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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17736–17747
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Studies of electromagnetically induced transparency in metamaterials

Hua Xu, Yuehui Lu, YoungPak Lee, and Byoung Seung Ham  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17736-17747 (2010)
http://dx.doi.org/10.1364/OE.18.017736


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Abstract

We have studied electromagnetically induced transparency (EIT) in metamaterials for various schemes corresponding to those in an atomic medium. We numerically calculate a symmetric dolmen scheme of metamaterials corresponding to a tripod model of EIT-based optical switching and illustrate plasmonic double dark resonances. Our study provides a fundamental understanding and useful guidelines in using metamaterials for plasmonic-based all-optical information processing.

© 2010 OSA

1. Introduction

Surface plasmon polaritons (SPPs) have been intensively studied from a simple transmission scheme of SPPs in a narrow metal strip interfacing with a dielectric material [1

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

] to complicated SPP device applications [2

2. M. Schnell, A. García-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3(5), 287–291 (2009). [CrossRef]

]. Localized surface plasmons have demonstrated interesting physics of resonance and fluorescence [3

3. T. Ha, Th. Enderle, D. F. Ogletree, D. S. Chemla, P. R. Selvin, and S. Weiss, “Probing the interaction between two single molecules: fluorescence resonance energy transfer between a single donor and a single acceptor,” Proc. Natl. Acad. Sci. U.S.A. 93(13), 6264–6268 (1996). [CrossRef] [PubMed]

,4

4. S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). [CrossRef] [PubMed]

]. Emerging research in plasmonic mode control by means of light offers a step toward light-controlled nano optics in metamaterials. A recent observation of optical field-based SPP switching is a good example of light controlled plasmonics, where the physics lies in the saturation phenomenon [5

5. D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007). [CrossRef]

]. The saturation phenomenon, however, is limited in the strong field. Nano photonics inherently works at a low power limit.

Electromagnetically induced transparency (EIT) is a direct result of destructive quantum interference between two pathways induced by another light [6

6. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]

8

8. M. C. Phillips and H. Wang, “Spin coherence and electromagnetically induced transparency via exciton correlations,” Phys. Rev. Lett. 89(18), 186401 (2002). [CrossRef] [PubMed]

], where zero absorption probability at line center is obtained [9

9. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherence media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]

]. Thus, EIT is free from the saturation phenomenon and is applied for ultra efficient nonlinear optics [10

10. P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. 20(9), 982–984 (1995). [CrossRef] [PubMed]

,11

11. S. E. Harris and L. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82(23), 4611–4614 (1999). [CrossRef]

]. Group velocity control based on EIT has also been intensively studied for fundamental physics [12

12. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

,13

13. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2002). [CrossRef] [PubMed]

], as well as for various applications such as entanglement generation [14

14. M. G. Payne and L. Deng, “Quantum entanglement of Fock states with perfectly efficient ultraslow single-probe photon four-wave mixing,” Phys. Rev. Lett. 91(12), 123602 (2003). [CrossRef] [PubMed]

] and photon logic gates [15

15. B. S. Ham and J. Hahn, “Observation of ultraslow light-based photon logic gates: NAND/OR,” Appl. Phys. Lett. 94(10), 101110 (2009). [CrossRef]

]. Hence, EIT can be applied for surface plasmon-based nano photonics requiring low light power.

Recently, classical EIT based on the localized SPPs in metamaterials has been demonstrated in both theory and experiment [16

16. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef] [PubMed]

,17

17. N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. 9(4), 1663–1667 (2009). [CrossRef] [PubMed]

]. Based on these results, we have studied EIT-like phenomena for two-, three- and four-level models using the finite-difference time-domain (FDTD) method. More interestingly, we illustrated plasmonic double dark resonances in our four-level tripod structure and this is explained as a result of perturbation by which the dark state destroys the transparency. It provides useful guidelines for plasmonic EIT in metamaterials and paves the way for its applications in delay lines, slow-light devices, selective storage, and optical switching.

2. Simulation method

For the study of EIT in metamaterials, we use the FDTD method [18

18. A. Taflove, and S. C. Hagness, Computational electrodynamics: The finite-difference time–domain method (Artech, 2000).

] for numerical calculation, using Meep, a free software package [19

19. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31(20), 2972–2974 (2006). [CrossRef] [PubMed]

,20

20. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]

]. The permittivity of silver was modeled using the Drude formula: εm=1ωp2/[ω(ω+iγ)], where the electric plasma frequency ωp and the scattering frequency γ are 1.366×1016 rad/s and 3.07×1013 Hz, respectively [21

21. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]

,22

22. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmonic metamaterial with coupling induced transparency,” arXiv: 0801.1536v1 [physics. optics].

]. The dispersion is implemented by the Drude-Lorentz model in Meep [23]: ε(ω,x)=[1+iσD(x)/ω][ε(x)+nσn(x)ωn2/(ωn2ω2iωγn)], where σD is the electric conductivity, ωn and γn are user-specified, and σn(x) is a user-specified function of position giving the strength of the n-th resonance. To implement the Drude model, we take n=1, σD=0, ε(x)=1, very small ω1, and the strength of resonance, σ1, large to cancel ω1 in the numerator. For the unit cell, the perfectly matched layer absorbing boundary conditions are imposed in the propagation direction of light, and the Bloch-periodic boundary conditions are imposed on the other two directions perpendicular to the propagation axis. We discretized the unit cell including the metal strips and the surroundings with a high resolution, corresponding to 158.3 pixels/wavelength, in order to access to a fine meshing and a satisfactory convergence. The unit cells are arranged periodically, and the spatial separation of the unit cells (e.g., 200 nm in the case of the four-level structure) are such that the in-plane couplings are negligible and only zeroth-order transmission is investigated free from diffraction.

For the atom systems, simulations were carried out with a density-matrix approach. The time-dependent density matrix equations were derived from the interaction Hamiltonian.

3. Two-level system

To study EIT-like features in a metamaterial, we study a dipole-allowed transition in a single metal strip as a basic building block. Hecht et al. proposed that a single metal strip is likely to be resonantly excited by incident light [24

24. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef] [PubMed]

]. The electrons of the metal strip are driven by the time-varying electric field of light into a collective oscillation (SPP) that formed a standing wave with the incident light which works as an optical dipole antenna [25

25. Q. H. Park, “Optical antennas and plasmonics,” Contemp. Phys. 50(2), 407–423 (2009). [CrossRef]

]. In other words, this type of antenna adequately couples with the optical field, where a two-level atomic system is resonantly excited by the optical field whose energy is the same as the atom system. The Hamiltonian and the time varying density matrix operator are given by:
H={δP|22|12ΩP|21|+H.c.},
(1)
ρ˙=i[H,ρ]+decay,
(2)
where ρ is a density matrix operator: ρ=|ΨΨ|; |Ψdenotes the state vector.

From the Liouville equation, we obtain the following equation for the density-matrix elements:
ρ˙21=i2ΩP(ρ11ρ22)+iδPρ12γ21ρ21,
(3)
whereδP=ω21ωP,Ωi is the Rabi frequency of the laser beam; ωi, andγij andρijare angular frequency, decay rate and density matrix element for the transition from |i to |j, respectively. Owing to the similarity of resonant excitation between the metal strip and the two-level atom system, we show that the resonant frequency of the metal strip ωres is analogous to the energy difference ω12 between states |1 and |2 in the two-level atomic system, where ωres is mainly determined by the length of the metal strip. Here we present a systematic mapping of the plasmon resonance of the rectangular dipole antenna onto a corresponding atomic model, illustrating the individual dependence of the width A and the length B over the typical range of the nanoantennas (A100  nm,  B350   nm), as described in Fig. 1
Fig. 1 (a) Schematic of the silver strip. The geometric parameters A and B are the width and length, respectively. An external light incident normal to the strip, and E-field polarized along the strip. (b) The dependence of the resonance wavelength on length B for different A. (c) The dependence of the resonance wavelength on width A for fixed lengthA=128  nm. The inset of Fig. 1 (c) shows the line shape for different widths: 20, 50, 100 nm. (d) The dependence of the resonance wavelength on aspect ratioB/Afor different A.
. The resonant wavelength of the rectangular antennas is shown in Fig. 1(b) as the function of the strips’ length B forA=20,50,80  nm. The plasmon resonance linearly depends on the length B. The dependence of the resonant wavelength on the width A is presented in Fig. 1(c), where the resonant wavelength is inversely proportional to A, resulting in a blue shift which agrees with the expectation that the resonant wavelength depends only on the aspect ratio in a quasistatic limit (i.e., the frequency of plasmon resonance only depends on the antenna shape via the aspect ratio, L/2R, if λL,R) [26

26. G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8(2), 631–636 (2008). [CrossRef] [PubMed]

]. As a general rule, the validity of quasistatic limit for the nanoantennas is also inspected, as shown in Fig. 1(d). Not all curves converge upon a common universal curve, which implies that the quasistatic limit does not hold perfectly here. However, we note that the divergence is relatively small at first and becomes larger with the increase of the aspect ratio. This discrepancy means that the approximation of the quasistatic limit is applicable only when the strips have a small aspect ratio.

For the studies of corresponding relations using a two-level atomic system as shown in Fig. 2(a)
Fig. 2 (a) Atom-field interaction diagram in a two-level system. (b) The imaginary part of ρ21spectra, in arbitrary units, for two level system with parameters:ΩP=1,Γ=0.5,δP=0andγ=4,5,6,7,8. (c) The imaginary part of ρ21spectra, in arbitrary units, for a two-level system with the same parameters as (b) exceptδP=0,2,4,5.5,6.5. All parameters are in units of Ω.
, we perform numerical simulations in Fig. 2. In Fig. 2(b), the absorption line becomes broadened as the decay rate γ increases, corresponding to that of atomic system in Fig. 1(c). Thus γ in the atomic system is analogous to A in rectangular nanoantennas. The blue shift corresponds to the increase of the detuning δ in Fig. 2(c). Thus, we can derive in the rectangular nanoantennas that a larger width A gives a bigger blue shift. As mentioned above, the decay of metallic nanoantennas results from both non-radiative and radiative damping [27

27. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]

], where the former is due to intrinsic metal loss, and the latter is due to strong dipole characteristics. The intrinsic metal loss cannot be altered by varying the shape of the nanoantennas. Therefore, the radiative damping shows the possibility of the broadening and attenuation, since the dipole characteristics are likely to change with increasing width A.

4. Three-level EIT system

The two-level system of Fig. 2(a) can be developed for a three-level scheme for EIT after adding a metastable state|3 (see Fig. 3(b)
Fig. 3 (a) The schematic of the plasmonic EIT system. A=60  nm, B=162  nm, L=118  nm, D=30  nm, Received 28 Jun 2010; revised 25 Jul 2010; accepted 27 Jul 2010; published 2 Aug 2010, s=30  nm. The thickness of each strip is 20 nm. (b) Atom-field interaction diagram of EIT atom system.
). The transitions|1|2 and |2|3 are dipole allowed, while |1|3 is dipole forbidden. Consequently, two possible pathways to dressed states induced by the coupling field ΩC lenders the probe field ΩP transparent via destructive quantum interference, otherwise absorptive. Cho et al. [28

28. D. J. Cho, F. Wang, X. Zhang, and Y. R. Shen, “Contribution of the electric quadrupole resonance in optical metamaterials,” Phys. Rev. B 78(12), 121101 (2008). [CrossRef]

] suggested that two magnetically excited parallel strips can work as the dark atom, since it cannot be excited directly by the external field but magnetically coupled with the bright state (the single strip). When the resonant wavelength of the two parallel strips is tuned to that of the single strip [16

16. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef] [PubMed]

], they interact with each other, which is analogous to two-photon resonance process: the excited atom fall down from state |1 to state |3, then pumped to state |1again. HereδP=δC=0, δP=ω12ωP and δC=ω13ωC are the detunings of the probeωP and the pump beam ωC, respectively .

For the three-level system, within the dipole approximation the atom-light interaction, Hint=μE, is often expressed in terms of the Rabi frequency Ω=μE0/, with E0 being the amplitude of the electric field E, and μ the transition electronic dipole moment. After introducing the rotating-wave approximation, the Hamiltonian of the three-level atomic system interacting with a coupling laser whose Rabi-frequency is ΩC and with a probe laser whose Rabi-frequency isΩPcan be expressed as:

H={δP|22|δC|33|12(ΩP|21|+ΩC|31|)+H.c.},
(4)
ρ˙12=i2ΩP(ρ11ρ22)+i2ΩCρ32+iδPρ12γ21ρ12,
(5)

Decreasing the length B(other parameters are fixed) in Fig. 3(a) resembles the enlarged energy difference between |2 and |1 (i.e., the elevated energy level|1) and the energy difference between |3 and |1 as well. As a result, both ω12 and ω13 are increased, which means an increase of both δP andδC. In contrast, the increased B gives rise to the decrease of both δP andδC, as shown in Figs. 4(a)
Fig. 4 The transmission spectra for (a) different single strip length B and (c) different strip width A, while other parameters are fixed. (b) and (d) Numerical simulations for the corresponding atomic EIT model.
and 4(b). The positions of the dip are nearly invariant, even though the feature of EIT becomes asymmetric. The position of the EIT feature is unambiguously determined by the frequency of the coupling field ωC in the atomic system, other thanω12 andω13. Furthermore, when the width A increases, the resonant peak, like a two-level system, the shifts are towards the higher frequencies (i.e. blue shift), and the absorption peak becomes broader, which resembles the increased detuning and decay, respectively, as shown in Figs. 4(c) and 4(d). Here, since the geometric parameters of the two parallel strips, behaving as the coupling field in the metamaterial, are not varied, the peak position of EIT remains unchanged. Thus, the length of the two parallel strips determines the peak position of EIT and they are linearly dependent, as shown in Figs. 5(a)
Fig. 5 (a) The transmission spectra for the different length of the two parallel strips, which is varied from 98 nm to 130 nm. (c) The transmission spectra for different separation D=30,50,70,90  nm. (b) and (d) Numerical simulation for the corresponding atomic EIT model.
and 5(b). Following this picture, stretching the two parallel strips gives rise to the red shift of EIT, resembling the decreased frequency of the coupling field ωC, which equivalently increased δC; shrinking these strips causes the blue shift as the increasedωC or the decreasedδC. The EIT-like feature in the metamaterial [see Fig. 3(a)] arising from the interaction between the single strip and the two parallel strips implies the possibility for the coupling strength to be adjusted by controlling their spatial distance D. The larger the separation D, the weaker the interaction becomes. The interaction between the single strip and parallel strips resembles the Rabi frequency of the coupling field ΩCin an atomic system of Fig. 3(b) [see Fig. 5 (d)]. In contrast, the smaller distance D represents a better transparency window, as shown in Fig. 5(c). To have an entire and quick view, the parametric comparison is carried out in Table 1

Table 1. Parametric Comparison between Metamaterial and Atom

table-icon
View This Table
.

5. Four-level tripod system

In metamaterials, not only can a three-level-like system be established, but also a four-level-like system with a tripod configuration [29

29. H. Xu, and B. S. Ham, “Plasmon-induced photonic switching in a metamaterial,” arXiv: 0905.3102v4 [quant-ph].

]. In a tripod system, the EIT-induced single dark resonance is swapped for another dark resonance when a third light ΩA is applied, known as coherence swapping [30

30. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: quantum switching,” Phys. Rev. Lett. 84(18), 4080–4083 (2000). [CrossRef] [PubMed]

,31

31. B. S. Ham, “Experimental demonstration of all-optical 1x2 quantum routing,” Appl. Phys. Lett. 85(6), 893–896 (2004). [CrossRef]

] or simply double dark resonance [32

32. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81(17), 3611–3614 (1998). [CrossRef]

,33

33. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

]. As a result, at line center of EIT, the quantum interference-based absorption cancellation results in strong absorption [30

30. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: quantum switching,” Phys. Rev. Lett. 84(18), 4080–4083 (2000). [CrossRef] [PubMed]

33

33. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

]. Here we consider replacing the third light ΩA with another magnetically excited metallic pair located at the right wing.

Figure 6
Fig. 6 (a) The schematic of the plasmonic tripod system; All parameters are same as Fig. 3(a) except the separation distance, D=40  nm, and the varied lengths of parallel strips. (b) Atom-field interaction diagram of tripod system. (c) Numerical analysis for the plasmonic tripod model for (a). (d) Numerical analysis for the corresponding atomic tripod model.
shows the numerical simulations for a tripod model of metamaterial and a corresponding atomic system. Figure 6(c) presents the simulation results for the metamaterial of Fig. 6(a). Figure 6(d) presents the corresponding tripod system of Fig. 6(b) using the following equations:
H={δC|11|δP|22|δA|33|12(ΩC|14|+ΩP|24|+ΩA|34|)+H.c.},
(6)
ρ˙24=iΩP(ρ22ρ44)iΩAρ23iΩCρ21iδPρ24γ24ρ24,
(7)
where δC=ω13ωC, δP=ω12ωP, δA=ω14ωA. By adding an additional laser beam ωA and a ground state|4 into the Λ type model of EIT composed of states |1,|2, and |3, we obtain degenerate dark resonance. Theoretical studies in a tripod model have been intensively performed [31

31. B. S. Ham, “Experimental demonstration of all-optical 1x2 quantum routing,” Appl. Phys. Lett. 85(6), 893–896 (2004). [CrossRef]

,32

32. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81(17), 3611–3614 (1998). [CrossRef]

]. For a special case of no detuningδA=δC=0, the degenerate dark state (|d1and |d2) at line center [see the first row of Fig. 6(d)] is obtained analytically as follows:
|d1=c1{ΩA|1ΩC|3},
(8)
|d2=c2{ΩpΩC|1+ΩpΩA|3(ΩC2+ΩA2)|2},
(9)
where the dark state |d2 becomes equivalent to the EIT mode or single dark state (|d) if the control light ΩA is removed: d|=c(ΩP|1ΩC|2). However, at both sidebands, the interaction Hamiltonian results in bright modes|b±, showing absorption:
|b±=1Ω(ΩC|1+ΩP|2+ΩA|3±Ω|4),
(10)
where Ω=ΩC2+ΩA2+ΩP2. The energy separation between two bright modes (i.e., absorption peaks) is determined by the generalized Rabi frequency Ω.

The metamaterial with equal length (L1=L2=L) of symmetric double parallel metal strips in the first row of Fig. 6(c) satisfies a degenerate tripod atomic system, which proves the degeneracy in Eqs. (9) and (10): see the red dash curves in the Fig. 0.6 (without the control fieldΩA). As expected, the peak-to-peak separation determined by Ω is wider than the separation determined by EIT by2 (Ω2ΩC). The general solution of Eq. (10) with a symmetric detuning (δC=δA=δ) can also be obtained from numerical simulations by introducing an asymptotic relationship with a detuning δ as follows:
e±=±Ω21+2(δΩ)2,
(11)
where the peak to peak energy separation must be greater than that obtained from Eq. (10): see Figs. 6 and 7
Fig. 7 Numerical analysis of double tripod systems composed of M1 and M2 for (a) metamaterials and (b) atomic system. For M1,L1=114  nm(blue dash dot)] ; For M2,L2=122  nm (red dot)]; Black line is for averaged sum.
.

We now discuss double dark resonance, which gives an absorption enhancement otherwise dark state of Eqs. (8) and (9) when detuning δ is introduced: the length of the double parallel metal strips L1 and L2 changes. Suppose a length change of the metallic pairs, L1 and L2, in an opposite direction: L1=LΔL andL2=L+ΔL, respectively. The fourth row in Fig. 6(c), shows respectively a single-dark mode (red dash line) and a double-dark mode (black line) of the localized plasmon interactions induced by a monochromatic light vertically incident on the metamaterial, where the scheme corresponds to a detuned tripod optical system interacting with three lights, as shown in Fig. 6(b). The double-dark mode [see black solid line in the fourth row of Fig. 6(d)] results in absorption enhancement at line center. The single dark modes appear at the side bands of the double-dark mode (probe resonance transition), where each position (energy splitting) is determined by each detuning ΩCor ΩA.

In the three-level system, we have already illustrated that the reduced length of L results in a blue shift of the detuning. For the second row of Fig. 6(c), where L1=LΔL and L2=L(ΔL=4  nm), a weak absorption enhancement appears near line center. Due to the detuning effect of the controlΩA, the spectral position of the double dark resonance shifts slightly to the right. Here the dark states (EIT transparency) appear at both δP=0 and δP=δC. The eigenvalue of the double dark state (enhanced absorption) is determined by the δC,ΩA, and ΩC. On the third row of Fig. 6(c), the length of the parallel double metal strips on the right wing is increased by ΔL(L2=L+ΔL). Because the detuning is still within the Rabi frequency (δA=0.3Ω;δC=0.3Ω; the corresponding Rabi frequency is analyzed through the simulations), the quantum coherence can still be sustained by inducing the double dark resonance resulting in absorption enhancement at line center. For the rest of Fig. 6(c), all the parameters remain the same unless otherwise indicated. Compared with the second row, the frequency shift of the probe on the third row is zero due to the balanced detuning. As the length of the double parallel metal strips continues to change, the system behaves noncoherently indicating a broadened absorption linewidth at line center (see the fifth row). All the corresponding figures in the right column for a tripod atomic model match well those in the left column.

In Fig. 7, we discuss whether the enhanced absorption results from the sum of two single classical oscillators. In the inset of Fig. 7 we assume that the tripod model (M) can be decomposed into two independently detuned EIT models (M1and M2). Figures 7(a) and 7(b) are for metamaterial and atomic systems, respectively. Then the average sum of both detuned EIT spectra is compared with that of the tripod model. As a result, each probe spectrum for the detuned model shows a mirror image of the other across line center [see the red dotted and blue dash-dot curves in Fig. 7(a) and 7(b)]. Compared with the third row of Fig. 6 (c), no absorption enhancement appears in the sum spectrum at line center in Fig. 7(a) and 7(b). Clearly, the enhancement cannot be attributed to the simple superposition of two single classical oscillators.

In order to provide a qualitative explanation of the enhanced absorption, the classical oscillator picture is applied to the four-level system. The model consists of three damped harmonic oscillators follows [34

34. J. Harden, A. Joshi, and J. D. Serna, arXiv:1006.5167v1 [quant-ph].

]: One of the oscillators of mass m1 (the particle 1) is driven by a harmonic force F=F0ei(ωx+ϕ) and attached to a wall by a spring with a force constant k1; the other two of m2 and m3 (the particles 2 and 3) are attached to it by springs with force constants k12 and k13, respectively; and they are fixed from the other side to a wall by springs with force constants k2 andk3, respectively. Alzar et al. indicated that the power, absorbed by the driven oscillator (i.e., the particle 1) as a function of the frequency ω, presents a EIT-like profile considering the particles 1 and 2, which resembles a three-level system [35

35. C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70(1), 37–41 (2002). [CrossRef]

]. Compared with their model, the particle 3 is involved here to take into account another dark mode when a third light ΩA is applied. Setting ϕ=0 in the driven force and m1=m2=m3=m for simplicity without any loss of generality, the equations of motion are written as follows,
x¨1(t)+γ1x˙1(t)+ω12x1(t)ΩC2x2(t)ΩA2x3(t)=F0meiωt,
(12a)
x¨2(t)+γ2x˙2(t)+ω22x2(t)ΩC2x1(t)=0,
(12b)
x¨3(t)+γ3x˙3(t)+ω32x3(t)ΩA2x1(t)=0,
(12c)
where the detunings of the probe, the coupling, and the pumping fields are defined as δP=ω1ω, δC=ω2ω, and δA=ω3ω, respectively. The coupling strengths are written as ΩC2=k12/m and ΩA2=k13/m. The friction constants associated with the energy dissipation are denoted to γ1, γ2, and γ3. Suppose the solutions have the time-harmonic form of x1=Nleiωt, where Nl are the constants and l=1,2,3. After some algebraic calculations on Eq. (12), finally, the absorbed power during one period of the driven force is expressed as

P(ω)=2πiF2ωm[(ω12ω2iγ1ω)(ΩC4ω22ω2iγ2ω+ΩA4ω32ω2iγ3ω)].
(13)

Equation (13) shows that the real part of P(ω) has the characteristic of Autler-Townes effect without any couplings. In contrast, the profile is dramatically modified when two couplings, ΩC and ΩA, take effect. As a result, a double dip-shape evolves into a triple dip-shape as shown in Fig. 6, arising from the double dark resonance. According to these numerical calculations, it is reasonable to assume ΩCΩA and γ2γ3. The enhanced absorption becomes evident when the detunings of the coupling and the pumping field have the opposite signs (i.e., δC=δA). This can be understood as a result of destructive interaction of two dark modes according to Eq. (13), since the sum of the last term in the denominator is close to zero considering the finite detunings (e.g., within the Rabi frequency), as shown in Fig. 8
Fig. 8 Sum of the last term in the denominator of Eq. (13). The parameters have the values: F/m=0.1 force per mass units, γ1=4×102, γ2=γ3=1×107, and ΩC=ΩA=0.8.
. Therefore, the effects of the coupling and the pumping fields on the bright atom can be negligible and the absorbed power at the resonant frequency has a similar value to that of Autler-Townes peak, as shown in Fig. 9
Fig. 9 Real part of the absorbed power, P(ω) of Eq. (10). The detunings of the coupling and the pump fields have the same amplitudes in opposite signs in the four-level tripod system, where deltaδC=+0.085 and δA=+0.085. Two individual three-level systems are presented for comparison purpose when δC=+0.085or δA=+0.085, as well as the two-level system without any couplings. Here other parameters are the same as Fig. 8.
. As a result, the transparency was destroyed by the perturbation.

The plasmonic tripod system discussed in Fig. 6 might be attractive for selective storage of excited energy into either the left dark cell or the right cell. Specifically, the field can be concentrated into the left cell if the frequency of incident light is equal to the resonant frequency of the left cell; if the frequency of the incidence light is tuned to the same frequency as that of the right cell, the field can be localized into the right cell. This outcome may have application in plasmon-based photonic switching.

6. Conclusion

We numerically analyzed single dark (EIT) and double dark (EIT-based photonic switching) states in metamaterials compared with corresponding atomic models. From numerical calculations, geometrical parameters of the dolmen shaped metal strips can be used for absorption control of incident light for plasmonic switching. Based on the comparison, a plasmon-induced tripod model is presented for a double dark resonance. This model may be applied for the selective storage of excited energy and plasmon-based all-optical information processing.

Acknowledgements

We acknowledge that this work was supported by the Center for Photon Information Processing of the Korean Ministry of Education, Science and Technology via KOSEF. We thank S. Zhang and X. Zhang of UC Berkeley for helpful discussions.

References and links

1.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

2.

M. Schnell, A. García-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3(5), 287–291 (2009). [CrossRef]

3.

T. Ha, Th. Enderle, D. F. Ogletree, D. S. Chemla, P. R. Selvin, and S. Weiss, “Probing the interaction between two single molecules: fluorescence resonance energy transfer between a single donor and a single acceptor,” Proc. Natl. Acad. Sci. U.S.A. 93(13), 6264–6268 (1996). [CrossRef] [PubMed]

4.

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). [CrossRef] [PubMed]

5.

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007). [CrossRef]

6.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]

7.

B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Enhanced nondegenerate four-wave mixing owing to electromagnetically induced transparency in a spectral hole-burning crystal,” Opt. Lett. 22(15), 1138–1140 (1997). [CrossRef] [PubMed]

8.

M. C. Phillips and H. Wang, “Spin coherence and electromagnetically induced transparency via exciton correlations,” Phys. Rev. Lett. 89(18), 186401 (2002). [CrossRef] [PubMed]

9.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherence media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]

10.

P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. 20(9), 982–984 (1995). [CrossRef] [PubMed]

11.

S. E. Harris and L. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82(23), 4611–4614 (1999). [CrossRef]

12.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

13.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2002). [CrossRef] [PubMed]

14.

M. G. Payne and L. Deng, “Quantum entanglement of Fock states with perfectly efficient ultraslow single-probe photon four-wave mixing,” Phys. Rev. Lett. 91(12), 123602 (2003). [CrossRef] [PubMed]

15.

B. S. Ham and J. Hahn, “Observation of ultraslow light-based photon logic gates: NAND/OR,” Appl. Phys. Lett. 94(10), 101110 (2009). [CrossRef]

16.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef] [PubMed]

17.

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. 9(4), 1663–1667 (2009). [CrossRef] [PubMed]

18.

A. Taflove, and S. C. Hagness, Computational electrodynamics: The finite-difference time–domain method (Artech, 2000).

19.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31(20), 2972–2974 (2006). [CrossRef] [PubMed]

20.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]

21.

M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]

22.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmonic metamaterial with coupling induced transparency,” arXiv: 0801.1536v1 [physics. optics].

23.

http://ab-initio.mit.edu/wiki/index.php/Dielectric_materials_in_Meep.

24.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef] [PubMed]

25.

Q. H. Park, “Optical antennas and plasmonics,” Contemp. Phys. 50(2), 407–423 (2009). [CrossRef]

26.

G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8(2), 631–636 (2008). [CrossRef] [PubMed]

27.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]

28.

D. J. Cho, F. Wang, X. Zhang, and Y. R. Shen, “Contribution of the electric quadrupole resonance in optical metamaterials,” Phys. Rev. B 78(12), 121101 (2008). [CrossRef]

29.

H. Xu, and B. S. Ham, “Plasmon-induced photonic switching in a metamaterial,” arXiv: 0905.3102v4 [quant-ph].

30.

B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: quantum switching,” Phys. Rev. Lett. 84(18), 4080–4083 (2000). [CrossRef] [PubMed]

31.

B. S. Ham, “Experimental demonstration of all-optical 1x2 quantum routing,” Appl. Phys. Lett. 85(6), 893–896 (2004). [CrossRef]

32.

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81(17), 3611–3614 (1998). [CrossRef]

33.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

34.

J. Harden, A. Joshi, and J. D. Serna, arXiv:1006.5167v1 [quant-ph].

35.

C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70(1), 37–41 (2002). [CrossRef]

OCIS Codes
(260.2110) Physical optics : Electromagnetic optics
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: June 28, 2010
Revised Manuscript: July 25, 2010
Manuscript Accepted: July 27, 2010
Published: August 2, 2010

Citation
Hua Xu, Yuehui Lu, YoungPak Lee, and Byoung Seung Ham, "Studies of electromagnetically induced transparency in metamaterials," Opt. Express 18, 17736-17747 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17736


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References

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
  2. M. Schnell, A. García-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nat. Photonics 3(5), 287–291 (2009). [CrossRef]
  3. T. Ha, Th. Enderle, D. F. Ogletree, D. S. Chemla, P. R. Selvin, and S. Weiss, “Probing the interaction between two single molecules: fluorescence resonance energy transfer between a single donor and a single acceptor,” Proc. Natl. Acad. Sci. U.S.A. 93(13), 6264–6268 (1996). [CrossRef] [PubMed]
  4. S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). [CrossRef] [PubMed]
  5. D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007). [CrossRef]
  6. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]
  7. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Enhanced nondegenerate four-wave mixing owing to electromagnetically induced transparency in a spectral hole-burning crystal,” Opt. Lett. 22(15), 1138–1140 (1997). [CrossRef] [PubMed]
  8. M. C. Phillips and H. Wang, “Spin coherence and electromagnetically induced transparency via exciton correlations,” Phys. Rev. Lett. 89(18), 186401 (2002). [CrossRef] [PubMed]
  9. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherence media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]
  10. P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. 20(9), 982–984 (1995). [CrossRef] [PubMed]
  11. S. E. Harris and L. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82(23), 4611–4614 (1999). [CrossRef]
  12. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
  13. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2002). [CrossRef] [PubMed]
  14. M. G. Payne and L. Deng, “Quantum entanglement of Fock states with perfectly efficient ultraslow single-probe photon four-wave mixing,” Phys. Rev. Lett. 91(12), 123602 (2003). [CrossRef] [PubMed]
  15. B. S. Ham and J. Hahn, “Observation of ultraslow light-based photon logic gates: NAND/OR,” Appl. Phys. Lett. 94(10), 101110 (2009). [CrossRef]
  16. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef] [PubMed]
  17. N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. 9(4), 1663–1667 (2009). [CrossRef] [PubMed]
  18. A. Taflove, and S. C. Hagness, Computational electrodynamics: The finite-difference time–domain method (Artech, 2000).
  19. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31(20), 2972–2974 (2006). [CrossRef] [PubMed]
  20. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]
  21. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]
  22. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmonic metamaterial with coupling induced transparency,” arXiv: 0801.1536v1 [physics. optics].
  23. http://ab-initio.mit.edu/wiki/index.php/Dielectric_materials_in_Meep .
  24. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef] [PubMed]
  25. Q. H. Park, “Optical antennas and plasmonics,” Contemp. Phys. 50(2), 407–423 (2009). [CrossRef]
  26. G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8(2), 631–636 (2008). [CrossRef] [PubMed]
  27. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]
  28. D. J. Cho, F. Wang, X. Zhang, and Y. R. Shen, “Contribution of the electric quadrupole resonance in optical metamaterials,” Phys. Rev. B 78(12), 121101 (2008). [CrossRef]
  29. H. Xu, and B. S. Ham, “Plasmon-induced photonic switching in a metamaterial,” arXiv: 0905.3102v4 [quant-ph].
  30. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: quantum switching,” Phys. Rev. Lett. 84(18), 4080–4083 (2000). [CrossRef] [PubMed]
  31. B. S. Ham, “Experimental demonstration of all-optical 1x2 quantum routing,” Appl. Phys. Lett. 85(6), 893–896 (2004). [CrossRef]
  32. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81(17), 3611–3614 (1998). [CrossRef]
  33. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]
  34. J. Harden, A. Joshi, and J. D. Serna, arXiv:1006.5167v1 [quant-ph].
  35. C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70(1), 37–41 (2002). [CrossRef]

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