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Analogy of electromagnetically induced transparency in plasmonic nanodisk with a square ring resonator
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Analogy of electromagnetically induced transparency in plasmonic nanodisk with a square ring resonator
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Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, School for Information and Optoelectronic Science and Engineering, South China Normal University, Guangzhou 510006, China
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Article history: Received 14 June 2015 Accepted 20 October 2015 Available online xxxx Communicated by F. Porcelli
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Xianping Li, Zhongchao Wei, Yuebo Liu, Nianfa Zhong, Xiaopei Tan, Songsong Shi, Hongzhan Liu, Ruisheng Liang
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Keywords: Surface plasmonic Electromagnetic optics Waveguide Integrated optics devices
We have demonstrated the analogy of electromagnetically induced transparency in plasmonic nanodisk with a square ring resonator. A reasonable analysis of the transmission feature based on the temporal coupled-mode theory is given and shows good agreement with the Finit-Difference Time-Domain simulation. The transparency window can be easily tuned by changing the geometrical parameters and the insulator filled in the resonator. The transmission of the resonator system is close to 80% and the full width at half maximum is less than 46 nm. The sensitivity of the structure is about 812 nm/RIU. These characteristics make the new system with potential to apply for optical storage, ultrafast plasmonic switch and slow-light devices. © 2015 Published by Elsevier B.V.
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1. Introduction
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Electromagnetically induced transparency (EIT) as a thrilling and counterintuitive phenomenon shows a narrow spectral transparency region in a broad absorption regime, which occurs in three energy level atomic systems due to the quantum destructive interference between the excitation path-ways to the atomic upper level [1,2]. This phenomenon has received much attention due to its interesting physics [3,4] and potential applications such as the transfer of quantum correlations [5], slow light propagation [6], and so on. However, the demands of stable gas lasers and low temperature environment severely hamper the implementation of EIT in chip-scale application. Recently, EIT-like has been observed in many systems such as photonic crystal nanocavity [7], hybrid plasmonic waveguides [8,9], symmetric planar metamaterial [10], and resonator [11–13]. Among these systems, the EIT-like effect based on the surface plasmon polaritons (SPPs) has been favored more and more by researchers. Surface plasmon polaritons are a kind of electromagnetic waves propagating along the metal–dielectric interface with an exponential decaying field on both sides [14]. Due to the capabilities of overcoming the classical diffraction limit and manipulating light in the nanoscale domain, SPPs have been considered as one of the most promising energy and information carriers [15,16]. So
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E-mail address: [email protected] (Z. Wei). http://dx.doi.org/10.1016/j.physleta.2015.10.035 0375-9601/© 2015 Published by Elsevier B.V.
far, numerous EIT-like devices based on SPPs such as Fabry–Perot resonator [17], Fano resonances [18,19], asymmetric dual sidecoupled cavities [20], stub waveguide with ring resonator [21] have been investigated theoretically and demonstrated experimentally. Among multitudinous plasmonic structures, the plasmonic analogous to EIT in the metal–insulator–metal (MIM) waveguide systems [22–24] draw much attention because of the deep-sub-wavelength confinement of light and relatively simple fabrication [25]. An intriguing potential application for such plasmonic analogue of EIT structures is their use in light manipulation and transmission in nanoscale devices. In this paper, a new kind of analogies of EIT in metal–insulator– metal plasmonic waveguide consisting of a square ring resonator coupled with a nanodisk is proposed and numerically investigated. We firstly briefly reviewed the properties of basic geometry, then theoretically analyzed this structure by the temporal coupled-model theory (CMT). Especially, in order to understand the relationship between the transmission characteristics and geometrical parameters of the structure and refractive index of insulator, we acquired the intuitive images by Finit-Difference Time-Domain (FDTD) simulation. The results show that the transparency window can be easily tuned by changing the geometrical parameters of the structure and the EIT-like resonant peak has a linear relationship with refractive index. These results may inspire interest in nanoscale wavelength-filter, slow light devices and optical switching elements in highly integrated optical circuit.
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Fig. 1. (a) Schematic of MIM waveguide directly coupled to a nanodisk resonator with gap = 0. (b) Transmission spectra for MIM waveguide directly coupled to nanodisk, with d = 50 nm and r = 57 nm.
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2. Theoretical analysis and discussion of MIM waveguide with one nanodisk resonator To start, let us firstly briefly review the properties of basic geometry as shown in Fig. 1(a), the nanoscale plasmonic resonator system with one nanodisk directly coupled to a bus waveguide. The dielectric layer here is assumed to be air (n = 1) and the background material in blue is silver. The permittivity of the metallic region can be described by the Drude mode [26]:
ω2p εm (ω) = ε∞ − ω(ω + i γ )
(1)
where ε∞ = 3.7 is the dielectric constant at infinite frequency, ω p = 9.1 eV is the bulk plasma frequency of free conduction electrons, γ = 0.018 eV is the electron collision frequency and ω is the angular frequency of incident light in vacuum. Fig. 1(b) shows the transmission spectra for the MIM waveguide directly with one nanodisk resonator. According to CMT [27,28], the transmission of the system supporting a resonant mode can be indicated as:
(ω − ω0 )2 + (1/τ0 ) T= (ω − ω0 )2 + (1/τ0 + 1/τe )
(2)
where τ0 and τe are the decay rate due to the intrinsic loss and waveguide coupling loss, respectively. It is obvious that the mini2
(1/τ )
mum transmission T min = (1/τ +10/τ )2 , when ω = ω0 . As the size e 0 of the structure is much smaller than incident wavelength, the intrinsic loss can be neglected, then T min = 0, which is accordance well with the numerical simulation.
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ring resonator, the transmission spectrum exhibits a spectral dip at the resonance wavelength of 825 nm due to the destructive interference between the incident wave and escaped power from the resonator [29]. However as the nanodisk and square ring are combined into a composite structure as shown in Fig. 2(a), the transmission spectrum exhibits a narrow transparency peak in the transmission dip near the resonant wavelength of the ring resonator, the EIT-like transparency peak is close to 80% and the full width at half maximum (FWHM) is less than 46 nm. In order to have an insight into the physical mechanism behind the EIT-like effect in the new proposed system, we plot the electric field. Figs. 2(c) and (d) show the field distributions at the EIT-like transparency peak of 825 m without and with the square ring resonator. As shown in Fig. 2(c), the incident lights are reflected in the nanodisk-shaped waveguide, so that the whole modes are locked in the nanodisk, whereas they pass through the waveguide with ring resonator, as can be seen in Fig. 2(d). The results are consistent with the spectral response in Fig. 2(b). It is worth noting that the electromagnetic field in the nanodisk is weakened due to the interference effect when the disk couples with the square ring cavity. In order to analyze the EIT-like phenomenon in detail, we introduce a CMT-based transmission line theory. As shown in Fig. 2(a), the coupling coefficient between the bus waveguide and the nanodisk is denoted by γ , β is the coupling coefficient between the nanodisk and the square ring. α , δ are the decay rates due to the internal loss of nanodisk and square ring, respectively. The amplitudes of the incoming and outing waves of disk are denoted by S +11 , S +12 , S −11 , and S −12 , as seen in Fig. 2(a). The temporal evolution of the amplitude A of the single nanodisk and B of the square ring. The characteristic equation can be given:
3. EIT-like in a square ring resonator coupled with a nanodisk
dA
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The schematic illustration of EIT-like in metal–insulator–metal plasmonic system we proposed is shown in Fig. 2(a), it is composed of a nanodisk coupled with a square ring resonator. The inner and outer length of the square ring are l1 = 75 nm and l2 = 240 nm, respectively. More, in order to explain the subsequent analysis clearly, we set center length of square ring l = (l1 + l2 )/2. The width of bus waveguide is set to be d = 50 nm. The radii of nanodisk is r = 57 nm and the distance between the nanodisk and square ring is g = 11 nm. Fig. 2(b) shows the transmission spectra without and with the square ring resonator. Without the square
dt dB dt
√
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= ( j ωt − α − β − γ ) A + j γ ( S +11 + S +12 ) + j β B
(3)
= ( j ωn − β − δ) B + j β A
(4)
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where ωt , ωn are the resonant frequency of the nanodisk and square ring, respectively. Due to energy conservation and the time reversal symmetry, the relationship of the incoming and outing waves in the bus waveguide can be denoted as:
S −11 = S +12 + j
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γA
(5)
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Fig. 2. (a) Schematic diagram of a nanodisk coupled with the square ring in MIM waveguide system; (b) Transmission spectra without (black curve) and with (red curve) a square resonator, the geometrical parameters of the structure are set as d = 50 nm, r = 57 nm, g = 11 nm, l1 = 75 nm, l2 = 240 nm; (c–d) The magnetic distribution without and with square ring for the incident wavelength of 825 nm (i.e. EIT-like peak wavelength).
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S −12 = S +11 + j
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(7)
From Eq. (7), we can quantitatively and clearly comprehend the EIT-like phenomenon. The peak transmission can be achieved at the resonance frequency ω = ωn = ωt with the value of T = α +β+β/(β+δ) 2 | α +β+ γ +β/(β+δ) | . It is obvious that the EIT-like transparency window is formed at resonance frequency ω = ωt as long as ωt and ωn are close to each other, and the transmission is derived as j (ω −ωn )+β+1 2 T = | j (ω −t ω )+β+ γ +1 | on condition that the intrinsic loss is igt n nored.
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4. Transmission properties of the proposed structure with different parameters
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γA
When the S +12 = 0, from Eqs. (3)–(6), the transmission coefficient T into the output port at steady state can be determined as:
S −12 2 T = = S
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As we know, the transmission characteristics of the plasmonics waveguide system can be affected by the structure parameters. First, we calculated the transmission spectra for different coupling distance g between the nanodisk and square ring. Fig. 3(a) exhibits transmission spectra with different coupling gap g (9–14 nm)
at d = 50 nm, r = 57 nm, l1 = 80 nm, l2 = 250 nm. It is found that the transmission peak and the full width at half-maximum (FHWM) decrease with increasing distance g, so we can select g to get suitable transmission and FHWM. The length of the square ring resonator is also an important factor influencing the transmission properties. As a ring resonator, the standing wave will be excited in the square ring according to the theory of ring resonator [30,31]. The character equation can be given by
J n (kl1 ) J n (kl2 )
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where k = ε c (εr )1/2 ,
(8)
εr = n2neff is the effective relative permittiv-
ity, nneff is the effective refractive index of square ring. J n and J n stand for the first kind Bessel function with the order n and its derivative. N n and N n are the second Bessel function with order n and its derivative. Fig. 3(b) exhibits the transmission spectra with different l, it is obvious that the transmission peak of various l stays at a high level and the transparency peak performs a redshift with increasing l. In Fig. 3(c), the wavelength of peak spectra exhibits an approximately linear relation with l, which is in accord with the solution of Eq. (8). The transmission spectra with different refractive indices of nanodisk and square ring are investigated and other parameters are the same as those for Fig. 2(a). As shown in Fig. 4(a), the peak exhibits a redshift with the increase of the refractive index in
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Fig. 3. (a) Transmission spectra with different g of the square ring resonator, the other parameters d = 50 nm, r = 57 nm, g = 11 nm, the thickness of the square ring waveguide is fixed as 170 nm. (b) EIT-like peak wavelength with different l. (c) Transmission spectra with different distances l, the other geometrical parameters are the same as Fig. 2(a).
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FOM =
S
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(9)
FWHM where S = δλ/δn is the refractive index sensitivity (i.e. spectral shifts per refractive index). λ is the resonance line width as the full FWHW centered at the resonance wavelength λ. The refractive index sensitivity and FOM of the new system is as high as 812 nm/RIU and 17.6, respectively.
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ng =
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As we know, EIT-like phenomenon is always accompanied by slow-light effect. While the slow-light effect can be depicted as the group index n g , which is denoted by [29]:
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5. Slow light effect of this new structure
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Fig. 5(a) shows the phase shift from the optical source to the detector. We can see that the phase spectrum has two frequency segments due to anomalous and normal phase dispersion and the phase slope is steepest at the location of the EIT-like peak. Examining Fig. 5(a), we note that the phase spectrum has glitch of phase. Based on the careful analysis, we think the phase dithering results from reduplicative reflection of light between the narrow square ring. It is obvious that nair < nsilver , the reflective light has a dispersion of phase comparing with incident light. As shown in the red area of Fig. 5(b), there is an optical delay as high as 0.14 ps in the new system at the EIT-like peak. Based on the above analysis of phase glitch and Eq. (10), it is easy to understand the dithering of group delay and group index. Moreover, the corresponding group index reach 41.1 at the EIT-like peak, as shown in the red area of Fig. 5(c). 6. Conclusion
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nanodisk and square ring. In Fig. 4(b), it is obvious that the EIT-like resonant peak nearly has a linear relation with refractive index, the transmission peaks of transmittances are about 80% at 835, 851, 867, 884, and 900 nm when n = 1.00, 1.02, 1.04, 1.06, and 1.08, respectively, so we can use the new system as a sensor when the square ring and nanodisk is filled with changed dielectric. For sensing application, figure of merit (FOM) is usually applied to further evaluate the sensing performance as the following [32],
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(10)
where c is the velocity of light in the free space, υ g is the group velocity in the plasmonic waveguide systems, τ g and ϕ (ω) represent the optical delay time and transmission phase shift, respectively, L stands for length of the new structure. The slow-light effect is investigated numerically in this system with d = 50 nm, r = 57 nm, g = 11 nm, l1 = 250 nm, l2 = 80 nm.
A new kind of analogies of EIT in metal–insulator–metal plasmonic waveguide consisting of a square ring resonator coupled with a nanodisk is proposed and numerically investigated. The transparency window can be easily tuned by changing the geometrical parameters of the structure and the insulator filled in resonator. The CMT-based transmission line theory is well in accordance with the FDTD results, what’s more, we find the EIT-like resonant peak still has a linear relationship with refractive index and the refractive index sensitivity and FOM of the new system is as high as 812 nm/RIU and 17.6, respectively. While group index reach 41.1 at the EIT-like peak with the transmission is close to 80% at d = 50 nm, r = 57 nm, g = 11 nm, l1 = 250 nm, l2 = 80 nm. These advantages of our analogy of electromagnetically induced
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Fig. 4. (a) Transmission spectra with different refractive index of a nanodisk and a square ring. The other parameters are d = 50 nm, r = 57 nm, g = 11 nm, l1 = 250 nm, l2 = 80 nm. (b) EIT-like peak wavelength with different refractive index.
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Fig. 5. Simulated transmission phase shift (a), optical delay time (b), group index (c) with d = 50 nm, r = 57 nm, g = 11 nm, l1 = 250 nm, l2 = 80 nm.
transparency based on plasmonic waveguide resonator system provide a potential application in highly integrated optical circuits and signal processing. Acknowledgements This work was supported by the National Natural Science Foundation of China (NSFC) under Grants Nos. 61275059 and 11374107. References
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Analogy of electromagnetically induced transparency in plasmonic nanodisk with a square ring resonator
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Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, School for Information and Optoelectronic Science and Engineering, South China Normal University, Guangzhou 510006, China
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a r t i c l e
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Article history: Received 14 June 2015 Accepted 20 October 2015 Available online xxxx Communicated by F. Porcelli
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Xianping Li, Zhongchao Wei, Yuebo Liu, Nianfa Zhong, Xiaopei Tan, Songsong Shi, Hongzhan Liu, Ruisheng Liang
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Keywords: Surface plasmonic Electromagnetic optics Waveguide Integrated optics devices
We have demonstrated the analogy of electromagnetically induced transparency in plasmonic nanodisk with a square ring resonator. A reasonable analysis of the transmission feature based on the temporal coupled-mode theory is given and shows good agreement with the Finit-Difference Time-Domain simulation. The transparency window can be easily tuned by changing the geometrical parameters and the insulator filled in the resonator. The transmission of the resonator system is close to 80% and the full width at half maximum is less than 46 nm. The sensitivity of the structure is about 812 nm/RIU. These characteristics make the new system with potential to apply for optical storage, ultrafast plasmonic switch and slow-light devices. © 2015 Published by Elsevier B.V.
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1. Introduction
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Electromagnetically induced transparency (EIT) as a thrilling and counterintuitive phenomenon shows a narrow spectral transparency region in a broad absorption regime, which occurs in three energy level atomic systems due to the quantum destructive interference between the excitation path-ways to the atomic upper level [1,2]. This phenomenon has received much attention due to its interesting physics [3,4] and potential applications such as the transfer of quantum correlations [5], slow light propagation [6], and so on. However, the demands of stable gas lasers and low temperature environment severely hamper the implementation of EIT in chip-scale application. Recently, EIT-like has been observed in many systems such as photonic crystal nanocavity [7], hybrid plasmonic waveguides [8,9], symmetric planar metamaterial [10], and resonator [11–13]. Among these systems, the EIT-like effect based on the surface plasmon polaritons (SPPs) has been favored more and more by researchers. Surface plasmon polaritons are a kind of electromagnetic waves propagating along the metal–dielectric interface with an exponential decaying field on both sides [14]. Due to the capabilities of overcoming the classical diffraction limit and manipulating light in the nanoscale domain, SPPs have been considered as one of the most promising energy and information carriers [15,16]. So
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E-mail address: [email protected] (Z. Wei). http://dx.doi.org/10.1016/j.physleta.2015.10.035 0375-9601/© 2015 Published by Elsevier B.V.
far, numerous EIT-like devices based on SPPs such as Fabry–Perot resonator [17], Fano resonances [18,19], asymmetric dual sidecoupled cavities [20], stub waveguide with ring resonator [21] have been investigated theoretically and demonstrated experimentally. Among multitudinous plasmonic structures, the plasmonic analogous to EIT in the metal–insulator–metal (MIM) waveguide systems [22–24] draw much attention because of the deep-sub-wavelength confinement of light and relatively simple fabrication [25]. An intriguing potential application for such plasmonic analogue of EIT structures is their use in light manipulation and transmission in nanoscale devices. In this paper, a new kind of analogies of EIT in metal–insulator– metal plasmonic waveguide consisting of a square ring resonator coupled with a nanodisk is proposed and numerically investigated. We firstly briefly reviewed the properties of basic geometry, then theoretically analyzed this structure by the temporal coupled-model theory (CMT). Especially, in order to understand the relationship between the transmission characteristics and geometrical parameters of the structure and refractive index of insulator, we acquired the intuitive images by Finit-Difference Time-Domain (FDTD) simulation. The results show that the transparency window can be easily tuned by changing the geometrical parameters of the structure and the EIT-like resonant peak has a linear relationship with refractive index. These results may inspire interest in nanoscale wavelength-filter, slow light devices and optical switching elements in highly integrated optical circuit.
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Fig. 1. (a) Schematic of MIM waveguide directly coupled to a nanodisk resonator with gap = 0. (b) Transmission spectra for MIM waveguide directly coupled to nanodisk, with d = 50 nm and r = 57 nm.
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2. Theoretical analysis and discussion of MIM waveguide with one nanodisk resonator To start, let us firstly briefly review the properties of basic geometry as shown in Fig. 1(a), the nanoscale plasmonic resonator system with one nanodisk directly coupled to a bus waveguide. The dielectric layer here is assumed to be air (n = 1) and the background material in blue is silver. The permittivity of the metallic region can be described by the Drude mode [26]:
ω2p εm (ω) = ε∞ − ω(ω + i γ )
(1)
where ε∞ = 3.7 is the dielectric constant at infinite frequency, ω p = 9.1 eV is the bulk plasma frequency of free conduction electrons, γ = 0.018 eV is the electron collision frequency and ω is the angular frequency of incident light in vacuum. Fig. 1(b) shows the transmission spectra for the MIM waveguide directly with one nanodisk resonator. According to CMT [27,28], the transmission of the system supporting a resonant mode can be indicated as:
(ω − ω0 )2 + (1/τ0 ) T= (ω − ω0 )2 + (1/τ0 + 1/τe )
(2)
where τ0 and τe are the decay rate due to the intrinsic loss and waveguide coupling loss, respectively. It is obvious that the mini2
(1/τ )
mum transmission T min = (1/τ +10/τ )2 , when ω = ω0 . As the size e 0 of the structure is much smaller than incident wavelength, the intrinsic loss can be neglected, then T min = 0, which is accordance well with the numerical simulation.
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ring resonator, the transmission spectrum exhibits a spectral dip at the resonance wavelength of 825 nm due to the destructive interference between the incident wave and escaped power from the resonator [29]. However as the nanodisk and square ring are combined into a composite structure as shown in Fig. 2(a), the transmission spectrum exhibits a narrow transparency peak in the transmission dip near the resonant wavelength of the ring resonator, the EIT-like transparency peak is close to 80% and the full width at half maximum (FWHM) is less than 46 nm. In order to have an insight into the physical mechanism behind the EIT-like effect in the new proposed system, we plot the electric field. Figs. 2(c) and (d) show the field distributions at the EIT-like transparency peak of 825 m without and with the square ring resonator. As shown in Fig. 2(c), the incident lights are reflected in the nanodisk-shaped waveguide, so that the whole modes are locked in the nanodisk, whereas they pass through the waveguide with ring resonator, as can be seen in Fig. 2(d). The results are consistent with the spectral response in Fig. 2(b). It is worth noting that the electromagnetic field in the nanodisk is weakened due to the interference effect when the disk couples with the square ring cavity. In order to analyze the EIT-like phenomenon in detail, we introduce a CMT-based transmission line theory. As shown in Fig. 2(a), the coupling coefficient between the bus waveguide and the nanodisk is denoted by γ , β is the coupling coefficient between the nanodisk and the square ring. α , δ are the decay rates due to the internal loss of nanodisk and square ring, respectively. The amplitudes of the incoming and outing waves of disk are denoted by S +11 , S +12 , S −11 , and S −12 , as seen in Fig. 2(a). The temporal evolution of the amplitude A of the single nanodisk and B of the square ring. The characteristic equation can be given:
3. EIT-like in a square ring resonator coupled with a nanodisk
dA
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The schematic illustration of EIT-like in metal–insulator–metal plasmonic system we proposed is shown in Fig. 2(a), it is composed of a nanodisk coupled with a square ring resonator. The inner and outer length of the square ring are l1 = 75 nm and l2 = 240 nm, respectively. More, in order to explain the subsequent analysis clearly, we set center length of square ring l = (l1 + l2 )/2. The width of bus waveguide is set to be d = 50 nm. The radii of nanodisk is r = 57 nm and the distance between the nanodisk and square ring is g = 11 nm. Fig. 2(b) shows the transmission spectra without and with the square ring resonator. Without the square
dt dB dt
√
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= ( j ωt − α − β − γ ) A + j γ ( S +11 + S +12 ) + j β B
(3)
= ( j ωn − β − δ) B + j β A
(4)
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where ωt , ωn are the resonant frequency of the nanodisk and square ring, respectively. Due to energy conservation and the time reversal symmetry, the relationship of the incoming and outing waves in the bus waveguide can be denoted as:
S −11 = S +12 + j
90
√
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γA
(5)
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Fig. 2. (a) Schematic diagram of a nanodisk coupled with the square ring in MIM waveguide system; (b) Transmission spectra without (black curve) and with (red curve) a square resonator, the geometrical parameters of the structure are set as d = 50 nm, r = 57 nm, g = 11 nm, l1 = 75 nm, l2 = 240 nm; (c–d) The magnetic distribution without and with square ring for the incident wavelength of 825 nm (i.e. EIT-like peak wavelength).
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S −12 = S +11 + j
√
+11
(6)
β
j (ω − ωt ) + α + β + j (ω−ω )+β+δ n β
j (ω − ωt ) + α + β + γ + j (ω−ω )+δ+β n
2
(7)
From Eq. (7), we can quantitatively and clearly comprehend the EIT-like phenomenon. The peak transmission can be achieved at the resonance frequency ω = ωn = ωt with the value of T = α +β+β/(β+δ) 2 | α +β+ γ +β/(β+δ) | . It is obvious that the EIT-like transparency window is formed at resonance frequency ω = ωt as long as ωt and ωn are close to each other, and the transmission is derived as j (ω −ωn )+β+1 2 T = | j (ω −t ω )+β+ γ +1 | on condition that the intrinsic loss is igt n nored.
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4. Transmission properties of the proposed structure with different parameters
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γA
When the S +12 = 0, from Eqs. (3)–(6), the transmission coefficient T into the output port at steady state can be determined as:
S −12 2 T = = S
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As we know, the transmission characteristics of the plasmonics waveguide system can be affected by the structure parameters. First, we calculated the transmission spectra for different coupling distance g between the nanodisk and square ring. Fig. 3(a) exhibits transmission spectra with different coupling gap g (9–14 nm)
at d = 50 nm, r = 57 nm, l1 = 80 nm, l2 = 250 nm. It is found that the transmission peak and the full width at half-maximum (FHWM) decrease with increasing distance g, so we can select g to get suitable transmission and FHWM. The length of the square ring resonator is also an important factor influencing the transmission properties. As a ring resonator, the standing wave will be excited in the square ring according to the theory of ring resonator [30,31]. The character equation can be given by
J n (kl1 ) J n (kl2 )
−
N n (kl1 ) N n (kl2 )
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=0
where k = ε c (εr )1/2 ,
(8)
εr = n2neff is the effective relative permittiv-
ity, nneff is the effective refractive index of square ring. J n and J n stand for the first kind Bessel function with the order n and its derivative. N n and N n are the second Bessel function with order n and its derivative. Fig. 3(b) exhibits the transmission spectra with different l, it is obvious that the transmission peak of various l stays at a high level and the transparency peak performs a redshift with increasing l. In Fig. 3(c), the wavelength of peak spectra exhibits an approximately linear relation with l, which is in accord with the solution of Eq. (8). The transmission spectra with different refractive indices of nanodisk and square ring are investigated and other parameters are the same as those for Fig. 2(a). As shown in Fig. 4(a), the peak exhibits a redshift with the increase of the refractive index in
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Fig. 3. (a) Transmission spectra with different g of the square ring resonator, the other parameters d = 50 nm, r = 57 nm, g = 11 nm, the thickness of the square ring waveguide is fixed as 170 nm. (b) EIT-like peak wavelength with different l. (c) Transmission spectra with different distances l, the other geometrical parameters are the same as Fig. 2(a).
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FOM =
S
=
δλ/δn
λ
(9)
FWHM where S = δλ/δn is the refractive index sensitivity (i.e. spectral shifts per refractive index). λ is the resonance line width as the full FWHW centered at the resonance wavelength λ. The refractive index sensitivity and FOM of the new system is as high as 812 nm/RIU and 17.6, respectively.
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ng =
c
υg
c
c dϕ (ω)
L
L
= τg = .
dω
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As we know, EIT-like phenomenon is always accompanied by slow-light effect. While the slow-light effect can be depicted as the group index n g , which is denoted by [29]:
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5. Slow light effect of this new structure
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Fig. 5(a) shows the phase shift from the optical source to the detector. We can see that the phase spectrum has two frequency segments due to anomalous and normal phase dispersion and the phase slope is steepest at the location of the EIT-like peak. Examining Fig. 5(a), we note that the phase spectrum has glitch of phase. Based on the careful analysis, we think the phase dithering results from reduplicative reflection of light between the narrow square ring. It is obvious that nair < nsilver , the reflective light has a dispersion of phase comparing with incident light. As shown in the red area of Fig. 5(b), there is an optical delay as high as 0.14 ps in the new system at the EIT-like peak. Based on the above analysis of phase glitch and Eq. (10), it is easy to understand the dithering of group delay and group index. Moreover, the corresponding group index reach 41.1 at the EIT-like peak, as shown in the red area of Fig. 5(c). 6. Conclusion
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nanodisk and square ring. In Fig. 4(b), it is obvious that the EIT-like resonant peak nearly has a linear relation with refractive index, the transmission peaks of transmittances are about 80% at 835, 851, 867, 884, and 900 nm when n = 1.00, 1.02, 1.04, 1.06, and 1.08, respectively, so we can use the new system as a sensor when the square ring and nanodisk is filled with changed dielectric. For sensing application, figure of merit (FOM) is usually applied to further evaluate the sensing performance as the following [32],
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(10)
where c is the velocity of light in the free space, υ g is the group velocity in the plasmonic waveguide systems, τ g and ϕ (ω) represent the optical delay time and transmission phase shift, respectively, L stands for length of the new structure. The slow-light effect is investigated numerically in this system with d = 50 nm, r = 57 nm, g = 11 nm, l1 = 250 nm, l2 = 80 nm.
A new kind of analogies of EIT in metal–insulator–metal plasmonic waveguide consisting of a square ring resonator coupled with a nanodisk is proposed and numerically investigated. The transparency window can be easily tuned by changing the geometrical parameters of the structure and the insulator filled in resonator. The CMT-based transmission line theory is well in accordance with the FDTD results, what’s more, we find the EIT-like resonant peak still has a linear relationship with refractive index and the refractive index sensitivity and FOM of the new system is as high as 812 nm/RIU and 17.6, respectively. While group index reach 41.1 at the EIT-like peak with the transmission is close to 80% at d = 50 nm, r = 57 nm, g = 11 nm, l1 = 250 nm, l2 = 80 nm. These advantages of our analogy of electromagnetically induced
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Fig. 4. (a) Transmission spectra with different refractive index of a nanodisk and a square ring. The other parameters are d = 50 nm, r = 57 nm, g = 11 nm, l1 = 250 nm, l2 = 80 nm. (b) EIT-like peak wavelength with different refractive index.
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Fig. 5. Simulated transmission phase shift (a), optical delay time (b), group index (c) with d = 50 nm, r = 57 nm, g = 11 nm, l1 = 250 nm, l2 = 80 nm.
transparency based on plasmonic waveguide resonator system provide a potential application in highly integrated optical circuits and signal processing. Acknowledgements This work was supported by the National Natural Science Foundation of China (NSFC) under Grants Nos. 61275059 and 11374107. References
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