l e
i n f o
Article history: Received 22 January 2010 Received in revised form 23 May 2010 Accepted 28 June 2010 Keywords: Critical coupling Fabry–Pérot Metamaterials
a b s t r a c t We study critical coupling (CC) in a system of a dielectric layer sandwiched between two metamaterial layers leading to near-total suppression of both reflection and transmission at specified frequencies. The tunability of the CC frequency is demonstrated by varying the angle of incidence retaining the full causal response for the metamaterials. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In recent years there has been a surge of activities on layered media [1,2]. The research is directed not only to well-studied guided/surface modes but also to novel applications ranging from sensors to slow light, from strong coupling regime of cavity QED to resonant tunneling [3–7]. One of the very interesting applications has been critical coupling (CC) [8–11]. In a critically coupled structure there is near-perfect absorption of incident light, leading to almost complete suppression of both reflection and transmission. CC at a given frequency was first reported by Tischler et al. [8]. They used a 5 nm thick layer of highly absorbing polymer, separated from a dielectric Bragg reflector (DBR) by a spacer layer under normal incidence. The polymer layer was later replaced by a metal colloid film to show the possibility of CC simultaneously at two distinct frequencies [10]. Subsequently these studies were extended to oblique incidence to show the feasibility of CC both for TE and TM-polarized light [11]. The choice and the tunability of the CC frequency are limited by the spectral properties of the absorbers and the DBR. CC occurs as a consequence of the fact that the absorption feature of the lossy film falls within the stop gap of the DBR. With an increase in the angle of incidence the absorption feature remains unaffected, while the stop gap shifts to larger frequencies. Thus a structure designed for CC for normal incidence loses critical coupling at larger angles, since the absorption band now falls beyond the stop gap. In this article we show that both these limitations can be overcome by using metamaterials (MMs). Under metamaterials one now understands a broad class of
⁎ Corresponding author. E-mail addresses: [email protected] (S. Deb), [email protected] (S. Dutta Gupta). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.06.085
“artificial effectively homogeneous electromagnetic structures not readily found in nature” with unusual properties [12]. Historically, research on metamaterials started with the theoretical paper by Veselago, where he investigated the counterintuitive properties of materials (not occurring in nature) with both the permittivity () and permeability (μ) negative [13]. These are now referred to as left handed materials or materials with negative refraction. Intensive research started off from the prediction of perfect lensing and the experimental realization of these materials [14,15]. Now the list of possible applications is truly vast and it is evergrowing. Very recently yet another application of a slab of MM was pointed out by Bloemer et al. [16] in a spectral domain where they do not exhibit negative refraction. It was shown that in the spectral domain between the magnetic and the electric plasma frequencies a MM slab can exhibit a stop-band much like in 1D photonic band gap (PBG) structures. The origin of stop bands in 1D PBG structures is due to constructive interference of the smaller reflections from each interface. In contrast, the stop gap in MMs between the plasma frequencies is due to the predominantly imaginary refractive index (a consequence of opposite signs of Re () and Re(μ)), forbidding the existence of propagating modes. In this article we show that such stop bands of the MM can be exploited effectively to achieve frequency tunable CC. In particular, we study a system of two MM layers (with two distinct sets of parameters) separated by a dielectric layer. The parameters are chosen such that the stop band of one falls inside the stop band of the other. The CC is mediated by the Fabry–Pérot (FP) resonances of the dielectric layer sandwiched between the MM layers. We demonstrate a great flexibility with respect to the choice of the CC frequency, even to the extent of simultaneous CC at two wellseparated frequencies. It may be noted that perfectly absorbing MM structures in the microwave regime have been proposed recently [17–19]. The absorbance in these studies were controlled
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
by tuning the properties of the MMs themselves (eg., matching the impedance to that of free space as in standard FDTD approach [20]). They have limitations of a narrow band response and lack the flexibility of tuning the absorbance at other frequencies/ angles. We achieve CC by changing the parameters of our structure like the angle of incidence and the thickness of the dielectric layer without changing the MM properties. The structure of the paper is as follows. In Section 2 we present a brief description of our system and the characteristic matrix approach to calculate the total scattering from the structure. In Section 3 we present the numerical results and finally we summarize the main results in the conclusions.
Consider a layered structure as shown in Fig. 1 illuminated by a plane polarized light at an angle. The top and bottom layers are made of MMs, while the middle one is a nonmagnetic dielectric. We first recall some of the important features of a MM slab in the context of omnidirectional reflection. We choose the frequency dependence of the permittivity and permeability of the MM from the experimental work of Shelby et al. [15] ðf Þ = 1−
2 2 fep −feo 2 f −feo + iγf 2
2
μðf Þ = 1−
layered structure. The amplitude reflection (r) and transmission (t) coefficients of such a system are given by r=
ðm11 + m12 pf Þpi −ðm21 + m22 pf Þ ; ðm11 + m12 pf Þpi + ðm21 + m22 pf Þ
t=
2pi ; + m12 pf Þpi + ðm21 + m22 pf Þ
ð1Þ
2
2 f 2 −fmo + iγf
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < i;f = μi;f cosθi;f = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : μ = cosθ i;f i;f i;f
ðfor TE polarizationÞ; ðfor TM polarizationÞ;
ð3Þ
and θi, f are the angles of incidence and emergence in the first and last medium, respectively. The intensity reflection and transmission of the structure are given by R= |r|2 and T = |t|2 (for identical media of incidence and emergence). From energy conservation R + T + A = 1, where A is the absorption in the structure. Thus R + T = 0 implies that all the incident energy is absorbed by the structure. This is what we refer to as critical coupling. 3. Numerical results
;
fmp −fmo
ðm11
ð2Þ
where mij (i, j = 1,2) are the elements of the total characteristic matrix of the structure (expressions for mij can be found, for example, in [23]), pi;f
2. Formulation
4765
;
where fep (fmp) is the electric (magnetic) plasma frequency, feo (fmo) is the electric (magnetic) resonance frequency and γ is the decay rate. Experimental demonstration of negative index behaviour [15] for TE polarization was made using a 2D anisotropic structure. We assume our MM to be 3D and isotropic with macroscopic properties given by Eq. (1). As mentioned in the Introduction, such a slab can exhibit omnidirectional reflectivity in the frequency band between the electric and magnetic plasma frequencies [16]. Within this band the real part of the refractive index was shown to be nearly zero. An increase in the angle of incidence leads to the broadening of the gap with the higher frequency edge shifting to larger frequencies with almost static lower frequency edge. In this paper we use two such MM slabs with distinct set of plasma frequencies, such that the stop band of one falls inside the stop band of the other. Note that the plasma frequencies can be tuned by controlling the filling fraction of the metal wires and split rings in the unit cell structure of the MM [21,22]. We now recall briefly the characteristic matrix approach [2,23] for calculating the intensity reflection (R) and transmission (T) of a
Fig. 1. Schematic of the layered structure.
Our structure (see Fig. 1) consists of two MM layers (top Mt and bottom Mb with thickness d1 and d3, respectively), separated by a dielectric slab of dielectric constant 2, permeability μ2 and thickness d2. The objective is to achieve CC within the smaller stop band of Mt. The structure is embedded in air with i = f = 1, μi = μf = 1. First, we consider the incidence of a TE-polarized plane wave at an angle θi and show later that similar results can be obtained for TM-polarized light also. Note that the spacer layer, due to its higher refractive index compared to its neighbors forms a Fabry–Pérot cavity with its own Airy resonances. These modes are affected by the high losses in the MMs and associated phase change at the interfaces. The signature of these modes are imprinted on the total scattering. The parameters for one of the MMs (labeled as Mb in Fig. 1) are chosen from the experiment of Shelby et al. [15], i.e., fep = 12.8 GHz, fmp = 10.95 GHz, fmo = 10.05 GHz, feo = 10.3 GHz and a = 0.01 GHz. For the other MM (labeled by Mt in Fig. 1) the parameters are the same, except for fep = 11.75 GHz, fmp = 11.4 GHz. The real and imaginary parts of the refractive index (n) of the two MMs are shown in Fig. 2(a) and (b), respectively. The intensity reflection and transmission profiles for slabs of MMs with width d = 15 cm (Mb), 9 cm (Mt) for normal incidence are shown in Fig. 2(c) and (d), respectively. The occurrence of the stop band of Mt inside that of Mb can easily be discerned from these plots. The feature that is crucial for our applications is the near-zero real part of the refractive index and broadband absorption over the stop bands. This is in contrast to the narrow band absorbers used in earlier realizations of CC [8,10,11,17]. Note that it is essential to have absorption for CC. Thus it is the broadband nature of metamaterial absorption that enables tunability of the CC frequency. We now demonstrate CC at single and simultaneously at two frequencies by a suitable choice of the spacer layer thickness. In Fig. 3(a) we show CC near the right band edge (at f = 11.74 GHz) for normal incidence with d2 ∼ 21 cm. As mentioned earlier, the dips in the reflectivity profile at the points P and P1 are due to FP resonances of the dielectric spacer layer. This can be checked in two ways. First, they broaden upon introduction of absorption in the dielectric spacer material (for example Fig. 6). Secondly, the feature at P1 can be recovered again by increasing d2 by (Δd2)1 = c / (2n2f1), which adds a phase of 2π to the reflected wave. Here f1 refers to the CC frequency at P1 and c is the speed of light in free space. Note that combining the FP
4766
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
Fig. 2. Left: (a) Real and (b) imaginary parts of the refractive indices of the two MMs considered — Mt (dashed) and Mb (solid). Right: The reflection (solid) and transmission (dashed) profiles of a single slab as a function of frequency for (c) Mb, d = 15 cm and (d) Mt, d = 9 cm.
resonances with the band edge features makes it easier to achieve CC. It is clear from above discussions that CC near the right band edge can be retained, say, at frequency f1 by increasing/decreasing d2 in steps of
(Δd2)1. This change in d2 alters the phase at the other frequencies. Thus by changing d2, CC can be achieved simultaneously at another frequency f2 when mc / (2n2f1) = pc/ (2n2f2) (m and p are integers).
Fig. 3. R + T (solid) showing CC at normal incidence (a) near the right band edge (d2 = 21.126 cm). (b) Simultaneously at two frequencies near the band edges (d2 = 28.992 cm) (dashed curves of T are provided for reference). R + T (TE polarization) for small change in θi at 0∘ (solid), 2∘ (dashed) and 4∘ (dash–dot) (c) near the right band edge (d) near the left band edge. (c) and (d) have the same parameters as (b).
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
4767
Fig. 4. CC at higher angles of incidence (TE polarization) for d2 = 21.126 cm near the (a) right band edge at θi = 18.5∘ and 29.4∘. (b) Left band edge at θi = 18∘ and 26.5∘. The curve at θi = 0∘ has been reproduced for reference. The CC frequencies at oblique angles have been marked by circles on the frequency axis. The structure parameters are the same as in Fig. 3(a).
This ensures phase change in multiples of π for a single pass through the spacer layer at both the frequencies f1 and f2. Simultaneous CC near the two band edges was achieved in this way (Fig. 3(b)). We now look at the possibility of fine-tuning the CC frequency by varying the angle of incidence θi. We define tunability in the context of certain allowed tolerance. If we take the tolerance level, say, as R + T = 10 − 3, then one can have a rangeof angle over which the total scattering is less than the specified tolerance. Referring to the case of simultaneous CC near both band edges (Fig. 3(b)), there is a fine tunability over a small frequency domain by varying θi within ± 20. Beyond this range, the total scattering rises quickly above the tolerance level. For example, at θi = 4∘, R + T ∼ 10 − 2 (dash–dot curves in Fig. 3(c) and (d)). Note that the plots in Fig. 3(c) and (d) are for TE
polarization. They show the total scattering profile around the higher (Fig. 3(c)) and lower (Fig. 3(d)) CC frequencies in Fig. 3(b). Recalling that a change in the angle of incidence causes a change in the width of the stop band of the MM slab, we show in Fig. 4(a) and (b) that CC can be achieved at different frequencies near the edges of the widened band. We choose the parameters as in Fig. 3(a) where CC occurs for normal incidence near the right band edge (reproduced as solid curves in Fig. 4). We show CC for TE polarization at two angles of incidence near the left (for θi = 18∘, 26.5∘) and right (for θi = 18.5∘, 29.4∘) band edges. We must note that at θi = 18∘ the total scattering (near the right band edge) rises to ∼ 10 − 2 which is higher than the chosen tolerance for R + T. The CC frequencies at the oblique angles have been marked by circles on the frequency axis. It must be noted
Fig. 5. T (dashed) and R + T (solid) showing CC for TM polarization at θi = 14∘ (a) near the right band edge (d2 = 21.126 cm) (b) simultaneously also near the left band edge (d2 = 22.9314 cm). The other parameters are the same as in Fig. 3(a).
4768
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
Fig. 6. R + T with a lossy cavity (2 = 3.8 + 0.05i) showing CC (a) for θi = 0∘ with d2 = 21.54 cm (b) TE (solid, θi = 15∘) and TM (dashed, θi = 12∘) polarizations with d2 = 21.126 cm. T in (a) (dashed) has been provided for reference.
that small changes (b 1∘) to θi (for larger angles of incidence) cause R + T to exceed the tolerance. The discrete tunability of the CC frequencies is a consequence of the discrete FP modes that are excited at discrete angles of incidence. Analogous results hold for TMpolarized light (distinguishable from TE polarization only for oblique incidence) also. Fig. 5(a) shows CC for TM polarization near the right band edge for one of the tunable angles of incidence, namely θi = 14∘ (the other parameters are the same as for Fig. 3(a)). Increasing d2 by eltad2 = 1.9188 cm ensures simultaneous CC of frequencies near both the band edges (see Fig. 5(b)). Figs. 4(a), (b) and 5 thus demonstrate the ease of changing the CC frequency by a mere tilt of the structure with respect to incident light for both polarizations. It is to be noted that CC at oblique incidence, as for normal incidence, originates again from the interplay of the FP resonances of the spacer layer with the stop band features of the MM. It is clear from Figs. 3 and 5 that the structure has a sharp response to the CC frequencies. In order to have a broader response we use a lossy dielectric slab (ε2 = 3.8 + 0.05i) and the FP resonances broaden as would be expected. The broadened FP resonances can be tuned for
CC either by choosing an appropriate cavity width or the angle of incidence. Fig. 6(a) shows CC for the lossy cavity at normal incidence with d2 = 21.54 cm. Retaining the cavity width at d2 = 21.126 cm, CC can be achieved by changing only the angle of incidence (see Fig. 6(b)) for TE (θi = 15∘, solid curve) as well as for TM (θi = 12∘, dashed curve). Absorption of more than 97% takes place for TM(TE) polarization over a frequency range of 12.08 GHz to 12.18 GHz (12.11 GHz to 12.15 GHz). Note that the width of the FP cavity and the angle of incidence were the only parameters needed to tune the CC frequencies rather than the electric or magnetic responses of the MMs. The fact that almost all the incident energy is absorbed in the structure at a CC frequency can be further understood by looking at the field distribution inside the layered structure (Fig. 7). In Fig. 7 we have plotted the magnitude of the electric field amplitude at and slightly away from the CC frequency (see the dip marked by P1 in Fig. 3(a)). The field amplitudes are normalized with respect to the incident field amplitude. It is clear from the figure that at the CC frequency there is an enhancement of the fields in the dielectric
Fig. 7. Field distribution at (f = 11.74 GHz, solid curve) and away (f = 11.8 GHz, dashed curve) from the CC frequency for the parameters in Fig. 3(a). The vertical dashed lines mark the interfaces of the layered structure. (Inset) Schematic of the refractive index distribution and an equvalent picture of a quantum mechanical potential.
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
layer. In order to gain some insight into the field distributions and the resonances of the structure we look at a simplified model (neglecting losses) of an equivalent quantum mechanical scattering problem. The equivalence of one-dimensional optical and quantum mechanical problems is now well understood [24,25]. The refractive index profile and the corresponding potential (ignoring losses) are shown in the inset of Fig. 7. In the frequency range where both the MMs have null (real part) refractive index, the MM layers act like barriers enclosing the well formed by the high index dielectric layer. One thus has the possibility of exciting the bound states of this well (these are the FP modes). The real scenario is much more complicated, because of the losses in the MMs. For example, the barrier represented by the bottom MM is practically impenetrable because of the very high losses. Thus there is practically no transmission through the structure. The ‘bound’ states of the well dig holes in the otherwise total reflection resulting in the CC. Finally, let us compare the present scheme of achieving CC with the earlier methods making use of the narrow absorption resonances of polymer or silver colloid films. The substrate used in the earlier structures was a DBR, while the current one uses a MM layer for broadband reflection. The advantages of using a MM was already pointed out by Bloemer et al. [16]. In the earlier scheme, the role of the spacer layer was mainly to control the phases so that there is destructive interference in the medium of incidence leading to the suppression of reflection. In the present scheme, the role of the spacer layer is twofold, namely it adjusts the phases. At the same time its resonances (there are multitudes of them) hold the key to the CC. The easy control of these resonances leads to the overall flexibility of the CC phenomenon of the total structure. Note that earlier schemes suffered from the rigidity that CC could be achieved only at or near the absorption frequency of the polymer or the colloid film. Moreover, these frequencies need to be within the stop band of the DBR. Now because of the broadband feature of the MM absorption such limitations are removed yielding a structure with tunable CC. 4. Conclusions We have shown critical coupling with TE and TM-polarized light in a layered structure with two metamaterial slabs separated by a dielectric spacer layer. Both the metamaterial slabs are modeled by a full causal response. CC at a single and a pair of frequencies have been demonstrated. Continuous tunability for small angles of incidence and discrete tunability at larger angles have been demonstrated. Tuning the system at different frequencies is achieved by a simple change of the angle of incidence. These flexible attributes are shown to follow
4769
from the omnidirectional stop band features of the metamaterial slabs. We also probed how to broaden the sharp CC features by introducing damping in the dielectric layer. Truly broadband CC is indeed a challenging problem, and is beyond the scope of this paper. CC should be useful for devices where near-perfect absorption of incident light is desired at one or more frequencies.
Acknowledgements The authors are thankful to the Department of Science and Technology, Government of India, for support. One of the authors (SDG) is also thankful to the Nano Initiative Program of University of Hyderabad for partial support.
References [1] P. Yeh, Optical Waves in Layered Media, Wiley, 1988. [2] S. Dutta Gupta, in: E. Wolf (Ed.), Progress in Optics, vol. 38, Elsevier Science, 1998, p. 1. [3] P. Debackere, S. Scheerlinck, P. Bienstman, R. Baets, Opt. Exp. 14 (2006) 7063. [4] V.S.C. Manga Rao, S. Dutta Gupta, G.S. Agarwal, Opt. Lett. 29 (2004) 307. [5] S. Dutta Gupta, Pramana — J. Phys. 72 (2009) 303. [6] J.R. Tischler, M.S. Bradley, V. Bulović, J.H. Song, A. Nurmikko, Phys. Rev. Lett. 95 (2005) 036401. [7] S. Tomita, et al., Opt. Exp. 16 (2008) 9942. [8] J.R. Tischler, M.S. Bradley, V. Bulović, Opt. Lett. 31 (2006) 2045. [9] J.R. Tischler, M.S. Bradley, Q. Zhang, T. Atay, A. Nurmikko, V. Bulović, Org. Electron. 8 (2007) 94. [10] S. Dutta Gupta, Opt. Lett. 32 (2007) 1483. [11] S. Deb, S. Dutta Gupta, J. Banerji, S. Dutta Gupta, J. Opt. A: Pure Appl. Opt. 9 (2007) 555. [12] C. Caloz, T. Itoh, Electromagnetic Metamaterials: Transmission Line theory and Microwave Applications, John Wiley & Sons, 2006. [13] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [14] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. [15] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77. [16] M. Bloemer, G. D'Aguanno, M. Scalora, App. Phys. Lett. 87 (2005) 261921. [17] N.I. Landy, S. Sajuyigbe, J.J. Mock, D.R. Smith, W.J. Padilla, Phys. Rev. Lett. 100 (2008) 207402. [18] Bo Zhu, Zheng-Bin Wang, Zhen-Zhong Yu, Qi Zhang, Jun-Ming Zhao, Yi-Jun Feng, Tian Jiang, Chin. Phys. Lett. 26 (2009) 114102. [19] Z. Zu, W. Lin, L. Kong, Microwave Opt. Technol. Lett. 49 (2007) 1616. [20] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edArtech House, 2005. [21] R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, S. Schultz Appl, Phys. Lett. 78 (2001) 489. [22] J.B. Pendry, A. Holden, D. Robbins, W. Stewart, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075. [23] M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Cambridge University Press, 2001. [24] I. Kay, H.E. Moses, J. Appl. Phys. 27 (1956) 1503. [25] S. Dutta Gupta, G.S. Agarwal, Opt. Exp. 15 (2007) 9614.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Critical coupling in a Fabry–Pérot cavity with metamaterial mirrors Subimal Deb ⁎, S. Dutta Gupta School of Physics, University of Hyderabad, Hyderabad 500046, India
a r t i c l e
i n f o
Article history: Received 22 January 2010 Received in revised form 23 May 2010 Accepted 28 June 2010 Keywords: Critical coupling Fabry–Pérot Metamaterials
a b s t r a c t We study critical coupling (CC) in a system of a dielectric layer sandwiched between two metamaterial layers leading to near-total suppression of both reflection and transmission at specified frequencies. The tunability of the CC frequency is demonstrated by varying the angle of incidence retaining the full causal response for the metamaterials. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In recent years there has been a surge of activities on layered media [1,2]. The research is directed not only to well-studied guided/surface modes but also to novel applications ranging from sensors to slow light, from strong coupling regime of cavity QED to resonant tunneling [3–7]. One of the very interesting applications has been critical coupling (CC) [8–11]. In a critically coupled structure there is near-perfect absorption of incident light, leading to almost complete suppression of both reflection and transmission. CC at a given frequency was first reported by Tischler et al. [8]. They used a 5 nm thick layer of highly absorbing polymer, separated from a dielectric Bragg reflector (DBR) by a spacer layer under normal incidence. The polymer layer was later replaced by a metal colloid film to show the possibility of CC simultaneously at two distinct frequencies [10]. Subsequently these studies were extended to oblique incidence to show the feasibility of CC both for TE and TM-polarized light [11]. The choice and the tunability of the CC frequency are limited by the spectral properties of the absorbers and the DBR. CC occurs as a consequence of the fact that the absorption feature of the lossy film falls within the stop gap of the DBR. With an increase in the angle of incidence the absorption feature remains unaffected, while the stop gap shifts to larger frequencies. Thus a structure designed for CC for normal incidence loses critical coupling at larger angles, since the absorption band now falls beyond the stop gap. In this article we show that both these limitations can be overcome by using metamaterials (MMs). Under metamaterials one now understands a broad class of
⁎ Corresponding author. E-mail addresses: [email protected] (S. Deb), [email protected] (S. Dutta Gupta). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.06.085
“artificial effectively homogeneous electromagnetic structures not readily found in nature” with unusual properties [12]. Historically, research on metamaterials started with the theoretical paper by Veselago, where he investigated the counterintuitive properties of materials (not occurring in nature) with both the permittivity () and permeability (μ) negative [13]. These are now referred to as left handed materials or materials with negative refraction. Intensive research started off from the prediction of perfect lensing and the experimental realization of these materials [14,15]. Now the list of possible applications is truly vast and it is evergrowing. Very recently yet another application of a slab of MM was pointed out by Bloemer et al. [16] in a spectral domain where they do not exhibit negative refraction. It was shown that in the spectral domain between the magnetic and the electric plasma frequencies a MM slab can exhibit a stop-band much like in 1D photonic band gap (PBG) structures. The origin of stop bands in 1D PBG structures is due to constructive interference of the smaller reflections from each interface. In contrast, the stop gap in MMs between the plasma frequencies is due to the predominantly imaginary refractive index (a consequence of opposite signs of Re () and Re(μ)), forbidding the existence of propagating modes. In this article we show that such stop bands of the MM can be exploited effectively to achieve frequency tunable CC. In particular, we study a system of two MM layers (with two distinct sets of parameters) separated by a dielectric layer. The parameters are chosen such that the stop band of one falls inside the stop band of the other. The CC is mediated by the Fabry–Pérot (FP) resonances of the dielectric layer sandwiched between the MM layers. We demonstrate a great flexibility with respect to the choice of the CC frequency, even to the extent of simultaneous CC at two wellseparated frequencies. It may be noted that perfectly absorbing MM structures in the microwave regime have been proposed recently [17–19]. The absorbance in these studies were controlled
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
by tuning the properties of the MMs themselves (eg., matching the impedance to that of free space as in standard FDTD approach [20]). They have limitations of a narrow band response and lack the flexibility of tuning the absorbance at other frequencies/ angles. We achieve CC by changing the parameters of our structure like the angle of incidence and the thickness of the dielectric layer without changing the MM properties. The structure of the paper is as follows. In Section 2 we present a brief description of our system and the characteristic matrix approach to calculate the total scattering from the structure. In Section 3 we present the numerical results and finally we summarize the main results in the conclusions.
Consider a layered structure as shown in Fig. 1 illuminated by a plane polarized light at an angle. The top and bottom layers are made of MMs, while the middle one is a nonmagnetic dielectric. We first recall some of the important features of a MM slab in the context of omnidirectional reflection. We choose the frequency dependence of the permittivity and permeability of the MM from the experimental work of Shelby et al. [15] ðf Þ = 1−
2 2 fep −feo 2 f −feo + iγf 2
2
μðf Þ = 1−
layered structure. The amplitude reflection (r) and transmission (t) coefficients of such a system are given by r=
ðm11 + m12 pf Þpi −ðm21 + m22 pf Þ ; ðm11 + m12 pf Þpi + ðm21 + m22 pf Þ
t=
2pi ; + m12 pf Þpi + ðm21 + m22 pf Þ
ð1Þ
2
2 f 2 −fmo + iγf
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < i;f = μi;f cosθi;f = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : μ = cosθ i;f i;f i;f
ðfor TE polarizationÞ; ðfor TM polarizationÞ;
ð3Þ
and θi, f are the angles of incidence and emergence in the first and last medium, respectively. The intensity reflection and transmission of the structure are given by R= |r|2 and T = |t|2 (for identical media of incidence and emergence). From energy conservation R + T + A = 1, where A is the absorption in the structure. Thus R + T = 0 implies that all the incident energy is absorbed by the structure. This is what we refer to as critical coupling. 3. Numerical results
;
fmp −fmo
ðm11
ð2Þ
where mij (i, j = 1,2) are the elements of the total characteristic matrix of the structure (expressions for mij can be found, for example, in [23]), pi;f
2. Formulation
4765
;
where fep (fmp) is the electric (magnetic) plasma frequency, feo (fmo) is the electric (magnetic) resonance frequency and γ is the decay rate. Experimental demonstration of negative index behaviour [15] for TE polarization was made using a 2D anisotropic structure. We assume our MM to be 3D and isotropic with macroscopic properties given by Eq. (1). As mentioned in the Introduction, such a slab can exhibit omnidirectional reflectivity in the frequency band between the electric and magnetic plasma frequencies [16]. Within this band the real part of the refractive index was shown to be nearly zero. An increase in the angle of incidence leads to the broadening of the gap with the higher frequency edge shifting to larger frequencies with almost static lower frequency edge. In this paper we use two such MM slabs with distinct set of plasma frequencies, such that the stop band of one falls inside the stop band of the other. Note that the plasma frequencies can be tuned by controlling the filling fraction of the metal wires and split rings in the unit cell structure of the MM [21,22]. We now recall briefly the characteristic matrix approach [2,23] for calculating the intensity reflection (R) and transmission (T) of a
Fig. 1. Schematic of the layered structure.
Our structure (see Fig. 1) consists of two MM layers (top Mt and bottom Mb with thickness d1 and d3, respectively), separated by a dielectric slab of dielectric constant 2, permeability μ2 and thickness d2. The objective is to achieve CC within the smaller stop band of Mt. The structure is embedded in air with i = f = 1, μi = μf = 1. First, we consider the incidence of a TE-polarized plane wave at an angle θi and show later that similar results can be obtained for TM-polarized light also. Note that the spacer layer, due to its higher refractive index compared to its neighbors forms a Fabry–Pérot cavity with its own Airy resonances. These modes are affected by the high losses in the MMs and associated phase change at the interfaces. The signature of these modes are imprinted on the total scattering. The parameters for one of the MMs (labeled as Mb in Fig. 1) are chosen from the experiment of Shelby et al. [15], i.e., fep = 12.8 GHz, fmp = 10.95 GHz, fmo = 10.05 GHz, feo = 10.3 GHz and a = 0.01 GHz. For the other MM (labeled by Mt in Fig. 1) the parameters are the same, except for fep = 11.75 GHz, fmp = 11.4 GHz. The real and imaginary parts of the refractive index (n) of the two MMs are shown in Fig. 2(a) and (b), respectively. The intensity reflection and transmission profiles for slabs of MMs with width d = 15 cm (Mb), 9 cm (Mt) for normal incidence are shown in Fig. 2(c) and (d), respectively. The occurrence of the stop band of Mt inside that of Mb can easily be discerned from these plots. The feature that is crucial for our applications is the near-zero real part of the refractive index and broadband absorption over the stop bands. This is in contrast to the narrow band absorbers used in earlier realizations of CC [8,10,11,17]. Note that it is essential to have absorption for CC. Thus it is the broadband nature of metamaterial absorption that enables tunability of the CC frequency. We now demonstrate CC at single and simultaneously at two frequencies by a suitable choice of the spacer layer thickness. In Fig. 3(a) we show CC near the right band edge (at f = 11.74 GHz) for normal incidence with d2 ∼ 21 cm. As mentioned earlier, the dips in the reflectivity profile at the points P and P1 are due to FP resonances of the dielectric spacer layer. This can be checked in two ways. First, they broaden upon introduction of absorption in the dielectric spacer material (for example Fig. 6). Secondly, the feature at P1 can be recovered again by increasing d2 by (Δd2)1 = c / (2n2f1), which adds a phase of 2π to the reflected wave. Here f1 refers to the CC frequency at P1 and c is the speed of light in free space. Note that combining the FP
4766
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
Fig. 2. Left: (a) Real and (b) imaginary parts of the refractive indices of the two MMs considered — Mt (dashed) and Mb (solid). Right: The reflection (solid) and transmission (dashed) profiles of a single slab as a function of frequency for (c) Mb, d = 15 cm and (d) Mt, d = 9 cm.
resonances with the band edge features makes it easier to achieve CC. It is clear from above discussions that CC near the right band edge can be retained, say, at frequency f1 by increasing/decreasing d2 in steps of
(Δd2)1. This change in d2 alters the phase at the other frequencies. Thus by changing d2, CC can be achieved simultaneously at another frequency f2 when mc / (2n2f1) = pc/ (2n2f2) (m and p are integers).
Fig. 3. R + T (solid) showing CC at normal incidence (a) near the right band edge (d2 = 21.126 cm). (b) Simultaneously at two frequencies near the band edges (d2 = 28.992 cm) (dashed curves of T are provided for reference). R + T (TE polarization) for small change in θi at 0∘ (solid), 2∘ (dashed) and 4∘ (dash–dot) (c) near the right band edge (d) near the left band edge. (c) and (d) have the same parameters as (b).
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
4767
Fig. 4. CC at higher angles of incidence (TE polarization) for d2 = 21.126 cm near the (a) right band edge at θi = 18.5∘ and 29.4∘. (b) Left band edge at θi = 18∘ and 26.5∘. The curve at θi = 0∘ has been reproduced for reference. The CC frequencies at oblique angles have been marked by circles on the frequency axis. The structure parameters are the same as in Fig. 3(a).
This ensures phase change in multiples of π for a single pass through the spacer layer at both the frequencies f1 and f2. Simultaneous CC near the two band edges was achieved in this way (Fig. 3(b)). We now look at the possibility of fine-tuning the CC frequency by varying the angle of incidence θi. We define tunability in the context of certain allowed tolerance. If we take the tolerance level, say, as R + T = 10 − 3, then one can have a rangeof angle over which the total scattering is less than the specified tolerance. Referring to the case of simultaneous CC near both band edges (Fig. 3(b)), there is a fine tunability over a small frequency domain by varying θi within ± 20. Beyond this range, the total scattering rises quickly above the tolerance level. For example, at θi = 4∘, R + T ∼ 10 − 2 (dash–dot curves in Fig. 3(c) and (d)). Note that the plots in Fig. 3(c) and (d) are for TE
polarization. They show the total scattering profile around the higher (Fig. 3(c)) and lower (Fig. 3(d)) CC frequencies in Fig. 3(b). Recalling that a change in the angle of incidence causes a change in the width of the stop band of the MM slab, we show in Fig. 4(a) and (b) that CC can be achieved at different frequencies near the edges of the widened band. We choose the parameters as in Fig. 3(a) where CC occurs for normal incidence near the right band edge (reproduced as solid curves in Fig. 4). We show CC for TE polarization at two angles of incidence near the left (for θi = 18∘, 26.5∘) and right (for θi = 18.5∘, 29.4∘) band edges. We must note that at θi = 18∘ the total scattering (near the right band edge) rises to ∼ 10 − 2 which is higher than the chosen tolerance for R + T. The CC frequencies at the oblique angles have been marked by circles on the frequency axis. It must be noted
Fig. 5. T (dashed) and R + T (solid) showing CC for TM polarization at θi = 14∘ (a) near the right band edge (d2 = 21.126 cm) (b) simultaneously also near the left band edge (d2 = 22.9314 cm). The other parameters are the same as in Fig. 3(a).
4768
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
Fig. 6. R + T with a lossy cavity (2 = 3.8 + 0.05i) showing CC (a) for θi = 0∘ with d2 = 21.54 cm (b) TE (solid, θi = 15∘) and TM (dashed, θi = 12∘) polarizations with d2 = 21.126 cm. T in (a) (dashed) has been provided for reference.
that small changes (b 1∘) to θi (for larger angles of incidence) cause R + T to exceed the tolerance. The discrete tunability of the CC frequencies is a consequence of the discrete FP modes that are excited at discrete angles of incidence. Analogous results hold for TMpolarized light (distinguishable from TE polarization only for oblique incidence) also. Fig. 5(a) shows CC for TM polarization near the right band edge for one of the tunable angles of incidence, namely θi = 14∘ (the other parameters are the same as for Fig. 3(a)). Increasing d2 by eltad2 = 1.9188 cm ensures simultaneous CC of frequencies near both the band edges (see Fig. 5(b)). Figs. 4(a), (b) and 5 thus demonstrate the ease of changing the CC frequency by a mere tilt of the structure with respect to incident light for both polarizations. It is to be noted that CC at oblique incidence, as for normal incidence, originates again from the interplay of the FP resonances of the spacer layer with the stop band features of the MM. It is clear from Figs. 3 and 5 that the structure has a sharp response to the CC frequencies. In order to have a broader response we use a lossy dielectric slab (ε2 = 3.8 + 0.05i) and the FP resonances broaden as would be expected. The broadened FP resonances can be tuned for
CC either by choosing an appropriate cavity width or the angle of incidence. Fig. 6(a) shows CC for the lossy cavity at normal incidence with d2 = 21.54 cm. Retaining the cavity width at d2 = 21.126 cm, CC can be achieved by changing only the angle of incidence (see Fig. 6(b)) for TE (θi = 15∘, solid curve) as well as for TM (θi = 12∘, dashed curve). Absorption of more than 97% takes place for TM(TE) polarization over a frequency range of 12.08 GHz to 12.18 GHz (12.11 GHz to 12.15 GHz). Note that the width of the FP cavity and the angle of incidence were the only parameters needed to tune the CC frequencies rather than the electric or magnetic responses of the MMs. The fact that almost all the incident energy is absorbed in the structure at a CC frequency can be further understood by looking at the field distribution inside the layered structure (Fig. 7). In Fig. 7 we have plotted the magnitude of the electric field amplitude at and slightly away from the CC frequency (see the dip marked by P1 in Fig. 3(a)). The field amplitudes are normalized with respect to the incident field amplitude. It is clear from the figure that at the CC frequency there is an enhancement of the fields in the dielectric
Fig. 7. Field distribution at (f = 11.74 GHz, solid curve) and away (f = 11.8 GHz, dashed curve) from the CC frequency for the parameters in Fig. 3(a). The vertical dashed lines mark the interfaces of the layered structure. (Inset) Schematic of the refractive index distribution and an equvalent picture of a quantum mechanical potential.
S. Deb, S. Dutta Gupta / Optics Communications 283 (2010) 4764–4769
layer. In order to gain some insight into the field distributions and the resonances of the structure we look at a simplified model (neglecting losses) of an equivalent quantum mechanical scattering problem. The equivalence of one-dimensional optical and quantum mechanical problems is now well understood [24,25]. The refractive index profile and the corresponding potential (ignoring losses) are shown in the inset of Fig. 7. In the frequency range where both the MMs have null (real part) refractive index, the MM layers act like barriers enclosing the well formed by the high index dielectric layer. One thus has the possibility of exciting the bound states of this well (these are the FP modes). The real scenario is much more complicated, because of the losses in the MMs. For example, the barrier represented by the bottom MM is practically impenetrable because of the very high losses. Thus there is practically no transmission through the structure. The ‘bound’ states of the well dig holes in the otherwise total reflection resulting in the CC. Finally, let us compare the present scheme of achieving CC with the earlier methods making use of the narrow absorption resonances of polymer or silver colloid films. The substrate used in the earlier structures was a DBR, while the current one uses a MM layer for broadband reflection. The advantages of using a MM was already pointed out by Bloemer et al. [16]. In the earlier scheme, the role of the spacer layer was mainly to control the phases so that there is destructive interference in the medium of incidence leading to the suppression of reflection. In the present scheme, the role of the spacer layer is twofold, namely it adjusts the phases. At the same time its resonances (there are multitudes of them) hold the key to the CC. The easy control of these resonances leads to the overall flexibility of the CC phenomenon of the total structure. Note that earlier schemes suffered from the rigidity that CC could be achieved only at or near the absorption frequency of the polymer or the colloid film. Moreover, these frequencies need to be within the stop band of the DBR. Now because of the broadband feature of the MM absorption such limitations are removed yielding a structure with tunable CC. 4. Conclusions We have shown critical coupling with TE and TM-polarized light in a layered structure with two metamaterial slabs separated by a dielectric spacer layer. Both the metamaterial slabs are modeled by a full causal response. CC at a single and a pair of frequencies have been demonstrated. Continuous tunability for small angles of incidence and discrete tunability at larger angles have been demonstrated. Tuning the system at different frequencies is achieved by a simple change of the angle of incidence. These flexible attributes are shown to follow
4769
from the omnidirectional stop band features of the metamaterial slabs. We also probed how to broaden the sharp CC features by introducing damping in the dielectric layer. Truly broadband CC is indeed a challenging problem, and is beyond the scope of this paper. CC should be useful for devices where near-perfect absorption of incident light is desired at one or more frequencies.
Acknowledgements The authors are thankful to the Department of Science and Technology, Government of India, for support. One of the authors (SDG) is also thankful to the Nano Initiative Program of University of Hyderabad for partial support.
References [1] P. Yeh, Optical Waves in Layered Media, Wiley, 1988. [2] S. Dutta Gupta, in: E. Wolf (Ed.), Progress in Optics, vol. 38, Elsevier Science, 1998, p. 1. [3] P. Debackere, S. Scheerlinck, P. Bienstman, R. Baets, Opt. Exp. 14 (2006) 7063. [4] V.S.C. Manga Rao, S. Dutta Gupta, G.S. Agarwal, Opt. Lett. 29 (2004) 307. [5] S. Dutta Gupta, Pramana — J. Phys. 72 (2009) 303. [6] J.R. Tischler, M.S. Bradley, V. Bulović, J.H. Song, A. Nurmikko, Phys. Rev. Lett. 95 (2005) 036401. [7] S. Tomita, et al., Opt. Exp. 16 (2008) 9942. [8] J.R. Tischler, M.S. Bradley, V. Bulović, Opt. Lett. 31 (2006) 2045. [9] J.R. Tischler, M.S. Bradley, Q. Zhang, T. Atay, A. Nurmikko, V. Bulović, Org. Electron. 8 (2007) 94. [10] S. Dutta Gupta, Opt. Lett. 32 (2007) 1483. [11] S. Deb, S. Dutta Gupta, J. Banerji, S. Dutta Gupta, J. Opt. A: Pure Appl. Opt. 9 (2007) 555. [12] C. Caloz, T. Itoh, Electromagnetic Metamaterials: Transmission Line theory and Microwave Applications, John Wiley & Sons, 2006. [13] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [14] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. [15] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77. [16] M. Bloemer, G. D'Aguanno, M. Scalora, App. Phys. Lett. 87 (2005) 261921. [17] N.I. Landy, S. Sajuyigbe, J.J. Mock, D.R. Smith, W.J. Padilla, Phys. Rev. Lett. 100 (2008) 207402. [18] Bo Zhu, Zheng-Bin Wang, Zhen-Zhong Yu, Qi Zhang, Jun-Ming Zhao, Yi-Jun Feng, Tian Jiang, Chin. Phys. Lett. 26 (2009) 114102. [19] Z. Zu, W. Lin, L. Kong, Microwave Opt. Technol. Lett. 49 (2007) 1616. [20] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edArtech House, 2005. [21] R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, S. Schultz Appl, Phys. Lett. 78 (2001) 489. [22] J.B. Pendry, A. Holden, D. Robbins, W. Stewart, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075. [23] M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Cambridge University Press, 2001. [24] I. Kay, H.E. Moses, J. Appl. Phys. 27 (1956) 1503. [25] S. Dutta Gupta, G.S. Agarwal, Opt. Exp. 15 (2007) 9614.