Asymmetric band-pass plasmonic nanodisk filter with mode inhibition and spectrally splitting capabilities

Guangzhi Zhan; Ruisheng Liang; Haitao Liang; Jie Luo; Ruitong Zhao

doi:10.1364/OE.22.009912

1. Introduction

Surface plasmon polaritons (SPPs) are kinds of waves that spread along a metal-dielectric interface with an index attenuation of electromagnetic field in both side, and because of the prominent features that it can overcome the limit of traditional optical diffraction and manipulate optical wave on sub-wavelength scales [1], it has been considered to be information carriers in highly integrated optical circuits. By now, a number of SPPs subwavelength optical waveguide devices, such as bend waveguide device [2], Bragg reflectors [3,4], sensors [5], nano-particles [6], Y-shaped combiners [7], Mach-Zehnder interferometers [8], V-grooves [9], and nanowires [10], have been investigated experimentally or theoretically. The MIM waveguides are the hottest spot in recent years because of miniaturization and high level of integration of optical circuits. At present, many researchers are committed to the development of nanoscale MIM waveguide devices such as apertures [11], wavelength selective waveguide [12], filters based on ring resonators [13], nanodisk resonator [14], and all-optical switching [15]. Among the SPP-guiding structures mentioned above, Band-pass and band-stop filters are vitall components of wavelength-selective devices because of their practicability, symmetry, and simple manufacturing [16].

In some systems such as the wavelength division multiplexing (WDM) system, the band-pass filter [17–20] is of great significance. WDM plays an important role in signal processing in optical communication technology. Therefore, how to realize wavelength selection is the priority mission in plasmon devices. In this case, Bragg reflectors of MIM structure [21] have been theoretically proposed. But most of them have a relatively large size and high transmittance loss. Recently, nanoring resonators, nanodisk resonators [22], and rectangular resonators [23] have been designed as bang-pass filters. However, almost all of the above structures can only change their resonance wavelength by modifying the geometry of their resonant cavity.

In this paper, a circular band-pass plasmonic filter based on a nanodisk resonant cavity is proposed and its transmission properties are investigated numerically and analytically. Comparing with previous filter researches, a new adjusting mechanism is applied to nanodisk structure to enhance the filtering characteristics in transmittance and pulse width. In this new filter structure, the resonance modes, inside the nanodisk, can be effectively suppressed by selecting proper length in input and output coupling area and separated by putting the output port in right place. Moreover, the transmission spectra (including the resonance wavelengths and bandwidths) of the filter is easily adjusted by changing the geometric dimensioning of the nanodisk and the coupling distance between the waveguides and the nanodisk. The completely matched layer absorbing boundary conditions of finite difference method is used in the simulation.

2. Device structure and theoretical analysis

The illustration in Fig. 1(a) shows that the structure is composed of two slits, a nanodisk resonator symmetrically placed between the two slits, and in a homogeneous metallic background. The width of waveguides t is set as 50nm, the distance between the boundaries of the waveguides and the cavity d is 8 nm and the radius of the disk cavity r is set as 410 nm. The medium of the slits and nanodisk is assumed to be air whose refractive index is set to 1. The surrounding metal is silver, whose frequency-dependent complex relative constant is characterized by the Drude model [2,24]

ε_{m} (ω) = ε_{\infty} - \frac{ω_{D}^{2}}{ω^{2} + i γ_{D} ω} - \sum_{m = 1}^{2} \frac{g_{L m} ω_{L m}^{2} Δ ε}{ω^{2} - ω_{L m}^{2} + i 2 γ_{L m} ω} .

Here

ε_{\infty}

is the large-frequency limit of the permittivity with the value of 2.3646, and

ω

is the angular frequency of the incident wavelength.

γ_{D}

= 0.07489eV is the frequency of the damped oscillation.

ω_{D}

= 8.7377eV is the frequency of the majority of the plasma.

ω_{L 1}

= 4.3802eV,

g_{L 1}

= 0.26663,

γ_{L 1}

= 0.28eV,

g_{L 2}

= 0.7337,

ω_{L 2}

= 5.183eV,

Δ ε

= 1.1831, and

γ_{L 2}

= 0.5482eV. These parameters listed above provide good description of permittivity data for silver [25].

Fig. 1 (a) The schematic diagram of nanodisk filter (b) The transmission spectrum of the nanodisk filter with r = 410nm, d = 8nm, t = 50nm.

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When the incident optical wave transmits through input waveguide, part of the energy will be reflected and part of it coupled into nanodisk segment. When resonance condition is reached, the stable standing wave in the nanodisk will be formed. The resonant condition can be obtained by solving following equation [16,26]

k_{d} \frac{H_{n}^{(1)'} (k_{m} r)}{H_{n}^{(1)} (k_{m} r)} = k_{m} \frac{J_{n}^{'} (k_{d} r)}{J_{n} (k_{d} r)} .

Here

k_{d, m} = k {(ε_{d, m})}^{1 / 2}

are the wave vectors in the metal and the dielectric nanodisk, respectively.

ε_{m}

is the relative dielectric constant of the metal and

ε_{d}

is the relative dielectric constant of the dielectric, which can be obtained from the Drude model.

k

stands for the wave number which includes a relatively small negative imaginary part, and the negative imaginary part here stands for the loss. r represents the radius of the nanodisk cavity.

H_{n}^{(1)}

and

H_{n}^{(1)'}

are the Hankel function with the order n.

J_{n}^{}

and

J_{n}^{'}

are the Bessel function with the order n, respectively. The first and second order of Bessel and Hankel functions correspond to the first and second order modes that resonate inside the nanodisk. From Eq. (2), one can find that the resonance wavelength

λ_{0}

is determined by refractive index and the radius. By using the coupled-mode theory [27], the transmission T near the resonant wavelength of the system is described below

T = \frac{{(\frac{1}{τ_{w}})}^{2}}{{(ω - ω_{0})}^{2} + {(\frac{1}{τ_{w}} + \frac{1}{τ_{i}})}^{2}} .

Where

ω

stands for the frequency of the incident light, and

ω_{0}

is the resonance frequency.

τ_{w}

stands for the energy escaping from the waveguide and

τ_{i}

is the decay rate of the electromagnetic field because of the internal loss in the nanodisk. The transmittance formula shows that if the incident light is far from resonance frequency, it will be completely reflected. Thus, there is a transmitted peak at resonance frequency. From Eq. (3), the resonance peak transmittance

T_{\max} = {(1 / τ_{w})}^{2} / {(1 / τ_{w} + 1 / τ_{i})}^{2}

is close to 1 when

1 / τ_{i}

is far less than

1 / τ_{w}

. Moreover, one can find that the transmission spectra around the resonant frequency shows Lorentzian profiles.

3. Simulation results and analysis

The FDTD method is used to simulate the transmission characteristics. The grid size in the x and z directions are selected as 5 $\times$ 5nm, which has enough precision in the following simulation. The SPPs enter into structure from left side. Figure 1(b) shows the numerical simulation results of the transmission spectra when the gap between the disk and waveguide is 8nm. From it, one can see that there are two resonance modes (Mode 1 and Mode 2) at the wavelengths of 956nm and 1550nm locating in the wavelength range from 0.75um to 2um, and the corresponding maximum transmittances are 81% and 75%, respectively. The full width at half-maximum (FWHM) of such two resonance modes are 37nm and 48nm, respectively. The transmission peaks do not reach unity due to the waveguide loss in the metal slits and the internal loss in the nanodisk. Both the maximum transmittance and FWHM are better than the structure based on the nano-cavity resonators proposed by Huang et al. [28]. As shown in Fig. 1(b), the transmittance of mode 2 has higher value than mode 1. This phenomenon happens because the resonant mode 2 has larger power escape than mode 1.

Figure 2(a) shows the transmission spectrum according to different radii of the nanodisk cavity. It is obvious that the transmitted peak has a red-shift with the increasing radius. As shown in Fig. 2(b), the wavelength-shifts of the resonant modes 1 and 2 almost have linear relationship with the radius of the nanodisk cavity. The results of FDTD simulation is in accordance with the solution of Eq. (2). According to the results and the above analysis, one can easily manipulate the wavelengths of resonant modes by modifying the radius of nanodisk resonator.

Fig. 2 (a) Transmission spectra about different radii of the nanodisk cavity with d = 8nm, t = 50nm. (b) Relationship between peak resonance wavelengths and the radius of the cavity.

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In the next step of the research, we focus on the distance between the boundaries of the waveguides and the cavity d, which is also an important factor influencing the characteristic of transmission spectra near the resonant wavelength. With increasing the distance of boundaries of the waveguides and the cavity, the external loss $1 / τ w$ will decrease rapidly while the internal loss $1 / τ i$ is almost unchanged. Therefore, the peak of transmission will decrease as the distance increasing. As shown in Figs. 3(a) and 3(b), the theoretic analysis is verified by FDTD simulations. The transmission peaks can be controlled by adjusting the distance between the boundaries of the waveguides and the cavity d. We also find that the center wavelength has a slight blue shift. In addition, it is obvious that the FWHM of the resonance spectrum markedly decrease with increasing the distance between the boundaries of the waveguides and the cavity as shown in Fig. 3(a).

Fig. 3 (a) Transmission spectra about different distance between the boundaries of the waveguides and the cavity d with r = 410, t = 50nm. (b) Relationship between the peak resonance wavelengths and the distance of d.

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4. Transmission properties of suppression mode band-pass filter

Figure 4 shows the schematic diagram of filter with the ability of inhibiting resonant mode. We can see that the only difference between Fig. 1(a) and Fig. 4 is a section of waveguide added on the output waveguide. In Fig. 4, we use L to stand for such segment of the waveguide. As shown in Fig. 5(a), the transmission spectra almost the same with this segment of the waveguide increasing from 0nm to 50nm. However, as this segment increasing and reaching to 135nm, the transmittance of mode2 (956nm) has remarkable attenuation while the transmittance of mode1 (1550nm) is almost unchanged. In Fig. 5(b), we can see that the transmission spectra of original structure and new structure have a huge difference at the wavelength of 956nm. This phenomenon can be explained by the following principle. When the incident wideband SPPs waves are coupled into output waveguide from nanodisk, the wave divided into two parts, the forward waves and backward waves. We use S1 and S2 to stand for them, respectively. After backward waves are reflected at the end of output port, the new coupled forward waves will encounter reflected waves (S2’). If such two waves satisfy the phase conditions, it will reduce the intensity of some certain wavelength. Defining $Δ φ$ to be the phase delay of per round trip in the segment of the waveguide, one gets $Δ φ = k (ω) \times 2 L + φ_{r}$ , where $ϕ_{r}$ is the phase shift of the wave reflected at the end of the output waveguide, and $k (ω)$ is the angular wavenumber of the wavelengths inside the waveguide at the frequency of the light in vacuum. Obviously, some wavelengths will disappear when the following condition is satisfied $Δ φ = (2 m - 1) π$ , As $Re [k (ω)] = 2 π n e f f / λ$ , the disappearing wavelength $λ_{m}$ is given as follows [28]

λ_{m} = \frac{4 n_{e f f} L}{(2 m - 1) - φ_{r} / π} .

Where

λ_{m}

is vacuum wavelength and n_eff is the real part of the effective index in the waveguide. According to Eq. (4), we substitute the data of m = 1, n_eff = 1.5(according with width of the waveguide t = 50nm), and L = 135nm into Eq. (4), then we can get the disappearing wavelength of 952nm when assuming the phase shift

ϕ_{r}

with 0.15

π

. In this way, it is quite reasonable to compare theoretical analysis with simulation. Due to the symmetry of the suppression mode band-pass filter, a similar result will get when using such inhibiting structure at input waveguide instead of output waveguide. Furthermore, the inhibiting structure can be used in the input and output waveguide at the same time to filter out two different wavelengths. Figures 6(a) and 6(b) show the propagation of the field |Hz| for incident light with the wavelength of 956nm and 1550nm. As for the output waveguide, it is coupled to the antinode when the incident light are 956nm and 1550nm. However, the wavelength of 956nm will be suppressed when the inhibiting structure is added in the output waveguide. We can find the entirely different result from the field images and significant effect by using the inhibiting structure at output waveguide.

Fig. 4 The schematic diagram of an asymmetric nanodisk filter with d = 8nm, r = 410nm, and t = 50nm.

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Fig. 5 (a) Transmission spectra of different length of L from 0nm to 50nm. (b) Transmittance contrast image between 0nm and 135nm of length of L.

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Fig. 6 Magnetic fields of the band-pass filter with inhibiting structure for monochromatic light at different wavelengths, (a) $λ$ = 956nm, (b) $λ$ = 1550nm, and all geometric parameters are the same as used in Fig. 5(a).

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5. The design of 1 $\times$ 2 wavelength demultiplexing

According to the above characteristics of the asymmetric plasmonic filter based on nanodisk resonators, a 1 $\times$ 2 wavelength demultiplexing structure based on nanodisk is designed. Figure 7(a) shows the schematic diagram. This structure consists of two output ports at the bottom and middle of the nanodisk, respectively. All of the distance between the boundaries of the waveguides and the cavity is set as 8 nm. And the bottom output waveguide uses the inhibiting structure. As shown in Fig. 7(b), the transmitted-peak wavelengths of two output channels are 956nm and 1550nm, respectively. This is to say, the two resonant modes inside the nanodisk are separated and exit from different channels, port 1 and port 2, respectively. Figures 8(a) and 8(b) show the propagation of the field |Hz| for incident light with the wavelength of 956nm and 1550nm. It is obvious that the incident light at 956nm and 1550 nm can only pass through the corresponding port, respectively. As for port 2 without the inhibiting structure, the wavelength of 956nm and 1550nm can be output and with a low transmittance, but there is only the wavelength of 1550nm can be output when using the inhibiting structure. This characteristic can be used to realize a band-pass filter with spectrally splitting function.

Fig. 7 (a) Schematic diagram of a 1 $\times$ 2 wavelength demultiplexing structure based on nanodisk cavity. (b) The transmission spectra of a 1 $\times$ 2 wavelength demultiplexing structure with d = 8nm, r = 410nm, and t = 50nm.

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Fig. 8 Magnetic fields of the 1 $\times$ 2 wavelength demultiplexing schematic diagram for monochromatic light at different wavelengths, (a) $λ$ = 956nm, (b) $λ$ = 1550nm, and all geometric parameters are the same as used in Fig. 5(a).

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At last, comparing of our proposed structure with those considered in Refs [17–20], the research in our work focuses on inhibition and separation of the modes inside the nanodisk. Although separating the mode in a nano-cavity technique has been achieved [20], it is the first time to put forward a feasible scheme to separate mode in a nanodisk cavity. Moreover, the proposed method can overcome the problem of low transmittance in the most of filters [17,18]. The transmittance in our proposed structure can be up to 80% without multiple wavelength responses.

6. Summary

In this paper, a subwavelength plasmonic nanodisk filter is proposed and numerically analyzed by utilizing FDTD method. The novel phenomena of suppressing resonance mode and spectrally splitting light have been both theoretically demonstrated and numerically verified in this paper. The desired wavelengths can be obtained by adjusting the radius of the nanodisk resonator. The FWHM and transmittance can be tuned by modifying distance of the coupling. In addition, the resonance modes inside nanodisk can be easily inhibited and separated by adding a segment of waveguide at the input/output waveguide. The proposed structures have important applications in highly integrated optical circuits and nanoscale optics.

Acknowledgments

This work was supported by the National Natural Science Foundation of China Grants No.61275059 and No.61307062.

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Asymmetric band-pass plasmonic nanodisk filter with mode inhibition and spectrally splitting capabilities

Abstract

1. Introduction

2. Device structure and theoretical analysis

3. Simulation results and analysis

4. Transmission properties of suppression mode band-pass filter

5. The design of 1 $\times$ 2 wavelength demultiplexing

6. Summary

Acknowledgments

References and links

Cited By

Figures (8)

Equations (4)

Optics Express

Abstract

1. Introduction

2. Device structure and theoretical analysis

3. Simulation results and analysis

4. Transmission properties of suppression mode band-pass filter

5. The design of 1×2 wavelength demultiplexing

6. Summary

Acknowledgments

References and links

Cited By

Figures (8)

Equations (4)

Optics Express

5. The design of 1 $\times$ 2 wavelength demultiplexing