Introduction

Surface plasmon polaritons have the potential to guide light at deep subwavelength scale [1, 2]. Therefore, the investigation of plasmonic waveguide structures has been paid great attentions and several different nanoscale plasmonic waveguide structures have been recently proposed, such as metallic nanowires [3, 4], metallic nanoparticle arrays [5, 6] as well as V grooves in metal substrates [7], plasmon slots [8], and metal wedges [9, 10]. Among those structures, waveguides consisted of an insulator sandwiched between two metals serve as metal–dielectric–metal (MDM) waveguides or metal–insulator–metal support propagating surface plasmon modes that are strongly confined in the insulator region with an acceptable propagation length [11]. Therefore, MDM waveguides are promising for the design of nanoscale all-optical devices with a relatively easy fabrication according to the current state of the art [12, 13]. A variety of functional plasmonic MDM structures have been designed and fabricated, such as splitters [14], Y-shaped combiners [15], couplers [16], M–Z interferometers [17, 18], and filters [19–21].

Integrated optical devices and circuits are being increasingly used for light routing and switching in the rapidly developing area of broadband optical communications. Optical switches based on different physical mechanisms with different structures have been proposed [22–25]. Different physical mechanisms can be used to control the refractive index, such as the electro-optical effect [22], the plasma dispersion effect [26], and the thermo-optic effect [27]. Wavelength switch is one of key technologies in fields of optical communication and computing. Especially, in many cases such as WDM systems, it is required to implement several specific wavelengths conversion function. In this letter, a circuit switch and a wavelength conversion switch based on electro-optical effect are proposed and demonstrated numerically by using the finite difference time domain (FDTD) method with perfectly matched layer absorbing boundary condition. An analytic model based on resonator theory is given firstly. The relationship between the peak wavelength and nanocavity length is also addressed. Finally, the characteristics of the circuit switch waveguide structure and the wavelength conversion switch waveguide are analyzed.

Nanocavity Theory Model

Figure 1a shows the structure schematic of the nanocavity composed of two wide waveguides and a narrow waveguide filled with SiO2 material. Obviously, the structure can be treated as two MDM waveguides with different widths. Because the width of the narrow MDM waveguide is much smaller than that of the wide MDM waveguide, here we call the narrower waveguide as nanocavity. The dielectric constant ε m of silver can be calculated based on Drude–Lorentzian model:

$$ {\varepsilon_m}(\omega ) = {\varepsilon_{\infty }} - \frac{{\omega_D^2}}{{{\omega^2} + i{\gamma_D}\omega }} - \sum\limits_{{m = 1}}^2 {\frac{{{g_{{Lm}}}\omega_{{Lm}}^2\Delta \varepsilon }}{{{\omega^2} - \omega_{{Lm}}^2 + i2{\gamma_{{Lm}}}\omega }}}, $$
(1)

where, ε ∞ = 2.3646, ω D  = 8.7377 eV, γ D  = 0.07489 eV, Δε = 1.1831, g L1 = 0.26663, ω L1 = 4.3802 eV, γ L1 = 0.28 eV, g L2 = 0.7337, ω L2 = 5.183 eV, and γ L2 = 0.5482 eV. Equation 1 gives a good description of empirical dielectric constant data for silver within the range of 500–1,800 nm [28]. The complex propagation constant \( \beta = {\beta_R} + j{\beta_I} \) of surface plasmon polaritons in every MDM waveguide can be obtained by solving the dispersive mode equation [13, 29]:

$$ {\varepsilon_D}{k_{{z2}}} + {\varepsilon_m}{k_{{z1}}}\tanh \left( { - \frac{{i{k_{{z1}}}}}{2}w} \right) = 0, $$
(2)

k z1 and k z2 defined as: \( k_{{z1}}^2 = {\varepsilon_D}k_0^2 - {\beta^2} \)and \( k_{{z2}}^2 = \varepsilon {}_mk_0^2 - {\beta^2} \), where ε D and ε m are, respectively, dielectric constants of the dielectric and the metal. k 0 = 2π/λ is the wave number of light with wavelength λ in vacuum. w is the width of the slit waveguide. The effective index is represented as \( {n_{\text{eff}}} = {\beta_R}/{k_0} \).

Fig. 1
figure 1figure 1

a Schematics of MDM nanocavity structures (colored). b The electric fields on the two reflection surfaces. c The schematics of estimating the value of t. d and e The space distributions of the mode

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The effective index of the waveguide decreases with the increasing of the width of waveguide w at the same wavelength [20]. Therefore, the effective index n eff2 of the nanocavity part can be larger than n eff1 of the wide MDM waveguide part due to the difference of the width of the two waveguides and mediums filled in. The wave transmitted into the nanocavity will be partly reflected at two ends of the nanocavity, because of the index differences between n eff2 and n eff1. Therefore, one can expect the nanocavity operates as a Fabre–Pérot resonator. Figure 1b shows the electric fields on the two reflection ends of the nanocavity. r and t are respectively the reflection and transmission coefficients of an incident beam in the structure. And \( E_1^{\text{in}} \) and \( E_2^{\text{out}} \) stand for the fields of incident and output beams, respectively. Based on Fabre–Pérot resonator properties, one has \( E_2^{\text{out}} = E_2^1 + E_2^2 + E_2^3 + E_2^4 + \ldots + E_2^n \)(shown in Fig. 1b). The transmitted electric fields at end 2 of \( E_2^1 \), \( E_2^2 \) and \( E_2^3 \) are respectively given as follows:

$$ E_2^1 = {t^2}E_1^{\text{in}}\exp (i\phi (\lambda ))\exp ( - {\beta_I}L), $$
(3)
$$ E_2^2 = {t^2}{r^2}E_1^{\text{in}}\exp (i3\phi (\lambda ))\exp ( - 3{\beta_I}L), $$
(4)
$$ E_2^3 = {t^2}{r^4}E_1^{\text{in}}\exp (i5\phi (\lambda ))\exp ( - 5{\beta_I}L), $$
(5)

here the phase delay is \( \phi (\lambda ) = (2\pi /\lambda ){n_{\text{eff}}} \cdot L + {\varphi_1} + {\varphi_2} \), where φ 1 and φ 2 are, respectively, the phase shifts of the beam reflected at both ends of the nanocavity. β I and L stand for the imaginary part of the propagation constant and the length of the nanocavity, respectively. Therefore, one has the total output electric field of \( E_2^{\text{out}} \):

$$ \begin{array}{*{20}{c}} {E_2^{\text{out}} = {t^2}E_1^{\text{in}}\exp (i\phi (\lambda ))\exp ( - {\beta_I}L) + } \hfill \\ {\quad \quad {t^2}{r^2}E_1^{\text{in}}\exp (i3\phi (\lambda ))\exp ( - 3{\beta_I}L) + } \hfill \\ {\quad \quad {t^2}{r^4}E_1^{\text{in}}\exp (i5\phi (\lambda ))\exp ( - 5{\beta_I}L) + } \hfill \\ {\quad \quad ...} \hfill \\ {\quad \quad = \frac{{{t^2}E_1^{\text{in}}\exp (i\phi (\lambda ))\exp ( - {\beta_I}L)}}{{1 - {r^2}\exp (i2\phi (\lambda ))\exp ( - 2{\beta_I}L)}}} \hfill \\ \end{array} . $$
(6)

Therefore, the transmittance T is given by

$$ \begin{array}{*{20}{c}} {T = {{\left| {\frac{{E_2^{\text{out}}}}{{E_1^{\text{in}}}}} \right|}^2} = \left| {\frac{{{t^2}\exp (i\phi (\lambda ))\exp ( - {\beta_I}L)}}{{1 - {r^2}\exp (i2\phi (\lambda ))\exp ( - 2{\beta_I}L)}}} \right|} \\ { = \frac{{{t^4}\exp ( - 2{\beta_I}L)}}{{1 + {r^4}\exp ( - 4{\beta_I}L) - 2{r^2}\exp ( - 2{\beta_I}L)\cos (2\phi (\lambda ))}}} \\ \end{array} . $$
(7)

It can be seen from Eq. 7 that the transmittance T will become maximum, only when the following condition is satisfied: \( 2\phi (\lambda ) = 2m\pi \)(m = 1, 2, 3…). Therefore, the peak wavelength λ max of the nanocavity structure is determined as follows:

$$ {\lambda_{{\max }}} = 2{n_{\text{eff}}}L/(m - ({\varphi_1} + {\varphi_2})/(2\pi )). $$
(8)

From Eq. 8, one can see that the wavelength λ max is proportional to the length and the effective index of the nanocavity, respectively. The maximum transmittance T can be expressed as follows:

$$ {T_{{\max }}} = \frac{{{t^4}\exp ( - 2{\beta_I}L)}}{{{{[1 - {r^2}\exp ( - 2{\beta_I}L)]}^2}}}. $$
(9)

Obviously, the maximum transmittance T max is determined by the reflection coefficient of r and transmission coefficient of t, as well as propagation loss of \( \exp ( - 2{\beta_I}L) \). Provided that \( {r^2} + {t^2} = 1 \)and propagation loss of \( \exp ( - 2{\beta_I}L) = 0 \), one would obtain the highest transmittance T max = 1.

On the other hand, if the following condition is satisfied: \( 2\phi (\lambda ) = (2m + 1)\pi \), the transmittance T will become minimum. The minimum transmittance T is given as follows:

$$ {T_{{\min }}} = \frac{{{t^4}\exp ( - 2{\beta_I}L)}}{{{{[1 + {r^2}\exp ( - 2{\beta_I}L)]}^2}}}. $$
(10)

It can be seen from Eq. 10 that T min = 0 only when the propagation loss of \( \exp ( - 2{\beta_I}L) \)can be ignored.

In order to estimate values of T max and T min, it is necessary to know the values of β I and t (or r). For a slit waveguide with a small width of w, solving Eq. 2 gives n eff and β I . Figure 1c shows the calculation of the value of t. One unit incident optical power signal from point S transmits along the waveguide. The power monitor is set at the position of P which is about 10 nm away from point S, to detect the transmitted power P t . Therefore, the value of t is equal to P t .

Take nanocavity structure (shown in Fig. 1a) with w s  = 250 nm, w = 15 nm, and L = 150 nm as an example, the values of n eff and β I are respectively calculated to be 2.99 and 0.17. The value of t is calculated to be 0.2. Thus, the peak wavelength of λ max ≈ 900 nm for m = 1 and \( {\varphi_1} = {\varphi_2} = 0 \), while the maximum transmittance of T max is simply estimated to be 66% according to the Eq. 9, namely, the insertion loss is about −1.8 dB. According to Eq. 10, the minimum transmittance of T min could be theoretically calculated to be 0.012, namely, the extinction ratio of the nanocavity structure as a wavelength shift switch will be aroud −19.1 dB.

The refractive index of SiO2 is set to be 1.44. In the following FDTD simulations, the grid sizes in the x and z directions are set to be 3 × 3 nm. The fundamental transverse magnetic (TM) mode of the plasmonic waveguide is launched from the left to the right of the waveguide. Because the width of the MDM plasmonic waveguide is much smaller than the wavelength, only the fundamental TM waveguide mode can propagate. Two power monitors are set at the locations of P (outside the resonator but just near its entrance) and Q (outside the resonator but just near its exit) to detect the incident power without reflection (replacing narrow waveguide with wide waveguide) and transmitted power of P in and P out. The transmittance is defined to be T = P out/P in. The space distributions of the modes are presented in Fig. 1d and e. For the sake of comparison, the wide waveguide width of w s , the nanocavity waveguide width of w, the distance of d, and the nanocavity waveguide length of L are kept the same as the above theoretical analysis, namely, respectively fixed to be 250, 15, 50, and 150 nm. A typical selective filtering transmission spectrum is shown in Fig. 2a. From Fig. 2a, one can see that the peak wavelength occurs at around the wavelength of 910 nm and its transmittance is about 64%, which agree with the above theoretical analysis well. The inset of Fig. 2a shows the peak wavelength as a function of the nanocavity length L. From it, it can be seen that the peak wavelength linearly increases with the increase of the length of the nanocavity as expected from Eq. 8. The selective wavelength can be easily tuned by choosing an appropriate nanocavity length. The spectrum of the nanocavity structure in a log scale is shown in Fig. 2b. It shows that the insertion loss of this filter is about −1.9 dB as expected from Eq. 9. The extinction ratio of the nanocavity is about −20 dB as expected from Eq. 10. Therefore, the simulated results agree with the theoretical analysis well.

Fig. 2
figure 2

a Transmission spectrum of the nanocavity structure with the length of L = 150 nm. Transmission spectra versus the nanocavity length L with the same width of w = 15 nm (inset). b The spectrum of the nanocavity structure in a log scale

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Discussion of Dual-Nanocavities Structure for Circuit Switch

Figure 3 shows a typical schematic of a 1 × 2 circuit switch structure based on MDM nanocavities. The electro-optical (EO) material, an organic material with the refractive index of 2.2, filled into the nanocavities of the channels 1 and 2 is assumed to be 4-dimethylamino-N-methyl-4-stilbazolium tosylate. It exhibits a large EO coefficient (dn/dE = 3.41 nm/V) compared with that of standard inorganic EO materials such as LiNbO3 (dn/dE = 0.16 nmV), because of a large delocalised π electron system [30]. Two power monitors are set at the locations of Q 1 and Q 2 (outside the resonator but just near its exit) to detect the output channels 1 and 2 transmitted powers of P 1 and P 2.The transmittance are defined to be T 1  = P 1/P in and T 2 = P 2/P in, where P in is the incident power without reflection of the beam. L 1 and L 2 are respectively the nanocavity length of the channels 1 and 2. The others parameters of the structure are kept the same as Section Nanocavity Theory Model. An electric field is added in order to control the refractive index of EO material.

Fig. 3
figure 3

Schematic of 1 × 2 circuit switch structure based on MDM nanocavities (colored)

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Figure 4a shows transmission spectra at the outputs of the two channels. One can see that channels 1 and 2 and respectively select 1,280 and 1,550 nm bands with the length of L 1 = 120 nm and L 2 = 142 nm, and the maximum transmittance in two bands can exceed 30% (−5.2 dB). Figure 4b shows transmission spectra at the outputs of the two channels after an eclectic field (V = 1.95 V, and the electric field intensity correspond to 0.13 KV/μm which is acceptable experimentally) is applied. One can see that channels 1 and 2 can respectively select 1,550 and 1,860 nm bands with the length of L 1 = 120 nm and L 2 = 142 nm. From Fig. 4a and b, one can see that the peak wavelength shift toward long wavelength after the electric field applied. It is noticed that the output wavelength at 1,550 nm can be switched from the channels 2 to 1 when the applied voltage is tuned from V = 0 to 1.95 V. Insertion loss is defined as the ratio between the output and input power. The insertion losses of the channels 1 and 2 are respectively −3.6 and −5.1 dB for 1,550 nm wavelength. Isolation is defined as the output power ratio between two separate channels. The isolation is about 18 dB between channels 1 and 2 for 1,550 nm wavelength at the absence of electric field. Extinction ratio is defined as insertion loss difference between the input and output ports in the states on and off. The extinction ratio of the channels 1 and 2 are respectively about 18 and 24 dB for 1,550 nm wavelength. From the inset of Fig. 2a, one can see that the switch wavelength can be easily tuned by choosing an appropriate nanocavity length. Therefore, this structure is suitable for wideband wavelengths switch.

Fig. 4
figure 4

a Transmission spectra of the 1 × 2 circuit switch structure in the absence of electric field with the lengths of L 1 = 120 and L 2 = 142 nm (colored). b Transmission spectra of the 1 × 2 circuit switch structure in the presence of electric field (V = 1.95 V) with the lengths of L 1 = 120 and L 2 = 142 nm

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Figure 5a and b show the propagation of field H y for a monochromatic wave with wavelength of 1,550 nm launched into circuit switch structure. The switch effect is clearly observed. From the figure, one can see the wave with wavelength of 1,550 nm is output through the channel 2 in the absence of electric filed shown in Fig. 5a and output through the channel 1 in the presence of electric filed (V = 1.95 V) shown in Fig. 5b.

Fig. 5
figure 5

The contour profiles of field Hy of the 1 × 2 circuit switch structure at wavelength λ = 1,550 nm (colored). All parameters of the structure are same as in Fig. 4

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Discussion of Dual-Nanocavities Structure for Wavelength Conversion Switch

Figure 6 shows a typical schematic of a 1 × 2 wavelength conversion switch structure based on MDM nanocavities. Two power monitors are set at the locations of Q 1 and Q 2 (outside the resonator but just near its exit) to detect the output channels 1 and 2 transmitted powers of P 1 and P 2.The transmittance are defined to be T 1  = P 1 /P in and T 2 = P 2/P in, where P in is the incident power without reflection of the beam. L 1 and L 2 are respectively the nanocavity length of the channels 1 and 2. The others parameters of the structure are kept the same as Section Nanocavity Theory Model. V 1 and V 2 are respectively voltages of the channels 1 and 2.

Fig. 6
figure 6

Schematic of 1 × 2 wavelength conversion switch structure based on MDM nanocavities (colored)

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Figure 7a shows transmission spectra at the outputs of the two channels with the voltage of V 1 = 0 V and V 2 = 1.8 V. One can see that channels 1 and 2 can respectively select 1,310 and 1,550 nm bands with the length of L 1 = L 2 = 120 nm Fig. 7b shows transmission spectra at the outputs of the two channels with the voltage of V 1 = 1.8 V and V 2 = 0 V. One can see that channels 1 and 2 can respectively select 1,550 and 1,310 nm bands with the length of L 1 = L 2 = 120 nm. From Fig. 7a and b, it is noticed that the output wavelength at 1,550 nm can be switched from the channels 2 to 1 when the applied voltages are tuned synchronously from V 1 = 0 to 1.8 V and V 2 = 1.8 to 0 V. And the output wavelength at 1,310 nm can be switched from channels 1 to 2 when the applied voltage is tuned synchronously from V 1 = 0 to 1.8 V and V 2 = 1.8 to 0 V Therefore, the waveguide structure is a typical wavelengths conversion switch structure. The insertion losses of both channels 1 and 2 are −5.7 and −6.8 dB for 1,310 and 1,550 nm wavelength, respectively. The extinction ratio of both channels 1 and 2 are respectively about 16 dB for 1,550 nm wavelength and 23 dB for 1,310 nm wavelength. Cross-talk is defined as the ratio between the power of the undesired and desired bands at the outputs. The cross-talk between channels 1 and 2 is around −23 dB for the 1,550 nm wavelength, and the cross-talk between them is −28 dB for the 1,310 nm wavelength. Based on electro-optical effect, the switching time is about several nanoseconds level.

Fig. 7
figure 7

a Transmission spectra of the 1 × 2 wavelengths conversion switch structure with the lengths of L 1 = L 2 = 120 nm, V 1 = 0 V and V 2 = 1.8 V (colored). b Transmission spectra of the 1 × 2 wavelengths conversion switch structure with the lengths of L 1 = L 2 = 120 nm, V 1 = 1.8 V and V 2 = 0 V

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Conclusion

In conclusion, we demonstrate electro-optic switches with a subwavelength device length based on MDM nanocavities waveguide combined with organic EO materials. The circuit switch and wavelength conversion switch are introduced and their performances are analyzed. The structure is suitable for wideband wavelengths switch. The wavelength conversion switch structure might become a choice for the design of integrated architectures for optical computing and communication, especially in WDM systems in the nanoscale.