## Abstract

Directional coupler (DC) and nonlinear Mach-Zehnder interferometer (MZI) based on metal-insulator-metal (MIM) plasmonic waveguide are investigated numerically. We show that the coupling length increases almost linearly with the wavelength and this property is utilized in the design of wavelength division multiplexer (WDM). A nonlinear MZI, with one branch filled with Kerr nonlinear medium, is built to ensure controlling light with light. Employing nonlinear processes including self-phase modulation (SPM) and cross-phase modulation (XPM), intensity-based router and all-optical switch are realized.

©2010 Optical Society of America

## 1. Introduction

The diffraction limit of light poses a great challenge to the miniaturization and high-density integration of photonic circuits. One solution to overcome this problem is to exploit the properties of surface plasmon polaritons (SPPs), which are bound waves at the interface between a metal and a dielectric [1]. In recent years, various configurations of SPPs waveguides such as metal-insulator-metal (MIM) waveguide, insulator-metal-insulator (IMI) waveguide and dielectric loaded surface plasmon polaritons waveguide (DLSPPW) have been suggested and demonstrated [2–7]. IMI waveguide and DLSPPW have much longer propagation length than MIM waveguide because the energy mostly propagates in the low loss dielectric layer with very little field intensity inside the lossy metal. To achieve nanoscale photonic functionality, however, it is preferable to utilize MIM waveguide due to its ability to confine light to deep subwavelength scale (<λ/15) [2]. The propagation length of MIM waveguide, defined as the distance after which the intensity decreases to 1/e of its original value, varies from several micrometers to several tens of micrometers, sufficiently large for nanophotonic applications.

Passive photonic circuits elements such as S bends, directional couplers (DC) and Mach–Zehnder interferometers (MZIs) based on MIM waveguide have been proposed [8–11]. In a recent experiment, Z. Han etc demonstrated an aperture-assisted coupling structure [12].These devices can provide some of the essential components for signal processing, but creating a truly all-optical circuit requires active devices [13]. One promising approach is to exploit nonlinear phenomena such as optical Kerr effect and optical bistability. Some theoretical investigations of all-optical switches using MIM waveguides have been proposed recently [14,15].

In this paper, we calculate the coupling length of plasmonic DC in a wide spectral range using coupled-mode theory and transfer matrix method. The DC is designed to be a wavelength division multiplexer (WDM), in which signals with different wavelengths are separated at the output ports due to the wavelength-division property. Nonlinear MZI is then presented by connecting two DCs and filling one branch arm with Kerr nonlinear medium. Intensity-based router and all-optical switch are realized via self-phase modulation (SPM) and cross-phase modulation (XPM) with the pump light “off” and “on”. Compared with previous works, the proposed nonlinear MZI may have great potential practical applications in all-optical signal processing.

## 2. Principle and the wavelength-selective direction coupler

The MIM slab waveguide is considered where a dielectric slab is surrounded by two semi-infinite silver layers. The numerical simulation can be reduced to two-dimensional problem if the silver film is thick enough such as 200nm [11]. In this case, the characteristic equation of the fundamental transverse magnetic mode TM_{0} is derived from the poles of transmission function in the evanescent region [3] as

*k*stands for the vacuum wave vector,

_{0}*ε*and

_{m}*ε*are the permittivities for the silver and dielectric media, respectively, and

_{d}*w*is the slab width. Drude model

*ε*(ω) =

_{m}*ε*-

_{∞}*ω*

_{p}^{2}/(

*ω*

^{2}+ i

*ωγ*) is employed in numerical simulations with

*ε*= 5.0,

_{∞}*γ*= 0.0987eV, and

*ω*= 9.5eV for silver [16]. It's worth to note that the errors between model and experimental data would be significant for some wavelengths. Measured or reported experimental data is recommended for practical applications. The effective mode refractive index

_{p}*n*is defined as the ratio of

_{eff}*β*and

*k*, which can be calculated by solving Eq. (1) over the complex plane. The propagation length, defined as the distance when the intensity decrease to 1/e of its original value, can be calculated by 1/(2Im(β)) and tends to decrease as the slab width and mode confinement increase, which is shown in Fig. 1 . Since we are more concerned about the deep sub-wavelength property of MIM waveguide, the slab width is chosen as 100nm even the propagation length is only 5.8μm at wavelength 1.064μm.

_{0}As a composite waveguide structure, two identical parallel slabs are separated by a distance *d*, forming four metal-dielectric interfaces (the inset of Fig. 2
). The slabs are further connected to parallel waveguides with 400nm center-to-center separation by 1μm-long S bends. If the gap width *d* is small enough, a significant fraction of power is transferred from one slab to the other slab, and for this reason this structure is called directional coupler (DC).

In the well-established coupled-mode theory (CMT) [17,18], the field distribution in the DC can be expressed as the superposition of the symmetrical and anti-symmetrical supermodes

where*H*(

_{s}*x*) and

*H*(

_{a}*x*) are the amplitudes of supermodes as illustrated in the inset of Fig. 2.

When the signal is injected from the upper waveguide at *z* = 0, the amplitudes of magnetic field for two waveguides at *z* are

*β*and

_{a}*β*are complex propagation constants of the anti-symmetrical and symmetrical supermodes,

_{s}*β*and

*κ =*(

*β*-

_{a}*β*)/2 are propagation constant of single plasmonic waveguide and coupling coefficient of two parallel waveguides, respectively.

_{s}Transfer matrix method [19,20] is employed to calculate the propagation constant of supermodes for two adjacent MIM plasmonic waveguide. The key point of the calculation is to find the proper parallel complex wave vector, at which the transmission and reflection approaches infinity for light incident on the two adjacent MIM waveguides. Since no approximation is included in transfer matrix equations, accurate results could be obtained by proper and sufficient iterative numerical solving process. The calculated complex effective mode refractive indexes of the supermodes, defined as the ratios of the propagation constants with vacuum vector *k _{0}*, are illustrated in Fig. 2.

The minimum interaction length required for a complete power crossover (i.e., coupling length) is determined by the beating length of the two supermodes as

The typical dependencies of the coupling length *L _{c}* on wavelength are shown in Fig. 3
for different gap widths. Unlike the traditional dielectric waveguide and dielectric-loaded long range plasmonic waveguide [7,18],

*L*increases almost linearly with wavelength in a wide spectral range. This difference can be intuitively understood through the frequency behavior of the coupling coefficient. In the condition of weak coupling, most of mode overlapping is in the silver gap and the coupling coefficient can be related to

_{c}*ε*,

_{m}*λ*and

*d*through [18] as

In the conventional DC, the permittivity of the dielectric is positive and remains almost unchanged and the coupling coefficient is determined mainly by the wavelength, resulting that the coupling length varies inversely with the wavelength in a narrow frequency range [18]. Intuitively one would also expect the coupling coefficient to increase with wavelength in the DC composed of MIM waveguides due to poorer mode confinement. However, as mode coupling occurs in the silver gap for the MIM plasmonic DC and the permittivity of silver *ε _{m}* with strong dispersion becomes more negative as the wavelength increases, $\sqrt{{n}_{eff}^{2}-{\epsilon}_{m}}/\lambda $ will increase slightly with wavelength and the overall tendency is reversed.

It is worth to note that the separation (20nm, 30nm and 40nm, for instance) seems to be very small from the view point of conventional dielectric waveguide. But the light is highly confined in the dielectric layer for the MIM plasmonic waveguide. The decay depth in silver wall at which light decreases to be 1/e of the value at metal dielectric interface is only 10.5nm. So this results not so strong coupling effect even for 20nm wide separation and CMT is applicable as approximate guidance for designing directional coupler structure.

In the case of lossy plasmonic waveguides, the propagation lengths of the supermodes should also be taken into account, which could be determined by the imaginary parts of the propagation constants (Fig. 2). The ratio of coupling length to average propagation length (2/Im(*β _{s}* +

*β*)) is of great importance for maximum power transfer [11]. In our case, this ratio at wavelength of 1064nm is 0.2566, which is small enough for power transfer.

_{a}In addition, the maximum and minimum power output at the two exit coupler arms cannot be obtained simultaneously at the same wavelength. The extinction ratio between the output power, |a(z)/b(z)|^{2} is degraded by this so-called extreme power position offset effect. This can be well understood by considering loss effect. As shown in Fig. 2, the symmetrical supermode has longer propagation length than the antisymmetrical supermode (Im(*β _{s}*)<(

*β*)). So

_{a}*κ*is a complex value and the phase shift between a(z) and b(z) is not 90° anymore and there is a position offset between the extreme values of |

*a*(z)| and |

*b*(z)|. That is to say, when |a(z)| gets to its maxima, |b(z)| does not reach its minima [8].

As the gap width *d* decreases, the real and imaginary parts of the coupling coefficient *κ* will increase, which results the decreased coupling length and enhanced extreme power position offset effect. The error of CMT will also become larger for such strong coupling. For these reasons, the refractive index of dielectric *n* and gap width *d* are fixed at 1.535 (corresponding to the refractive index of BCB (Benzocyclobutene) polymer at the light wavelength of 1.55 μm) and 20nm (a typical value chosen in literature [8,9]) in the following.

In order to prove the analysis above, a 600nm-long DC is simulated by 2D-FDTD method with pulse excitation. The mesh sizes are set to be Δ*x* = 2 nm, Δ*z* = 4 nm, which are small enough to capture the change of the field at the interfaces between the dielectric layers and the silver layers. The error due to staircase approximation of the S bends also vanishes in this case. Perfect Matched Layer (PML) boundary condition is employed for all the boundaries. The transverse magnetic mode was excited in the waveguide and observation points and lines are added at the four ports as detectors of magnetic field and power. The spectral response is calculated by Fast Fourier Transform (FFT) of time-domain results at the output ports (Fig. 4
). Coupling ratio near 50% at a wavelength of 1064nm is observed. Here the coupling ratio is defined as the ratio of output power in the coupled branch to total output power in both branches. Since signal is injected into Port1 in our simulations, the coupling ratio is 100% if the energy is fully coupled to Port-4 (corresponding to the minimum power at Port-3) and 50% if the energy is spitted equally between Port-3 and Port-4. In the second case, the directional coupler is said to be a 3-dB DC. The coupling length evaluated from FDTD (e.g., 1.2μm at wavelength of 1064nm) is shorter than that calculated using CMT (e.g., 1.9μm at 1064nm in Fig. 3) due to the additional mode coupling in the S bends and error resulting from the approximation of CMT [17].

## 3. Nonlinear Mach-Zehnder interferometer

The schematic of the nonlinear Mach-Zehnder structure is shown in Fig. 5
. Two plasmonic DCs are connected by two branch arms while Arm-2 is filled with Kerr nonlinear medium. Most of the parameters for the DCs are the same as previous, with *D* = 300nm, *d* = 20nm, *l _{1}* = 1000nm. We choose MEH-PPV [poly(2-methoxy-5-(28-ethylhexyloxy)-PPV)] as the nonlinear material in our structure due to its high optical nonlinearity and good patterning behavior [21]. 2D-FDTD method is utilized to account for the nonlinear processes, while the mesh setting is the same as previous. The response time of this material is not considered in our simulations due to its ultrafast electronic nonlinearity.

#### 3.1 Intensity-based router

In the absence of pump, a TM-polarized continuous wave (CW) at a wavelength of *λ _{s}* = 1.064μm is injected into port-1, and is split equally by the first coupler (i.e., the coupler is 3dB) into two branch arms with the coupler length

*l*= 600nm. After propagating through Arm-1 and Arm-2, the optical waves interact again in the second 3dB coupler. As a result of self-phase modulation (SPM) induced by optical Kerr effect, the phase retardation of Arm-2 is changed by

_{2}*n*is related to

_{eff}*n*through Eq. (1). The intensity-dependent refractive index is

*n = n*, where

_{0}+ n_{2}I*I*is the intensity of the signal,

*n*and

_{0}*n*are the linear refractive index and nonlinear Kerr coefficient. For MEH-PPV,

_{2}*n*= 1.6494 and

_{0}*n*= 1.8 × 10

_{2}^{13}cm

^{2}/W at 1064nm are used in our study. The arm length is chosen as

*l*= 3.77μm corresponding to a phase shift of Δ

_{3}*φ*= π between Arm-1 and Arm-2 at low-intensity input when the effect refractive indexes for dielectric and Kerr medium are 1.872 and 2.016, respectively.

In the FDTD simulations, the signal power is increased from 1e8W/m to 1.1e10W/m. As shown in Fig. 6 , when the input power is 5e8W/m, the normalized power for Port-3 and Port-4 is 0.096 and 0.0002 (the extinction ratio is 16.8dB), respectively. When the power is increased to 4.45e9W/m, the normalized power for Port-3 and Port-4 is 0.001512 and 0.09528 (the extinction ratio is 18dB). The magnetic field distributions for the two input powers are also shown in Fig. 6. The intensity-dependent output property can be utilized in passive optical devices such as a 1 × 2 intensity-based router, with two levels of input signal power 5e8W/m and 4.45e9W/m.

#### 3.2 All-optical switch

To accomplish all-optical switch, a pump CW at a wavelength of *λ _{p}* = 1064nm is injected into port-1 and is coupled to the lower branch Arm-2 totally by the first coupler (i.e., the coupler exhibits a coupling ratio 100% at

*λ*) and then coupled to Port-3 completely by the second identical coupler with

_{p}*l*= 1.6μm (less than the calculated coupling length 1.9μm as shown in Fig. 3, the difference attributes to the error of CMT and additional coupling in the S bends). At the same time, a signal CW at a wavelength of

_{2}*λ*= 860nm is injected into port-2 and is split equally between both arms (the calculated coupling length for

_{s}*λ*= 860nm is 1.39μm as shown in Fig. 3). The phase shift of signal light due to cross-phase modulation (XPM) is the same as Eq. (7) while the refractive index change is Δ

_{s}*n = n*(

_{2}*I*), where

_{1}+ 2I_{2}*I*and

_{1}*I*are the intensity of signal and pump light. Here,

_{2}*l*is chosen as 3μm corresponding to a signal phase shift of Δ

_{3}*φ*= π between Arm-1 and Arm-2 at low-intensity pump light input. Note that the refractive index change induced by XPM is much larger than that induced by SPM at the same input power intensity because the near 100% coupling ratio of the pump light.

In our simulations, the pump power is scanned from 1e7W/m to 1.43e9W/m. As shown in Fig. 7 , when the pump power is 7.6e7W/m, the normalized signal power is 0.00065 and 0.0184 for Port-4 and Port-3 (the extinction ratio is 14.5dB), respectively; When the pump power is increased to 9.9e8W/m, the normalized signal power is 0.0188 and 0.00048 for Port-4 and Port-3 (the extinction ratio is 15.9dB). As a result, the ‘off’ and ‘on’ state can be defined as the states where pump powers are 7.6e7W/m (or zero) and 9.9e8W/m, respectively. As we expected, the transmittances of pump power at Port-3 (varying from 0.048 to 0.052) and Port-4 (varying from 0.00125 to 0.0056) remain almost unchanged, with an extinction ratio about 15dB. The transmittances of signal light are much less than that of pump light because the propagation length at 860nm is shorter than that at 1064nm (Fig. 1). The transmittances of pump light are also less than that of the intensity-based router (Fig. 6) because the couplers here are 1.6μm, which is much larger than 0.6μm for the couplers in the router.

As the arm length is quite short, very high pump intensity is necessary to induce a sufficiently high phase shift (i.e., a phase shift of π) even the total power is not so large due to plasmonic enhancement. Longer branch arms and nonlinear medium with higher nonlinearity may further reduce the power requirement dramatically while the former is only feasible in long range plasmonic waveguides and the latter relies on the development of nonlinear optical materials.

## 4. Conclusion

In this paper, we have numerically investigated direction coupler based on metal-insulator-metal plasmonic waveguide using coupled-mode theory and transfer matrix method. The directional coupler is designed as a wavelength division multiplexer due to the near linear dependence of coupling length with wavelength. The physical explanation of extreme power position offset effect is also given. Employing self-phase modulation and cross-phase modulation, nonlinear Mach-Zehnder interferometers are proposed and studied using finite-difference time-domain method. Applications such as intensity-based router and all-optical switch are discussed in detail.

## Acknowledgments

This work was supported by 973 Program of China (No. 2006-CB302900) and the National Natural Science Foundation of China (NNSFC) (No. 60778018).

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