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Sci Rep. 2017; 7: 46508.

Published online 2017 April 18. doi: 10.1038/srep46508

PMCID: PMC5394464

Ki Young Lee,^{1,}^{*} Jae Woong Yoon,^{a,}^{1,}^{*} Seok Ho Song,^{b,}^{1} and Robert Magnusson^{2}

Received 2017 January 13; Accepted 2017 March 15.

Copyright © 2017, The Author(s)

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

We propose a free-space electro-optic transmission modulator based on multiple *p-n*-junction semiconductor subwavelength gratings. The proposed device operates with a high-Q guided-mode resonance undergoing electro-optic resonance shift due to direct electrical control. Using rigorous electrical and optical modeling methods, we theoretically demonstrate a modulation depth of 84%, on-state efficiency 85%, and on-off extinction ratio of 19dB at 1,550nm wavelength under electrical control signals within a favorably low bias voltage range from −4V to +1V. This functionality operates in the transmission mode and sustainable in the high-speed operation regime up to a 10-GHz-scale modulation bandwidth in principle. The theoretical performance prediction is remarkably advantageous over plasmonic tunable metasurfaces in the power-efficiency and absolute modulation-depth aspects. Therefore, further experimental study is of great interest for creating practical-level metasurface components in various application areas.

Leaky-mode resonances in nanopatterned thin-film structures are of interest owing to their great potential for creating integration-compatible, multifunctional devices harnessing desired spectral, polarization, intensity, and phase properties^{1}^{,2}^{,3}. Guided-mode-resonance elements^{1}, high-contrast gratings^{2}, and plasmonic metasurfaces^{3} have been extensively studied within this context. Adding active tunability to these device classes for applications in practice, various approaches have been suggested using thermo-optic effects^{4}, micro-electro-mechanical system architectures^{5}, and liquid-crystal-based index-tuning methods^{6}. Further expanding the application areas and innovating classical device counterparts, a major line of research is presently pursuing higher tuning speed, smaller device footprint area, and better long-term stability in order to secure essential requirements in potential application areas including ultra-broadband optical signal processing, high-power laser machining, and compact LIDAR systems.

To this end, free-carrier-induced electro-optic (EO) effects in heavily doped semiconductors and transparent conducting oxides at epsilon-near-zero (ENZ) conditions have been extensively studied as an efficient tuning mechanism. In particular, ENZ nanofilms incorporated in metal-oxide-semiconductor (MOS) capacitor arrays have showed remarkable intensity and phase modulation properties driven by field-effect free-carrier accumulation and depletion^{7}^{,8}^{,9}^{,10}. In this approach, highly dissipative, deep-subwavelength plasmonic resonances are necessary to induce significant optical interaction with sub-10-nm-thick EO-active layers. Consequently, strong absorption and low resonance Q factor result in performance restrictions such as shallow signal modulation depth, low absolute efficiency, and poor high-power durability. In addition, it is presently unclear whether or not the plasmonic MOS capacitor approach using highly reflective metallic components as an indispensable constituent material can operate in the transmission mode which is desirable for variety of applications.

Pursuing high-performance tunable leaky-mode resonance devices operating in the transmission mode in this paper, we propose an approach based on high-Q guided-mode resonances (GMRs) in low-loss semiconductor nanogratings. The proposed device class consists of moderately doped, low-loss semiconductor *p-n* junctions in a resonant subwavelength grating structure as shown in Fig. 1(a–d) where basic operation scheme, device structure, and electrical connections, and a possible architecture for integrated modulator arrays are illustrated, respectively. This structure is designed such that a high-Q GMR is supported in the optical domain while in the electrical domain bias voltage across the multiple *p-n* junctions effectively control density of mobile electrons and holes, resulting in the associated tuning of the Drude-type optical dielectric constant^{11}. The major advantage of this configuration over the plasmonic tunable metasurfaces is to sustain a low-loss, high-Q resonance feature that experiences a remarkable resonance center shift under a favorable bias-voltage region. Hence, desired properties such as high efficiency and high on/off extinction ratio can be supported in the transmission mode operation. While the proposed operation principle is applicable for various group IV and III-V-compound semiconductor materials, here we select an interleaved Si *p-n* junction nanograting architecture as one promising example. Using well-established electrical and optical modeling methods, we theoretically demonstrate a robust transmission modulator with on/off power ratio of 18.9dB, on-state efficiency of 85.2%, and modulation bandwidth of 54.3GHz at an operation wavelength of 1550nm. These performance characteristics are driven by favorably small bias voltage values in a range of −4V ~ +1V and possibly maintained in the high-speed operation regime up to 50GHz when appropriate in-plane miniaturization schemes are incorporated.

The EO effect of our interest in this paper is based on change in mobile electron and hole densities that we denote by *N*_{e} and *N*_{h}, respectively, in response to an applied bias voltage *V*_{a} across the *p-n* junctions. Dielectric constant *ε*_{Si} of Si in the optical domain is determined by a Drude formula

where *ω*_{e,h} and Γ_{e,h} denote plasma and collision frequencies, respectively, for electron (*e*) and hole (*h*) plasmas, and *ε*_{∞}=11.7. Following the canonical Drude model, *ω*_{e,h}^{2}(*N*_{e,h})=*N*_{e,h}*e*^{2}(*ε*_{0}*m*^{*}_{e,h})^{−1} with the elementary electric charge *e*, vacuum permittivity *ε*_{0}, and effective mass *m*^{*}_{e,h}. In moderately doped Si with donor and acceptor doping concentration of *N*_{0}=10^{18}cm^{−3} (=*N*_{e}=*N*_{h}), for example, values for these parameters are *m*^{*}_{e}=0.27*m*_{e}, *m*^{*}_{h}=0.39*m*_{e}, *ω*_{e}=0.449eV/*ħ*, Γ_{e}=4.95×10^{−2}eV/*ħ, ω*_{h}=0.374eV/*ħ*, and Γ_{h}=5.36×10^{−2}eV/*ħ*^{12}. The two frequency-dependent parts in the Drude-part dielectric constant *ε*_{Drude} describe optical response of the mobile electron and hole plasmas, respectively. Under applied bias voltage *V*_{a} across the *p-n* junctions in the configuration in Fig. 1(b,c), mobile electrons and holes are redistributed to modify *N*_{e} and *N*_{h}, and hence the *ω*_{e} and *ω*_{h} values until the internal electric field energy becomes minimal. Therefore, the desired bias-voltage-dependent dielectric constant is obtained in this interaction process. Previously, a similar effect provides a robust tuning mechanism for high-speed, low power consuming in-line Si-photonic modulators with device footprint length scales in the order of 1mm to 100μm^{13}^{,14}. In our case, the effect is used to create an electrically tunable low-loss GMR element taking advantage of the resonant light confinement in a subwavelength-thick zero-order grating layer.

Rigorously treating this free-carrier-induced EO effect, the carrier-density *N*_{e} and *N*_{h} distributions in thermal equilibrium follow the Poisson-Boltzmann distribution^{15}. We use a 2-dimensional finite-element-method model^{16} of the Poisson-Boltzmann equation to calculate bias-voltage-dependent *N*_{e}(*y, z*) and *N*_{h}(*y, z*). We assume *d*_{1}=55nm, *d*_{2}=2*d*_{1}=110nm, and identical donor (*N*_{n}) and acceptor (*N*_{p}) concentrations such that *N*_{p}=*N*_{n}=*N*_{0}=10^{18}cm^{−3} in this calculation. In this multiple-junction structure, we include five pairs of alternating 110-nm-thick *p-n* junction cells in the order of *np-pn-np-pn-np* from bottom. The thickness of 110nm for a single *p-n* junction unit cell is chosen such that the depletion layer at a bias voltage *V*_{a}=−4V well below the breakdown voltage of −5.54V for the given doping concentration completely covers the entire device volume and consequently the mobile-carrier-induced EO effect occurs over the entire device. The peculiar feature of *d*_{2}=2*d*_{1} is a natural consequence of this layer-design rule to maximize the EO effect with a favorably minimal material embodiment in the proposed device concept.

Results for three bias-voltage values of *V*_{a}=−4V (reverse bias), 0V (neutral), and +1V (forward bias) are shown in Fig. 2(a–c), respectively. Therein, we indicate distribution of an effective compound-carrier density *N*^{*}=(*m*^{*}_{e}*m*^{*}_{h})^{−1/2}(*m*^{*}_{h}*N*_{e}+*m*^{*}_{e}*N*_{h}) that we define as an intuitive, single density parameter directly relevant to the free-carrier EO effect. This definition is followed by a simpler expression for *ε*_{Drude} as

where *m*^{*}=(*m*^{*}_{e}*m*^{*}_{h})^{1/2} and Γ′=(*m*^{*}_{h}*N*_{e}Γ_{e}+*m*^{*}_{e}*N*_{h}Γ_{h})(*m*^{*}_{h}*N*_{e}+*m*^{*}_{e}*N*_{h})^{−1} in the weak collision regime where Γ_{e,h}*ω* and Γ_{e,h}*ω*_{e,h}. Obviously, *ε*_{Drude} is linearly proportional to *N*^{*}. The *V*_{a}-dependent *N*^{*} distributions in Fig. 2(a–c) show that *N*^{*} is effectively modified from 0 to 2×10^{18}cm^{−3} over the whole structure under *V*_{a} adjustment in a range from −4V to 1V. In addition, *N*^{*} is almost independent of in-plane position *y*, confirming that the electrical signal-injection configuration envisioned in Fig. 1(c) should be feasible for efficiently controlling *N*^{*} over the whole isolated device region. Subsequent change in *ε*_{Drude} at a wavelength of 1,550nm is in a range from 0 to −1.3×10^{−2} for Re(*ε*_{Drude}) and from 0 to 0.86×10^{−4} for Im(*ε*_{Drude}) as shown in Fig. 2(d–f). Therefore, the proposed configuration provides a 10^{−2}-order EO change in Re(*ε*_{Drude}) over a 550-nm-thick Si film under a favorably low bias-voltage tuning range from −4V to +1V while keeping acceptably low material absorption levels of Im(*ε*_{Si})<10^{−3}.

We apply the obtained bias-voltage-induced mobile-carrier effect to an example GMR element optimized for the transmission-mode optical modulation in the telecommunications C-band around 1,550nm. Figure 3(a) shows the *V*_{a}–dependent transmission spectra in dB (a linear scale in the inset) under transverse-electric (TE) polarized planewave incidence at surface-normal angle (*θ*=0). The optimized grating-design parameter values are given in the caption. Following the standard convention, the TE polarization refers to electric field oscillating in the axis of grating lines (*y*-axis). We use the finite-element method^{16} in this calculation involving *ε*_{Si}(*z*) profiles obtained by the method described in the previous section. The transmission spectra show an asymmetric Fano-resonance profile as a result of the configuration interference between resonant and non-resonant pathways^{17}. The resonant pathway is created by coupling of the incident wave with a leaky TE_{0} mode and its radiation decay toward the transmitted zero-order planewave channel through dominant first-order diffraction processes. This resonance feature possesses remarkably high resonance Q factor ~3.69×10^{3}. Consequently, the design yields a very high field enhancement factor ~1.2×10^{3} in the 550-nm-thick EO-active Si-*p-n*-junction layers as confirmed in Fig. 3(b) showing an electric-field intensity distribution at the resonance center wavelength.

Subtle interaction between the highly enhanced resonant optical fields and bias-voltage-induced mobile-carrier effect results in a resonance-center (λ_{c}) shift Δλ_{c} as shown in Fig. 3(a). Basically, the observed resonance shift is directly resulting from the *V*_{a}-dependent change in *ε*_{Si}. Although there is no exact closed-form expression for the dielectric-constant-dependent resonance shift known in general, a plausible estimation can be found by taking the Wentzel-Kramers-Brillouin (WKB) approximation on the resonance condition. In our case, the change in *ε*_{Si} leads to the change in the optical path length inside the Si grating bars while there is no optical-path-length difference outside. For a small dielectric-constant change, i.e., Δ*ε*_{Si} 1, that further implies no significant modification in the field distribution of the leaky resonance mode, the WKB approximation for the eigenvalue determination^{18} dictates that the optical phase accumulation inside the Si bars should remain constant under small change in *ε*_{Si}. This condition directly yields a relation (*n*_{Si}+Δ*n*_{Si})^{−1}(λ_{c}+Δλ_{c})=*n*_{Si}^{−1}λ_{c}, where *n*_{Si}=*ε*_{Si}^{1/2} and consequently Δ*n*_{Si}=(2*n*_{Si})^{−1}Re(Δ*ε*_{Si})=(2*n*_{Si})^{−1} Re(Δ*ε*_{Drude}) as the dielectric constant change is solely in the Drude part. Including Eq. (2) and the standard phase-matching condition Λ^{−1}=*n*_{eff}λ_{c}^{−1} for a GMR at normal incidence, where *n*_{eff} is effective index of the leaky guided mode, the constant optical-phase-accumulation condition is rewritten by

where *g*(*ω*)=*e*^{2}[*ε*_{0}*m*^{*}(*ω*^{2}+Γ′^{2})]^{−1}. Obviously, increase in *N*^{*} (Δ*N*^{*}>0) with *V*_{a} leads to a corresponding linear blue shift of the resonance feature or vice versa.

From Fig. 3(a), we find that the resonance-center shift in response to the applied bias voltage *V*_{a} has two different regimes. In Fig. 3(c), we show Δλ_{c}(*V*_{a}) that reveals slow blue shift with increasing *V*_{a} in the low-voltage region of *V*_{a}<1V and abrupt increase in the differential resonance shift Δλ_{c}/Δ*V*_{a} in the high-voltage region of *V*_{a}>1V. According to Eq. (3), this peculiar property is associated with the dependence of *N*^{*} on *V*_{a}. *V*_{a}–dependent volume average *N*^{*}(*V*_{a}) of the effective compound-carrier density is plotted in Fig. 3(d) and it is in exact correlation with −Δλ_{c}(*V*_{a}) in Fig. 3(c). Explaining the dependence of *N*^{*} on *V*_{a} in Fig. 3(d), a key factor is built-in potential *V*_{built-in} across the *p-n* junction. Applying the Poisson-Boltzmann equation for *V*_{a}=0 in our case with *N*_{p}=*N*_{n}=*N*_{0}=10^{18}cm^{−3}, we obtain *V*_{built-in}=0.934V. In the low bias-voltage region of *V*_{a}<*V*_{built-in}, increase in *N*^{*} with *V*_{a} is led by the decrease in the depletion layer thickness without significant growth in the carrier-density level. In contrast, in the high bias-voltage region of *V*_{a}>*V*_{built-in}, the depletion region is closed and the excessive electrons and holes injected from the electrodes lead to mobile-carrier density level growth in the whole Si regions to result in a more rapid increase in *N*^{*} with *V*_{a}. In Fig. 3(d), we provide the *V*_{a}-dependent depletion layer thickness and illustrations showing the two regimes of carrier distribution statics.

Importantly, the obtained resonance-shift tunability in Fig. 3(c) implies a full λ_{c} tuning range over 2.3nm that is remarkably larger than the resonance bandwidth of 0.43nm by a factor 5.3. Therefore, we can fully utilize the spectral maximum and minimum as the on-state and off-state transmittance levels, respectively. For our particular design at an optimal wavelength of λ_{0}=1550.02nm which corresponds to the transmittance minimum at *V*_{a}≈*V*_{built-in}, sharp transmission modulation is obtained as shown in Fig. 4(a). In this case, the intensity modulation is induced mainly in the closed depletion-layer regime and thereby small bias-voltage tuning induces rapid intensity change as confirmed again in the total electric field patterns for *V*_{a}=0.9V and −4.0V in Fig. 4(b,c). Key performance parameters in this case are transmittance modulation depth of 83.9%, on-state efficiency of 85.2%, and on-off extinction ratio of 18.9dB under remarkably low control bias-voltage signals within the −4 ~ +0.9V range. Importantly, the obtained performance parameters are highly desirable for variety of applications when compared with LiNbO_{3}-crystal-based EO modulators requiring 100V-scale control signals for similar performances.

For a different operation wavelength which corresponds to the transmittance minimum at *V*_{a}<*V*_{built-in}, we have much slower intensity tuning as the device operates in the open depletion-layer regime. For example, we select λ_{0}=1550.29nm and the corresponding *V*_{a}-dependent transmittance is indicated by red dashed curve in Fig. 4(a). Such slow intensity modulation is desirable for continuous modulation or precise control of light intensity for analog signal processing systems while the rapid, closed depletion-layer regime is more appropriate for digital signal processing applications.

The proposed device is basically a 1D-periodic GMR element and thereby has a characteristic angular dispersion. In our case, a primary angular dispersion of the resonance location appears for the angle of incidence with respect to *x* axis on which the discrete light diffraction processes take place. As the resonance location follows the dispersion curve of the leaky guided mode, the angular shift of the resonance wavelength as a function of polar angle *θ* of incidence can be found from the definition of group velocity, i.e., *V*_{x}=*ω*(**k**)/*k*_{x}, where *V*_{x}, *ω*(**k**), and *k*_{x} denote group velocity in *x* axis, dispersion frequency surface, and *x*-component wavevector of the leaky guided mode, respectively. A simple calculus with basic relations of *ω*(**k**)=2π*c*λ_{c}^{−1}, *k*_{x}=2π*c*λ_{c}^{−1}sin*θ*, and the diffractive phase matching condition λ_{c}^{−1}sin*θ*=*n*_{eff}λ_{c}^{−1}−Λ^{−1} results in

where *n*_{G}=*c*/*V*_{x} is group index of the leaky guided mode. For near-normal incidence (*θ*1), Eq. (4) reduces to λ_{c}/*θ* ≈−*n*_{G}^{−1}*n*_{eff}Λ. This property explicitly appears in the angle-dependent transmission spectrum as shown in Fig. 5(a) where the angle dependent λ_{c} loci are identified as being along the dark transmission dip. Therein, λ_{c}/*θ*=0 at *θ*=0 as the group index *n*_{G} diverges to infinity for the laterally standing guided-mode with *V*_{x}=0 at normal incidence. For off-normal incidence (*θ* ≠ 0) for which the two counter-propagating guided modes are not coincidental anymore and the resonance is driven by a single leaky-guided mode. Consequently, the group-to-effective-index ratio *n*_{G}^{−1}*n*_{eff} tends to a constant value and so is λ_{c}/*θ*. Estimating from Fig. 5(a), the off-normal angle-tunability λ_{c}/*θ*≈−2nm/deg. at *θ*=5°. Importantly, the angle-dependent shift of λ_{c} does not significantly affect the resonance profile and the EO-tunable resonance shift as shown in Fig. 5(b). In particular, the EO tunability of 0.17nm/V persists for all three cases and the transmission modulation depth values are 85%, 83%, and 77% for *θ*=1°, 3°, and 5°, respectively.

The angular dispersion of λ_{c} and persistent EO tuning properties suggest important information for operation of the proposed device class in practice. First, highly collimated light beam should be used to fully utilize the proposed device functionality. Angular full-width-at-half-maximum bandwidth for the GMR in our case is estimated from Fig. 5(a) as 2.6° at *θ*=0 and 0.18° at *θ*=5°. Divergence angle of the incident light beam should be well within these values. In another consideration, the angle tunability of λ_{c} provides an efficient way to precisely shoot the desired operation wavelength. In practice, fabrication errors and imperfections present. Although exact tolerance values depend on fabrication steps and specific tools selected for device production, one may accept a-few-nm scale errors in the spatial device parameters and grating period and a-few-Å scale errors in the layer thickness values. A combination of these errors in period, grating linewidth, and layer thicknesses might result in an imperfect λ_{c} off from the desired value by an amount even in a 10-nm scale. Considering 0.1nm (distributed feedback type) ~10nm (Fabry-Pérot type) for typical diode laser line width around 1,550nm, the anticipated fabrication errors can be critically problematic if there is no tuning method available for matching λ_{c} to the source-laser wavelength. Suppose that we control *θ* within a ±10° range and with an accuracy of 10^{−3} deg. The angle tunability of 2nm/deg. implies a full tuning range of ~40nm with an accuracy of ~2pm. We note that typical angle precision of commercially available rotary or tilting stages for optomechanical controls is in 10^{−4} deg. scales.

Further considering applicability of the proposed device concept in practice, modulation bandwidth is an important measure. The GMR bandwidth and RC time constant are two major factors in this consideration. For the analyzed example device in Figs 3 and and4,4, the estimated GMR bandwidth is Δ*f*_{opt}~54.3GHz. Therefore, stable 10-GHz-scale optical signals can be generated in a purely optical property aspect. However, the final modulation bandwidth should be also restricted by electrical response characteristics, i.e., a bias-voltage signal modulation bandwidth Δ*f*_{bias}=*τ*_{RC}^{−1}, where *τ*_{RC} denotes the RC time constant determined by the junction capacitance and termination impedance. Assuming the standard radio-frequency termination impedance of 50Ω and the design configuration used in Figs 2 and and4,4, estimated Δ*f*_{bias} values are 14.7MHz for a device footprint area of 1×1mm^{2}. Since Δ*f*_{bias} is inversely proportional to device footprint area, GHz-scale modulation bandwidth should be feasible for small devices with the footprint area <260×260μm^{2}. We note that Inoue *et al*.^{19} recently demonstrated a high-Q GMR filter with a device footprint area reduced down to 10×10μm^{2} without significant degradation in the spectral performance characteristics by using graded-parametric design approach combined with integrated first-order Bragg reflection boundaries. Introducing such miniaturization strategy to the proposed concept, Δ*f*_{bias}>Δ*f*_{opt} and the full resonance bandwidth should be available for the final optical signal-modulation bandwidth.

In summary, we proposed a multiple-*p-n*-junction subwavelength grating structure that enables high-performance EO modulation in the transmission mode. The proposed device operates under high-Q GMRs interacting with electric signals through the Drude-type optical free-carrier effect. Using rigorous electrical and optical modeling methods, we theoretically demonstrated highly efficient transmission modulation generated by remarkably low-voltage control signals with modulation speed in possibly 10GHz scales. Notably, the obtained properties are supported by the low-loss free-carrier-induced EO effect occurring in the whole device region with 500-nm-thick Si layers as opposed to the transparent-conducting-oxide-based plasmonic metasurface approaches involving a sub-10-nm-thick EO-active region and strong ohmic absorption. In another similar approach, a low-loss GMR modulator was suggested using a combination of the Burstein-Moss effect, Pockels effect, and Fraz-Keldysh effect in a weakly-modulated InGaAsP waveguide grating structure^{20}^{,21}. Therein, a robust reflection modulation with an extinction ratio of 17dB and modulation bandwidth of 5MHz was experimentally demonstrated.

Experimental realization of the proposed device concept is definitely the next step. In potential fabrication, crucial parts are to establish 100-nm-thick multiple *p-n* junction cells and subwavelength grating structure with a critical dimension in a few 100nm scale. First, the multiple *p-n* junction cells can be generated by the standard deposition processes based on chemical vapor deposition and sputtering techniques. We note that the present state-of-the-art deposition methods easily produce such multiple-junction semiconductor layer structures as established well in tandem solar cells and in vertical-cavity surface-emitting lasers^{22}^{,23}^{,24}. Second, the subwavelength periodic structure for a high-Q resonance excitation is also well-established using standard nanolithography techniques including the laser-interference lithography^{4} and electron-beam lithography^{24}.

Considering further study, applicability and limitations of the proposed concept over other spectral domains are of key importance. Although direct application of the concept to the visible domain is unclear because of highly lossy nature of the group IV or III-V-compound semiconductors, longer-wavelength applications in the mid-infrared (mid-IR) and THz domains are intriguing in several aspects. First, we notice from Eq. (2) that the EO modulation of *ε*_{Drude} scales with λ^{2}. This implies that the modulation amplitude Δ*ε*_{Drude} that is in the order of 10^{−2} around λ=1.5μm under Δ*V*_{a}=5V is amplified up to the unity order around λ=15μm in the mid-IR domain and even further up to the order of 10^{2} around λ=150μm in the THz domain. Therefore, the proposed device class can take advantage of the far stronger EO effect in the mid-IR and THz domains. In addition, fabrication errors and imperfections with respect to the operation wavelength are substantially lower and consequently precise device fabrication is much more feasible in the longer wavelength domains. Therefore, further in-depth study on the available materials, parametric optimization, and experimental realization in the telecommunications IR and longer wavelength domains is of great interest to develop compact, low driving power, and high-speed modulators for applications in telecommunications, optical information processing, LIDARs, laser machining, and many others.

**How to cite this article:** Lee, K. Y. *et al*. Multiple *p-n* junction subwavelength gratings for transmission-mode electro-optic modulators. *Sci. Rep.*
**7**, 46508; doi: 10.1038/srep46508 (2017).

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This research was supported in part by the Basic Science Research Program (NRF-2015R1A2A2A01007553) and by the Global Frontier Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT & Future Planning (NRF-2014M3A6B3063708). We thank Hyun Jae Lee from University of Seoul for helpful discussions on electrical properties of semiconductor thin films.

The authors declare no competing financial interests.

**Author Contributions** The original concept leading to this result was conceived by J.W.Y., K.Y.L., and S.H.S. K.Y.L. performed the theoretical analyses under supervision by J.W.Y. and S.H.S. All authors discussed the results. J.W.Y., K.Y.L., S.H.S., and R.M. wrote the manuscript.

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