Abstract
We propose a freespace electrooptic transmission modulator based on multiple pnjunction semiconductor subwavelength gratings. The proposed device operates with a highQ guidedmode resonance undergoing electrooptic resonance shift due to direct electrical control. Using rigorous electrical and optical modeling methods, we theoretically demonstrate a modulation depth of 84%, onstate efficiency 85%, and onoff extinction ratio of 19 dB at 1,550 nm wavelength under electrical control signals within a favorably low bias voltage range from −4 V to +1 V. This functionality operates in the transmission mode and sustainable in the highspeed operation regime up to a 10GHzscale modulation bandwidth in principle. The theoretical performance prediction is remarkably advantageous over plasmonic tunable metasurfaces in the powerefficiency and absolute modulationdepth aspects. Therefore, further experimental study is of great interest for creating practicallevel metasurface components in various application areas.
Introduction
Leakymode resonances in nanopatterned thinfilm structures are of interest owing to their great potential for creating integrationcompatible, multifunctional devices harnessing desired spectral, polarization, intensity, and phase properties^{1,2,3}. Guidedmoderesonance elements^{1}, highcontrast 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 thermooptic effects^{4}, microelectromechanical system architectures^{5}, and liquidcrystalbased indextuning 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 longterm stability in order to secure essential requirements in potential application areas including ultrabroadband optical signal processing, highpower laser machining, and compact LIDAR systems.
To this end, freecarrierinduced electrooptic (EO) effects in heavily doped semiconductors and transparent conducting oxides at epsilonnearzero (ENZ) conditions have been extensively studied as an efficient tuning mechanism. In particular, ENZ nanofilms incorporated in metaloxidesemiconductor (MOS) capacitor arrays have showed remarkable intensity and phase modulation properties driven by fieldeffect freecarrier accumulation and depletion^{7,8,9,10}. In this approach, highly dissipative, deepsubwavelength plasmonic resonances are necessary to induce significant optical interaction with sub10nmthick EOactive layers. Consequently, strong absorption and low resonance Q factor result in performance restrictions such as shallow signal modulation depth, low absolute efficiency, and poor highpower 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 highperformance tunable leakymode resonance devices operating in the transmission mode in this paper, we propose an approach based on highQ guidedmode resonances (GMRs) in lowloss semiconductor nanogratings. The proposed device class consists of moderately doped, lowloss semiconductor pn 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 highQ GMR is supported in the optical domain while in the electrical domain bias voltage across the multiple pn junctions effectively control density of mobile electrons and holes, resulting in the associated tuning of the Drudetype optical dielectric constant^{11}. The major advantage of this configuration over the plasmonic tunable metasurfaces is to sustain a lowloss, highQ resonance feature that experiences a remarkable resonance center shift under a favorable biasvoltage 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 IIIVcompound semiconductor materials, here we select an interleaved Si pn junction nanograting architecture as one promising example. Using wellestablished electrical and optical modeling methods, we theoretically demonstrate a robust transmission modulator with on/off power ratio of 18.9 dB, onstate efficiency of 85.2%, and modulation bandwidth of 54.3 GHz at an operation wavelength of 1550 nm. These performance characteristics are driven by favorably small bias voltage values in a range of −4 V ~ +1 V and possibly maintained in the highspeed operation regime up to 50 GHz when appropriate inplane miniaturization schemes are incorporated.
Results
Freecarrierinduced EO effect in multiple pn junction layers
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 pn 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} = 0.27m_{e}, m^{*}_{h} = 0.39m_{e}, ω_{e} = 0.449 eV/ħ, Γ_{e} = 4.95 × 10^{−2} eV/ħ, ω_{h} = 0.374 eV/ħ, and Γ_{h} = 5.36 × 10^{−2} eV/ħ^{12}. The two frequencydependent parts in the Drudepart dielectric constant ε_{Drude} describe optical response of the mobile electron and hole plasmas, respectively. Under applied bias voltage V_{a} across the pn 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 biasvoltagedependent dielectric constant is obtained in this interaction process. Previously, a similar effect provides a robust tuning mechanism for highspeed, low power consuming inline Siphotonic modulators with device footprint length scales in the order of 1 mm to 100 μm^{13,14}. In our case, the effect is used to create an electrically tunable lowloss GMR element taking advantage of the resonant light confinement in a subwavelengththick zeroorder grating layer.
Rigorously treating this freecarrierinduced EO effect, the carrierdensity N_{e} and N_{h} distributions in thermal equilibrium follow the PoissonBoltzmann distribution^{15}. We use a 2dimensional finiteelementmethod model^{16} of the PoissonBoltzmann equation to calculate biasvoltagedependent N_{e}(y, z) and N_{h}(y, z). We assume d_{1} = 55 nm, d_{2} = 2d_{1} = 110 nm, 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 multiplejunction structure, we include five pairs of alternating 110nmthick pn junction cells in the order of nppnnppnnp from bottom. The thickness of 110 nm for a single pn junction unit cell is chosen such that the depletion layer at a bias voltage V_{a} = −4 V well below the breakdown voltage of −5.54 V for the given doping concentration completely covers the entire device volume and consequently the mobilecarrierinduced EO effect occurs over the entire device. The peculiar feature of d_{2} = 2d_{1} is a natural consequence of this layerdesign rule to maximize the EO effect with a favorably minimal material embodiment in the proposed device concept.
Results for three biasvoltage values of V_{a} = −4 V (reverse bias), 0 V (neutral), and +1 V (forward bias) are shown in Fig. 2(a–c), respectively. Therein, we indicate distribution of an effective compoundcarrier 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 freecarrier 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 −4 V to 1 V. In addition, N^{*} is almost independent of inplane position y, confirming that the electrical signalinjection 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 1,550 nm 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 550nmthick Si film under a favorably low biasvoltage tuning range from −4 V to +1 V while keeping acceptably low material absorption levels of Im(ε_{Si}) < 10^{−3}.
EOtunable GMR excitation
We apply the obtained biasvoltageinduced mobilecarrier effect to an example GMR element optimized for the transmissionmode optical modulation in the telecommunications Cband around 1,550 nm. Figure 3(a) shows the V_{a}–dependent transmission spectra in dB (a linear scale in the inset) under transverseelectric (TE) polarized planewave incidence at surfacenormal angle (θ = 0). The optimized gratingdesign 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 (yaxis). We use the finiteelement 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 Fanoresonance profile as a result of the configuration interference between resonant and nonresonant 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 zeroorder planewave channel through dominant firstorder 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 550nmthick EOactive Sipnjunction layers as confirmed in Fig. 3(b) showing an electricfield intensity distribution at the resonance center wavelength.
Subtle interaction between the highly enhanced resonant optical fields and biasvoltageinduced mobilecarrier effect results in a resonancecenter (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 closedform expression for the dielectricconstantdependent resonance shift known in general, a plausible estimation can be found by taking the WentzelKramersBrillouin (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 opticalpathlength difference outside. For a small dielectricconstant change, i.e., Δε_{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} + Δn_{Si})^{−1}(λ_{c} + Δλ_{c}) = n_{Si}^{−1}λ_{c}, where n_{Si} = ε_{Si}^{1/2} and consequently Δn_{Si} = (2n_{Si})^{−1}Re(Δε_{Si}) = (2n_{Si})^{−1} Re(Δε_{Drude}) as the dielectric constant change is solely in the Drude part. Including Eq. (2) and the standard phasematching 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 opticalphaseaccumulation condition is rewritten by
where g(ω) = e^{2}[ε_{0}m^{*}(ω^{2} + Γ′^{2})]^{−1}. Obviously, increase in N^{*} (ΔN^{*} > 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 resonancecenter 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 lowvoltage region of V_{a} < 1 V and abrupt increase in the differential resonance shift Δλ_{c}/ΔV_{a} in the highvoltage region of V_{a} > 1 V. 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 compoundcarrier 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 builtin potential V_{builtin} across the pn junction. Applying the PoissonBoltzmann equation for V_{a} = 0 in our case with N_{p} = N_{n} = N_{0} = 10^{18} cm^{−3}, we obtain V_{builtin} = 0.934 V. In the low biasvoltage region of V_{a} < V_{builtin}, increase in 〈N^{*}〉 with V_{a} is led by the decrease in the depletion layer thickness without significant growth in the carrierdensity level. In contrast, in the high biasvoltage region of V_{a} > V_{builtin}, the depletion region is closed and the excessive electrons and holes injected from the electrodes lead to mobilecarrier 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 resonanceshift tunability in Fig. 3(c) implies a full λ_{c} tuning range over 2.3 nm that is remarkably larger than the resonance bandwidth of 0.43 nm by a factor 5.3. Therefore, we can fully utilize the spectral maximum and minimum as the onstate and offstate transmittance levels, respectively. For our particular design at an optimal wavelength of λ_{0} = 1550.02 nm which corresponds to the transmittance minimum at V_{a} ≈ V_{builtin}, sharp transmission modulation is obtained as shown in Fig. 4(a). In this case, the intensity modulation is induced mainly in the closed depletionlayer regime and thereby small biasvoltage tuning induces rapid intensity change as confirmed again in the total electric field patterns for V_{a} = 0.9 V and −4.0 V in Fig. 4(b,c). Key performance parameters in this case are transmittance modulation depth of 83.9%, onstate efficiency of 85.2%, and onoff extinction ratio of 18.9 dB under remarkably low control biasvoltage signals within the −4 ~ +0.9 V range. Importantly, the obtained performance parameters are highly desirable for variety of applications when compared with LiNbO_{3}crystalbased EO modulators requiring 100 Vscale control signals for similar performances.
For a different operation wavelength which corresponds to the transmittance minimum at V_{a} < V_{builtin}, we have much slower intensity tuning as the device operates in the open depletionlayer regime. For example, we select λ_{0} = 1550.29 nm 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 depletionlayer regime is more appropriate for digital signal processing applications.
The proposed device is basically a 1Dperiodic 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 xcomponent wavevector of the leaky guided mode, respectively. A simple calculus with basic relations of ω(k) = 2πcλ_{c}^{−1}, k_{x} = 2πcλ_{c}^{−1}sinθ, and the diffractive phase matching condition λ_{c}^{−1}sinθ = n_{eff}λ_{c}^{−1}−Λ^{−1} results in
where n_{G} = c/V_{x} is group index of the leaky guided mode. For nearnormal incidence (θ ≪ 1), Eq. (4) reduces to ∂λ_{c}/∂θ ≈ − n_{G}^{−1}n_{eff}Λ. This property explicitly appears in the angledependent 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 guidedmode with V_{x} = 0 at normal incidence. For offnormal incidence (θ ≠ 0) for which the two counterpropagating guided modes are not coincidental anymore and the resonance is driven by a single leakyguided mode. Consequently, the grouptoeffectiveindex ratio n_{G}^{−1}n_{eff} tends to a constant value and so is ∂λ_{c}/∂θ. Estimating from Fig. 5(a), the offnormal angletunability ∂λ_{c}/∂θ ≈ −2 nm/deg. at θ = 5°. Importantly, the angledependent shift of λ_{c} does not significantly affect the resonance profile and the EOtunable resonance shift as shown in Fig. 5(b). In particular, the EO tunability of 0.17 nm/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 fullwidthathalfmaximum 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 afewnm scale errors in the spatial device parameters and grating period and afewÅ 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 10nm scale. Considering 0.1 nm (distributed feedback type) ~10 nm (FabryPérot type) for typical diode laser line width around 1,550 nm, the anticipated fabrication errors can be critically problematic if there is no tuning method available for matching λ_{c} to the sourcelaser wavelength. Suppose that we control θ within a ±10° range and with an accuracy of 10^{−3} deg. The angle tunability of 2 nm/deg. implies a full tuning range of ~40 nm with an accuracy of ~2 pm. 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 4, the estimated GMR bandwidth is Δf_{opt} ~ 54.3 GHz. Therefore, stable 10GHzscale 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 biasvoltage signal modulation bandwidth Δf_{bias} = τ_{RC}^{−1}, where τ_{RC} denotes the RC time constant determined by the junction capacitance and termination impedance. Assuming the standard radiofrequency termination impedance of 50 Ω and the design configuration used in Figs 2 and 4, estimated Δf_{bias} values are 14.7 MHz for a device footprint area of 1 × 1 mm^{2}. Since Δf_{bias} is inversely proportional to device footprint area, GHzscale modulation bandwidth should be feasible for small devices with the footprint area <260 × 260 μm^{2}. We note that Inoue et al.^{19} recently demonstrated a highQ GMR filter with a device footprint area reduced down to 10 × 10 μm^{2} without significant degradation in the spectral performance characteristics by using gradedparametric design approach combined with integrated firstorder Bragg reflection boundaries. Introducing such miniaturization strategy to the proposed concept, Δf_{bias} > Δf_{opt} and the full resonance bandwidth should be available for the final optical signalmodulation bandwidth.
Discussion
In summary, we proposed a multiplepnjunction subwavelength grating structure that enables highperformance EO modulation in the transmission mode. The proposed device operates under highQ GMRs interacting with electric signals through the Drudetype optical freecarrier effect. Using rigorous electrical and optical modeling methods, we theoretically demonstrated highly efficient transmission modulation generated by remarkably lowvoltage control signals with modulation speed in possibly 10 GHz scales. Notably, the obtained properties are supported by the lowloss freecarrierinduced EO effect occurring in the whole device region with 500nmthick Si layers as opposed to the transparentconductingoxidebased plasmonic metasurface approaches involving a sub10nmthick EOactive region and strong ohmic absorption. In another similar approach, a lowloss GMR modulator was suggested using a combination of the BursteinMoss effect, Pockels effect, and FrazKeldysh effect in a weaklymodulated InGaAsP waveguide grating structure^{20,21}. Therein, a robust reflection modulation with an extinction ratio of 17 dB and modulation bandwidth of 5 MHz was experimentally demonstrated.
Experimental realization of the proposed device concept is definitely the next step. In potential fabrication, crucial parts are to establish 100nmthick multiple pn junction cells and subwavelength grating structure with a critical dimension in a few 100 nm scale. First, the multiple pn junction cells can be generated by the standard deposition processes based on chemical vapor deposition and sputtering techniques. We note that the present stateoftheart deposition methods easily produce such multiplejunction semiconductor layer structures as established well in tandem solar cells and in verticalcavity surfaceemitting lasers^{22,23,24}. Second, the subwavelength periodic structure for a highQ resonance excitation is also wellestablished using standard nanolithography techniques including the laserinterference lithography^{4} and electronbeam 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 IIIVcompound semiconductors, longerwavelength applications in the midinfrared (midIR) 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 Δε_{Drude} that is in the order of 10^{−2} around λ = 1.5 μm under ΔV_{a} = 5 V is amplified up to the unity order around λ = 15 μm in the midIR domain and even further up to the order of 10^{2} around λ = 150 μm in the THz domain. Therefore, the proposed device class can take advantage of the far stronger EO effect in the midIR 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 indepth 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 highspeed modulators for applications in telecommunications, optical information processing, LIDARs, laser machining, and many others.
Additional Information
How to cite this article: Lee, K. Y. et al. Multiple pn junction subwavelength gratings for transmissionmode electrooptic modulators. Sci. Rep. 7, 46508; doi: 10.1038/srep46508 (2017).
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Acknowledgements
This research was supported in part by the Basic Science Research Program (NRF2015R1A2A2A01007553) and by the Global Frontier Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT & Future Planning (NRF2014M3A6B3063708). We thank Hyun Jae Lee from University of Seoul for helpful discussions on electrical properties of semiconductor thin films.
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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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Lee, K., Yoon, J., Song, S. et al. Multiple pn junction subwavelength gratings for transmissionmode electrooptic modulators. Sci Rep 7, 46508 (2017). https://doi.org/10.1038/srep46508
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