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Admittance matching analysis and optimization of transmission through an ultracompact hybrid plasmonic waveguide Bragg grating
Optics Communications 444 (2019) 39–44
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Admittance matching analysis and optimization of transmission through an ultracompact hybrid plasmonic waveguide Bragg grating Ji Xu a ,∗, Xinyi Lu a , YiLin Chen a , Yunqing Lu a , Ning Liu a , Jin Wang b , Baifu Zhang c a
College of Electronic and Optical Engineering & College of Microelectronics, Nanjing University of Posts and Telecommunications, Nanjing 210023, China College of Telecommunications & Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China c School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China b
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Keywords: Integrated optical devices Waveguides Surface plasmons Bragg reflectors Admittance matching analysis
ABSTRACT Hybrid plasmonic waveguide Bragg gratings (HPWBGs) are promising and essential components in ultracompact integrated circuits. In this paper, the transmission spectra of HPWBGs composed of two types of HPWs with different low-index dielectric layers are calculated through the transfer matrix method. Admittance match theory is used to analyze and improve transmission properties. Two admittance-matching conditions (the complete matching and the conjugate matching) are illustrated to optimize the transmission spectra effectively in specific passband, and the properties of bandgaps are also influenced. The results demonstrate the feasibility of the admittance match theory in modulating the transmission spectra of HPWBGs and this work provide a design approach for HPWBGs with a small footprint.
1. Introduction Several different optical waveguide structures have been developed as essential components in ultracompact integrated circuits, such as photonic crystal waveguides [1], and plasmonic waveguides [2–4]. In particular, subwavelength optical confinement due to surface plasmon polaritons (SPPs), and the potential for photoelectric integration with appropriate material composition have been widely studied in plasmonic waveguides research [5]. However, the loss caused by the metal results in a small propagation distance of mode in the waveguide, which limits the application of plasmonic-waveguide-based devices. To overcome this drawback, hybrid plasmonic waveguides (HPWs) have been proposed during the last decade. One low-index layer sandwiched between the metal and the high-index medium leads to mode energy concentration of the surface plasmonic waveguide, which is associated with low loss of the dielectric waveguide. Various integrated photonic devices based on HPWs have been developed, such as plasmonic nanolasers [6], high efficient optical modulators [7], and polarization beam splitters [8]. Bragg gratings have been extensively studied as wavelengthdependent photonic devices. In recent years, devices designed on HPWs with properties of high transmittance and low loss have shown applicability in a wide range of optical systems and have attracted great attention. Jiansheng Liu et al. designed a nanostructure grating based on a hybrid plasmonic slot waveguide [9], and this device can be applied as a broadband transverse magnetic (TM) polarization mode filter. Daoxin Dai et al. designed an ultracompact broadband TM-pass ∗
polarizer with a simple structure. It exhibited excellent performance by using a silicon hybrid plasmonic waveguide grating [10]. However, further research on the transmission properties were not carried out in detail. It is noteworthy that the passband of the transmission spectrum exhibits significant oscillations, which are unfavorable to the properties of the optical devices. Therefore, it is necessary to reduce the oscillations and improve the transmission spectra of the hybrid plasmonic waveguide Bragg gratings. Fortunately, admittance matching theory has been proven to be effective [11–13]. In these reported works above, the outermost dielectric layer acts as a matching coating in the whole structure, which reduces the admittance mismatch between the stack and the external medium, and transmittance optimization has been confirmed theoretically and experimentally [14,15]. A hybrid plasmonic waveguide Bragg grating (HPWBG) could be modeled and analyzed as a multilayered structure to improve the transmission properties via the same method. In this paper, an HPWBG composed of an alternating arrangement of two types of HPWs with different low-index dielectric layers is presented. According to the mode analysis of HPWs, the HPWBG operating in the optical communication band is constructed and discussed. Admittance matching theory is used to analyze and improve the transmission spectra. It demonstrates that the transmission properties of the HPWBG can be optimized by simply tuning the thickness of the outermost layer.
Corresponding author. E-mail address: [email protected] (J. Xu).
https://doi.org/10.1016/j.optcom.2019.03.054 Received 6 October 2018; Received in revised form 15 March 2019; Accepted 22 March 2019 Available online 27 March 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
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Optics Communications 444 (2019) 39–44
Fig. 3. Schematic diagram of Structure-TiO2 and Structure-SiO2 . Fig. 1. Cross-section of the HPW with a metal strip cap.
2. Mode analysis of the HPW and construction of the HPWBG The cross-section of the HPW with a metal strip cap is shown in Fig. 1, and it is covered by PMMA cladding. The thickness and width of silicon, and the thickness of the low-index dielectric layer are set as ℎSi = 230 nm and 𝑤Si = 400 nm, and ℎl = 50 nm, respectively. Both the width 𝑤m and height ℎm of the Ag top cap are 100 nm. For design and simulations, optical constants of Ag used in this paper are obtained from the Lorentz–Drude model described by Rakic et al. [16], the optical constants of TiO2 are from the model proposed by Kim [17], 𝑛Si = 3.455 for the silicon core, 𝑛SiO2 = 1.445 for the silica insulator, and 𝑛PMMA = 1.481 for the PMMA cladding. The mode properties of the proposed HPW are numerically analyzed through COMSOL Multiphysics based on the finite element method (FEM). Two HPWs (HPWSiO2 and HPWTiO2 ) are constructed with SiO2 and TiO2 as low-index dielectric layers, respectively. Figs. 2(a) and (b) show the calculated effective refractive indices 𝑛ef f of the transverse magnetic (TM) and transverse electric (TE) polarization fundamental modes in the two HPWs. Here, the frequencies are expressed as photon energy with unit of eV, i.e., 𝜔 = 1240/𝜆 (eV). Fig. 2(c) displays |E| distributions of TE and TM modes in HPWTiO2 at an optical communication wavelength of 1550 nm. As can be confirmed from Fig. 2(a), a large gap can be observed between the two curves of Re(𝑛ef f ) for TM modes in the two HPWs. And the two dispersion relations for TE modes almost coincide with each other. As shown in Fig. 2(c), the mode energy is mainly concentrated in the silicon core for TE polarization, thus, changing the low-index dielectric leads to weak influence on the mode properties. In contrast, hybrid plasmonic mode exists for TM polarization. The mode field is confined around the metal strip in the region of low-index layer and the energy propagates along the metal strip. A large difference between the indices of the two low-index materials contributes to the large difference between the 𝑛ef f of TM-polarized modes. Therefore, by modifying the structural parameters and selecting the low-index materials properly, mode properties of different polarizations can be adjusted with different 𝑛ef f . As a wavelength-dependent device, the structure of Bragg grating is actually a periodic modulation of the refractive index. It is well known
Fig. 4. Transmission spectra of SS and ST .
that larger index contrast leads to wider bandgap. Therefore, when HPWSiO2 and HPWTiO2 are placed in an alternating arrangement, 𝑛ef f is periodically modulated in order to form a Bragg grating manipulating the TM-polarized mode. At 𝜆 = 1550 nm, the values of 𝑛ef f for TM modes in HPWSiO2 and HPWTiO2 are 2.1523 − 0.0106⋅i and 2.7006 − 0.0193⋅i, respectively. Structural parameters of Bragg grating are determined by Bragg equation: q𝜆/2 = Re(𝑛nef f,1 )⋅ 𝑑B + Re(𝑛nef f,2 )⋅ (𝛬 − dB ), where q is the order of the Bragg reflection and set to be 1. The periodic length 𝛬 is calculated as 320 nm for Bragg angular frequency of 0.8 eV (𝜆 = 1550 nm), and the length of each low-index dielectric 𝑑B is set as 𝛬/2. The number of periods is chosen as 𝑁 = 10.5, which means that the two ends of the waveguide have the same construction. The total length of the HPWBG is only 3.4 μm with this design, so the proposed structure is compact. By alternating the order of the two lowindex dielectrics, two kinds of Bragg grating are formed and labeled as Structure-TiO2 (ST ) and Structure-SiO2 (SS ) respectively, shown in Fig. 3. Fig. 4 shows the transmission spectra of the TM and TE modes for the two structures, calculated by the transfer matrix method (TMM) with the assumption of normal-incidence for simplicity. For TE modes, the transmission spectra are almost two overlapping curves, and there is no obvious bandgap in the transmission spectra. This is because the
Fig. 2. (a) Real part and (b) imaginary part of the effective refractive indices 𝑛ef f of HPWSiO2 and HPWTiO2 ; (c) |E| distributions of TE and TM modes in HPWTiO2 at wavelength of 1550 nm. 40
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Optics Communications 444 (2019) 39–44
the obvious difference in transmission properties of the two structures. Additionally, the transmission spectra of the two structures exhibit oscillations in the passband, and the bandgap is not deep enough. It is necessary to improve the transmission properties. 3. Admittance matching analysis and optimization of the transmission of HPWBG Electromagnetic waves in a Bragg structure reflect at the layer interfaces, which leads to coherent superposition at particular wavelengths [18]. This is analogous to the transmission resonance phenomena in a one-dimensional metallic–dielectric (M–D) photonic crystal even given the unavoidable absorptive loss within the passband. Admittance matching analysis has been reported to study and improve the transmission of an absorbing film at a specified wavelength [19– 21]. Considering the improvement of the transmittance is related to the suppression of the reflectivity, maximum transmittance (𝑇MAX ) only occurs when the reflectance approaches zero for both ends incidences, which means that the admittance matching between the film and the surrounding media is reached. Notably, it can be achieved by modifying the materials and geometry parameters of the structure [21]. Therefore, this is also a promising method to analyze and optimize the transmission properties of the HPWBG. For the HPWBG with a layered sequence of m lossy assemblies with transmittances of 𝑇1 , 𝑇2 , . . . , 𝑇𝑚 , the total transmittance 𝑇total is the product of the transmittance values for each assembly [21], i.e., 𝑇total = 𝑇1 ⋅ 𝑇2 ⋅ . . . ⋅ 𝑇𝑚 . In order to maintain the function of the Bragg grating, the internal components of ST and SS are considered to be unmodified, which means 𝑇2 , 𝑇3 , . . . , and 𝑇𝑚−1 remaining unchanged. 𝑇total can be improved through optimizing the values of 𝑇1 and 𝑇𝑚 , by changing the length of the outermost matching layer. As an example illustrated in Fig. 5, ST ’ (optimized structure of ST with matching layers) is divided into two parts: one is the internal periodic Bragg structure, and the other is the outermost HPWTiO2 acting as two matching layers [14]. As the lengths of the outermost matching layers change, the optical admittances of the two matching layers (𝑌out and Y ’out ) change as well. When the admittance matching condition between the matching layers and the adjacent HPWSiO2 is satisfied, the reflectance from the two ends of the structure is reduced to a minimum, and 𝑇total is improved.
Fig. 5. Structure diagram of ST ’.
Fig. 6. Optical admittances of HPWSiO2 and HPWTiO2 .
𝑛ef f values of TE modes in HPWSiO2 and HPWTiO2 are almost the same, and the influence of the periodic modulation of the structure is weak or even negligible. Moreover, the TE modes will be cut off as the frequency decreases gradually, which is consistent with the behavior of traditional waveguides. However, for TM modes, there is a bandgap covering the Bragg angular frequency 0.8 eV (𝜆 = 1550 nm) as predicted, and ST has a bandgap deeper than that of SS . In spite of the same internal construction of the Bragg grating, the different ends of ST and SS lead to
∑ Fig. 7. (a) Transmission spectra, (b) |Δ|Y ||, (c) |Δ𝜑| and (d) | 𝜑| for TM mode in ST ’. 41
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Optics Communications 444 (2019) 39–44
∑ Fig. 8. Transmission spectra and |Δ|Y || for (a) 𝑑M = 160 nm, (b) 𝑑M = 80 nm, (c) 𝑑M = 210 nm, respectively. (d) |Δ𝜑| and | 𝜑| for 𝑑M = 80 nm and 160 nm, (e) transmission spectra in low and high frequency passbands, (f) transmittance and |Δ|Y || with the increasing 𝑑M at 1550 nm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
𝑌B is the optical admittance determined by the parameters of HPWSiO2 adjacent to the matching layer. It is expressed as 𝑌B = 𝑋B + 𝑍B ⋅i. The formulas calculating 𝑋B and 𝑍B are described as follows [15,20,21]: (( 2 )( ) 𝜂R + 𝜂I2 𝜂R sinh 𝛽 cosh 𝛽 + 𝜂I sin 𝛼 cos 𝛼 𝑋B = ( ) 𝜂R sinh 𝛽 cosh 𝛽 − 𝜂I sin 𝛼 cos 𝛼 ( )2 1∕2 𝜂 2 𝜂 2 sin2 𝛼 cos ℎ2 𝛽 + cos2 𝛼 sin ℎ2 𝛽 ⎞ ⎟ − R( I (1) )2 ⎟ , 𝜂R sinh 𝛽 cosh 𝛽 − 𝜂I sin 𝛼 cos 𝛼 ⎠ and 𝑍B =
) ( 𝜂R 𝜂I sin2 𝛼 cos ℎ2 𝛽 + cos2 𝛼 sin ℎ2 𝛽 ( ) . 𝜂R sinh 𝛽 cosh 𝛽 − 𝜂I sin 𝛼 cos 𝛼
(2)
Here, 𝜂R and 𝜂I are the real and imaginary parts of 𝜂B , and 𝜂B is the optical admittance of the HPWSiO2 adjacent to the matching layer. 𝑁B = 𝑛B – 𝜅B ⋅i is the complex refractive index of HPWSiO2 , which is exactly the value of 𝑛ef f . Under normal incidence, 𝜂B = 𝜂R − 𝜂I ⋅ i = 𝑛B − 𝜅B ⋅i. 𝛿B is the effective complex phase thickness of HPWSiO2 and expressed as 𝛿B = 𝛼 – 𝛽⋅i = (2𝜋∕𝜆)⋅ (𝑛B – 𝜅B ⋅i)⋅ 𝑑B [20]. The calculated optical admittances of HPWSiO2 and HPWTiO2 are shown in Fig. 6. It is easy to realize minimal reflectance when the two admittance matching layers have the same construction [14,21]. The admittances 𝑌out and Y ’out are synchronously modified by adjusting the thickness of the matching layers at each end and set as 𝑌M = 𝑌out = Y ’out . In this model, 𝑌M is calculated from the parameters of the matching layer with thickness 𝑑M and complex index 𝑛M on a massive substrate (the part of incident medium) with index 𝑛sub . 𝑌M is expressed as 𝑌M =
𝑛sub cos 𝛿M + 𝑛M sin 𝛿M ⋅ i , ( ) cos 𝛿M + 𝑛sub ∕𝑛M sin 𝛿M ⋅ i
Fig. 9. Transmission spectra of TE and TM modes in ST ’ for 𝑑M = 80 nm and 210 nm.
where 𝛿M = (2𝜋 / 𝜆) 𝑛M 𝑑M is the phase thickness of the matching layer, and 𝑛sub is set to be 1.0 for simplification. In order to analyze the admittance condition, the difference of admittance modulus |Δ|Y || = ||𝑌B | – |𝑌M || is calculated. When 𝑑M = 𝑑B = 160 nm, the two structures are exactly the same as SS and ST . At wavelength of 1550 nm (𝜔 = 0.8 eV), |Δ|Y || = 1.86 for SS , which is much smaller than |Δ|Y || = 3.95 for ST . It is conceivable that the transmittance around 0.8 eV is higher with a better matching condition, i.e., 𝑆S exhibits a worse bandgap correspondingly, as shown in Fig. 4. Actually, two matching conditions of optical admittance should be studied here: one is the complete matching of admittance, and it means the simultaneous matching of modulus and argument; the other is the
(3) 42
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Optics Communications 444 (2019) 39–44
Fig. 10. (a), (b) and (c) |E| distributions of TM mode in ST at 𝜆 = 1550 nm; (d) Transmission spectra in low frequency passband.
conjugate matching of admittance, and it means the matching of the modulus and the opposite signs of arguments [20]. Thus, difference of admittance arguments |Δ𝜑| = |𝜑B – 𝜑M | and sum of admittance ∑ arguments | 𝜑| = |𝜑B + 𝜑M | are defined and analyzed. Here, 𝜑B and 𝜑M are the arguments of the complex admittance 𝑌B and 𝑌M , respectively. Fig. 7 shows the transmission spectra, |Δ|Y ||, |Δ𝜑| and ∑ | 𝜑| for TM mode in ST ’ in detail, and the behaviors of which with the parameter variation are similar to that of SS ’. It can be observed from Figs. 7(a) and (b) that, the optimization and smoothness of the transmission spectra correspond to the matching of admittance modulus, i.e., the minimum of |Δ|Y ||, marked with white dash lines for 𝑑M = 80 nm, and white dash–dot lines for 𝑑M = 210 nm, respectively. Periodic changes in the transmission spectra occur with the increasing 𝑑M , especially in the bandgap, marked by a dot line in Fig. 7(a). More importantly, it can be observed from Figs. 7(b)–(d) that, not all the minimums of |Δ|Y || lead to the optimization and smoothness in Fig. 7(a), and the matching condition of admittance argument is essential. Only when the complete matching or the conjugate matching occurs, the optimization of transmission happens, which means that ∑ the minimal value of |Δ𝜑| or | 𝜑| is required, as shown in Figs. 7(c) and (d). For example, when 𝑑M = 80 nm, |Δ𝜑| approaches to zero, the complete matching condition is satisfied in the low frequency passband, meanwhile, the optimization and smoothness can be observed. ∑ In contrast, when 𝑑M = 210 nm, | 𝜑| approaches to a small value, which leads to a conjugate matching condition, thus the optimization and smoothness of transmission is then realized in the high frequency passband. The two situations alternately appear as 𝑑M increases. To further illustrate and prove the effectiveness of the two admittance matching conditions for influencing the transmission properties, Fig. 8 details the properties of transmission spectra in case of different matching layers. As shown in Fig. 8(a), when 𝑑M = 𝑑B = 160 nm, the oscillations of transmission spectrum are obvious in both the lower and higher frequency passbands, marked by the pink and blue regions, respectively. In addition, the curve of transmittance reaches a nadir value in the bandgap, as |Δ|Y || becomes a relatively large value. It can be observed from Figs. 8(b) an (d) that, when 𝑑M = 80 nm, |Δ|Y || and |Δ𝜑| approach minimum around the frequency of 0.73 eV simultaneously. And it can be concluded from the comparison between the two situations of 𝑑M = 160 nm and 𝑑M = 80 nm, depicted in Fig. 8(e) that, the apparent optimization and smoothness of transmission in the lower frequency passband prove the effectiveness of complete matching ∑ of admittance. In contrast, for 𝑑M = 210 nm, |Δ|Y || and | 𝜑| approach minimum in the higher frequency passband marked by the blue region, and corresponding optimization and smoothness of transmission due to the conjugate matching of admittance is also evidently demonstrated in Fig. 8(e). Beyond that, Fig. 8(f) displays the transmittance and |Δ|Y || with the increasing 𝑑M at 1550 nm, which further demonstrates that, deeper bandgap appears when larger admittance mismatch occurs. Taking into account all of these factors, it can be concluded that, the matching layer not only affects the passband but also the bandgap, and larger mismatch leads to the lower transmission and results in a more ideal bandgap. A matching layer with certain thickness exhibits specific effects on the transmittance because of the different conditions of admittance matching, and it is easy to optimize the transmission
of the low or the high frequency passbands and even the bandgap by modulating the admittance of matching layer. Fig. 9 shows the transmission spectra of TE and TM modes in ST ’ for 𝑑M = 80 nm and 210 nm. It can be seen that the optimization process still maintains a high transmittance of TE modes. Modulating the value of 𝑑M is an easy way to optimize the transmission spectra in different wavebands for TM polarization. Here, simulations are carried out by FEM, and results are shown in Fig. 10 as a comparison with the results of TMM. Form the field distributions at 𝜆 = 1550 nm illustrated in Figs. 10(a)–(c), it can be found that the field distribution at the cross-section of the waveguide exhibits the property of TM mode, and field distributions along with the propagation direction demonstrate the propagation attenuation of modes in bandgap, and the field limitation in low-index region. Fig. 10(d) depicts the transmission spectra in low frequency passband as 𝑑M = 160 nm and 80 nm. With the change of 𝑑M optimization of transmission can be observed, showing the same tendency with the results of TMM. Additionally, it would be more accurate to investigate the properties of spectrum numerically, by a 3D full vector simulation method. In our future work, we will do further research on the admittance matching theory to make it more general and more convenient. 4. Conclusions This paper proposes a type of Bragg waveguide grating designed on HPWs, which combines the abilities of mode energy concentration, polarization manipulation, and wavelength selection. More importantly, admittance matching theory is utilized to analyze and optimize the transmission properties of the HPWBGs. Two kinds of optical admittance matching condition are discussed in this work: one is the complete matching of admittance, and the other is the conjugate matching of admittance. The results prove that admittance matching theory is effective in improving the transmission spectra of HPWBGs around specific frequencies and providing a design approach for HPWBGs with a small footprint. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 11404170, 61604073), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20160839), and the Scientific Research Project of Nanjing University of Posts and Telecommunications, China (Grant no. NY217110). References [1] Q. Hu, J.Z. Zhao, R.W. Peng, Y. Zhou, Q.L. Yang, M. Wang, ‘‘Rainbow’’ trapped in a self-similar coaxial optical waveguide, Appl. Phys. Lett. 96.16 (2010) 143. [2] Q. Hu, D.H. Xu, Y. Zhou, R.W. Peng, R.H. Fan, N.X. Fang, Q.J. Wang, X.R. Huang, M. Wang, Position-sensitive spectral splitting with a plasmonic nanowire on silicon chip, Sci. Rep. 3.10 (2013) 3095. [3] T.W. Lee, E.L. Da, Y.J. Lee, S.H. Kwon, Low cross-talk, deep subwavelength plasmonic metal/insulator/metal waveguide intersections with broadband tunability, Photonics Res. 4.6 (2016) 272–276. 43
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Admittance matching analysis and optimization of transmission through an ultracompact hybrid plasmonic waveguide Bragg grating Ji Xu a ,∗, Xinyi Lu a , YiLin Chen a , Yunqing Lu a , Ning Liu a , Jin Wang b , Baifu Zhang c a
College of Electronic and Optical Engineering & College of Microelectronics, Nanjing University of Posts and Telecommunications, Nanjing 210023, China College of Telecommunications & Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China c School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China b
ARTICLE
INFO
Keywords: Integrated optical devices Waveguides Surface plasmons Bragg reflectors Admittance matching analysis
ABSTRACT Hybrid plasmonic waveguide Bragg gratings (HPWBGs) are promising and essential components in ultracompact integrated circuits. In this paper, the transmission spectra of HPWBGs composed of two types of HPWs with different low-index dielectric layers are calculated through the transfer matrix method. Admittance match theory is used to analyze and improve transmission properties. Two admittance-matching conditions (the complete matching and the conjugate matching) are illustrated to optimize the transmission spectra effectively in specific passband, and the properties of bandgaps are also influenced. The results demonstrate the feasibility of the admittance match theory in modulating the transmission spectra of HPWBGs and this work provide a design approach for HPWBGs with a small footprint.
1. Introduction Several different optical waveguide structures have been developed as essential components in ultracompact integrated circuits, such as photonic crystal waveguides [1], and plasmonic waveguides [2–4]. In particular, subwavelength optical confinement due to surface plasmon polaritons (SPPs), and the potential for photoelectric integration with appropriate material composition have been widely studied in plasmonic waveguides research [5]. However, the loss caused by the metal results in a small propagation distance of mode in the waveguide, which limits the application of plasmonic-waveguide-based devices. To overcome this drawback, hybrid plasmonic waveguides (HPWs) have been proposed during the last decade. One low-index layer sandwiched between the metal and the high-index medium leads to mode energy concentration of the surface plasmonic waveguide, which is associated with low loss of the dielectric waveguide. Various integrated photonic devices based on HPWs have been developed, such as plasmonic nanolasers [6], high efficient optical modulators [7], and polarization beam splitters [8]. Bragg gratings have been extensively studied as wavelengthdependent photonic devices. In recent years, devices designed on HPWs with properties of high transmittance and low loss have shown applicability in a wide range of optical systems and have attracted great attention. Jiansheng Liu et al. designed a nanostructure grating based on a hybrid plasmonic slot waveguide [9], and this device can be applied as a broadband transverse magnetic (TM) polarization mode filter. Daoxin Dai et al. designed an ultracompact broadband TM-pass ∗
polarizer with a simple structure. It exhibited excellent performance by using a silicon hybrid plasmonic waveguide grating [10]. However, further research on the transmission properties were not carried out in detail. It is noteworthy that the passband of the transmission spectrum exhibits significant oscillations, which are unfavorable to the properties of the optical devices. Therefore, it is necessary to reduce the oscillations and improve the transmission spectra of the hybrid plasmonic waveguide Bragg gratings. Fortunately, admittance matching theory has been proven to be effective [11–13]. In these reported works above, the outermost dielectric layer acts as a matching coating in the whole structure, which reduces the admittance mismatch between the stack and the external medium, and transmittance optimization has been confirmed theoretically and experimentally [14,15]. A hybrid plasmonic waveguide Bragg grating (HPWBG) could be modeled and analyzed as a multilayered structure to improve the transmission properties via the same method. In this paper, an HPWBG composed of an alternating arrangement of two types of HPWs with different low-index dielectric layers is presented. According to the mode analysis of HPWs, the HPWBG operating in the optical communication band is constructed and discussed. Admittance matching theory is used to analyze and improve the transmission spectra. It demonstrates that the transmission properties of the HPWBG can be optimized by simply tuning the thickness of the outermost layer.
Corresponding author. E-mail address: [email protected] (J. Xu).
https://doi.org/10.1016/j.optcom.2019.03.054 Received 6 October 2018; Received in revised form 15 March 2019; Accepted 22 March 2019 Available online 27 March 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
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Fig. 3. Schematic diagram of Structure-TiO2 and Structure-SiO2 . Fig. 1. Cross-section of the HPW with a metal strip cap.
2. Mode analysis of the HPW and construction of the HPWBG The cross-section of the HPW with a metal strip cap is shown in Fig. 1, and it is covered by PMMA cladding. The thickness and width of silicon, and the thickness of the low-index dielectric layer are set as ℎSi = 230 nm and 𝑤Si = 400 nm, and ℎl = 50 nm, respectively. Both the width 𝑤m and height ℎm of the Ag top cap are 100 nm. For design and simulations, optical constants of Ag used in this paper are obtained from the Lorentz–Drude model described by Rakic et al. [16], the optical constants of TiO2 are from the model proposed by Kim [17], 𝑛Si = 3.455 for the silicon core, 𝑛SiO2 = 1.445 for the silica insulator, and 𝑛PMMA = 1.481 for the PMMA cladding. The mode properties of the proposed HPW are numerically analyzed through COMSOL Multiphysics based on the finite element method (FEM). Two HPWs (HPWSiO2 and HPWTiO2 ) are constructed with SiO2 and TiO2 as low-index dielectric layers, respectively. Figs. 2(a) and (b) show the calculated effective refractive indices 𝑛ef f of the transverse magnetic (TM) and transverse electric (TE) polarization fundamental modes in the two HPWs. Here, the frequencies are expressed as photon energy with unit of eV, i.e., 𝜔 = 1240/𝜆 (eV). Fig. 2(c) displays |E| distributions of TE and TM modes in HPWTiO2 at an optical communication wavelength of 1550 nm. As can be confirmed from Fig. 2(a), a large gap can be observed between the two curves of Re(𝑛ef f ) for TM modes in the two HPWs. And the two dispersion relations for TE modes almost coincide with each other. As shown in Fig. 2(c), the mode energy is mainly concentrated in the silicon core for TE polarization, thus, changing the low-index dielectric leads to weak influence on the mode properties. In contrast, hybrid plasmonic mode exists for TM polarization. The mode field is confined around the metal strip in the region of low-index layer and the energy propagates along the metal strip. A large difference between the indices of the two low-index materials contributes to the large difference between the 𝑛ef f of TM-polarized modes. Therefore, by modifying the structural parameters and selecting the low-index materials properly, mode properties of different polarizations can be adjusted with different 𝑛ef f . As a wavelength-dependent device, the structure of Bragg grating is actually a periodic modulation of the refractive index. It is well known
Fig. 4. Transmission spectra of SS and ST .
that larger index contrast leads to wider bandgap. Therefore, when HPWSiO2 and HPWTiO2 are placed in an alternating arrangement, 𝑛ef f is periodically modulated in order to form a Bragg grating manipulating the TM-polarized mode. At 𝜆 = 1550 nm, the values of 𝑛ef f for TM modes in HPWSiO2 and HPWTiO2 are 2.1523 − 0.0106⋅i and 2.7006 − 0.0193⋅i, respectively. Structural parameters of Bragg grating are determined by Bragg equation: q𝜆/2 = Re(𝑛nef f,1 )⋅ 𝑑B + Re(𝑛nef f,2 )⋅ (𝛬 − dB ), where q is the order of the Bragg reflection and set to be 1. The periodic length 𝛬 is calculated as 320 nm for Bragg angular frequency of 0.8 eV (𝜆 = 1550 nm), and the length of each low-index dielectric 𝑑B is set as 𝛬/2. The number of periods is chosen as 𝑁 = 10.5, which means that the two ends of the waveguide have the same construction. The total length of the HPWBG is only 3.4 μm with this design, so the proposed structure is compact. By alternating the order of the two lowindex dielectrics, two kinds of Bragg grating are formed and labeled as Structure-TiO2 (ST ) and Structure-SiO2 (SS ) respectively, shown in Fig. 3. Fig. 4 shows the transmission spectra of the TM and TE modes for the two structures, calculated by the transfer matrix method (TMM) with the assumption of normal-incidence for simplicity. For TE modes, the transmission spectra are almost two overlapping curves, and there is no obvious bandgap in the transmission spectra. This is because the
Fig. 2. (a) Real part and (b) imaginary part of the effective refractive indices 𝑛ef f of HPWSiO2 and HPWTiO2 ; (c) |E| distributions of TE and TM modes in HPWTiO2 at wavelength of 1550 nm. 40
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the obvious difference in transmission properties of the two structures. Additionally, the transmission spectra of the two structures exhibit oscillations in the passband, and the bandgap is not deep enough. It is necessary to improve the transmission properties. 3. Admittance matching analysis and optimization of the transmission of HPWBG Electromagnetic waves in a Bragg structure reflect at the layer interfaces, which leads to coherent superposition at particular wavelengths [18]. This is analogous to the transmission resonance phenomena in a one-dimensional metallic–dielectric (M–D) photonic crystal even given the unavoidable absorptive loss within the passband. Admittance matching analysis has been reported to study and improve the transmission of an absorbing film at a specified wavelength [19– 21]. Considering the improvement of the transmittance is related to the suppression of the reflectivity, maximum transmittance (𝑇MAX ) only occurs when the reflectance approaches zero for both ends incidences, which means that the admittance matching between the film and the surrounding media is reached. Notably, it can be achieved by modifying the materials and geometry parameters of the structure [21]. Therefore, this is also a promising method to analyze and optimize the transmission properties of the HPWBG. For the HPWBG with a layered sequence of m lossy assemblies with transmittances of 𝑇1 , 𝑇2 , . . . , 𝑇𝑚 , the total transmittance 𝑇total is the product of the transmittance values for each assembly [21], i.e., 𝑇total = 𝑇1 ⋅ 𝑇2 ⋅ . . . ⋅ 𝑇𝑚 . In order to maintain the function of the Bragg grating, the internal components of ST and SS are considered to be unmodified, which means 𝑇2 , 𝑇3 , . . . , and 𝑇𝑚−1 remaining unchanged. 𝑇total can be improved through optimizing the values of 𝑇1 and 𝑇𝑚 , by changing the length of the outermost matching layer. As an example illustrated in Fig. 5, ST ’ (optimized structure of ST with matching layers) is divided into two parts: one is the internal periodic Bragg structure, and the other is the outermost HPWTiO2 acting as two matching layers [14]. As the lengths of the outermost matching layers change, the optical admittances of the two matching layers (𝑌out and Y ’out ) change as well. When the admittance matching condition between the matching layers and the adjacent HPWSiO2 is satisfied, the reflectance from the two ends of the structure is reduced to a minimum, and 𝑇total is improved.
Fig. 5. Structure diagram of ST ’.
Fig. 6. Optical admittances of HPWSiO2 and HPWTiO2 .
𝑛ef f values of TE modes in HPWSiO2 and HPWTiO2 are almost the same, and the influence of the periodic modulation of the structure is weak or even negligible. Moreover, the TE modes will be cut off as the frequency decreases gradually, which is consistent with the behavior of traditional waveguides. However, for TM modes, there is a bandgap covering the Bragg angular frequency 0.8 eV (𝜆 = 1550 nm) as predicted, and ST has a bandgap deeper than that of SS . In spite of the same internal construction of the Bragg grating, the different ends of ST and SS lead to
∑ Fig. 7. (a) Transmission spectra, (b) |Δ|Y ||, (c) |Δ𝜑| and (d) | 𝜑| for TM mode in ST ’. 41
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∑ Fig. 8. Transmission spectra and |Δ|Y || for (a) 𝑑M = 160 nm, (b) 𝑑M = 80 nm, (c) 𝑑M = 210 nm, respectively. (d) |Δ𝜑| and | 𝜑| for 𝑑M = 80 nm and 160 nm, (e) transmission spectra in low and high frequency passbands, (f) transmittance and |Δ|Y || with the increasing 𝑑M at 1550 nm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
𝑌B is the optical admittance determined by the parameters of HPWSiO2 adjacent to the matching layer. It is expressed as 𝑌B = 𝑋B + 𝑍B ⋅i. The formulas calculating 𝑋B and 𝑍B are described as follows [15,20,21]: (( 2 )( ) 𝜂R + 𝜂I2 𝜂R sinh 𝛽 cosh 𝛽 + 𝜂I sin 𝛼 cos 𝛼 𝑋B = ( ) 𝜂R sinh 𝛽 cosh 𝛽 − 𝜂I sin 𝛼 cos 𝛼 ( )2 1∕2 𝜂 2 𝜂 2 sin2 𝛼 cos ℎ2 𝛽 + cos2 𝛼 sin ℎ2 𝛽 ⎞ ⎟ − R( I (1) )2 ⎟ , 𝜂R sinh 𝛽 cosh 𝛽 − 𝜂I sin 𝛼 cos 𝛼 ⎠ and 𝑍B =
) ( 𝜂R 𝜂I sin2 𝛼 cos ℎ2 𝛽 + cos2 𝛼 sin ℎ2 𝛽 ( ) . 𝜂R sinh 𝛽 cosh 𝛽 − 𝜂I sin 𝛼 cos 𝛼
(2)
Here, 𝜂R and 𝜂I are the real and imaginary parts of 𝜂B , and 𝜂B is the optical admittance of the HPWSiO2 adjacent to the matching layer. 𝑁B = 𝑛B – 𝜅B ⋅i is the complex refractive index of HPWSiO2 , which is exactly the value of 𝑛ef f . Under normal incidence, 𝜂B = 𝜂R − 𝜂I ⋅ i = 𝑛B − 𝜅B ⋅i. 𝛿B is the effective complex phase thickness of HPWSiO2 and expressed as 𝛿B = 𝛼 – 𝛽⋅i = (2𝜋∕𝜆)⋅ (𝑛B – 𝜅B ⋅i)⋅ 𝑑B [20]. The calculated optical admittances of HPWSiO2 and HPWTiO2 are shown in Fig. 6. It is easy to realize minimal reflectance when the two admittance matching layers have the same construction [14,21]. The admittances 𝑌out and Y ’out are synchronously modified by adjusting the thickness of the matching layers at each end and set as 𝑌M = 𝑌out = Y ’out . In this model, 𝑌M is calculated from the parameters of the matching layer with thickness 𝑑M and complex index 𝑛M on a massive substrate (the part of incident medium) with index 𝑛sub . 𝑌M is expressed as 𝑌M =
𝑛sub cos 𝛿M + 𝑛M sin 𝛿M ⋅ i , ( ) cos 𝛿M + 𝑛sub ∕𝑛M sin 𝛿M ⋅ i
Fig. 9. Transmission spectra of TE and TM modes in ST ’ for 𝑑M = 80 nm and 210 nm.
where 𝛿M = (2𝜋 / 𝜆) 𝑛M 𝑑M is the phase thickness of the matching layer, and 𝑛sub is set to be 1.0 for simplification. In order to analyze the admittance condition, the difference of admittance modulus |Δ|Y || = ||𝑌B | – |𝑌M || is calculated. When 𝑑M = 𝑑B = 160 nm, the two structures are exactly the same as SS and ST . At wavelength of 1550 nm (𝜔 = 0.8 eV), |Δ|Y || = 1.86 for SS , which is much smaller than |Δ|Y || = 3.95 for ST . It is conceivable that the transmittance around 0.8 eV is higher with a better matching condition, i.e., 𝑆S exhibits a worse bandgap correspondingly, as shown in Fig. 4. Actually, two matching conditions of optical admittance should be studied here: one is the complete matching of admittance, and it means the simultaneous matching of modulus and argument; the other is the
(3) 42
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Optics Communications 444 (2019) 39–44
Fig. 10. (a), (b) and (c) |E| distributions of TM mode in ST at 𝜆 = 1550 nm; (d) Transmission spectra in low frequency passband.
conjugate matching of admittance, and it means the matching of the modulus and the opposite signs of arguments [20]. Thus, difference of admittance arguments |Δ𝜑| = |𝜑B – 𝜑M | and sum of admittance ∑ arguments | 𝜑| = |𝜑B + 𝜑M | are defined and analyzed. Here, 𝜑B and 𝜑M are the arguments of the complex admittance 𝑌B and 𝑌M , respectively. Fig. 7 shows the transmission spectra, |Δ|Y ||, |Δ𝜑| and ∑ | 𝜑| for TM mode in ST ’ in detail, and the behaviors of which with the parameter variation are similar to that of SS ’. It can be observed from Figs. 7(a) and (b) that, the optimization and smoothness of the transmission spectra correspond to the matching of admittance modulus, i.e., the minimum of |Δ|Y ||, marked with white dash lines for 𝑑M = 80 nm, and white dash–dot lines for 𝑑M = 210 nm, respectively. Periodic changes in the transmission spectra occur with the increasing 𝑑M , especially in the bandgap, marked by a dot line in Fig. 7(a). More importantly, it can be observed from Figs. 7(b)–(d) that, not all the minimums of |Δ|Y || lead to the optimization and smoothness in Fig. 7(a), and the matching condition of admittance argument is essential. Only when the complete matching or the conjugate matching occurs, the optimization of transmission happens, which means that ∑ the minimal value of |Δ𝜑| or | 𝜑| is required, as shown in Figs. 7(c) and (d). For example, when 𝑑M = 80 nm, |Δ𝜑| approaches to zero, the complete matching condition is satisfied in the low frequency passband, meanwhile, the optimization and smoothness can be observed. ∑ In contrast, when 𝑑M = 210 nm, | 𝜑| approaches to a small value, which leads to a conjugate matching condition, thus the optimization and smoothness of transmission is then realized in the high frequency passband. The two situations alternately appear as 𝑑M increases. To further illustrate and prove the effectiveness of the two admittance matching conditions for influencing the transmission properties, Fig. 8 details the properties of transmission spectra in case of different matching layers. As shown in Fig. 8(a), when 𝑑M = 𝑑B = 160 nm, the oscillations of transmission spectrum are obvious in both the lower and higher frequency passbands, marked by the pink and blue regions, respectively. In addition, the curve of transmittance reaches a nadir value in the bandgap, as |Δ|Y || becomes a relatively large value. It can be observed from Figs. 8(b) an (d) that, when 𝑑M = 80 nm, |Δ|Y || and |Δ𝜑| approach minimum around the frequency of 0.73 eV simultaneously. And it can be concluded from the comparison between the two situations of 𝑑M = 160 nm and 𝑑M = 80 nm, depicted in Fig. 8(e) that, the apparent optimization and smoothness of transmission in the lower frequency passband prove the effectiveness of complete matching ∑ of admittance. In contrast, for 𝑑M = 210 nm, |Δ|Y || and | 𝜑| approach minimum in the higher frequency passband marked by the blue region, and corresponding optimization and smoothness of transmission due to the conjugate matching of admittance is also evidently demonstrated in Fig. 8(e). Beyond that, Fig. 8(f) displays the transmittance and |Δ|Y || with the increasing 𝑑M at 1550 nm, which further demonstrates that, deeper bandgap appears when larger admittance mismatch occurs. Taking into account all of these factors, it can be concluded that, the matching layer not only affects the passband but also the bandgap, and larger mismatch leads to the lower transmission and results in a more ideal bandgap. A matching layer with certain thickness exhibits specific effects on the transmittance because of the different conditions of admittance matching, and it is easy to optimize the transmission
of the low or the high frequency passbands and even the bandgap by modulating the admittance of matching layer. Fig. 9 shows the transmission spectra of TE and TM modes in ST ’ for 𝑑M = 80 nm and 210 nm. It can be seen that the optimization process still maintains a high transmittance of TE modes. Modulating the value of 𝑑M is an easy way to optimize the transmission spectra in different wavebands for TM polarization. Here, simulations are carried out by FEM, and results are shown in Fig. 10 as a comparison with the results of TMM. Form the field distributions at 𝜆 = 1550 nm illustrated in Figs. 10(a)–(c), it can be found that the field distribution at the cross-section of the waveguide exhibits the property of TM mode, and field distributions along with the propagation direction demonstrate the propagation attenuation of modes in bandgap, and the field limitation in low-index region. Fig. 10(d) depicts the transmission spectra in low frequency passband as 𝑑M = 160 nm and 80 nm. With the change of 𝑑M optimization of transmission can be observed, showing the same tendency with the results of TMM. Additionally, it would be more accurate to investigate the properties of spectrum numerically, by a 3D full vector simulation method. In our future work, we will do further research on the admittance matching theory to make it more general and more convenient. 4. Conclusions This paper proposes a type of Bragg waveguide grating designed on HPWs, which combines the abilities of mode energy concentration, polarization manipulation, and wavelength selection. More importantly, admittance matching theory is utilized to analyze and optimize the transmission properties of the HPWBGs. Two kinds of optical admittance matching condition are discussed in this work: one is the complete matching of admittance, and the other is the conjugate matching of admittance. The results prove that admittance matching theory is effective in improving the transmission spectra of HPWBGs around specific frequencies and providing a design approach for HPWBGs with a small footprint. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 11404170, 61604073), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20160839), and the Scientific Research Project of Nanjing University of Posts and Telecommunications, China (Grant no. NY217110). References [1] Q. Hu, J.Z. Zhao, R.W. Peng, Y. Zhou, Q.L. Yang, M. Wang, ‘‘Rainbow’’ trapped in a self-similar coaxial optical waveguide, Appl. Phys. Lett. 96.16 (2010) 143. [2] Q. Hu, D.H. Xu, Y. Zhou, R.W. Peng, R.H. Fan, N.X. Fang, Q.J. Wang, X.R. Huang, M. Wang, Position-sensitive spectral splitting with a plasmonic nanowire on silicon chip, Sci. Rep. 3.10 (2013) 3095. [3] T.W. Lee, E.L. Da, Y.J. Lee, S.H. Kwon, Low cross-talk, deep subwavelength plasmonic metal/insulator/metal waveguide intersections with broadband tunability, Photonics Res. 4.6 (2016) 272–276. 43
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