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Design of broadband power splitters using two-mode interference in slot waveguides
Optics Communications 355 (2015) 367–375
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design of broadband power splitters using two-mode interference in slot waveguides Bing Chen a,b,n, Chunliang Liu a, Jinhai Si b a b
Key Laboratory of Physical Electronics and Devices of the Ministry of Education, Xi’an Jiaotong University, Xi’an 710049, China Shaanxi Key Laboratory of Information Photonic Technique, Xi’an Jiaotong University, Xi’an 710049, China
art ic l e i nf o
a b s t r a c t
Article history: Received 5 May 2015 Received in revised form 3 July 2015 Accepted 5 July 2015
Two-mode interference effects in slot waveguides have been predicted and investigated through plane wave expansion and finite-difference time-domain methods. A broadband 1 2 power splitter (PS) based on this effect is presented, and its transmission characteristics are investigated by using finite-difference time-domain method. Calculated results indicate that, for the 1 2 PS whose interaction length is as short as 0.509 mm, the high transmission ( Z95%) is observed within a wavelength range of 1.389– 1.747 mm, corresponding to a relative bandwidth of 22.87%. Combining three 1 2 PS, we construct a 1 4 PS with excellent transmission characteristics. Those simple PS are expected to be applied to highly dense photonic integrated circuits. & 2015 Published by Elsevier B.V.
Keywords: Power splitters Slot waveguides Two-mode interferences
1. Introduction Power splitters (PS) are the most commonly used power division devices, and are key components in integrated optics. The development of highly dense photonic integrated circuits has raised the need for compact PS. Based on the novel optical waveguides such as photonic crystal waveguides [1–8], Bragg reflection waveguides [9–11], periodic dielectric waveguides [12] and plasmonic subwavelength waveguides [13–15], a considerable number of compact PS have been presented. Take 1 2 PS as an example, they can be approximately classified into two kinds according to their working mechanisms: T/Y branch type [5–9,14,15] and two-mode interference (TMI) type [1–4,10–13]. In General, the former is more compact, while the optimization of their branch region is complicated [5–9]. The latter is simpler, while it has large interaction length of TMI [1–4,10–12]. In 2004, Almeida et al. present a novel waveguide geometry named as a slot waveguide, where the guided light is strongly confined within a narrow low-index gap (slot region) between two high-index photonic wires [16,17]. This enables high interaction between a low-index material and guided light, which may lead to the implementation of compact and high-performance photonic components such as resonators [18] and polarization beam splitters [19–23]. n Corresponding author at: Key Laboratory of Physical Electronics and Devices of the Ministry of Education, Xi’an Jiaotong University, Xi’an 710049, China. E-mail address: [email protected] (B. Chen).
http://dx.doi.org/10.1016/j.optcom.2015.07.009 0030-4018/& 2015 Published by Elsevier B.V.
Inspired by these ideas, for the first time to our knowledge, an 1 2 PS based on TMI in slot waveguides is proposed in this paper. The interaction length of TMI (denoted as L) in the proposed PS is one order of magnitude less than that of the PS based on TMI in photonic crystal waveguides [1–4], or Bragg reflection waveguides [10,11], or periodic dielectric waveguides [12], and is almost equivalent to that of the PS based on TMI in plasmonic subwavelength waveguides [13]. The interaction length (L) of the 1 2 PS based on TMI in different waveguides at working wavelength 1.55 μm are listed in Table 1. It can be seen that L of PS based on slot waveguides (0.509 μm) is one order of magnitude less than that of PS based on TMI in either photonic crystal waveguides (3.100–13.64 μm) or Bragg reflection waveguides (3.901– 8.235 μm) or periodic dielectric waveguides (2.716 μm), and is basically equivalent to that of PS based on TMI in plasmonic subwavelength waveguides (0.440 μm). In this paper, the dispersion characteristics of single- and trichannel slot waveguides are firstly investigated. Secondly, TMI effects in tri-channel slot waveguides are discussed, and the interaction length of TMI (L) is obtained. Then, the 1 2 PS based on TMI is constructed, and its transmission characteristics with different L are studied. Next, a compact 1 4 PS is constructed on the basis of three 1 2 PS, and its transmission characteristic is also investigated. Finally, these numerically computed results are discussed and some important conclusions are drawn.
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Table 1 Comparison of interaction lengths L for the 1 2 power splitter based on two-mode interference in different waveguides at working wavelength 1.55 μm. Waveguide types
Materials/index
Interaction modes
L (unit: μm)
References
Photonic crystal waveguides
GaAs /n¼ 3.46, air/n¼ 1 GaAs/n ¼3, air/n¼ 1 GaAs/n ¼3.46, air/n¼ 1 Si/n¼ 3.4, air/n¼ 1 Tellurium/n¼4.6, polystyrene/n¼ 1.6, air/n¼1 Tellurium/n¼4.6, polystyrene/n¼ 1.6, air/n¼1 GaAs/n ¼3.46, air/n¼ 1 Si/n¼ 3.5, Ag/n ¼11.214þ 0.144i, air/n¼ 1 Si/n¼ 3.5, air/n¼ 1
TM0, TM2 TM0, TM2 TE0, TE2 TM0, TM1 TE0, TE2 TE0, TE2 TM0, TM2 TM0, TM2 TM0, TM2
3.720 7.998 3.100 13.640 8.235 3.901 2.716 0.440 0.509
2004-JLT 2004-OE 2009-AO 2011-PTL 2011-JOSAB 2012-COL 2007-OE 2007-OC This work
Bragg reflection waveguides Periodic dielectric waveguides Plasmonic waveguides Slot waveguides
2. Dispersion characteristics of slot waveguides The inset in Fig. 1 shows a schematic drawing of a singlechannel slot waveguide, which consists of a dielectric layer of lower index sandwiched by two high-index photonic wires in the X-direction, where a is artificial period. The refractive index for two materials are n0 ¼1 (air) and n1 ¼ 3.5 (silicon), and the corresponding thicknesses are h0 ¼ 0.10a and h1 ¼ 0.3a, respectively. The light is confined in and transmitted along the air channel (Z-direction), and the transmission constant is β. The geometry is uniform in the Y-direction. The dispersion curves of TM modes (magnetic field H normal to the incident plane (X–Z plane)) are calculated by using the plane-wave expansion method based on a super-cell [24]. The rectangle region surrounded by the dotted lines in the inset of Fig. 1 is a schematic drawing of super-cell. In our simulations, the size of a super-cell is taken as 19a a, and 40 grid points are used to represent each artificial period a. Fig. 1 gives TM mode dispersion curves of the single-channel slot waveguide. X-axis is normalized propagation constant. Y-axis is normalized frequency. It can be seen from Fig. 1 that in the frequency range of 0–0.28[2πc/a] (c is the speed of light in vacuum), there is only one TM0 (the zero-order TM mode, even mode); while in the frequency range of 0.28–0.55[2πc/a], there appears TM1 (the first-order TM mode, odd mode) besides TM0. Two additional identical high-index photonic wires are symmetrically located on the upper and lower side of the single-
[1] [2] [3] [4] [10] [11] [12] [13]
channel slot waveguide; and a tri-channel slot waveguide is formed, as shown in the inset of Fig. 2. TM mode dispersion curves of the tri-channel slot waveguide are given in Fig. 2, where X- and Y-axis are normalized propagation constant and normalized frequency, respectively. It can be seen from Fig. 2 that there are four dispersion curves, denoted as TM0, TM1, TM2 (the second order TM mode, even mode) and TM3 (the third order TM mode, odd mode). In Figs. 1 and 2, points A (0.7484[2π/a], 0.41[2πc/a]), B (0.4501 [2π/a], 0.41[2πc/a]), C (0.8253[2π/a], 0.41[2πc/a]), D (0.7321[2π/a], 0.41[2πc/a]), E (0.5663[2π/a], 0.41[2πc/a]), F (0.4162[2π/a], 0.41 [2πc/a]) were chosen at the normalized frequency of 0.41[2πc/a]. Points A and B are located in TM0 and TM1 dispersion curves of the single-channel slot waveguide, respectively. Points B, C, D and F are located in TM0, TM1, TM2 and TM3 dispersion curves of the trichannel slot waveguide, respectively. The distributions of both electric field Ex and magnetic field Hy at points A, B, C, D, E, and F are given in Fig. 3. It can be seen from Fig. 3 that light enhancement and confinement is caused by large discontinuity of the electric field (Ex) at high-index-contrast interfaces [16,17]. For the single-channel slot waveguide, Ex of TM0 mode is mainly concentrated in the air channel with low-index, while Ex of TM1 mode is nearly zero in the air channel. For the tri-channel slot waveguide, Ex of TM0 (or TM2) mode is concentrated in three air channels, and the maximum peak is located in the central air channel; while Ex of TM1 (or TM3) mode is mainly concentrated in both the upper and lower air channel, and is nearly zero in the
Fig. 1. TM mode dispersion curves of the slot waveguide shown in the inset. X-axis is normalized propagation constant. Y-axis is normalized frequency. The slot waveguide consists of a dielectric layer of lower index sandwiched by two high-index photonic wires in the X-direction, where a is artificial period. The refractive index for two materials are n0 ¼ 1.0 (air) and n1 ¼ 3.5 (silicon), and the corresponding thickness are h0 ¼ 0.10a and h1 ¼ 0.3a, respectively. The light is confined in and transmitted along air channel, and the transmission constant is β. The geometry is uniform in the Y-direction.
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Fig. 2. TM mode dispersion curves of the tri-channel slot waveguide shown in the inset. X-axis is normalized propagation constant. Y-axis is normalized frequency. The waveguide parameters are n0 ¼1.0(air), n1 ¼3.5 (silicon), h0 ¼ 0.10a and h1 ¼0.3a, where a is artificial period.
central air channel.
3. Transmission characteristics of 180° arc bends in singlechannel slot waveguides Since the single-channel slot waveguide bends will be used in
following studies, their transmission characteristics are firstly discussed here. Generally, the optimization of slot waveguide bend is a complex question [25,26]. For the sake of simplicity, we only consider a simple 180° arc slot waveguide bend shown in inset of Fig. 4(a), whose transmission characteristics is calculated by using the finite-difference time-domain method. In our computation models, the perfectly-matched-layer absorbing boundary
Fig. 3. Electric field Ex and magnetic field Hy distributions at A, B (in Fig. 1 for single-channel slot waveguide), C, D, E and F points (in Fig. 2 for tri-channel slot waveguide).
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Fig. 4. (a) The transmission spectra for the 180° arc bends with the different radii of curvature R. A schematic drawing of the waveguide with 180° arc bend is shown in the inset. The waveguide parameters are the same as those shown in Fig. 1. For 180o arc bend with R ¼ 4a, steady-state field distributions Ex (b) and Hy (c) at a frequency of 0.41 [2πc/a].
conditions are implemented at the boundaries of the computation region [27], and 50 grid points are used to represent each artificial period a. A Gaussian optical pulse is incident from the input port. In order to record the incidence pulse, the observation point 1# is placed in the middle of input waveguide. In order to record the transmission pulse, the observation point 2# are placed in the middle of output waveguide. The incidence and transmission frequency spectra are calculated from the Fourier transform of corresponding pulses. The transmission frequency spectrum normalized by the incidence frequency spectrum, and the normalized transmission spectrum is obtained. Fig. 4(a) gives the relationship curves between the transmittance of TM0 and radius of curvature (denoted as R). It can be
observed from Fig. 4(a) that with the increase of R, there are upward shifts in the transmittance curves. When R ¼2a, the normalized transmittance is higher than 0.9 in a frequency range of 0.3934–0.4992[2πc/a]. When R¼4a, the transmittance is higher than 0.98 in a frequency range of 0.3477–0.4661[2πc/a], corresponding to a relative bandwidth of 29.10%. When R is increased to 7a, the transmittance is beyond 0.98 in a frequency range of 0.2970–0.4838[2πc/a], corresponding to a relative bandwidth of 47.85%. In view of device compactness, R is taken as 4a in the following studies. For the 180° arc bend with R¼4a shown in the inset of Fig. 4(a), the steady-state Ex and Hy field distributions at a frequency of 0.41[2πc/a] are given in Fig. 4(b) and (c), respectively.
Fig. 5. The interaction length L between TM0 and TM2 in tri-channel slot waveguide. The inset is a schematic drawing of two-mode interference in tri-channel slot waveguides, which is symmetrically connected with a sing-channel slot waveguide. The waveguide parameters are the same as Figs. 1 and 2.
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4. Two-mode interference in tri-channel slot waveguides Multimode interference (or self-imaging) is a property of multimode waveguides by which an input field profile is reproduced in single or multiple images at periodic intervals along the propagation direction of the guide [28]. Multimode interference has also been used in the semiconductor waveguide for SLED power improvement [29,30]. As the simplest multimode interference, two-mode interference (TMI) is widely used to construct compact power splitters [1–4,10–13]. The inset in Fig. 5 gives a schematic drawing of TMI in the tri-channel slot waveguide, which is symmetrically connected with a sing-channel slot waveguide. When TM0 (even mode) in the single-channel slot waveguide goes into the tri-channel slot waveguide, both TM0 and TM2 modes (even modes) into the tri-channel slot waveguide can be excited simultaneously, while both TM1 and TM3 modes (odd mode) cannot be excited due to different mode symmetry. When TM0 and TM2 modes are transmitted simultaneously in the trichannel slot waveguide, TMI (the interference between TM0 and TM2 modes) occurs, which leads to the alternative appearance of two-fold image and single image at the same periodical interval, which is named as the interaction length (denoted as L). If the propagation constants of the TM0 and TM2 modes in the trichannel slot waveguide are denoted as β0 and β2 (unit: 2π/a), respectively, then L can be expressed as π/(β0–β2). According to Fig. 2, Fig. 5 gives the relationship curve between the normalized interaction length (L/a) and the normalized frequency. It can be seen from Fig. 5 that L/a is 1.861–2.938 in a frequency range of 0.25–0.50[2πc/a]. In particular, L/a is less than 2 in a frequency range of 0.3152–0.4302[2πc/a]. Additionally, for the structure shown in the inset in Fig. 5, Fig. 6 gives steady-state Ex, Hy and time-averaged Poynting vector Pz distributions at a frequency of 0.41[2πc/a] by using the finite-difference time-domain method [25]. It can be observed from Fig. 6 that L is as short as 1.93a, which is in good agreement with the result shown in Fig. 5.
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5. An 1 2 power splitter based on two-mode interference in slot waveguides The structure shown in the inset in Fig. 5 is combined with flexible bends of slot waveguides, and the 1 2 power splitter (PS) based on TMI in slot waveguides is constructed, as shown in Fig. 7. According to the calculated results shown in Fig. 4(a), all the radius of curvature R of waveguide bends are taken as 4a. All angles of waveguide bends (denoted as θ) are taken as 50°. The transmission characteristics of the PS shown in Fig. 7 are investigated by using the finite-difference time-domain method. In our computation models, the perfectly-matched-layer absorbing boundary conditions are implemented at the boundaries of the computation region [27], and 50 grid points are used to represent each artificial period a. A Gaussian optical pulse is incident from the input port. In order to record the incidence and reflection pulses, the observation point 1# is placed in the middle of input waveguide. In order to record the transmission pulse, the observation points 2# and 3# are placed in the middle of two output waveguides, respectively. The incidence, transmission and reflection frequency spectra are calculated from the Fourier transform of corresponding pulses, respectively. Next, the reflection and transmission frequency spectra are normalized by the incidence frequency spectrum, and the normalized reflection and transmission spectra are obtained, respectively. Finally, the normalized reflectance is denoted as Γ, and the normalized transmittance at output ports 1 and 2 are denoted as T1 and T2, respectively. The total transmittance (denoted as Tt) is equal to T1 þT2. A high transmission band (denoted as HTB) is defined as the frequency band corresponding to Tt Z0.95. If the calculation errors are ignored, the normalized transmission loss is equal to 1 Γ Tt. For the 1 2PS shown in Fig. 7, Fig. 8(a), (b), (c) and (d) gives transmission (T1, T2 and Tt), reflection (Γ) and loss spectra, corresponding to L ¼1.9a, 1.2a, 0.8a and 0.6a, respectively. It can be observed from Fig. 8(a) that there is no HTB for L ¼1.9a. There is the maximum Tt of 0.9456 at a frequency of 0.4978[2πc/a], and the
Fig. 6. For two-mode interference in the tri-channel slot waveguide shown in the inset in Fig. 5, steady-state Ex, Hy and time-averaged Poynting vector Pz distributions at a frequency of 0.41[2πc/a].
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Fig. 7. A schematic drawing 1 2 power splitter based on two-mode interference in slot waveguides.
corresponding reflectance (Γ) and loss are 0.0013 and 0.0531, respectively. It can be observed from Fig. 8(b) that for L ¼1.2a, there is a HTB of 0.4239–0.4848[2πc/a], corresponding to a relative bandwidth of 13.40%. There is the maximum Tt of 0.9768 at a frequency of 0.4586[2πc/a], the corresponding loss is decreased to 0.0220. When L is as short as 0.8a, HTB’s frequency range is 0.3637–0.4576[2πc/a] marked in Fig. 8(c), the corresponding relative bandwidth is increased to 22.87%. For example, at a frequency of 0.4100[2πc/a], Tt is 0.9638, and transmittances at output
ports 1 (T1) and 2 (T2) are 0.4800 and 0.4838, respectively; the reflectance (Γ) and loss are 0.0001 and 0.0361, respectively. When L is decreased to 0.6a, HTB’s relative bandwidth is decreased to 9.74%, corresponding to the frequency range of 0.3789–0.4177[2πc/ a] marked in Fig. 8(d). There is the maximum Tt of 0.9564 at a frequency of 0.3953[2πc/a]. For comparison, the transmittance, reflectance, and loss at the four frequencies mentioned above are listed in Table 2. Additionally, it can be observed from Fig. 8 and Table 2 that for
Fig. 8. Transmission at output ports 1 and 2 (T1 and T2), total transmission (Tt), reflection (Γ) and loss spectra of the 1 2 power splitter shown in Fig. 7 with different L: (a) L ¼ 1.9a, (b) L ¼1.2a, (c) L ¼ 0.8a and (d) L ¼ 0.6a.
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Table 2 Comparison of transmittance, reflectance, and loss of the 1 2 PS with different L. [n0, n1, h0, h1, R, θ]¼ [1, 3.5, 0.1a, 0.3a, 4a, 50o]
L 1.9a
1.2a
0.8a
0.6a
Normalized Frequency (unit: 2πc/a) Normalized T1 T2 transmittance Tt*1 Normalized reflectance (Γ) Normalized loss (1 Tt Γ) Frequency rang of HTB*2
0.4978 0.4586
0.4100
0.3953
0.4784 0.4672 0.9456 0.0013 0.0531 –
HTB’s relative bandwidth (%)
0
0.4800 0.4838 0.9638 0.0001 0.0361 0.3637– 0.4575 22.87
0.4826 0.4738 0.9564 0.0017 0.0437 0.3789– 0.4177 9.74
0.4772 0.4996 0.9768 0.0012 0.0220 0.4239– 0.4848 13.40
Tt*1: Tt ¼ T1 þ T2. HTB*2: high transmission band for Tt Z 0.95.
the PS based on slot waveguides, there is no obvious back-reflection from the TMI region (tri-channel waveguide), which is because there is continuous air channel from single- to tri-channel slot waveguides, and is no reflection interface between two different materials. However, there is obvious radiation loss in branch region when the two light beams are separated into two singlechannel slot waveguides from tri-channel slot waveguides. The 1 2 PS based on slot waveguides composes of two parts: a tri-channel slot waveguide and three slot waveguides (one straight waveguide at input and two curved ones at outputs). A tri-channel slot waveguide can be regarded as a resonant cavity. The
Fig. 10. A schematic drawing 1 4 power splitter, which is composed of three 1 2 power splitter with L ¼0.8a. The curvature radii of all arc are set as 4a.
transmission property of the PS mainly depends on the structure and its geometry size of resonant cavity. At a certain frequency, the eigen-modes in resonant cavity are very complicated. In the symmetric cavity, there are both symmetric and asymmetric eigen-modes simultaneously. The transmission difference between
Fig. 9. For the 1 2 power splitter with L ¼ 0.8a shown in Fig. 7, at a frequency of 0.41[2πc/a], (a) steady-state electrical field Ex distribution; (b) steady-state magnetic field Hy distribution.
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Fig. 11. For the 1 4 power splitter shown in Fig. 8, at a frequency of 0.41[2πc/a]: (a) steady-state electric field Ex distribution; (b) steady-state magnetic field Hy distribution.
two outputs is mainly caused by the existence of asymmetric modes. For example, let us assume Am is the amplitude of symmetric mode, and Bm is the amplitude of asymmetric mode. Then Am þBm and Am Bm are amplitudes at two output ports of symmetric cavity. There is the difference of 2Bm between two ports. In fact, the electromagnetic field is vector but not scalar field, and electric/magnetic fields meet the right hand relation, so two outputs are not totally symmetric. Therefore, The difference between the transmittance of two outputs ports is likely caused by the symmetry of quasi-guided or resonant modes [31]. It is necessary to illustrate that there is a large difference between L in Figs. 4 and 7. For example, at a frequency of 0.41[2πc/a], L in Fig. 4 is 1.93a, while L in Fig. 7 is only 0.8a. It is because there still exists the strong coupling between two output single-channel
waveguides at the branch region in Fig. 7 when two light beams are just spatially separated. Therefore, the L in Fig. 7 is much smaller than that in Fig. 4. For the 1 2PS with L¼ 0.8a, at a frequency of 0.41[2πc/a], the steady-state field Ex and Hy are shown in Fig. 9(a) and (b), respectively.
6. An 1 4 power splitter based on two-mode interference in slot waveguides By combing three 1 2PS shown in Fig. 7 with flexible bends of slot waveguides, we construct a 1 4PS, as shown in Fig. 10. The 1 4PS consists of three 1 2PS with L¼ 0.8a: the first 1 2PS with θ ¼ 76° is defined as the first stage, and the upper and lower
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1 2PS with θ ¼50° are defined as the second stage. The two stages are cascaded through two groups of 76° arc bends. All of R are set as 4a. The incident light goes into the input port of the first stage 1 2PS, and is equally divided into two parts for the first time, which are input into the two second stage PS through two groups of 76° arc bends; and then are equally divided into two parts for the second time; and finally the incident light is equally divided into four parts through three 1 2PS. The transmission characteristics of the 1 4PS are also studied by using the finite-difference time-domain method. The numerical results indicate that in the frequency range of 0.3637–0.4576[2πc/ a], the total transmission is larger than 0.9. The transmittances at four output ports also have a good consistency. Steady-state field Ex and Hy distributions at the frequency 0.41[2πc/a] in the 1 4PS, are shown in Fig. 11 (b) and (c), respectively. Finally, it is necessary to illustrate that the PS presented in this paper are compact in size. For example, if the optical communication wavelength of 1.55 mm is located at the frequency 0.41 [2πc/a], the corresponding artificial period a is 636 nm. For the 1 2PS shown in Fig. 7, [n0, n1, h0, h1, L, R] ¼[1, 3.5, 64 nm, 191 nm, 509 nm, 2.542 mm]. If Li ¼5 a (Li denotes the length of straight waveguides on input and output ports), the size of the 1 2PS shown in Fig. 7 is 10.25 mm 6.69 mm (16.13a 10.52a), while the total size of the 1 4PS shown in Fig. 10 is 22.05 mm 14.87 mm (34.69a 23.44a), and the wavelength range corresponding to high transmission is 1.389–1.747 mm.
[5] [6]
[7] [8]
[9] [10]
[11]
[12] [13]
[14]
[15]
[16] [17]
[18]
7. Conclusions [19]
Power splitters (PS) based on two-mode interference effects in slot waveguides are proposed and demonstrated numerically. Through studies of their transmission characteristics, for the PS with the interaction length of 0.509 mm, the high efficient transmission is observed within a wavelength range of 1.389–1.747 mm, corresponding to a device size of 10.25 mm 6.69 mm (1 2PS) or 22.05 mm 14.87 mm (1 4PS). Those broadband PS are expected to be applied to highly dense photonic integrated circuits.
[20] [21] [22]
[23]
[24]
Acknowledgments
[25]
The research has been partially supported by Natural Science Basic Research Plan in Shannxi Province of China (2013JM8002).
[26] [27]
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design of broadband power splitters using two-mode interference in slot waveguides Bing Chen a,b,n, Chunliang Liu a, Jinhai Si b a b
Key Laboratory of Physical Electronics and Devices of the Ministry of Education, Xi’an Jiaotong University, Xi’an 710049, China Shaanxi Key Laboratory of Information Photonic Technique, Xi’an Jiaotong University, Xi’an 710049, China
art ic l e i nf o
a b s t r a c t
Article history: Received 5 May 2015 Received in revised form 3 July 2015 Accepted 5 July 2015
Two-mode interference effects in slot waveguides have been predicted and investigated through plane wave expansion and finite-difference time-domain methods. A broadband 1 2 power splitter (PS) based on this effect is presented, and its transmission characteristics are investigated by using finite-difference time-domain method. Calculated results indicate that, for the 1 2 PS whose interaction length is as short as 0.509 mm, the high transmission ( Z95%) is observed within a wavelength range of 1.389– 1.747 mm, corresponding to a relative bandwidth of 22.87%. Combining three 1 2 PS, we construct a 1 4 PS with excellent transmission characteristics. Those simple PS are expected to be applied to highly dense photonic integrated circuits. & 2015 Published by Elsevier B.V.
Keywords: Power splitters Slot waveguides Two-mode interferences
1. Introduction Power splitters (PS) are the most commonly used power division devices, and are key components in integrated optics. The development of highly dense photonic integrated circuits has raised the need for compact PS. Based on the novel optical waveguides such as photonic crystal waveguides [1–8], Bragg reflection waveguides [9–11], periodic dielectric waveguides [12] and plasmonic subwavelength waveguides [13–15], a considerable number of compact PS have been presented. Take 1 2 PS as an example, they can be approximately classified into two kinds according to their working mechanisms: T/Y branch type [5–9,14,15] and two-mode interference (TMI) type [1–4,10–13]. In General, the former is more compact, while the optimization of their branch region is complicated [5–9]. The latter is simpler, while it has large interaction length of TMI [1–4,10–12]. In 2004, Almeida et al. present a novel waveguide geometry named as a slot waveguide, where the guided light is strongly confined within a narrow low-index gap (slot region) between two high-index photonic wires [16,17]. This enables high interaction between a low-index material and guided light, which may lead to the implementation of compact and high-performance photonic components such as resonators [18] and polarization beam splitters [19–23]. n Corresponding author at: Key Laboratory of Physical Electronics and Devices of the Ministry of Education, Xi’an Jiaotong University, Xi’an 710049, China. E-mail address: [email protected] (B. Chen).
http://dx.doi.org/10.1016/j.optcom.2015.07.009 0030-4018/& 2015 Published by Elsevier B.V.
Inspired by these ideas, for the first time to our knowledge, an 1 2 PS based on TMI in slot waveguides is proposed in this paper. The interaction length of TMI (denoted as L) in the proposed PS is one order of magnitude less than that of the PS based on TMI in photonic crystal waveguides [1–4], or Bragg reflection waveguides [10,11], or periodic dielectric waveguides [12], and is almost equivalent to that of the PS based on TMI in plasmonic subwavelength waveguides [13]. The interaction length (L) of the 1 2 PS based on TMI in different waveguides at working wavelength 1.55 μm are listed in Table 1. It can be seen that L of PS based on slot waveguides (0.509 μm) is one order of magnitude less than that of PS based on TMI in either photonic crystal waveguides (3.100–13.64 μm) or Bragg reflection waveguides (3.901– 8.235 μm) or periodic dielectric waveguides (2.716 μm), and is basically equivalent to that of PS based on TMI in plasmonic subwavelength waveguides (0.440 μm). In this paper, the dispersion characteristics of single- and trichannel slot waveguides are firstly investigated. Secondly, TMI effects in tri-channel slot waveguides are discussed, and the interaction length of TMI (L) is obtained. Then, the 1 2 PS based on TMI is constructed, and its transmission characteristics with different L are studied. Next, a compact 1 4 PS is constructed on the basis of three 1 2 PS, and its transmission characteristic is also investigated. Finally, these numerically computed results are discussed and some important conclusions are drawn.
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Table 1 Comparison of interaction lengths L for the 1 2 power splitter based on two-mode interference in different waveguides at working wavelength 1.55 μm. Waveguide types
Materials/index
Interaction modes
L (unit: μm)
References
Photonic crystal waveguides
GaAs /n¼ 3.46, air/n¼ 1 GaAs/n ¼3, air/n¼ 1 GaAs/n ¼3.46, air/n¼ 1 Si/n¼ 3.4, air/n¼ 1 Tellurium/n¼4.6, polystyrene/n¼ 1.6, air/n¼1 Tellurium/n¼4.6, polystyrene/n¼ 1.6, air/n¼1 GaAs/n ¼3.46, air/n¼ 1 Si/n¼ 3.5, Ag/n ¼11.214þ 0.144i, air/n¼ 1 Si/n¼ 3.5, air/n¼ 1
TM0, TM2 TM0, TM2 TE0, TE2 TM0, TM1 TE0, TE2 TE0, TE2 TM0, TM2 TM0, TM2 TM0, TM2
3.720 7.998 3.100 13.640 8.235 3.901 2.716 0.440 0.509
2004-JLT 2004-OE 2009-AO 2011-PTL 2011-JOSAB 2012-COL 2007-OE 2007-OC This work
Bragg reflection waveguides Periodic dielectric waveguides Plasmonic waveguides Slot waveguides
2. Dispersion characteristics of slot waveguides The inset in Fig. 1 shows a schematic drawing of a singlechannel slot waveguide, which consists of a dielectric layer of lower index sandwiched by two high-index photonic wires in the X-direction, where a is artificial period. The refractive index for two materials are n0 ¼1 (air) and n1 ¼ 3.5 (silicon), and the corresponding thicknesses are h0 ¼ 0.10a and h1 ¼ 0.3a, respectively. The light is confined in and transmitted along the air channel (Z-direction), and the transmission constant is β. The geometry is uniform in the Y-direction. The dispersion curves of TM modes (magnetic field H normal to the incident plane (X–Z plane)) are calculated by using the plane-wave expansion method based on a super-cell [24]. The rectangle region surrounded by the dotted lines in the inset of Fig. 1 is a schematic drawing of super-cell. In our simulations, the size of a super-cell is taken as 19a a, and 40 grid points are used to represent each artificial period a. Fig. 1 gives TM mode dispersion curves of the single-channel slot waveguide. X-axis is normalized propagation constant. Y-axis is normalized frequency. It can be seen from Fig. 1 that in the frequency range of 0–0.28[2πc/a] (c is the speed of light in vacuum), there is only one TM0 (the zero-order TM mode, even mode); while in the frequency range of 0.28–0.55[2πc/a], there appears TM1 (the first-order TM mode, odd mode) besides TM0. Two additional identical high-index photonic wires are symmetrically located on the upper and lower side of the single-
[1] [2] [3] [4] [10] [11] [12] [13]
channel slot waveguide; and a tri-channel slot waveguide is formed, as shown in the inset of Fig. 2. TM mode dispersion curves of the tri-channel slot waveguide are given in Fig. 2, where X- and Y-axis are normalized propagation constant and normalized frequency, respectively. It can be seen from Fig. 2 that there are four dispersion curves, denoted as TM0, TM1, TM2 (the second order TM mode, even mode) and TM3 (the third order TM mode, odd mode). In Figs. 1 and 2, points A (0.7484[2π/a], 0.41[2πc/a]), B (0.4501 [2π/a], 0.41[2πc/a]), C (0.8253[2π/a], 0.41[2πc/a]), D (0.7321[2π/a], 0.41[2πc/a]), E (0.5663[2π/a], 0.41[2πc/a]), F (0.4162[2π/a], 0.41 [2πc/a]) were chosen at the normalized frequency of 0.41[2πc/a]. Points A and B are located in TM0 and TM1 dispersion curves of the single-channel slot waveguide, respectively. Points B, C, D and F are located in TM0, TM1, TM2 and TM3 dispersion curves of the trichannel slot waveguide, respectively. The distributions of both electric field Ex and magnetic field Hy at points A, B, C, D, E, and F are given in Fig. 3. It can be seen from Fig. 3 that light enhancement and confinement is caused by large discontinuity of the electric field (Ex) at high-index-contrast interfaces [16,17]. For the single-channel slot waveguide, Ex of TM0 mode is mainly concentrated in the air channel with low-index, while Ex of TM1 mode is nearly zero in the air channel. For the tri-channel slot waveguide, Ex of TM0 (or TM2) mode is concentrated in three air channels, and the maximum peak is located in the central air channel; while Ex of TM1 (or TM3) mode is mainly concentrated in both the upper and lower air channel, and is nearly zero in the
Fig. 1. TM mode dispersion curves of the slot waveguide shown in the inset. X-axis is normalized propagation constant. Y-axis is normalized frequency. The slot waveguide consists of a dielectric layer of lower index sandwiched by two high-index photonic wires in the X-direction, where a is artificial period. The refractive index for two materials are n0 ¼ 1.0 (air) and n1 ¼ 3.5 (silicon), and the corresponding thickness are h0 ¼ 0.10a and h1 ¼ 0.3a, respectively. The light is confined in and transmitted along air channel, and the transmission constant is β. The geometry is uniform in the Y-direction.
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Fig. 2. TM mode dispersion curves of the tri-channel slot waveguide shown in the inset. X-axis is normalized propagation constant. Y-axis is normalized frequency. The waveguide parameters are n0 ¼1.0(air), n1 ¼3.5 (silicon), h0 ¼ 0.10a and h1 ¼0.3a, where a is artificial period.
central air channel.
3. Transmission characteristics of 180° arc bends in singlechannel slot waveguides Since the single-channel slot waveguide bends will be used in
following studies, their transmission characteristics are firstly discussed here. Generally, the optimization of slot waveguide bend is a complex question [25,26]. For the sake of simplicity, we only consider a simple 180° arc slot waveguide bend shown in inset of Fig. 4(a), whose transmission characteristics is calculated by using the finite-difference time-domain method. In our computation models, the perfectly-matched-layer absorbing boundary
Fig. 3. Electric field Ex and magnetic field Hy distributions at A, B (in Fig. 1 for single-channel slot waveguide), C, D, E and F points (in Fig. 2 for tri-channel slot waveguide).
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Fig. 4. (a) The transmission spectra for the 180° arc bends with the different radii of curvature R. A schematic drawing of the waveguide with 180° arc bend is shown in the inset. The waveguide parameters are the same as those shown in Fig. 1. For 180o arc bend with R ¼ 4a, steady-state field distributions Ex (b) and Hy (c) at a frequency of 0.41 [2πc/a].
conditions are implemented at the boundaries of the computation region [27], and 50 grid points are used to represent each artificial period a. A Gaussian optical pulse is incident from the input port. In order to record the incidence pulse, the observation point 1# is placed in the middle of input waveguide. In order to record the transmission pulse, the observation point 2# are placed in the middle of output waveguide. The incidence and transmission frequency spectra are calculated from the Fourier transform of corresponding pulses. The transmission frequency spectrum normalized by the incidence frequency spectrum, and the normalized transmission spectrum is obtained. Fig. 4(a) gives the relationship curves between the transmittance of TM0 and radius of curvature (denoted as R). It can be
observed from Fig. 4(a) that with the increase of R, there are upward shifts in the transmittance curves. When R ¼2a, the normalized transmittance is higher than 0.9 in a frequency range of 0.3934–0.4992[2πc/a]. When R¼4a, the transmittance is higher than 0.98 in a frequency range of 0.3477–0.4661[2πc/a], corresponding to a relative bandwidth of 29.10%. When R is increased to 7a, the transmittance is beyond 0.98 in a frequency range of 0.2970–0.4838[2πc/a], corresponding to a relative bandwidth of 47.85%. In view of device compactness, R is taken as 4a in the following studies. For the 180° arc bend with R¼4a shown in the inset of Fig. 4(a), the steady-state Ex and Hy field distributions at a frequency of 0.41[2πc/a] are given in Fig. 4(b) and (c), respectively.
Fig. 5. The interaction length L between TM0 and TM2 in tri-channel slot waveguide. The inset is a schematic drawing of two-mode interference in tri-channel slot waveguides, which is symmetrically connected with a sing-channel slot waveguide. The waveguide parameters are the same as Figs. 1 and 2.
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4. Two-mode interference in tri-channel slot waveguides Multimode interference (or self-imaging) is a property of multimode waveguides by which an input field profile is reproduced in single or multiple images at periodic intervals along the propagation direction of the guide [28]. Multimode interference has also been used in the semiconductor waveguide for SLED power improvement [29,30]. As the simplest multimode interference, two-mode interference (TMI) is widely used to construct compact power splitters [1–4,10–13]. The inset in Fig. 5 gives a schematic drawing of TMI in the tri-channel slot waveguide, which is symmetrically connected with a sing-channel slot waveguide. When TM0 (even mode) in the single-channel slot waveguide goes into the tri-channel slot waveguide, both TM0 and TM2 modes (even modes) into the tri-channel slot waveguide can be excited simultaneously, while both TM1 and TM3 modes (odd mode) cannot be excited due to different mode symmetry. When TM0 and TM2 modes are transmitted simultaneously in the trichannel slot waveguide, TMI (the interference between TM0 and TM2 modes) occurs, which leads to the alternative appearance of two-fold image and single image at the same periodical interval, which is named as the interaction length (denoted as L). If the propagation constants of the TM0 and TM2 modes in the trichannel slot waveguide are denoted as β0 and β2 (unit: 2π/a), respectively, then L can be expressed as π/(β0–β2). According to Fig. 2, Fig. 5 gives the relationship curve between the normalized interaction length (L/a) and the normalized frequency. It can be seen from Fig. 5 that L/a is 1.861–2.938 in a frequency range of 0.25–0.50[2πc/a]. In particular, L/a is less than 2 in a frequency range of 0.3152–0.4302[2πc/a]. Additionally, for the structure shown in the inset in Fig. 5, Fig. 6 gives steady-state Ex, Hy and time-averaged Poynting vector Pz distributions at a frequency of 0.41[2πc/a] by using the finite-difference time-domain method [25]. It can be observed from Fig. 6 that L is as short as 1.93a, which is in good agreement with the result shown in Fig. 5.
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5. An 1 2 power splitter based on two-mode interference in slot waveguides The structure shown in the inset in Fig. 5 is combined with flexible bends of slot waveguides, and the 1 2 power splitter (PS) based on TMI in slot waveguides is constructed, as shown in Fig. 7. According to the calculated results shown in Fig. 4(a), all the radius of curvature R of waveguide bends are taken as 4a. All angles of waveguide bends (denoted as θ) are taken as 50°. The transmission characteristics of the PS shown in Fig. 7 are investigated by using the finite-difference time-domain method. In our computation models, the perfectly-matched-layer absorbing boundary conditions are implemented at the boundaries of the computation region [27], and 50 grid points are used to represent each artificial period a. A Gaussian optical pulse is incident from the input port. In order to record the incidence and reflection pulses, the observation point 1# is placed in the middle of input waveguide. In order to record the transmission pulse, the observation points 2# and 3# are placed in the middle of two output waveguides, respectively. The incidence, transmission and reflection frequency spectra are calculated from the Fourier transform of corresponding pulses, respectively. Next, the reflection and transmission frequency spectra are normalized by the incidence frequency spectrum, and the normalized reflection and transmission spectra are obtained, respectively. Finally, the normalized reflectance is denoted as Γ, and the normalized transmittance at output ports 1 and 2 are denoted as T1 and T2, respectively. The total transmittance (denoted as Tt) is equal to T1 þT2. A high transmission band (denoted as HTB) is defined as the frequency band corresponding to Tt Z0.95. If the calculation errors are ignored, the normalized transmission loss is equal to 1 Γ Tt. For the 1 2PS shown in Fig. 7, Fig. 8(a), (b), (c) and (d) gives transmission (T1, T2 and Tt), reflection (Γ) and loss spectra, corresponding to L ¼1.9a, 1.2a, 0.8a and 0.6a, respectively. It can be observed from Fig. 8(a) that there is no HTB for L ¼1.9a. There is the maximum Tt of 0.9456 at a frequency of 0.4978[2πc/a], and the
Fig. 6. For two-mode interference in the tri-channel slot waveguide shown in the inset in Fig. 5, steady-state Ex, Hy and time-averaged Poynting vector Pz distributions at a frequency of 0.41[2πc/a].
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Fig. 7. A schematic drawing 1 2 power splitter based on two-mode interference in slot waveguides.
corresponding reflectance (Γ) and loss are 0.0013 and 0.0531, respectively. It can be observed from Fig. 8(b) that for L ¼1.2a, there is a HTB of 0.4239–0.4848[2πc/a], corresponding to a relative bandwidth of 13.40%. There is the maximum Tt of 0.9768 at a frequency of 0.4586[2πc/a], the corresponding loss is decreased to 0.0220. When L is as short as 0.8a, HTB’s frequency range is 0.3637–0.4576[2πc/a] marked in Fig. 8(c), the corresponding relative bandwidth is increased to 22.87%. For example, at a frequency of 0.4100[2πc/a], Tt is 0.9638, and transmittances at output
ports 1 (T1) and 2 (T2) are 0.4800 and 0.4838, respectively; the reflectance (Γ) and loss are 0.0001 and 0.0361, respectively. When L is decreased to 0.6a, HTB’s relative bandwidth is decreased to 9.74%, corresponding to the frequency range of 0.3789–0.4177[2πc/ a] marked in Fig. 8(d). There is the maximum Tt of 0.9564 at a frequency of 0.3953[2πc/a]. For comparison, the transmittance, reflectance, and loss at the four frequencies mentioned above are listed in Table 2. Additionally, it can be observed from Fig. 8 and Table 2 that for
Fig. 8. Transmission at output ports 1 and 2 (T1 and T2), total transmission (Tt), reflection (Γ) and loss spectra of the 1 2 power splitter shown in Fig. 7 with different L: (a) L ¼ 1.9a, (b) L ¼1.2a, (c) L ¼ 0.8a and (d) L ¼ 0.6a.
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Table 2 Comparison of transmittance, reflectance, and loss of the 1 2 PS with different L. [n0, n1, h0, h1, R, θ]¼ [1, 3.5, 0.1a, 0.3a, 4a, 50o]
L 1.9a
1.2a
0.8a
0.6a
Normalized Frequency (unit: 2πc/a) Normalized T1 T2 transmittance Tt*1 Normalized reflectance (Γ) Normalized loss (1 Tt Γ) Frequency rang of HTB*2
0.4978 0.4586
0.4100
0.3953
0.4784 0.4672 0.9456 0.0013 0.0531 –
HTB’s relative bandwidth (%)
0
0.4800 0.4838 0.9638 0.0001 0.0361 0.3637– 0.4575 22.87
0.4826 0.4738 0.9564 0.0017 0.0437 0.3789– 0.4177 9.74
0.4772 0.4996 0.9768 0.0012 0.0220 0.4239– 0.4848 13.40
Tt*1: Tt ¼ T1 þ T2. HTB*2: high transmission band for Tt Z 0.95.
the PS based on slot waveguides, there is no obvious back-reflection from the TMI region (tri-channel waveguide), which is because there is continuous air channel from single- to tri-channel slot waveguides, and is no reflection interface between two different materials. However, there is obvious radiation loss in branch region when the two light beams are separated into two singlechannel slot waveguides from tri-channel slot waveguides. The 1 2 PS based on slot waveguides composes of two parts: a tri-channel slot waveguide and three slot waveguides (one straight waveguide at input and two curved ones at outputs). A tri-channel slot waveguide can be regarded as a resonant cavity. The
Fig. 10. A schematic drawing 1 4 power splitter, which is composed of three 1 2 power splitter with L ¼0.8a. The curvature radii of all arc are set as 4a.
transmission property of the PS mainly depends on the structure and its geometry size of resonant cavity. At a certain frequency, the eigen-modes in resonant cavity are very complicated. In the symmetric cavity, there are both symmetric and asymmetric eigen-modes simultaneously. The transmission difference between
Fig. 9. For the 1 2 power splitter with L ¼ 0.8a shown in Fig. 7, at a frequency of 0.41[2πc/a], (a) steady-state electrical field Ex distribution; (b) steady-state magnetic field Hy distribution.
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Fig. 11. For the 1 4 power splitter shown in Fig. 8, at a frequency of 0.41[2πc/a]: (a) steady-state electric field Ex distribution; (b) steady-state magnetic field Hy distribution.
two outputs is mainly caused by the existence of asymmetric modes. For example, let us assume Am is the amplitude of symmetric mode, and Bm is the amplitude of asymmetric mode. Then Am þBm and Am Bm are amplitudes at two output ports of symmetric cavity. There is the difference of 2Bm between two ports. In fact, the electromagnetic field is vector but not scalar field, and electric/magnetic fields meet the right hand relation, so two outputs are not totally symmetric. Therefore, The difference between the transmittance of two outputs ports is likely caused by the symmetry of quasi-guided or resonant modes [31]. It is necessary to illustrate that there is a large difference between L in Figs. 4 and 7. For example, at a frequency of 0.41[2πc/a], L in Fig. 4 is 1.93a, while L in Fig. 7 is only 0.8a. It is because there still exists the strong coupling between two output single-channel
waveguides at the branch region in Fig. 7 when two light beams are just spatially separated. Therefore, the L in Fig. 7 is much smaller than that in Fig. 4. For the 1 2PS with L¼ 0.8a, at a frequency of 0.41[2πc/a], the steady-state field Ex and Hy are shown in Fig. 9(a) and (b), respectively.
6. An 1 4 power splitter based on two-mode interference in slot waveguides By combing three 1 2PS shown in Fig. 7 with flexible bends of slot waveguides, we construct a 1 4PS, as shown in Fig. 10. The 1 4PS consists of three 1 2PS with L¼ 0.8a: the first 1 2PS with θ ¼ 76° is defined as the first stage, and the upper and lower
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1 2PS with θ ¼50° are defined as the second stage. The two stages are cascaded through two groups of 76° arc bends. All of R are set as 4a. The incident light goes into the input port of the first stage 1 2PS, and is equally divided into two parts for the first time, which are input into the two second stage PS through two groups of 76° arc bends; and then are equally divided into two parts for the second time; and finally the incident light is equally divided into four parts through three 1 2PS. The transmission characteristics of the 1 4PS are also studied by using the finite-difference time-domain method. The numerical results indicate that in the frequency range of 0.3637–0.4576[2πc/ a], the total transmission is larger than 0.9. The transmittances at four output ports also have a good consistency. Steady-state field Ex and Hy distributions at the frequency 0.41[2πc/a] in the 1 4PS, are shown in Fig. 11 (b) and (c), respectively. Finally, it is necessary to illustrate that the PS presented in this paper are compact in size. For example, if the optical communication wavelength of 1.55 mm is located at the frequency 0.41 [2πc/a], the corresponding artificial period a is 636 nm. For the 1 2PS shown in Fig. 7, [n0, n1, h0, h1, L, R] ¼[1, 3.5, 64 nm, 191 nm, 509 nm, 2.542 mm]. If Li ¼5 a (Li denotes the length of straight waveguides on input and output ports), the size of the 1 2PS shown in Fig. 7 is 10.25 mm 6.69 mm (16.13a 10.52a), while the total size of the 1 4PS shown in Fig. 10 is 22.05 mm 14.87 mm (34.69a 23.44a), and the wavelength range corresponding to high transmission is 1.389–1.747 mm.
[5] [6]
[7] [8]
[9] [10]
[11]
[12] [13]
[14]
[15]
[16] [17]
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7. Conclusions [19]
Power splitters (PS) based on two-mode interference effects in slot waveguides are proposed and demonstrated numerically. Through studies of their transmission characteristics, for the PS with the interaction length of 0.509 mm, the high efficient transmission is observed within a wavelength range of 1.389–1.747 mm, corresponding to a device size of 10.25 mm 6.69 mm (1 2PS) or 22.05 mm 14.87 mm (1 4PS). Those broadband PS are expected to be applied to highly dense photonic integrated circuits.
[20] [21] [22]
[23]
[24]
Acknowledgments
[25]
The research has been partially supported by Natural Science Basic Research Plan in Shannxi Province of China (2013JM8002).
[26] [27]
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