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We found that metal-dielectric-metal plasmon waveguides with a stub structure, i.e. a branch of the waveguide with a finite length, can function as wavelength selective filters of a submicron size. It was found that the transmission characteristics of such structures depend on the phase relationship between the plasmon wave passing through the stub and the one returning to the waveguide from the stub. We also propose structures with a lossless 90o bend in a plasmon waveguide, utilizing a stub structure. Furthermore, we present a functional stub structure, e.g., a 1:1 demultiplexer and a wavelength selective demultiplexer.
©2008 Optical Society of America
1. Introduction
Recently, plasmon waveguides have attracted much attention because they have the potential to guide light in a region beyond the so-called diffraction limit. In consequence, they can become a strong candidate in realizing integrated optical circuits including sub-wavelength and/or nanometer-size optical devices [1–2]. A variety of plasmon waveguides, e.g. arrayed rods [1], wedges [2,3], V-grooves [4–6], gaps [7–12], etc., have been proposed as waveguides that are applicable even in nanophotonic circuits. Among them, gap plasmon waveguides [7–12] possess remarkable advantages when considering realistic applications: 1) plasmon fields are strongly confined within the nanometer-size metal gaps, 2) the characteristics of the gap plasmons may be insensitive to the surface roughness of the metal, and 3) the structure is simple, its fabrication being easy. On the basis of these reasons, we focused our attention on gap plasmon waveguides in this paper.
In integrated optical circuits, various optical devices are required, e.g., wavelength selective filters, demultiplexers, etc. Recently, the applications of a Bragg reflector [13], ring resonator [6,14], and Fabry-Perot resonator [15] have been demonstrated in the field of plasmonics. The development of other plasmon devices such as band-pass or block filters may be necessary for fabricating integrated optical circuits with a high density.
While developing a variety of plasmon optical devices, the device size is required to be minimized in order to reduce the propagating loss. Note that the propagating length of surface plasmons is considerably small owing to the loss due to metals as compared with that of guided waves propagating in dielectric waveguides. In the case of the application of rings or Bragg reflectors in plasmon waveguides, the structure size relatively increases, and hence, the range to be employed may be limited. Therefore, developing plasmon structures in order to solve these problems is required.
A stub structure [16] is one of the key elements in microwave engineering and is employed in various microwave devices to reduce their size. Some research groups have proposed a wavelength filter by using a stub structure in a photonic crystal waveguide [17,18]. They numerically demonstrated in a successful manner that a compact and simple structure with stubs can function as a wavelength filter. Such a structure may be employed in a plasmon waveguide to perform as a wavelength selective filter.
In this paper, we have numerically presented the characteristics of stub structures in a gap plasmon waveguide. Note that stubs have the advantages of a small size, a simple structure, and easy fabrication. The first objective of this paper is to present the characteristics of a single- and a double-stub structure in a plasmon waveguide and interpret those characteristics. The second objective is to demonstrate that the transmissions characteristics of a 90° bend and a T-splitter in a gap plasmon waveguide can be remarkably improved by adding a single-stub structure to these structures.
2. Numerical configuration
In this paper, we adopted a two-dimensional gap plasmon waveguide, i.e., a metal/dielectric/metal (MDM) structure. Note that characteristics of a gap plasmon propagating in an MDM structure is similar to those of the second mode of gap plasmons propagating in a three-dimensional gap plasmon waveguide structure. Figure 1 illustrates two typical configurations for a numerical simulation in order to analyze the transmission characteristics of a single- and a double-stub structure in the MDM structure. One has a double-stub structure (Fig. 1(a)), and the other has a single-stub structure (Fig. 1(b)). We employed the FDTD method for the numerical simulation. The two-dimensional gap plasmon waveguide with a gap width of w was made of silver with the dielectric constant formulated by using the Drude model; the dielectric constant for the high plasmon frequency, ε inf, was 5.0, and the plasmon energy corresponding to the plasmon frequency was 9.216 eV. In order to elucidate the effect of the stub structure, damping due to the metal was ignored in this case. The calculated real part of the dielectric constant was −127.9 at a wavelength of 1.550 µm, and this value agreed with that presented by Johnson and Christy [19]. The refractive index of the gap was 1.00. The gap width w and the width of the stub structure were set to be equal. A TE-polarized light, i.e. the electric field of the incident light, E, being parallel to the y axis, was launched along the x axis from a light source, and the gap plasmon was excited by means of end-fire coupling. The vacuum wavelength of the light, λbulk, was 1.550 µm. The calculated area was divided by Yee’s mesh with a size of 5 nm and surrounded by first-order Mur’s absorbing boundary.
Transmission was evaluated from the electric field intensity measured at an observing point positioned 2 µm away from the right edge of the stub in the gap plasmon waveguide divided by that estimated at the same position in a structure without the stub. Note that all the variations in the transmission characteristics originated from the stub structure because we ignored the optical loss due to the imaginary part of the dielectric constant of silver.
3. Results
3.1 Filtering function of stub structures
In Fig. 2, the transmission characteristics are illustrated as a function of the stub length L for the double-stub structure (a) and the single-stub structure (b). The red and black lines correspond to w=50 nm and 100 nm, respectively. In Fig. 2(a), the minimum transmissions for w=50 nm and 100 nm are −52 dB at L=1.44 µm and −46 dB at L=0.32 µm, respectively. In Fig. 2(b), the minimum transmissions for w=50 nm and 100 nm are −33 dB at L=0.27 µm and/or 0.83 µm and −43 dB at L=0.30 µm, respectively. The maximum transmissions are nearly equal to 0 dB. Zero transmission implies that there are no optical loss in the structure employed here. When we employed the dielectric constant of silver, including the loss, in our simulation for w=100 nm, where the dielectric constant of silver was −127.9+i3.0 at λbulk=1.550 µm, the maximum transmission at L=1.3 µm decreases to −0.1 dB, and the minimum transmission at L=325 nm increased to −44 dB. Namely, for our configuration, i.e., for L ranging from 0 to 1.6 µm and λbulk=1.550 µm, the loss due to silver had no significant contribution in the transmission characteristics of the double-stub structure.
Fig. 2. Transmissions characteristics vs. stub length for the double stub (a) and the single stub (b). The red and the black lines are for w=50 nm and 100 nm, respectively. The FWHM (full width at half maximum) of the dips for the single- and double-stub structures were evaluated from the transmission spectra for w=50 nm and 100 nm. For the single-stub structure, these values were 175 nm and 190 nm for w=50 nm and 100 nm, respectively. For the double-stub structure, these values were 285 nm and 325 nm for w=50 nm and 100 nm, respectively. Namely, the FWHM of the single-stub structure was smaller than that of the double-stub structure. The single stub may be a superior filter in terms of blocking a specific wavelength as compared to the double stub because of the narrow FWHM of the dips. The transmission characteristics varied periodically with the stub length. The intervals in the change in the transmission characteristics for w=50 nm and 100 nm were 575 nm and 650 nm, respectively. These lengths corresponded to half the plasmon wavelength in the respective gap waveguides, as mentioned in the following paragraph.
The distributions of the light intensity associated with the gap plasmons propagating in the MDM structures with and without the stubs for w=100 nm are indicated in Fig. 3. Figure 3(a) corresponds to L=0, i.e. without the stub, and Figs. 3(b) and (c) correspond to L=325 nm and 650 nm, respectively. Note that the transmissions for L=325 nm and 650 nm are nearly equal to 0% and 100%, respectively. From Fig. 3(a), it can be observed that the wavelength of the propagating plasmon, λGP, is 1300 nm for w=100 nm. In Fig. 3(b), for L=325 nm, the transmission is 0%. The gap plasmon enters the stub and the plasmon field abruptly disappears after passing through the stub part. In Fig. 3(c), for L=660 nm, the transmission is 100%. The field oscillates periodically over the entire waveguide and the light intensity distribution is the same as that for the waveguide without the stub, as shown in Fig. 3(a). λGP is twice that of the interval in transmission characteristics for the double-stub structure with w=100 nm. For the single-stub structure, the distributions of the light intensity are presented in Fig. 3(d) for L=325 nm and in Fig. 3(e) for L=650 nm. The behavior is quite similar to that observed in the case of the double-stub structure. The respective transmissions observed in Fig. 3(d) and Fig. 3(e) are nearly equal to 0% and 100%. Our conclusion is that the stub structures employed here function as good resonators for the gap plasmons and can be utilized as band pass or block filters such as Fabry-Perot resonators.
Fig. 3. Light intensity distribution of gap plasmons with and without the stubs for w=100 nm. (a) corresponds to the case of no stub. (b) and (c) correspond to the double-stub structure with L=325 nm and 650 nm, respectively. (d) and (e) correspond to the single-stub structure with L=325 nm and 650 nm, respectively.
In the following section, we will present the operation principle of the double–stub structure, i.e., the origin of the variation in the transmission characteristics with L for the double–stub structure. While observing the transmission characteristics at a specific point, we found two waves: one was the gap plasmon passing through the stub with a phase change of θT, and the other was the gap plasmon returning from the stub with a phase change of θB1+θB2+2θL+θR, as shown in Fig. 4. The dependence of the transmission characteristics on L originates from the destructive and constructive interferences between the two waves, based on the phase changes.
In Fig. 4, the gap plasmon enters the stub, bringing about a phase change of θB1. An additional phase change occurs by traveling inside the stub, θL, and reflecting back at the end face of the stub, θR. Further phase changes θL and θB2 occur when the gap plasmon reflected back at the end face of the stub travels through the stub and returns to the waveguide. θL is equal to the product of the propagation constant of the gap plasmon β (=2π/λGP) and L. On the other hand, the gap plasmon passing through the stub is accompanied by the phase change of θT.
Fig. 4. Phase changes in the gap plasmons for the double-stub structure (a), a simplified transmission line expression of the double-stub structure (b), and an equivalent circuit of the double-stub structure (c).
In the present configuration, w is considerably smaller than the wavelength of the gap plasmon, λ GP. We, therefore, made a quasistatic approximation [20], assuming that no retardation of the gap plasmons occurs at the junction of the waveguide and the stub. In such a case, the plasmon enters in and out of the stub and passes through the stub without any phase shift. Namely, θT, θB1, and θB2 become 0. Since silver was assumed to be a loss-free conductor, θR should be also zero. From these assumptions, the transmission characteristics of the double-stub structure were discussed by using the concept of a distributed constant circuit including a loss-free transmission line with a characteristic impedance of Z0.
A double-stub structure can be expressed as a transmission line with two stubs, as shown in Fig. 4(b) [16]. Note that the end of the stub can be regarded as an open circuit, taking into account θR=0. As is well known, the admittance of a single-stub structure with length L, Ys, is expressed as follows.
Note that the width of the stub is the same as that of the waveguide. An equivalent circuit of the double-stub structure is presented in Fig. 4(c).
From Fig. 4(c), the amplitude transmission t and the amplitude reflection r for the electric field can be obtained as
where Y0=1/Z0. The energy transmission by the double stub, i.e., |t|2, Td, therefore, is finally expressed as
Equation (3) provides the transmission characteristics depicted by the red and black lines for w=50 nm and 100 nm in Fig. 5, respectively. These results agree very well with those obtained by the FDTD method (open circles). The insertion loss due to the stub structure is very low. That is because the gap plasmon reflected back by the stub and propagating toward the incident side destructively interferes with the incident gap plasmon. Namely, the reflection becomes zero, guaranteeing 100% transmission under a certain condition.
Fig. 5. Transmission characteristics as a function of the stub length for the double-stub structure. The circles and the solid lines denote the numerical and the analytical results, respectively. The red and black lines correspond to w=50 nm and 100 nm, respectively.
For the single-stub structure, similarly, the energy transmission, Ts, can be obtained as
Equations (3) and (4) imply that the transmission characteristics can be controlled by the stub length L and can be governed by the interference between the gap plasmons passing through the stub and returning from the stub. Note that eqs. (3) and (4) were obtained under the condition of w≪λGP, i.e., the quasistatic approximation and a loss-free material. When w≪λGP is not satisfied, all the phase changes, i.e., θT, θB1, θB2, and θR, are required to be considered in addition to the multiple reflections inside the stub. It may, therefore, be difficult to obtain a simple equation of the transmission characteristics. In this paper, we discussed the transmission characteristics in the stub portion. Similarly, the reflection characteristics in the stub portion can be discussed and obtained.
3.2 Effect of rounded corner
When we fabricated a stub structure using a silver film by using physical processes, e.g. a focused ion beam (FIB) method, the four corners at the crossing of the stub and the waveguide were rounded due to a finite ion beam diameter and a drift in the beam position. Figure 6(a) presents an SEM image of the stub structure fabricated using an evaporated silver film on a glass substrate by using the FIB method. The film thickness was 60 nm and the width of the gap was 50 nm. From Fig. 6(a), the radius of the rounded corners r was estimated to be approximately 50 nm. When other physical processes are used to fabricate stubs, it may again be difficult to obtain sharp corners at the crossing. We, therefore, explored the effect of the rounded corners at the crossing on the transmission characteristics.
Fig. 6. SEM image of a stub structure fabricated using a silver film (a) and transmission vs. stub length for various radii of the corners r (b). Black, blue, and red lines represent r=0, 25, and 50 nm, respectively.
Figure 6(b) illustrates the simulated transmission characteristics as a function of L for the double-stub structure having rounded corners with a radius r. We assumed w to be 100 nm. The black, blue, and red lines correspond to r=0, 25, and 50 nm, respectively. The other parameters used for the calculations were the same as those used in the previous subsection.
In Fig. 6(b), the stub length providing the minimum transmission shifts toward a longer length with increasing r. The shift in the stub length for the minimum transmission was 2% with the radius ranging from 0 nm to 50 nm. The shift may have resulted from the phase changes. In the previous subsection, θT, θB1, and θB2 were assumed to be zero but they were not perfectly zero. For the rounded corners, the gap width w around the junction becomes larger than that in the case of sharp corners, and thus, θT, θB1, and θB2 slightly deviate from zero. θR, moreover, may have also changed as compared with that observed in the case of the sharp corners. Near the end of the stub having rounded corners, the width of the stub was modified gradually. Namely, the reflection conditions were changed near the end of the stub. In such a situation, when the effect of roundness is transferred onto θR, the effective value of θR does not become zero even if the metal is a loss-free conductor. θL may, of course, change as well. Fortunately, it was apparent that the absolute values and the shapes of the respective transmission characteristics were insensitive to the radius of the rounded corners. Such insensitivity arises from the fact that the transmission characteristics are determined by only the interference between the gap plasmons passing through the stub and returning from the stub, as already described. Namely, stub structures can be designed and be actually applied to a variety of optical devices, e.g. wavelength selective filters.
3.3 90° bend with stubs
One of the advantages of a gap plasmon waveguide is the high transmission at a 90o bend [20]. For example, the transmissions at the 90° bend for w=50 nm and 200 nm were estimated to be approximately 0.99 and 0.90 by using our FDTD simulations at λbulk=1550 nm, respectively. High transmission at the bend with a narrow w was a result of the small retardation at the bend with a narrow w, as already reported by Veronis and Fan [20]. The stub structures proposed here have the potential to provide optical devices with useful functions. In this section, therefore, we will discuss stub structures applied to a 90° bend and a T-splitting waveguide in order to make these structures functional.
Figure 7 illustrates four structures of the 90o bend. (a) is a 90° bend without a stub, (b) is a 90° bend with two stubs (Form 0), and (c) and (d) are 90o bends with one stub (Form 1 and Form 2). For Form 0 in Fig. 7(b), the stub lengths are equal. For calculations, w was set to be 200 nm. The other parameters used for the calculations were the same as those used in subsection 3.1.
Figure 8 presents the simulated dependences of the transmission characteristics on the stub length L. The black open circles represent Form 0. The red and blue open circles represent Form 1 and Form 2, respectively. The solid lines are only to aid visualization. Note that most parts of the red and blue lines overlap each other. In the case of (a), i.e. L=0, the transmission at the bend is approximately 0.9. On the other hand, for a suitable stub length, the maximum transmission at the bends becomes 1.0, i.e. no transmission loss. The transmissions become 1.00 at L=660 nm and 1380 nm for Form 0 and at L=620 nm and 1340 nm for Forms 1 and 2. At L=260 nm and 970 nm, the transmissions become 0 in all the structures with the stub.
Fig. 8. Dependence of transmission characteristics on stub length for three types of bends. Black, red, and blue lines represent Form 0, Form 1, and Form 2, respectively.
Figure 9 presents the light intensity distributions. (b), (c), and (d) indicate the minimum transmissions, and (e), (f), and (g) indicate the maximum ones. In (b), (c), and (d), the stub length was set to be 260 nm. (e), (f), and (g) correspond to L=660, 620, and 620 nm, respectively. In the case of (b), (c), and (d), the intensity at the output port was nearly equal to 0. On the other hand, for (e), (f), and (g), the light intensity at the output port was almost equal to the incident intensity. Namely, loss-free 90° bends can be readily fabricated by adjusting the stub length, and such a structure also functions as a band pass filter. The transmissions at 90° bends with a stub vary with L, and the interval in the variations is ~λGP/2. It should be significant to note that 100% transmission, i.e. 0% reflection, can be obtained when the phase difference between the gap plasmons reflected directly at the bend and returning to the waveguide is equal to Nπ.
The transmission spectra of Forms 1 and 2 are identical, as shown in Fig. 8. This result is reasonable because the phase change of the gap plasmon returning from the stub to the waveguide in Form 1 is the same as that in Form 2. Even if the quasistatic approximation is not a good approximation, the transmission spectra of Forms 1 and 2 should be the same, judging from the behavior of the gap plasmons.
In Form 0, the dependence of the transmission characteristics on the stub length differs from that in Forms 1 and 2. Namely, a notch at L=0.3 µm and a shoulder at L=1.05 µm can be observed in Fig. 8. This may be predicted to occur due to a complicated interference between the gap plasmons passing through and returning from the stubs in Form 0. The two stubs strongly coupled, and thus, the effective cavity length of the coupled stubs had to be taken into account. In consequence, new phase interference may have occurred, producing the notch and the shoulder.
As presented here, the effect of the stubs on the transmission characteristics at 90° bends is rather significant. Considering the role of plasmon waveguides in an optical integrated circuit, it can be noted that 90° bends with a stub could function as key optical devices.
Fig. 9. Light intensity distributions of gap plasmons for four types of bends. (b), (c), and (d): L=260 nm. (e): L=660 nm. (f) and (g): L=620 nm.
3.4 T-splitting waveguide with a stub
In the following segment, we will discuss the transmission characteristics of a T-splitting waveguide with a stub, as shown in Figs. 10(a) and (b). The transmission characteristics of the T-splitter were simulated by the FDTD method for a gap plasmon incident from port A. The transmissions were evaluated at ports B and C. We assumed that the width of the waveguide was 200 nm and the wavelength of the incident light in vacuum was 1.550 µm. The other numerical parameters used for the FDTD calculation were the same as those employed in the previous calculations.
Figure 11 reveals the dependence of the transmission characteristics on the stub length L. The black and red lines correspond to the transmissions at ports B and C, respectively. At L=0, the transmissions at ports B and C are 0.46 and 0.43 respectively. Namely, the reflection of the T-splitter without a stub is 0.11 because the damping due to the metal was assumed to be 0 in our calculations. The transmission characteristics at ports B and C periodically vary with L. The interval ~λGP/2 is the same as that in the structures discussed in the previous sections. Note, however, that we were not able to estimate the stub length required to obtain 100% transmission because the gap plasmons departing the stub never return to the stub.
Fig. 11. Dependence of transmission characteristics on stub length for a T-splitter with a stub. Black and red lines indicate transmission characteristics observed at output ports B and C, respectively.
At L=260 nm and 320 nm, we can observe the output light only at ports B and C, respectively. The transmissions at L=260 nm and 320 nm are 0.20 and 0.16, respectively. The T-splitter with a stub could function as a wavelength selective demultiplexer. At L=650 nm, the transmissions are the same for ports B and C, i.e., 0.45. In that case, it functions as a 1:1 demultiplexer.
In order to confirm the performance of a wavelength selective demultiplexer, we evaluated the transmission characteristics of the T-splitter with L=300 nm at excited wavelengths of 1.495 µm and 1.700 µm in a geometry illustrated in Fig. 10(b). The light intensity distribution for λbulk=1.495 µm and 1.700 µm is presented in Figs. 12(a) and (b), respectively. At 1.495 µm, the transmissions at ports B and C were evaluated to be 0.16 and 0.00, respectively. At 1.700 µm, the transmissions at ports B and C were evaluated to be 0.00 and 0.20, respectively. The extinction ratio was more than 16 dB although the loss due to the stub was 7 dB. This result clearly verifies that the T-splitter with a stub has a high potential as a wavelength selective demultiplexer of a sub-micrometer size.
Fig. 12. Light intensity distribution of the T-splitter with w=200 nm and a stub with L=300 nm. (a) and (b) represent λbulk=1.495 µ and 1.700 µm, respectively.
4. Conclusions
We have numerically described that single- and double-stub structures in a plasmon waveguide play a significant role in developing a wavelength selective filter of a sub-micrometer size, and stubs with an appropriate length provide no optical loss. We have also discussed the phase relationship of gap plasmons in structures including stubs. It has been found that rounded corners of a stub do not lead to any fundamental change in the transmission characteristics of the stub as compared to sharp corners. Such tolerance of the stub structure should be preferred in realizing optical devices because it promises a large margin in fabrication accuracy. By applying a stub to a 90o bend or a T-splitter in the gap plasmon waveguides, we can enhance the performances of these structures, e.g., a 90° bend with a low loss, a wavelength selective demultiplexer with a high extinction ratio, etc. We believe that the stub structure will become a key device in high density optical circuits in the near future.
Acknowledgments
This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Exploratory Research 2005-2006 and by Japan Science and Technology Agency Grant-in-Aid for Research for Promotion Technological Seeds in 2007.
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