Introduction

In view of the development of integrated photonics, various approaches have been proposed to design micro-nano optical components or systems, such as silicon-based waveguides, photonic crystal, plasmonic waveguides, and so on [1–4]. Among these technologies, it is worthy to point out that the future development of all optical communications and highly integrated photonic circuits may benefit from the research of surface plasmon polaritons (SPPs), which can overcome the classical optical diffraction limits. Particularly, plasmonic metal-dielectric-metal (MDM) waveguides have attracted considerable interest because of their advantages of strong confinement of SPPs, low propagation loss, and simple manufacture process. Thus, numerous MDM-based devices have been developed, including optical switches, Mach-Zehnder interferometers, biomedical sensors, wavelength filters, and so on [5–17]. In an integrated optical circuit, it is quite necessary to control the SPP propagation direction or split SPPs into multiple output waveguides with expected wavelengths or power ratios. For example, SPPs at different wavelengths could travel in the desired directions by setting different notches or cavity resonators on the opposite sides of an MDM waveguide. Y-shaped MDM waveguides were also employed to split SPPs into its two arms [18, 19]. Besides, unidirectional excitation of SPPs was achieved by the interference effects at the exits of multiple slits [20–24]. Recently, directional SPP propagation and coupling phenomena were also demonstrated through the polarization-sensitive apertures in a gold film and the near-field interference effects of a circularly polarized light, respectively [25, 26]. However, the performances, such as the transmissions, cross-talk isolations, wavelength manipulation, etc., are not satisfactory enough to favorably compare with that of the classical fiber equipments.

In this paper, a high-performance subwavelength MDM splitter is proposed by using an asymmetric rectangular waveguide. As SPPs at the exits of the arms are contributed from two optical paths with unequal widths, interferences will arise due to the different phase delays. SPPs at the desired wavelengths will transmit through Port1 or Port2 close to the arm where constructive interference occurs, and high cross-talk isolations will be simultaneously obtained at Port2 or Port1, respectively. Besides, by adding a notch in the arm to modulate the phase delay, the propagation directions for both SPP modes can be changed to be identical. The performances of the structure are investigated through the finite-difference time-domain (FDTD) method with perfectly matched layer (PML) boundary conditions. The results show that favorable performances are available for all the cases. As both SPP modes will separately propagate through two ports or they will simultaneously transmit towards only one port, the device can serve as a bidirectional/unidirectional wavelength splitter.

Bidirectional Wavelength Splitter

The proposed wavelength splitter is shown in Fig. 1. The metal and dielectric are assumed to be silver and air, respectively. An MDM bus waveguide is connected to an asymmetric rectangular structure, which provides two output ports in opposite directions. According to the dispersion equation of TM mode, the effective refractive index n eff will be significantly affected by the width of the waveguide. Therefore, the width of two arms (named as Arm1 and Arm2) are set to be different as W and D to obtain different phase delays. Usually, the effective refractive index n eff of the MDM structure can be obtained through [27]

$$ {\varepsilon}_i{k}_m+{\varepsilon}_m{k}_i \tanh \left(-j{k}_id/2\right)=0 $$
(1)
$$ {n}_{\mathrm{eff}}=\beta /{k}_0 $$
(2)

where k 0 = 2π/λ and β are the wave numbers of light in vacuum and waveguide, respectively, \( {k}_{i,m}=\sqrt{\varepsilon_{i,m}{k_0}^2-{\beta}^2} \) is the transverse propagation constant in air and silver, respectively, d is the general definition of the waveguide width (W or D can be substituted for it), and ε i and ε m are the dielectric constants of air and silver, respectively.

Fig. 1
figure 1

Scheme of bidirectional wavelength splitter

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After launching into the bus waveguide, SPPs will be separated into two parts at the intersection of the asymmetric rectangular structure and then propagate along its two arms, respectively. To make the bidirectional SPP splitting available, one should guarantee that destructive interferences and constructive interferences of SPPs at the desired wavelengths must occur at the exits of the two arms, respectively. For example, SPPs at the exit of Arm2 are contributed from two parts, one of which propagates along Arm1 and then through the middle output waveguide and the other from Arm2 directly. When the matched output port for SPPs at λ 1 is assumed to be Port1, constructive interference and destructive interference must arise at the exit of Arm1 and Arm2, respectively. As such, a high transmission for Port1 and a high cross-talk isolation for Port2 are obtained synchronously. In this case, the phase delays must satisfy the condition:

$$ {\theta}_2+{n}_wL\frac{2\pi }{\lambda_1}={\theta}_1+2N\pi $$
(3)
$$ {\theta}_1+{n}_wL\frac{2\pi }{\lambda_1}={\theta}_2+\left(2M+1\right)\pi $$
(4)

where θ 1 = (L 1 + L a )2πn w /λ 1 + ϕ 0 and θ 2 = (L 2 + L a1)2πn w /λ 1 + 2πn s L a2/λ 1are the phases at the exits of Arm1 and Arm2, respectively, ϕ 0 denotes the initial phase difference between the directly generated up-going and down-going SPPs by the bus waveguide, n w, s are the real parts of the effective indices of Arm1 and Arm2, L = L 1 + L 2 is the total length of the vertical side of the rectangular structure, L 1, 2 are the corresponding lengths of upper and below parts, L a1, a2 are the lengths of two parts of Arm2, L a is the total length of an arm, and N and M are integers.

According to Eqs. (3) and (4), the following phase condition can be obtained:

$$ \begin{array}{c}\hfill {\theta}_1-{\theta}_2=\frac{2\pi }{\lambda_1}\left[\left({L}_1-{L}_2\right){n}_w+\left({n}_w-{n}_s\right){L}_{a2}\right]+{\phi}_0\hfill \\ {}\hfill =K\pi +\frac{\pi }{2}\hfill \end{array} $$
(5)

where K is a positive or negative integer. When Eqs. (3)–(5) are achieved, SPPs at λ 1 will only propagate towards Port1. Likewise, when the phase conditions of SPPs at λ 2 are reversed at the exits of Arm1 and Arm2, the matched output port for λ 2 turns to be Port2. Specifically, Eqs. (3) and (4) should be rewritten as \( {\theta}_1^{\prime }+{n}_wL\frac{2\pi }{\lambda_2}={\theta}_2^{\prime }+2N\pi +{\phi}_0^{\prime } \) and \( {\theta}_2^{\prime }+{n}_wL\frac{2\pi }{\lambda_2}={\theta}_1^{\prime }+\left(2M+1\right)\pi \), respectively. Obviously, the supported transmission wavelengths for two output ports are different, and thus, the proposed structure can serve as a bidirectional wavelength splitter.

In the following, the FDTD method with PML boundary conditions is employed to investigate the performances of the structure. PML layers with reflection of 0.0001 are set to surround the whole structure, and the grid sizes along x- and y-directions are chosen to be Δx = Δy = 5 nm. The fundamental TM mode of the plasmonic waveguide is excited by a plane wave incident from the input waveguide. Two monitors are placed in Port1 and Port2 to obtain the transmission. By comparing the transmittances in two output ports, we can directly obtain the isolation at the specific wavelength. Other parameters are specified as follows: the width of Arm2 is D = 20 nm, the widths of all other waveguides are defined as W = 60 nm, the lengths of two parts of the structure’s left side are L 1 = 155 nm and L 2 = 305 nm, and the beginning length of Arm2 is L a1 = 100 nm. In addition, the detailed values of the optical constant of silver [28] are used during the simulations. The simulated spectra are shown in Fig. 2, where the length of the right side of Arm2 is considered as L a2 = 400 nm or L a2 = 450 nm, respectively. Detailed results, including the wavelengths and the transmissions, are shown in Table 1. By taking λ 2 = 1412.2nm and L a2 = 400 nm for example, we can obtain n w  = 1.298 and n s  = 1.713 through Eqs. (1) and (2). Furthermore, θ 2 − θ 1 ≈ 0.51π, which generally agrees with the analysis result, is obtained according to Eq. (5).

Fig. 2
figure 2

Transmission spectra for Port1 and Port2 with a L a2 = 400 nm and b L a2 = 450 nm

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Table 1 Simulation results with different L a2
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Obviously, the supported transmission wavelengths for two output ports are demonstrated to be completely different. A high transmittance will be achieved for the matched output waveguide while a high cross-talk isolation will be available for the other port in its opposite direction.

To analyze the electromagnetic responses of the structure, the magnetic fields for SPPs at the transmission wavelengths that correspond to the ones in Fig. 2 are provided in Fig. 3. In the rectangular circle, there is a slight resonance effect that is analogous to the ring cavity effect. However, since both sides of the circle are connected to the input and output waveguides directly, thus, the resonance effect will be weaker than those ring cavity filters, which have a coupling distance to the bus waveguide. Moreover, by taking Fig. 3a for example, the matched output port for SPPs at 883.8 nm should be Port1 based on the above transmission spectrum. It can be found that constructive interference and destructive interference occur at the exits of Arm1 and Arm2 in Fig. 3a, respectively. Therefore, most of the SPP energy directly passes through Port1, and there are almost no SPPs distributed in the output waveguide towards Port2. On the contrary, the propagation status for SPPs at 1412.2 nm is completely different in Fig. 3b, which shows that SPPs only transmit through Port2. These results are highly in accordance with that presented in Fig. 2a. Moreover, the magnetic fields of SPP modes at 948.2 and 1515.6 nm of Fig. 2b are also shown in Fig. 3c, d, which demonstrate that SPPs propagate only towards Port1 and Port2, respectively.

Fig. 3
figure 3

The magnetic fields for SPPs at the transmission wavelengths corresponding to that in Fig. 2: a 883.8 nm, b 1412.2 nm, c 948.2 nm, d 1515.6 nm

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Being compared with the results in Fig. 2a, b, a red shift phenomenon for the transmission wavelengths will arise at both ports by increasing L a2. To find out more details, L a2 is increased from 400 to 500 nm with a step of 10 nm while other parameters are fixed. The simulation results are shown in Fig. 4. The wavelength increase steps for Port1 and Port2 are fixed as about 18.6 and 11.8 nm, respectively, and thus, a linear relationship between the transmission wavelength and L a2 is confirmed. The simulated result agrees well with the theory analysis in Eq. (5), which indicates a linear influence of L a2 on the wavelength. Actually, the working frequency of our proposed structure can cover the range from visible light to infrared light that supports the SPPs generation, provided that the phase conditions are matched to Eqs. (3) and (4). This character may offer great flexibility to design the structure and one can easily obtain the desired transmission wavelength for the expected output port.

Fig. 4
figure 4

The transmission wavelength variation with respect to L a2

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Besides, the left side of the rectangular structure is also lengthened, while other parameters are the same as that in Fig. 2b. We can obtain the length differences of ΔL = L 2 ‐ L 1 = 70, 90, 110 nm, as L 1 = 165, 175, 185 nm and L 2 = 335, 365, 395 nm are defined in Fig. 5a–c, respectively. According to Eq. (5), the length differences of ΔL will also affect the transmission wavelength linearly, which is further confirmed by the simulated transmission spectra in Fig. 5. The detailed results are shown in Table 2. Obviously, both the SPP mode at λ 1 for Port1 and the mode at λ 2 for Port2 increase linearly with ΔL. Besides, it can be found that the cross-talk isolation for the unmatched output port receives a little tolerable penalty by increasing ΔL, but we still consider that the structure can successfully serve as a bidirectional wavelength splitter. More details about the magnetic field distributions can be seen in the insets of Fig. 5.

Fig. 5
figure 5

Transmission spectra for Port1 and Port2 with a L 1 = 165 nm, L 2 = 335 nm, b L 1 = 175 nm, L 2 = 365 nm, and c L 1 = 185 nm, L 2 = 395 nm

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Table 2 Simulation results with different L 1 and L 2
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Unidirectional Wavelength Splitter

Based on the above analyses and simulations, it can be understood that the transmission wavelengths can be manipulated by changing ΔL or L a2. However, the propagation directions for both modes are separated to two opposite sides in this structure. It is interesting to point out that unidirectional propagation for all the modes may be desired in the integrated photonic circuits. In this case, one should change the phase delay of one optical path to guarantee that destructive interferences for both modes will occur at the exit of Arm1 or Arm2 simultaneously.

On the basis of the scheme in Fig. 1, a notch is added in Arm1 to modulate the phase delay, as shown in Fig. 6. Note that the position of the notch, which is denoted by D s , is a significant factor to obtain the unidirectional propagation for both modes. Specifically, in Fig. 2, the matched output ports for SPP modes at λ 1 and λ 2 are Port1 and Port2, respectively. If both modes are expected to transmit through Port1 only, the notch must locate at the node of magnetic field of λ 1 (i.e., |H y | = 0) and at the anti-node of λ 2 (i.e., |H y | = Max) simultaneously. In this case, SPPs at λ 1 will pass the notch directly without additional phase delay. However, SPPs at λ 2 will be captured into the notch, and thus, the phase delay is modulated. By properly designing the length of the notch, destructive interferences and constructive interferences for both modes are achieved at the exits of Arm2 and Arm1, respectively. In this case, the phase delays of SPP modes at λ 1 and λ 2 will together obey Eqs. (3) and (4). Besides, Eq. (5) is still suitable for the mode at λ 1 but should be rewritten for the mode at λ 2 as

$$ \begin{array}{l}{\theta}_2-{\theta}_1=\frac{2\pi }{\lambda_2}\left[\left({L}_2-{L}_1\right){n}_w+\left({n}_s-{n}_w\right){L}_{a2}-2{L}_s{n}_w\right]+{\phi}_0+\varphi \\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill =K\pi +\frac{\pi }{2}\hfill \end{array}\end{array} $$
(6)

where L s is the length of the notch and φ is the phase delay caused by the reflection on the end of the notch. Then, SPPs at λ 1 and λ 2 will only propagate towards Port1. Interestingly, when the notch locates at the anti-node of H y of λ 1 and at the node of λ 2 simultaneously, the matched output ports for both SPPs modes turn to be Port2.

Fig. 6
figure 6

Scheme of unidirectional propagation for both modes

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In the following simulation, the width of the notch is W = 60 nm and other parameters are the same as that in Fig. 2b. The transmission spectra are shown in Fig. 7a with D s  = 305 nm and L s  = 365 nm and in Fig. 7b with D s  = 0 nm and L s  = 220 nm. Obviously, the matched output ports for both SPP modes are Port1 or Port2 in Fig. 7a, b, respectively. The detailed results are shown in Table 3. High transmission for the matched output port and high cross-talk isolation for the other port are also achieved for both modes.

Fig. 7
figure 7

Transmission spectra for Port1 and Port2 with a D s  = 305 nm and L s  = 365 nm and b D s  = 0 nm and L s  = 220 nm

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Table 3 Simulation results with different D s and L s
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The magnetic field distributions in Fig. 8a, b correspond to the transmission wavelengths in Fig. 7a, while the ones in Fig. 8c, d correspond to the transmission wavelengths in Fig. 7b. For example, it can be seen that most of the SPPs at 945.9 nm directly pass through the notch in Fig. 8a as the notch locates at the node of H y . However, different situations arise for SPPs at 1435.0 nm in Fig. 8b, where SPPs are captured into the notch for the reason that the magnetic field reaches almost the maximum value at the position of the notch. According to the interference effects at the exits of the arms, SPPs at both wavelengths only transmit through Port1. Moreover, the output ports for both modes are Port2 only in Fig. 8c, d because the SPP distributions in the notch are opposite to that in Fig. 8a, b, respectively. As SPPs at all the desired wavelengths will transmit only through the matched port in one direction, the proposed structure can act as a unidirectional wavelength splitter as well.

Fig. 8
figure 8

The magnetic fields for SPPs at the transmission wavelengths corresponding to those in Fig. 7: a 945.9 nm, b 1435.0 nm, c 919.9 nm, and d 1514.1 nm

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Conclusion

A bidirectional or unidirectional wavelength splitter has been proposed and investigated by an MDM structure with two asymmetric arms. According to the interference effects at the exits of the arms of the asymmetric structure, two SPP modes at the desired wavelengths will transmit through Port1 and Port2 respectively. The bidirectional wavelength splitter was with impressive performances, as the highest transmission and cross-talk isolation were ∼−0.60 and ∼32.9 dB at 883.8 nm for Port1 and Port2, respectively, and favorable performances were also achieved for all other cases. The transmission wavelengths have been demonstrated to be linear with the lengths of the arms, which might offer flexibility to design the device. Besides, unidirectional wavelength splitter was achieved by properly adding a notch in the arm of the structure. Both SPP modes would only transmit through Port1 or Port2.