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Enhanced microwave transmission and focusing effect through step structure with subwavelength slit arrays
Optics Communications 314 (2014) 31–35
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Enhanced microwave transmission and focusing effect through step structure with subwavelength slit arrays Liang Xu a,n, Jinxiang Cao a, Junming Liu b, Yu Liu a, Jian Wang a, Zhe Zheng a, Yinchang Du a, Xiao Zhang a, Pi Wang a a b
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Department of Physics, Rutgers, The State University of New Jersey, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 8 January 2013 Received in revised form 3 April 2013 Accepted 2 May 2013 Available online 23 May 2013
We explore the transmission and focusing properties of a step structure with subwavelength slit arrays in metallic aluminum plates. We show that the microcavities formed between adjacent step can greatly enhance the coupling of the incident wave into the slit by improving the impedance matching between the incident wave and the waveguide mode, leading to X-band (8–12 GHz) transmission enhancement and efficient focusing. In addition, the electric field is localized and enhanced at the slit exits, in contrast with non-step slit gratings. By putting the time gate in the time domain of transmission spectra, we are able to accurately calculate the improved coupling efficiency to illustrate the enhancement at both of the resonant and nonresonant frequencies. The simulation by a finite element code agrees well with the experimental results. Potential applications of this effect can be anticipated in microwave devices. & 2013 Elsevier B.V. All rights reserved.
Keywords: Subwavelength structure Waveguides Spoof surface plasmon
1. Introduction Since the discovery of extraordinary transmission phenomenon [1], a variety of plasmonic devices [2–4], which are based on subwavelength plasmonic waveguide [5] and diffractive wave, have been proposed. Besides hole arrays [6,7], the subwavelength slit arrays have become one of the basic structure element to achieve the minimization of optical devices, for example, the plasmonic lens [2], and to take the place of the refractive lens since the focusing capability of the conventional lens deteriorate with the miniaturization of their size. But the superiority of the refractive lens, which makes them the most ubiquitous optical and microwave component, is that they are transparent to broadband frequencies while many plasmonic devices are only useful when the surface plasmons (or spoof surface plasmons at microwave frequency) is excited at the resonant frequencies. There have been, so far, several experimental works that investigate EOT by modulating the size [8], the materials [9] or by changing the geometries [10–13] of subwavelength MIM waveguide. As we know, surface plasmons (SPs) are surface-bound waves that only propagate along the surface of a conductor with finite conductivity in the visible domain, By contrast at the microwave regime, metals can be often assumed to be perfectly conducting, the surface wave supported by a planar metal-dielectric interface is unconfined in
n
Corresponding author. Tel.: +86 18256900861. E-mail address: [email protected] (L. Xu).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.05.011
the dielectric. However, it has been shown that structured PEC surfaces can support surface-bound waves [14–17], which means that the structured surface now behaves much more like a planar metallic one at optical frequencies with tight confinement of the electromagnetic fields. This is the concept of “spoof” or “designer” surface plasmons. However, the coupling between the incident plane wave and the waveguide mode is not significantly improved at both of the resonant frequencies and none-resonant frequencies, the performance of plasmonic devices are therefore limited. Recently, a kind of double resonance is analytically and experimentally demonstrated to help improve the coupling process through dielectric-metal-dielectric sandwich hole arrays and a 100% transmission is observed when the incident wave couples to standing waves in both dielectric layers [18,19]. Furthermore, using microcavity structure [20–23], enhanced transmission with increased efficiency was demonstrated due to the interaction between the surface plasmon microcavity standing waves and the slit mode, and nanofocusing phenomenon was achieved based on the constructive interference in the microcavity. However, these experiments and simulations were performed in the optical regime and near infrared regime. While microcavity structure based on slit arrays has not been investigated at microwave regime. In this paper, we develop a 1D step structure and achieve a nearly full X-band (8–12 GHz) transmission enhancement and efficient focusing. We demonstrate that the microcavities formed between adjacent step can greatly enhance the coupling of the incident wave into the slit by improving the impedance matching
32
L. Xu et al. / Optics Communications 314 (2014) 31–35
Fig. 2. Experimental and simulated transmission spectra of sample A with step width d ¼12.5 mm and sample B. Fig. 1. (a) Schematic of the two samples with w¼ 3 mm, P ¼25 mm, H¼ 20 mm, h ¼ 10 mm. The incident normal p-polarized plane wave transmits directly through a slit and couples to surface mode. (b) Setup of the measurement system, and the coordinate system.
between the incident wave and the slit mode. Thus, the performance of the plasmonic devices based upon isolate slit arrays is improved at microwaves. A basic diffractive wave mode is proposed and an accurate coupling efficiency is calculated in this paper.
2. Experimental setup and simulation method In our experiment, we manufacture a much-studied aluminum slit array structure (sample B) as a comparison which was optimized to reach its highest transmission through various design principles [24]. The step structure (sample A) consists of 1D (onedimension) subwavelength slit arrays structure (10 mm thick and 3 mm width slit) with periodic step arrays (5 mm thick) loading (see Fig. 1(a)). The distance between the centers of adjacent slit is 25 mm. The area of each sample is 287.5 300 mm which is large enough to minimize the effects of finite sample size. The transmission characteristic was measured by microwave vector network analyzer E8362B which supports two identical horn antennas at the frequencies of 8–12 GHz. The receiver antenna was mounted on the computer-control rail for accuracy 2D position measurements (Fig. 1(b)). Through our work, normal p-polarized plane microwave (H field parallel to the slits) was illuminated on the center slit and the transmission enhancement of incident wave was normalized by 1. All of the structures were measured under the same conditions and the results were repeatable. Finite element method (FEM) was used to calculate electromagnetic transmission characteristics of the different samples. The dispersion of aluminum was captured using a Drude model. Perfect electric conditions were used on all sides of the calculation domain at the microwave frequencies. Near-field and transmission monitors were used to visualize the electric field localization and enhancement.
3. Experimental results and discussion By measuring transmission spectra at the normal direction 40 cm away from the central slit, we observed the X-band transmission enhancement through the step structure with step width d ¼ 12.5 mm (Fig. 2). At the surface wave resonant frequency (10 GHz), the transmission efficiency reaches its highest intensity 1.25, and the transmission efficiency is greater than that of the
Fig. 3. Experimental results of the electric intensity for the planar lens based on the different structure, the above for the isolate slit structure and the below for the step structure (d ¼ 12.5 mm), with a p-polarized plane wave incident on the center slit at the frequency of 10 GHz. The vertical white line denotes the position of the focal point.
incident wave ranging from 8.7 GHz to 10.6 GHz. Particularly, at the nonresonant frequency 8.7 GHz, the transmission efficiency of the step structure is improved greatly from 30% to 100%, with contrast to sample B. The validity of the experiment was justified by our simulation results (Fig. 2). Then, we compared the focusing effect of the Sample A to the Sample B, When the plate was illuminated by a plane wave at the resonant frequency (10 GHz), the energy will focus and we can see that the step structure and the isolate slit arrays have the same focal length of about 40 cm, approximate to 10 times of the resonant wavelength (Fig. 3). The focused energy was improved by step structure while the focusing effect and focal length remain unchanged. Fig. 4 shows the measured and simulated transmission with different step width. The transmission for a non-step slit arrays in a 20 mm thick aluminum plate is also shown for direct quantitative comparison. It is noted that the sample of step width d¼ 22 mm corresponds the non-step sample with thickness h¼20 mm. For different parameters, the step structure leads to a dramatic enhancement of the transmission over the most X-band region, comparing the non-step structure case (Fig. 4, black solid curve). The resonant transmission enhancement around 10 GHz of the samples is attributed to the
L. Xu et al. / Optics Communications 314 (2014) 31–35
33
Fig. 4. (a) measured and (b) simulated transmission spectra through the subwavelength slit array structure with the different step width. The parameters are same as the sample A in Fig. 1(a), and the sample is illuminated with normal incident p-polarized light.
Fig. 5. Profile of simulated magnitude of the electric field of the structure (a) step structure with step width d ¼ 12.5 mm, (b) non-step structure, both being illuminated at the same frequency of 10 GHz.
excitation of spoof surface plasmon when the dispersion of the spoof spps and the periodicity of the diffraction grating satisfy the conservation of momentum [25] ksp ¼ k0 sinθ 7 nkg
ð1Þ
where kg ¼ 2π=λg is the Bragg reciprocal-lattice vector of the periodic structure, θ is the angle of incidence with wave number k0 in free space, and n is the coupling order that can only be integers. We notice that the step structure makes the nonresonant transmission enhancement, especially at the low frequency. However, the transmission enhancement does not increase monotonously as the microcavity width becomes larger, and the simulation in Fig. 4 (b) agrees well with the experimental results. Based on transmission line theory [26,27], The proposed step structure can be modeled as three cascaded transmission lines with the microcavities, formed between adjacent step, and slit characterized by local width and thickness. The characteristic impedance of the microcavity transmission lines at the entrance and exit are Z 1 ¼ ðβ1 =jωε0 Þd and Z 3 ¼ ðβ3 =jωε0 Þd, respectively, and the characteristic impedance of the slit transmission line is Z 2 ¼ ðβ2 =jωε0 Þw where βi ¼ β′i þ β″i i (i¼1,2,3) is the complex wave vector of the propagating TM mode in the MIM waveguide. The wavenumber βi by the dispersion relation of a metal-insulator-metal waveguide: [2] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 a β2i −k0 − β2i −εm k0 A¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ tanh@ 2 2 εm β2i −k0 where a represents the width of MIM waveguide (d in microcavity and w in slit) and εm ≈−104 þ 107 i p[28] ffiffiffiffiffiffiffiffiffiffiffiis ffi the permittivity of aluminum at this wavelength. Z 0 ¼ μ0 =ε0 p is the surrounding characteristic impedance in our experiment. Thus, the characteristic impedance matching between surrounding air and microcavity MIM waveguide can be improved as the microcavity width increases. In turn, the characteristic impedance matching between microcavity MIM waveguide and slit MIM waveguide will be decreased as the microcavity width increases. More specifically, In Fig. 4 the optimized
step width is d¼12.5 mm. when the microcavity width is below 12.5 mm, the characteristic impedance matching between the surrounding air and microcavity waveguide would decrease, leading to the reduce of the transmission efficiency. However, when the microcavity width is above 12.5 mm, the characteristic impedance matching between the microcavity waveguide and slit waveguide would decrease rapidly, therefore the transmission efficiency would decrease as well. That is, the transmission efficiency is limited by the trade-off between the incident power at the surrounding air-cavity waveguide junction and the transmission coefficient of the cavity waveguide- slit waveguide junction. The step structure of step width d¼12.5 mm achieve an optimized impedance matching of trade-off, leading to the largest transmission enhancement. Based on the large transmission enhancement by step structure, the near field intensity is expected to be enhanced at the exit slit. To confirm, we plot the electric field spatial distribution (Fig. 5) at 10 GHz for the non-step structure with h¼20 mm and step structure with width d ¼12.5 mm. For other gratings, the field spatial distribution profiles are very similar. We notice that indeed the steps provide strong enhancement and localization of electric field at the exit of the slits, comparing to the non-step structure. It is noted that the step structure can achieve non-resonant behavior, the transmission spectra would tend to be broadband. To further investigate the transmission process, it is necessary to identify the change of the waveguide after forming a step shape as the final transmission comes from two different processes [29]. By offering a single incident pulse, the light is separated into two channels: direct transmission or coupling to the surface wave (see Fig. 1(b)). Since the surface wave channel has a time delay, which is purely decided by the phase retardation across the distance between slits, to reach its final transmission. The pulses with different arrival time become differentiated in time domain showed in Fig. 6. In Fig. 6, the initial transmission pulse (left gray region) corresponds to the wave transmission directly through the slits with the help of waveguide and the subsequent pulses (right gray region)
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L. Xu et al. / Optics Communications 314 (2014) 31–35
correspond to indirect diffractive waves. The origin of the pulses can be theoretically analyzed by considering time-dependent component to the electric field in simplified couple-mode formalization [30]. Through inverse Fourier Transform, temporal dependence of the transmitted fields can be expressed by a series of impulse functions: E′0 ðtÞ ¼ ηE0 ðtÞδðt−t 0 Þ þ γ 2 η ∑ E0 ðtÞδðt−ðt 0 þ 2n ΔtÞÞ
ð3Þ
n≠0
where t 0 represents the waveguide delay and Δt is the time for a surface wave passing across one slit. From Eq. (2), the imaginary part of the complex propagating wave vector in the MIM waveguide β″≈3:3 10−4 k0 which is negligible, therefore the propagating loss in the slit is neglected. Based on this property, we assume η and γ represent the coupling efficiency of waveguide mode and surface
wave mode with the incident wave respectively. Obviously, the transmission efficiency of the main pulse in time domain shows a higher coupling efficiency than that of the isolate slit arrays. In order to distinguish the role of the diffractive wave in focusing and certify the accurate coupling efficiency, we put a time gate between the main pulse and the diffractive pulses in time domain and refuse any signals later than gated time, therefore all of the diffractive waves derived from other slits are not received. Without the interference of the diffractive waves, the focusing effect disappears and the electric field decays with the value of Z axis (Fig. 7). It is interesting to see that in the near field, the electric intensity is still slightly greater than the incident wave, which is attributed to the interference between the plane wave transmitting directly through the center slit and the diffractive wave derived from the exit surface of the center slit. Due to the oscillating dipole formed near the central slit corners [31] and the electric field intensity of dipole mode in the far field [32], combining with the coupling efficiency, the measured electric field intensity in the far field can be expressed as Eq. (4) in our experiment: Etotal ðzÞ ¼ κ=z þ η
Fig. 6. Transmission pulses in time domain. The black solid line denotes the exciting pulse. The red dashed line and the blue dotted line denotes the transmission of step structure A (d ¼ 12.5 mm) and isolate slit structure B, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ð4Þ
where κ is an undetermined coefficient representing the excitation degree of dipole mode and η represents the coupling efficiency of the waveguide mode. After fitting the field distribution along the value of Z axis, A1 (step structure at 10 GHz), A2 (step structure at 9 GHz), B1 (isolate slit arrays 10 GHz) and B2 (isolate slit arrays 9 GHz) of Fig. 7 to Eq. (4), we can obtain the degree of coupling and excitation of different structures at resonant and nonresonant frequency (see Table 1). For the same structure, the intensity of the diffractive wave is much stronger at the resonance frequency (10 GHz) than that at the frequency of 9 GHz. The coupling efficiency of the step structure is enhanced to 97% at the frequencies of resonance and none-resonance, compared to the isolate slit structure with a coupling efficiency of 80% at 10 GHz and 65% at 9 GHz. Thus, the transmission enhancement is mainly attributed to the improved coupling process. Although the resonance in high frequency might be much complex, the slit mode and oscillating dipole mode can give basic information about the coupling efficiency and the excitation degree of the oscillating charges since the standard error of fitting is small.
4. Conclusion
Fig. 7. Electric intensity along the value of Z axis. A represents the step structure (d¼ 12.5 mm) and B represents the isolate slit arrays structure. The line without label denotes the transmission without time gate while label 1 denotes gated transmission at frequency of 10 GHz and label 2 denotes gated transmission at 9 GHz.
In conclusion, we presented and demonstrated transmission enhancement and focusing in X-band by sandwich-like step structure with subwavelength slit arrays experimentally and numerically. The enhanced microwave transmission and efficient focusing are achieved due to the improved coupling efficiency between incident wave and waveguide mode, and the simulation results agree well with the experimental results. By putting time gate between the main pulse and diffractive pulses in time domain, the focusing effect disappeared and the electric intensity decayed with the value of Z axis. Through the measured electric field intensity with the value of Z axis, the coupling efficiency between incidence wave and waveguide mode was qualitatively
Table 1 Fitting results of the coupling efficiency and excitation degree. Fitted line
A1
A2
B1
B2
Frequency κ (Standard error) η (Standard error)
10 GHz 0.86870 (0.02307) 0.97998 (0.00813)
9 GHz 0.38069 (0.02315) 0.97255 (0.00801)
10 GHz 0.57722 (0.01750) 0.80143 (0.00618)
9 GHz 0.14542 (0.06260) 0.65752 (0.00204)
L. Xu et al. / Optics Communications 314 (2014) 31–35
calculated to illustrate resonant and nonresonant transmission enhancement. As the improved performance based on step structure and ease of sample fabrication at the microwave frequencies, the step structure provides the possibility of improving the performance of many other microwave devices including plasmonic lens, sensing and so on. References [1]
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Enhanced microwave transmission and focusing effect through step structure with subwavelength slit arrays Liang Xu a,n, Jinxiang Cao a, Junming Liu b, Yu Liu a, Jian Wang a, Zhe Zheng a, Yinchang Du a, Xiao Zhang a, Pi Wang a a b
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Department of Physics, Rutgers, The State University of New Jersey, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 8 January 2013 Received in revised form 3 April 2013 Accepted 2 May 2013 Available online 23 May 2013
We explore the transmission and focusing properties of a step structure with subwavelength slit arrays in metallic aluminum plates. We show that the microcavities formed between adjacent step can greatly enhance the coupling of the incident wave into the slit by improving the impedance matching between the incident wave and the waveguide mode, leading to X-band (8–12 GHz) transmission enhancement and efficient focusing. In addition, the electric field is localized and enhanced at the slit exits, in contrast with non-step slit gratings. By putting the time gate in the time domain of transmission spectra, we are able to accurately calculate the improved coupling efficiency to illustrate the enhancement at both of the resonant and nonresonant frequencies. The simulation by a finite element code agrees well with the experimental results. Potential applications of this effect can be anticipated in microwave devices. & 2013 Elsevier B.V. All rights reserved.
Keywords: Subwavelength structure Waveguides Spoof surface plasmon
1. Introduction Since the discovery of extraordinary transmission phenomenon [1], a variety of plasmonic devices [2–4], which are based on subwavelength plasmonic waveguide [5] and diffractive wave, have been proposed. Besides hole arrays [6,7], the subwavelength slit arrays have become one of the basic structure element to achieve the minimization of optical devices, for example, the plasmonic lens [2], and to take the place of the refractive lens since the focusing capability of the conventional lens deteriorate with the miniaturization of their size. But the superiority of the refractive lens, which makes them the most ubiquitous optical and microwave component, is that they are transparent to broadband frequencies while many plasmonic devices are only useful when the surface plasmons (or spoof surface plasmons at microwave frequency) is excited at the resonant frequencies. There have been, so far, several experimental works that investigate EOT by modulating the size [8], the materials [9] or by changing the geometries [10–13] of subwavelength MIM waveguide. As we know, surface plasmons (SPs) are surface-bound waves that only propagate along the surface of a conductor with finite conductivity in the visible domain, By contrast at the microwave regime, metals can be often assumed to be perfectly conducting, the surface wave supported by a planar metal-dielectric interface is unconfined in
n
Corresponding author. Tel.: +86 18256900861. E-mail address: [email protected] (L. Xu).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.05.011
the dielectric. However, it has been shown that structured PEC surfaces can support surface-bound waves [14–17], which means that the structured surface now behaves much more like a planar metallic one at optical frequencies with tight confinement of the electromagnetic fields. This is the concept of “spoof” or “designer” surface plasmons. However, the coupling between the incident plane wave and the waveguide mode is not significantly improved at both of the resonant frequencies and none-resonant frequencies, the performance of plasmonic devices are therefore limited. Recently, a kind of double resonance is analytically and experimentally demonstrated to help improve the coupling process through dielectric-metal-dielectric sandwich hole arrays and a 100% transmission is observed when the incident wave couples to standing waves in both dielectric layers [18,19]. Furthermore, using microcavity structure [20–23], enhanced transmission with increased efficiency was demonstrated due to the interaction between the surface plasmon microcavity standing waves and the slit mode, and nanofocusing phenomenon was achieved based on the constructive interference in the microcavity. However, these experiments and simulations were performed in the optical regime and near infrared regime. While microcavity structure based on slit arrays has not been investigated at microwave regime. In this paper, we develop a 1D step structure and achieve a nearly full X-band (8–12 GHz) transmission enhancement and efficient focusing. We demonstrate that the microcavities formed between adjacent step can greatly enhance the coupling of the incident wave into the slit by improving the impedance matching
32
L. Xu et al. / Optics Communications 314 (2014) 31–35
Fig. 2. Experimental and simulated transmission spectra of sample A with step width d ¼12.5 mm and sample B. Fig. 1. (a) Schematic of the two samples with w¼ 3 mm, P ¼25 mm, H¼ 20 mm, h ¼ 10 mm. The incident normal p-polarized plane wave transmits directly through a slit and couples to surface mode. (b) Setup of the measurement system, and the coordinate system.
between the incident wave and the slit mode. Thus, the performance of the plasmonic devices based upon isolate slit arrays is improved at microwaves. A basic diffractive wave mode is proposed and an accurate coupling efficiency is calculated in this paper.
2. Experimental setup and simulation method In our experiment, we manufacture a much-studied aluminum slit array structure (sample B) as a comparison which was optimized to reach its highest transmission through various design principles [24]. The step structure (sample A) consists of 1D (onedimension) subwavelength slit arrays structure (10 mm thick and 3 mm width slit) with periodic step arrays (5 mm thick) loading (see Fig. 1(a)). The distance between the centers of adjacent slit is 25 mm. The area of each sample is 287.5 300 mm which is large enough to minimize the effects of finite sample size. The transmission characteristic was measured by microwave vector network analyzer E8362B which supports two identical horn antennas at the frequencies of 8–12 GHz. The receiver antenna was mounted on the computer-control rail for accuracy 2D position measurements (Fig. 1(b)). Through our work, normal p-polarized plane microwave (H field parallel to the slits) was illuminated on the center slit and the transmission enhancement of incident wave was normalized by 1. All of the structures were measured under the same conditions and the results were repeatable. Finite element method (FEM) was used to calculate electromagnetic transmission characteristics of the different samples. The dispersion of aluminum was captured using a Drude model. Perfect electric conditions were used on all sides of the calculation domain at the microwave frequencies. Near-field and transmission monitors were used to visualize the electric field localization and enhancement.
3. Experimental results and discussion By measuring transmission spectra at the normal direction 40 cm away from the central slit, we observed the X-band transmission enhancement through the step structure with step width d ¼ 12.5 mm (Fig. 2). At the surface wave resonant frequency (10 GHz), the transmission efficiency reaches its highest intensity 1.25, and the transmission efficiency is greater than that of the
Fig. 3. Experimental results of the electric intensity for the planar lens based on the different structure, the above for the isolate slit structure and the below for the step structure (d ¼ 12.5 mm), with a p-polarized plane wave incident on the center slit at the frequency of 10 GHz. The vertical white line denotes the position of the focal point.
incident wave ranging from 8.7 GHz to 10.6 GHz. Particularly, at the nonresonant frequency 8.7 GHz, the transmission efficiency of the step structure is improved greatly from 30% to 100%, with contrast to sample B. The validity of the experiment was justified by our simulation results (Fig. 2). Then, we compared the focusing effect of the Sample A to the Sample B, When the plate was illuminated by a plane wave at the resonant frequency (10 GHz), the energy will focus and we can see that the step structure and the isolate slit arrays have the same focal length of about 40 cm, approximate to 10 times of the resonant wavelength (Fig. 3). The focused energy was improved by step structure while the focusing effect and focal length remain unchanged. Fig. 4 shows the measured and simulated transmission with different step width. The transmission for a non-step slit arrays in a 20 mm thick aluminum plate is also shown for direct quantitative comparison. It is noted that the sample of step width d¼ 22 mm corresponds the non-step sample with thickness h¼20 mm. For different parameters, the step structure leads to a dramatic enhancement of the transmission over the most X-band region, comparing the non-step structure case (Fig. 4, black solid curve). The resonant transmission enhancement around 10 GHz of the samples is attributed to the
L. Xu et al. / Optics Communications 314 (2014) 31–35
33
Fig. 4. (a) measured and (b) simulated transmission spectra through the subwavelength slit array structure with the different step width. The parameters are same as the sample A in Fig. 1(a), and the sample is illuminated with normal incident p-polarized light.
Fig. 5. Profile of simulated magnitude of the electric field of the structure (a) step structure with step width d ¼ 12.5 mm, (b) non-step structure, both being illuminated at the same frequency of 10 GHz.
excitation of spoof surface plasmon when the dispersion of the spoof spps and the periodicity of the diffraction grating satisfy the conservation of momentum [25] ksp ¼ k0 sinθ 7 nkg
ð1Þ
where kg ¼ 2π=λg is the Bragg reciprocal-lattice vector of the periodic structure, θ is the angle of incidence with wave number k0 in free space, and n is the coupling order that can only be integers. We notice that the step structure makes the nonresonant transmission enhancement, especially at the low frequency. However, the transmission enhancement does not increase monotonously as the microcavity width becomes larger, and the simulation in Fig. 4 (b) agrees well with the experimental results. Based on transmission line theory [26,27], The proposed step structure can be modeled as three cascaded transmission lines with the microcavities, formed between adjacent step, and slit characterized by local width and thickness. The characteristic impedance of the microcavity transmission lines at the entrance and exit are Z 1 ¼ ðβ1 =jωε0 Þd and Z 3 ¼ ðβ3 =jωε0 Þd, respectively, and the characteristic impedance of the slit transmission line is Z 2 ¼ ðβ2 =jωε0 Þw where βi ¼ β′i þ β″i i (i¼1,2,3) is the complex wave vector of the propagating TM mode in the MIM waveguide. The wavenumber βi by the dispersion relation of a metal-insulator-metal waveguide: [2] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 a β2i −k0 − β2i −εm k0 A¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ tanh@ 2 2 εm β2i −k0 where a represents the width of MIM waveguide (d in microcavity and w in slit) and εm ≈−104 þ 107 i p[28] ffiffiffiffiffiffiffiffiffiffiffiis ffi the permittivity of aluminum at this wavelength. Z 0 ¼ μ0 =ε0 p is the surrounding characteristic impedance in our experiment. Thus, the characteristic impedance matching between surrounding air and microcavity MIM waveguide can be improved as the microcavity width increases. In turn, the characteristic impedance matching between microcavity MIM waveguide and slit MIM waveguide will be decreased as the microcavity width increases. More specifically, In Fig. 4 the optimized
step width is d¼12.5 mm. when the microcavity width is below 12.5 mm, the characteristic impedance matching between the surrounding air and microcavity waveguide would decrease, leading to the reduce of the transmission efficiency. However, when the microcavity width is above 12.5 mm, the characteristic impedance matching between the microcavity waveguide and slit waveguide would decrease rapidly, therefore the transmission efficiency would decrease as well. That is, the transmission efficiency is limited by the trade-off between the incident power at the surrounding air-cavity waveguide junction and the transmission coefficient of the cavity waveguide- slit waveguide junction. The step structure of step width d¼12.5 mm achieve an optimized impedance matching of trade-off, leading to the largest transmission enhancement. Based on the large transmission enhancement by step structure, the near field intensity is expected to be enhanced at the exit slit. To confirm, we plot the electric field spatial distribution (Fig. 5) at 10 GHz for the non-step structure with h¼20 mm and step structure with width d ¼12.5 mm. For other gratings, the field spatial distribution profiles are very similar. We notice that indeed the steps provide strong enhancement and localization of electric field at the exit of the slits, comparing to the non-step structure. It is noted that the step structure can achieve non-resonant behavior, the transmission spectra would tend to be broadband. To further investigate the transmission process, it is necessary to identify the change of the waveguide after forming a step shape as the final transmission comes from two different processes [29]. By offering a single incident pulse, the light is separated into two channels: direct transmission or coupling to the surface wave (see Fig. 1(b)). Since the surface wave channel has a time delay, which is purely decided by the phase retardation across the distance between slits, to reach its final transmission. The pulses with different arrival time become differentiated in time domain showed in Fig. 6. In Fig. 6, the initial transmission pulse (left gray region) corresponds to the wave transmission directly through the slits with the help of waveguide and the subsequent pulses (right gray region)
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L. Xu et al. / Optics Communications 314 (2014) 31–35
correspond to indirect diffractive waves. The origin of the pulses can be theoretically analyzed by considering time-dependent component to the electric field in simplified couple-mode formalization [30]. Through inverse Fourier Transform, temporal dependence of the transmitted fields can be expressed by a series of impulse functions: E′0 ðtÞ ¼ ηE0 ðtÞδðt−t 0 Þ þ γ 2 η ∑ E0 ðtÞδðt−ðt 0 þ 2n ΔtÞÞ
ð3Þ
n≠0
where t 0 represents the waveguide delay and Δt is the time for a surface wave passing across one slit. From Eq. (2), the imaginary part of the complex propagating wave vector in the MIM waveguide β″≈3:3 10−4 k0 which is negligible, therefore the propagating loss in the slit is neglected. Based on this property, we assume η and γ represent the coupling efficiency of waveguide mode and surface
wave mode with the incident wave respectively. Obviously, the transmission efficiency of the main pulse in time domain shows a higher coupling efficiency than that of the isolate slit arrays. In order to distinguish the role of the diffractive wave in focusing and certify the accurate coupling efficiency, we put a time gate between the main pulse and the diffractive pulses in time domain and refuse any signals later than gated time, therefore all of the diffractive waves derived from other slits are not received. Without the interference of the diffractive waves, the focusing effect disappears and the electric field decays with the value of Z axis (Fig. 7). It is interesting to see that in the near field, the electric intensity is still slightly greater than the incident wave, which is attributed to the interference between the plane wave transmitting directly through the center slit and the diffractive wave derived from the exit surface of the center slit. Due to the oscillating dipole formed near the central slit corners [31] and the electric field intensity of dipole mode in the far field [32], combining with the coupling efficiency, the measured electric field intensity in the far field can be expressed as Eq. (4) in our experiment: Etotal ðzÞ ¼ κ=z þ η
Fig. 6. Transmission pulses in time domain. The black solid line denotes the exciting pulse. The red dashed line and the blue dotted line denotes the transmission of step structure A (d ¼ 12.5 mm) and isolate slit structure B, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ð4Þ
where κ is an undetermined coefficient representing the excitation degree of dipole mode and η represents the coupling efficiency of the waveguide mode. After fitting the field distribution along the value of Z axis, A1 (step structure at 10 GHz), A2 (step structure at 9 GHz), B1 (isolate slit arrays 10 GHz) and B2 (isolate slit arrays 9 GHz) of Fig. 7 to Eq. (4), we can obtain the degree of coupling and excitation of different structures at resonant and nonresonant frequency (see Table 1). For the same structure, the intensity of the diffractive wave is much stronger at the resonance frequency (10 GHz) than that at the frequency of 9 GHz. The coupling efficiency of the step structure is enhanced to 97% at the frequencies of resonance and none-resonance, compared to the isolate slit structure with a coupling efficiency of 80% at 10 GHz and 65% at 9 GHz. Thus, the transmission enhancement is mainly attributed to the improved coupling process. Although the resonance in high frequency might be much complex, the slit mode and oscillating dipole mode can give basic information about the coupling efficiency and the excitation degree of the oscillating charges since the standard error of fitting is small.
4. Conclusion
Fig. 7. Electric intensity along the value of Z axis. A represents the step structure (d¼ 12.5 mm) and B represents the isolate slit arrays structure. The line without label denotes the transmission without time gate while label 1 denotes gated transmission at frequency of 10 GHz and label 2 denotes gated transmission at 9 GHz.
In conclusion, we presented and demonstrated transmission enhancement and focusing in X-band by sandwich-like step structure with subwavelength slit arrays experimentally and numerically. The enhanced microwave transmission and efficient focusing are achieved due to the improved coupling efficiency between incident wave and waveguide mode, and the simulation results agree well with the experimental results. By putting time gate between the main pulse and diffractive pulses in time domain, the focusing effect disappeared and the electric intensity decayed with the value of Z axis. Through the measured electric field intensity with the value of Z axis, the coupling efficiency between incidence wave and waveguide mode was qualitatively
Table 1 Fitting results of the coupling efficiency and excitation degree. Fitted line
A1
A2
B1
B2
Frequency κ (Standard error) η (Standard error)
10 GHz 0.86870 (0.02307) 0.97998 (0.00813)
9 GHz 0.38069 (0.02315) 0.97255 (0.00801)
10 GHz 0.57722 (0.01750) 0.80143 (0.00618)
9 GHz 0.14542 (0.06260) 0.65752 (0.00204)
L. Xu et al. / Optics Communications 314 (2014) 31–35
calculated to illustrate resonant and nonresonant transmission enhancement. As the improved performance based on step structure and ease of sample fabrication at the microwave frequencies, the step structure provides the possibility of improving the performance of many other microwave devices including plasmonic lens, sensing and so on. References [1]
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