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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 16 — Aug. 2, 2010
  • pp: 16902–16928
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Design of transmission line driven slot waveguide Mach-Zehnder interferometers and application to analog optical links

Jeremy Witzens, Thomas Baehr-Jones, and Michael Hochberg  »View Author Affiliations


Optics Express, Vol. 18, Issue 16, pp. 16902-16928 (2010)
http://dx.doi.org/10.1364/OE.18.016902


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Abstract

Slot waveguides allow joint confinement of the driving electrical radio frequency field and of the optical waveguide mode in a narrow slot, allowing for highly efficient polymer based interferometers. We show that the optical confinement can be simply explained by a perturbation theoretical approach taking into account the continuity of the electric displacement field. We design phase matched transmission lines and show that their impedance and RF losses can be modeled by an equivalent circuit and linked to slot waveguide properties by a simple set of equations, thus allowing optimization of the device without iterative simulations. We optimize the interferometers for analog optical links and predict record performance metrics (Vπ = 200 mV @ 10 GHz in push-pull configuration) assuming a modest second order nonlinear coefficient (r33 = 50 pm/V) and slot width (100 nm). Using high performance optical polymers (r33 = 150 pm/V), noise figures of state of the art analog optical links can be matched while reducing optical power levels by approximately 30 times. With required optical laser power levels predicted at 50 mW, this could be a game changing improvement by bringing high performance optical analog link power requirements in the reach of laser diodes. A modified transmitter architecture allows shot noise limited performance, while reducing power levels in the slot waveguides and enhancing reliability.

© 2010 OSA

1. Introduction

Recent advances in non-linear optical polymers make them extremely promising materials for high performance optical modulators. Electro-optic coefficients as high as r33 = 300 pm/V have been demonstrated [1

1. J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007). [CrossRef] [PubMed]

], and polymer based modulators with high reliability have been made commercially available [2

2. D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010). [CrossRef]

]. These advances have been enhanced by novel device design. In particular silicon waveguides with a narrow slot in the middle, so called slot waveguides (SWG), have been shown to yield high confinement of the optical field inside the slot [3

3. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

]. The same structures also allow dropping the entire radio-frequency (RF) driving voltage across the narrow slot [4

4. T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008). [CrossRef]

], thus yielding very high RF E-fields with high optical overlaps. This allows modulators with ultra low drive voltages to be realized.

2. Device overview

Figure 1
Fig. 1 Overview of the device geometry (not to scale). Gray areas represent aluminum, green areas silicon, blue silicon dioxide and orange the electro-optic polymer. The green shading indicates the implant concentrations. The inset shows the whole structure with both arms of the MZI operated in push-pull operation (arrows indicate the relative orientation of the E-field during operation).
shows a cross-section of a SWG and of half the TL. The complete TL is a coplanar line of the form Ground-Signal-Ground (GSG) that applies a push-pull signal to the two arms of the MZI, assuming the polymers are poled in the same direction in both slots. In this paper, Vπ is the value for the MZI operated in push-pull configuration, i.e., it is the aggregate effect of both arms. In addition to defining the geometry, Fig. 1 also defines three implant regions: Low, moderate and high density implants. Table 1

Table 1. Summery of device parameters.

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summarizes numerical values.

The devices are assumed to be fabricated in silicon-on-insulator (SOI) wafers. Not shown in Fig. 1 are the buried oxide (BOX) thickness and the silicon wafer handle. In order to suppress optical coupling from the waveguides to the silicon handle, the BOX has to be fairly thick and is assumed to be 3 μm in the rest of this paper. In order to avoid TL losses associated to currents in the silicon handle, it has to be made out of high resistivity silicon. In sections 4 and 5 the silicon handle is assumed to have a resistance of 100 Ω × cm. The reader can jump ahead to Fig. 18
Fig. 18 Schematic of an optical analog link. SWGs are represented in green and regular waveguides (on-chip) or fibers (off-chip) in orange.
for a complete device schematic. The narrow slots are the most challenging aspect of SWG fabrication. However, advances in resist shrinking methods allow fabricating very narrow slots, even with optical lithography. Figure 2
Fig. 2 SEM micrograph of a SWG with a 120 nm slot fabricated with optical lithography at BAE Systems. The slot was purposely defined off center, it was found that this increases the single mode frequency region.
shows an SEM micrograph of a 120 nm slot SWG fabricated with optical lithography.

3. Waveguide properties

This section describes the modeling of the stand alone, implanted waveguides. Section 3.1. describes the overlap of the optical field with the electrical RF driving field. Section 3.2. focuses on secondary metrics, such as implant induced optical losses, series resistance and capacitance.

3.1. Overlap

The strength of SWGs lies in the fact that the optical field is confined is a small volume (the slot) filled with polymer (Fig. 3(a)
Fig. 3 (a) Optical Ex field distribution in a.u. for a transverse electric (x-polarized) mode and (b) RF Ex field distribution. The waveguide geometry is described in Table 1. The very high E-field regions at the corners are artifacts of the finite-elements mode solver.
). Since the RF voltage can be entirely dropped across the same region (Fig. 3(b)), the resulting RF electrical field strength is very high relative to what would be obtained with wider cross-section modes.

Figure 4
Fig. 4 (a) optical field overlap with the slot region as a function of slot width, and (b) overlap multiplied by the Ex RF field strength for an applied bias of 1V. wg_rh = 200 nm and wg_ch = 50 nm. wg_rw is adjusted to maintain optimum overlap (Fig. 5). The inset shows that the FOM in (b) converges to a finite value.
shows the overlap of the optical field with the slot, as well as the optical overlap multiplied by the RF field strength (assuming a 1V bias), the actual FOM for modulation. Interestingly, the FOM keeps growing for smaller slots, down to the smallest simulated (0.2 nm). Of course there will be other limitations in practice such as fabrication, but also excessive capacitance (which scales as the inverse of the slot width) and dielectric breakdown of the polymer under excessive field strengths. High dielectric strengths in excess of 1.5 MV/cm have been demonstrated [17

17. Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008). [CrossRef]

], however there have been no reliability studies done under continuous exposure to such extreme fields. High capacitance makes phase matching more challenging and leads to higher TL losses (section 4). Finally, small slot widths lead to higher optical power densities that might reduce device lifetime. In the following, 100 nm slots are considered a safe practical dimension, since we have already realized high-quality 100 nm slots with optical lithography and since RF electrical field strengths are below 20 kV/cm (assuming a Vπ of 200 mV). The data shown in Fig. 4(b) can be described in intuitive terms by inverting it: A FOM of 20 μm−1 is the equivalent of perfectly confining the optical field inside a parallel plate capacitor with plate to plate spacing of 50 nm. The FOM for a 100 nm slot is almost an order of magnitude lower, at 2.7 μm−1, equivalent to a perfect field confinement between two plates spaced by 370 nm. Figure 5 shows the ridge width wg_rw optimized for highest slot overlap as a function of slot width wg_sw. It can be fitted as wg_rw = −0.1052 wg_sw2 + 0.1831 wg_sw + 0.4247, where all dimensions are in μm.

Interestingly, even though a cursory visual inspection of Fig. 4(b) could lead to the erroneous conclusion that the theoretical FOM diverges for vanishing slot size, it actually converges to a finite value easily predictable by perturbation theory. By applying the parallel plate approximation to the two internal waveguide edges defining the slot, it can be easily understood how the RF field is confined in the slot region. As already pointed out in [3

3. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

], the high optical field in the slot region is a consequence of the continuity of the normal electric displacement field at the silicon-slot interface.

Figure 7(a)
Fig. 7 (a) Comparison of the optical E-fields across SWGs with 2nm and 100 nm slots. Both modes have been normalized to carry equal flux. (b) Comparison between the simulated FOM (blue line) and the semi-analytical model (stars).
gives some insight into why the FOM drops so quickly for increasing slot width: The field inside the silicon rapidly diverges from the field of the slot-less waveguide, with field maxima moving away from the slot surface into the core of the two silicon regions and resembling the field distribution expected of two coupled waveguides cores (even though this SWG remains single mode). This results in a decrease of the optical field strength at the edges of the slot for a given maximum field in the silicon (max(ESi)). This weakening of the surface E-field can be quantified by introducing the parameter φ, as defined in Fig. 7(a). The decrease of the surface field is further worsened by reduction of max(ESi) itself. This is caused by the fact that the increased flux carried by the widened slot region has to be compensated by a reduction of the flux in the rest of the waveguide after field normalization.

Figure 7(b) compares the simulated FOM to K × max(ESi)2 × cos(φ)2, where K is a normalization constant and max(ESi) and φ were extracted from the field profiles.

The parameter φ can be predicted by a semi-analytical model. If one defines neff as the effective index of the waveguide, nslab as the effective index of the unetched silicon slab (of thickness wg_rh and clad by oxide on one side and polymer on the other, like the waveguide), α as the decay coefficient of the evanescent field inside the slot, β as the wave number of the waveguide, and if one approximates the field inside the silicon waveguide cores as max(ESi) × cos(k(x-wg_sw/2) + φ) × cos(ly + θ):
α = neff2+(nSi2-nslab2)-npolymer2λ
(3)
β = neffλ
(4)
k = nslab2-neff2λ
(5)
E = 0leads to ikEx= -iβEz in the silicon core regions and ikExαEzin the slot regions. The continuity equations at the slot silicon interface then lead to

φ = arctan(αktanh(αwg_sw2)n2Sin2polymer)
(6)

Figure 8
Fig. 8 φ as a function of slot width as extracted from mode profiles (black line) and as predicted with Eq. (6).
compares φ as predicted by Eq. (6) to the value extracted from simulated mode profiles. The rapid increase of φ with slot width accounts for about half the reduction of the overlap field-strength product, with the other half accounted for by the reduction of max(ESi).

Decreasing the slot width is not the only method allowing an increase of the slot overlap. A thicker silicon film, or equivalently larger ridge heights (wg_rh), result in higher overlaps as shown in Table 2

Table 2. SWG characteristics for different ridge and cladding thicknesses.

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. This is due to the fact that for thicker silicon films and a fixed wg_ch, the width of the silicon waveguide cores wg_rw can be reduced while maintaining good field confinement inside the waveguide, so that the slot makes up for a larger portion of the entire waveguide. It should also be noted that the overlap is very sensitive to lateral confinement driven by the cladding height wg_ch. While the overlap field-strength product can be increased by either a higher ridge height and thinner ridge width (increasing the overlap) or a thinner slot width (increasing the field strength), the two methods fundamentally differ in some aspects. Both result in increased SWG capacitance, but reducing the slot width also increases the optical power density inside the slot and is thus ultimately reliability limited, while increasing the ridge height reduces the power densities (section 5). Nonetheless, the analysis in this paper focuses on 200 nm silicon, since this was our process development focus at the time of publication.

3.2. Optical losses, resistance and capacitance

In order to optimize the MZI, several other waveguide characteristics have to be parameterized: Implant induced SWG losses, as well as SWG capacitance and series resistance.

The optical field has a very clear exponential decay in the cladding region, so that the overlap with the implants and the resulting optical losses can be characterized by three numbers: The overlap with the core silicon area (the unetched areas of thickness wg_rh), the overlap with the cladding silicon, and the decay length of the field in the cladding (Fig. 9
Fig. 9 Overlap of the optical field with the core silicon, cladding silicon, and decay length of the optical field inside the cladding silicon. The SWG ridge width, wg_rw, is intrinsically varied with wg_sw in order to maintain optimum slot overlap. wg_rh = 200 nm and wg_ch = 50 nm.
). The latter is described as the distance between the edge of the waveguide and the onset of moderate implants (imp_off), such that the overlap of the moderate implant region (imp_c2) with the cladding field is less than 10% of the total cladding optical field overlap.

For example for a 100 nm slot, and an implant offset imp_off = 330 nm (90% decay length), 43% (core overlap) + 0.9 × 7.7% (90% of cladding overlap) of the field overlaps with the low implant region, while 0.1 × 7.7% (10% of cladding overlap) of the field overlaps with the moderate implant region.

Figure 10
Fig. 10 SWG capacitance as a function of slot width. The continuous line is simulated (PISCES), while the dashed line corresponds to the parallel plate approximation.
shows the capacitance of the SWG as a function of slot width. A RF relative dielectric constant of 4 was assumed for both silicon dioxide and for the polymer. The SWG capacitance was simulated by solving Poisson’s equation with a finite-elements solver (PISCES) and compared to the parallel plate approximation. Due to the fringe fields, simulated values exceed the parallel plate approximation by an approximately constant amount. For a typical slot width of 100 nm, the fringe fields account for almost half the total capacitance (136 pF/m). Hence, it is very important to base calculations on simulated results.

Series resistance was also simulated with PISCES. However, in this case the obtained resistances were almost exactly equal to estimates based on simple film thickness approximations, even for the core waveguide region where the directions of current flow significantly diverge from the horizontal. Hence these results are not further discussed.

4. Transmission line design

In section 4.1 a typical TL is designed by solving Maxwell’s equations explicitly with a finite-elements based eigensolver (HFSS). Based on inspection of the TL modes (in particular current distributions inside the silicon) an equivalent lumped element circuit for infinitesimal section lengths is proposed (sections 4.2 and 4.3). Predictive analytical models are then derived that allow calculating the impedance and the excess loss of phase-matched TLs from the SWG group index, capacitance and series resistance (section 4.4). These models are validated by comparison to TL simulation results. It was found that the TL characteristics predicted by the analytical models match the simulation results very well in the appropriate regime.

4.1. Geometrical parameters

In this section primary geometrical parameters are set based on simulation results in order to achieve phase matching and reasonable TL losses. Phase matching could be expected to be a problem due to the polymer and silicon dioxide index being around 2 at RF frequencies, and thus significantly lower than the group index of the SWG (3.1). As it turns out, the substantial capacitive loading induced by the SWG slows down the RF signal to such an extent that the opposite becomes true: It becomes challenging to maintain a low enough TL index. For thin slots below 100 nm it becomes increasingly challenging to phase match the TL while maintaining easy to fabricate geometries. It can be done up to a certain point with thick metal films and small tl_sp, but these parameters are respectively limited by fabrication and waveguide losses.

Figure 11
Fig. 11 TL effective index and propagation losses as a function of design parameters at a frequency of 10 GHz. Each parameter is varied around the geometry given by Table 1. The propagation losses shown here include excess losses from the high and moderate resistivity silicon regions. These curves were obtained by iterative simulations with a finite-elements mode solver.
shows the sensitivity of the TL losses and effective index on design parameters (centered on the parameters given in Table 1) as extracted from finite-elements simulations. The group index of the SWG is 3.1, hence this should be the target for the TL effective index in order to obtain phase matching. The design resulted in a TL effective index neff = 3.36, which is slightly off target, but results in a negligible efficiency drop (4%) since the typical MZI length is expected to be relatively short (~9 mm for devices optimized for operation at 10 GHz, see section 5). An index of 3.1 could have been obtained by using thicker metal, or reducing the metal-to-metal spacing. However, both are expected to result in more challenging fabrication, and in the case of reduced metal spacing also higher waveguide losses since it brings the metal closer to the waveguides.

The 2 μm metal thickness was primarily driven by TL losses. Below 2 μm, the skin depth of the top and bottom of the metal overlap at 10 GHz and the series resistance of the aluminum lines quickly increases. In general, in order to achieve phase matching to the SWG a large metal stripe width (tl_w) and height (tl_h), and a small metal to metal spacing (tl_sp) were desirable (all of them decrease the linear inductance and thus also decrease the effective index of the loaded TL as explained at the end of section 4.4). Increasing the metal width becomes ineffective past a certain point, both in terms of losses and of phase matching, since the current concentrates in the vicinity of the metal edge facing towards the opposite metal strip.

4.2. Equivalent circuit for infinitesimal transmission line segments

Figure 12(a)
Fig. 12 (a) Equivalent circuit of the loaded TL and (b) simplified equivalent circuit. Elements in red correspond to the SWG. In (c) the series RC model for the SWG is transformed into a parallel model to allow using the standard telegraph line equations.
shows the equivalent circuit for an infinitesimal section of loaded TL.

The classic telegraph line equivalent circuit (LTL, CTL and RTL) is complemented by a model for the SWG (red). The horizontal branch of the SWG model corresponds to currents flowing inside the silicon along the axis of the TL (Jz) while RSWG corresponds to currents flowing from the metal lines to the slot (Jx) across the series resistance of the SWG (axes are labeled in Fig. 13
Fig. 13 (a) RF H-field distribution and (b) RF E-field distribution. The scale in (a) is 0 to 2e4 A/m. The scale in (b) is 0 to 1.15e8 V/m. The mode is normalized to carry 1W of power. Solutions are based on fully vectorial solutions of Maxwell’s equations at 10 GHz.
). It should be noted that for a quasi-TEM mode, such as would be obtained for highly conductive silicon, the current would predominantly flow along the axis of the TL and the horizontal branch would be the dominant contributor to the SWG model. On the other hand, a dominant vertical branch (RSWG) would correspond to currents inside the silicon flowing predominantly along the x-direction, resulting in a substantial deviation from the quasi-TEM approximation. In this case, Fig. 12(a) could be simplified into Fig. 12(b). The SWG could then be simply modeled by an additional capacitive loading of the TL (C¯SWG = CSWG/(1 + RSWG 2CSWG 2ω2)) combined with a shunt conductance (expressed as R¯1SWG = RSWGCSWG 2ω2/(1 + RSWG 2CSWG 2ω2)) resulting in excess TL losses (Fig. 12(c)).

In order to motivate the model shown in Fig. 12(b), the mode profile of a typical loaded TL is first obtained with a finite-elements mode solver and qualitatively inspected. Mostly, it is verified that with silicon implant concentrations compatible with acceptable waveguide losses, the current flow inside the silicon is predominantly along the x-direction, and that the H-field, and thus the self-inductance of the TL, are only slightly perturbed by the presence of the SWG, while the E-field, and thus the linear capacitance, are fundamentally modified.

The motivation behind first fitting the curves and then comparing the fitted parameters, rather than directly comparing the finite-elements results to curves estimated based on the PISCES numbers, was to accommodate small changes in the parameter values either due to changes in the meshing, or to more fundamental reasons, while demonstrating quasi-perfect agreement of the functional relationship. As it turned out, the discrepancy between fitted values and independently estimated values was only 4% for the SWG capacitance and 15% for the SWG series resistance. It was unclear prior to detailed numerical analysis whether the capacitance of the unloaded TL and of the SWG could be simply taken as is in the combined model (as would be expected in a geometry with spatially cleanly separated E-fields), or whether spatial overlap of the E-field generated by the metal lines with the conductive silicon would induce sufficient interaction to break this simple model. As it turns out, CTL extracted from the simulated loaded TL properties agrees within 13% with the value extracted from the unloaded TL (with unimplanted, non-conductive silicon), and LTL agrees within 1%.

Another open question was whether the excess TL transmission losses, that is the TL losses generated by current flow inside the implanted silicon, is truly dominated by laterally flowing Jx, or whether longitudinal currents (Jz) are sufficiently high inside the silicon to have a significant impact on TL losses. Here too, in the investigated implant density regime losses can be almost entirely ascribed to laterally flowing currents. Had this not been the case, it would have been interesting to investigate whether forcing lateral only currents with a segmented geometry would have helped, however this turned out to be a moot point for the reasons stated above.

4.3. Transmission line mode profile properties

It is very educational to inspect the field and current profiles of the TL mode since they provide a qualitative justification for the simplified equivalent circuit model and for the fitting models applied later in this section.

Figure 13 shows the E- and H-fields of the loaded TL mode. It is apparent that the silicon has very little impact on the H-field. In fact, the H-field closely resembles the distribution of the stand-alone TL. This can be explained by the fact that the current densities in the silicon are more than an order of magnitude lower than in the metal (modeled as aluminum). Hence it is expected that the linear inductance of the loaded TL will be very close to the inductance of the unloaded TL. This conclusion is further reinforced by the fact that the currents in the silicon are mostly along the transverse x-direction, rather than the propagation direction (z), and are thus mostly generating z-components of the H-field that do not contribute to the self-inductance of the TL. The currents in the silicon are breaking the quasi-TE symmetry since they are predominantly transverse. This explains why the TL phase velocity can be significantly different from the cladding material.

The E-field on the other hand is heavily influenced by the presence of the silicon. In fact, most of the voltage is dropped across the slot of the SWG, as required in order to maintain high opto-electronic device performance. It is thus expected for the capacitance of the SWG to be a large portion of the loaded TL capacitance. We will show that the loaded TL capacitance can be closely approximated by the sum of the unloaded TL capacitance and of the SWG capacitance. This increased capacitance slows down the phase velocity of the TL mode, and allows phase matching to the optical SWG mode.

Figure 14
Fig. 14 (a) RF loss volume density (b) Jz current distribution (c) Jx current distribution and (d) rescaled Jz current distribution. The scale in (a) is 0 to 2.2e15 W/m3, the scale in (b) 0 to 6e10 A/m2, the scale in (c) 0 to 2.5e9 A/m2, and the scale in (d) 0 to 1e8 A/m2. The mode is normalized to carry 1W of power.
shows RF loss distributions and current distributions in the TL. As previously mentioned, it can be seen that Jz in the metal is an order of magnitude larger than the current in the silicon, and the latter is predominantly (again by more than an order of magnitude) along the x-direction. It can also be seen in Fig. 14(a) that the losses are dominated by resistive losses in the low doping density silicon close to the waveguide.

It might be surprising that the sharp maximum of the E-field in Fig. 13(b) does not correlate to a corresponding feature in the H-field, as would be expected from a quasi-TEM mode. In such a mode Ez and Hz are zero, such that the following holds:
×H = εEt Hyz = εExt  Ex = 2πnλHyεω = Z0nHy                         Hxz = εEyt  Ey = 2πnλHxεω = Z0nHx
(7)
In other words, Ex and Hy strongly correlate with each other. The discrepancy between the E- and H-fields profiles is thus only possible due to the non-TEM nature of the mode. The relationship between the E- and H-fields in the loaded TL can be established by inspecting the fields generated by the current and charge distributions of the SWG alone. Since the currents inside the SWG are mostly along the x-direction, they can be approximated, and independently solved for, by considering a lumped capacitor with the SWG geometry driven by an equal voltage applied along its entire length. For symmetry reasons, when the SWG is driven in such a fashion the internal H-field is along the z-direction and the E-field confined to the xy plane with zero derivatives along the z-direction. The resulting fields are shown in Fig. 15
Fig. 15 (a) E-field in V/m and (b) H-field in A/m for an SWG driven as a lumped element at 10 GHz, i.e., with a homogeneous AC voltage applied along the entire length.
. Then
×H = εEt  Hzy = εExt  Ex = ilyHzεω                        Hzx = εEyt  Ey = ilxHzεω
(8)
where lx and ly are the characteristic length scales over which Hz varies in the x and y directions. In other words, the E-field in the SWG correlates to Hz, the non-TEM component of the H-field. It can be seen in Fig. 15 that the gradient of Hz is mostly confined to the slot. Hz does not register in the H-field magnitude shown in Fig. 13(a) because it is two orders of magnitude smaller than the ambient Hy field generated by the metal stripes. This is a consequence of the fact that the length scale over which Hz varies, the slot height wg_rh = 0.2 μm is much smaller than the wavelength (~1 cm), so that a very small Hz component can still generate a strong enough gradient to account for Ex inside the slot. Even though Ex is approximately 40 times larger in the slot than outside (the spacing between the metal lines is 40 times the slot width), the corresponding Hz-field is more than two orders of magnitude smaller.

4.4. Equivalent circuit validation and loaded transmission line modeling

Figure 16
Fig. 16 Real parts of the TL impedance and effective index. Black dots correspond to simulation data for the loaded TL. The black curves correspond to the fits described in the text. As a reference, the blue curves show the simulation results for the unloaded TL (silicon conductance uniformly set to zero). The red curves correspond to the analytical models described in the text, with independently evaluated values for SWG and TL characteristics applied to the formulas (“independently simulated” column of Table 3).
shows the simulated impedance and effective index of the loaded TL, as well as fitting results assuming functional relationships derived from the equivalent circuit model. The impedance and the effective index were independently fitted, with excellent consistency between the two resulting sets of fitting parameters, and between the fitting parameters and a-priori calculated capacitance and resistance values, thus validating the equivalent circuit model. The slight discrepancy between fit and simulation data at low frequencies is due to the linear resistance of the metal lines that was not included in the analytical model. At low frequencies the series resistance of the aluminum lines becomes comparable to iLω, which leads to a divergence of the TL impedance Z = (iωL+RMetal Series)/iωC from its ideal high-speed value Z = L/C), and of the TL index neff = c0(L+RMetal Series/iω)C from its ideal high-speed value neff = c0LC. It can be seen in both sets of curves that at high frequencies the characteristics asymptotically converge towards those of the unloaded TL, as the operation frequency exceeds the RC time constant of the SWG and the SWG capacitance is screened.

The following relation for the loaded TL impedance can be easily derived from the equivalent circuit and was taken as a basis for the functional form of the fit:
Z = 12LC+G/iω = 12LTLCTL+CSWG11+ifBW                             = 121(CTLLTL)+(CSWGLTL)11+ifBW
(9)
where the left side of the equation is the standard equation for the telegraph line impedance (with shunt conductance G). In the right side LTL is the inductance of the unloaded TL (assumed to be equal to the inductance of the loaded TL), CTL is the capacitance of the unloaded TL, CSWG is the capacitance of the SWG and BW is the intrinsic electrical bandwidth of the SWG, 1/2πRSWGCSWG, where RSWG is the series resistance (note that BW is the bandwidth of the SWG and not of the complete MZI, since the latter also suffers from frequency dependent TL losses). The additional factor ½ is to take into account that the complete TL of the push-pull MZI is of the form GSG, while the modeled TL is half the structure, GS, corresponding to a single waveguide arm (i.e., LTL and CTL correspond to the single arm structure, while the system impedance Z is for the complete push-pull structure). The fitted free parameters are [CTL/LTL]Z-fit, [CSWG/LTL]Z-fit, and [BW]Z-fit. Following model can be derived from the equivalent circuit for the TL effective index:
neff = c0L(C+G/iω) = c0CTLLTL+CSWGLTL11+ifBW
(10)
where c0 is the speed of light in vacuum. The fitted parameters are [CTLLTL]n-fit, [CSWGLTL]n-fit and [BW]n-fit (fitted independently from the impedance fit).

Table 3

Table 3. Summary of fitted parameters.

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summarizes the fitted parameters and compares the results of the two fits, as well as the fits to expected results obtained independently with PISCES.

In order to obtain phase matching, the effective index of the TL given by Eq. (10) has to equate the group index of the SWG. In a system with relatively homogeneous dielectric indices such as here (the RF indices of the polymer and of silicon dioxide are both ~2), LTLCTL is constrained by the phase velocity of the unloaded TEM TL mode (LTLCTL~ncladding 2/c0 2), so that the second term, LTLCSWG, has to be adjusted to obtain phase matching. In other words, Eq. (10) determines the target for LTL, and the latter can be obtained by adjusting the geometry of the metal lines. For example in the case of the coplanar waveguide geometry used here, increasing the line to line spacing, reducing the metal thickness or the metal width, all increase the self-inductance of the line, the consequences of which on the TL index can be seen in Fig. 11.

5. Device optimization and analog link performance

5.1. Analog link figure of merit

5.2. Device optimization

In order to derive device performance, a couple of assumptions have to be made on material properties and process quality. Here we assume relatively conservative numbers of r33 = 50 pm/V and unimplanted (baseline) waveguide losses of 6 dB/cm. To place this in context, electro-optic coefficients up to 300 pm/V have been recently demonstrated for advanced organic nonlinear polymers, and we recently measured baseline SWG losses of 8 dB/cm. An r33 of 30 pm/V has been previously experimentally demonstrated [4

4. T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008). [CrossRef]

] in SWGs using polymers with an ideal, fully poled coefficient of 100 pm/V (the discrepancy was attributed to partial poling). We are also confident that waveguide losses can be improved with some process development. Vπ for higher r33 can be easily derived by linearly rescaling results reported here.

Lopt=2αlog(1+αβ)
(17)

It can be seen that the input impedance of the GSG TL is very mismatched from 50Ω. However the device can be driven with two 50Ω inputs in a dual RF drive configuration, since each of the individual SG lines are approximately 50Ω (and exactly so for a 120 nm slot). This dual drive could for example be provided by an RF power splitter. Assuming a 50 pm/V polymer and a 100 nm slot, a DC drive voltage of Vπ < 200 mV is predicted for a device optimized for and operated at 10 GHz (with a 26 GHz bandwidth). This can be compared to the device referred to in [10

10. PhotonicSystems, part number PSI-3600-MOD-D1.

] with a Vπ of 1.1 V at 12 GHz. In other words, even assuming modest nonlinear coefficients, the polymer device is predicted to beat the best lithium niobate based devices by a factor 6 in drive voltage, and approximately a factor 30 in drive power (Vπ 2/Zin). Assuming best of class polymers (300 pm/V), the SWG drive voltage would be reduced to < 35 mV, or a factor ~1000 improvement in drive power. In order to obtain a fair comparison, the device in [10

10. PhotonicSystems, part number PSI-3600-MOD-D1.

] is assumed to be driven in dual RF drive configuration, i.e both arms of the MZI are supplied with an RF voltage so that the reported Vπ is halved relative to the single RF drive voltage. It should be noted that here a halved Vπ is assumed rather than the 1/2 improvement mentioned in the datasheet, since in the datasheet the Vπ is referred to the input of the RF power splitter supplying the dual RF drive, while in this paper the Vπ is referred to the input ports of the MZI (the power penalty is taken into account at the system level via Zin in Eq. (14).

One might wonder how dependant these results are on the specific choice of FOM. Figure 21
Fig. 21 TL propagation losses (red) and SWG propagation losses (blue) as a function of the lowest density implant concentration when varying around the optimum (dashed line) at 10 GHz.
shows TL and SWG propagation losses as a function of the lowest concentration implant (inside and in the direct vicinity of the SWG) that primarily drives SWG and TL losses. It can be seen that the optimum concentration is located at the elbow of both transmission loss curves. It is thus expected for the optimized implant concentration to be relatively insensitive on the particular FOM. Indeed, optimizing devices for a FOM of IL/Vπ X results in implant concentrations for the lowest density implant imp_c1 ranging from 4.8e16 cm−3 for X = 1 to 6.3e16 cm−3 for X = 3 at 10 GHz, and from 2.3e17 cm−3 for X = 1 to 4.0e17 cm−3 for X = 3 at 100 GHz. The device length 2log(1 + Xα/2β)/α also stays within a factor ~2, ranging from 5.3 mm to 12.3 mm at 10 GHz and from 2.0 mm to 3.0 mm at 100 GHz.

5.3. Analog link performance

Based on existing commercial reliability studies with long term reliability (extrapolated 25 years / 85C), we conservatively estimate that a 100 nm slot SWG should be able to handle at least a few mW of power (at the beginning of the SWG). This should be really understood as a lower bound to the power handling capability, since the estimate is based on reproducing similar optical power densities in the polymer as in studies that showed no degradation in accelerated testing. In this section we also assume an additional 3 dB of insertion losses to account for chip to fiber transitions (typical single mode laser packaging losses are taken as a baseline, since they also consist of submicron waveguides) and 2 dB of insertion losses for other on-chip optics on either side of the MZI (y-junctions or directional couplers, as well as SWG to ridge waveguide mode converters). In other words, there are an additional 10 dB total additional Tx losses assumed here, with 5 dB before the MZI and 5 dB after the MZI. In the graphs in this section, optical power levels refer to the optical power at the beginning of individual SWGs, since it is assumed that power levels are reliability limited, but schemes are also compared at the end of the section in terms of required overall laser power. Semiconductor lasers with ultra-low RIN of −160 dB/Hz can be obtained (e.g. JDS Uniphase CQF938 series) and are assumed to be used here. Finally, nonlinear coefficients of 150 pm/V are assumed in this section, which is close to the coefficients demonstrated for best of class polymers (300 pm/V), but slightly reduced to allow for suboptimal poling, small polymer instability or more conventional materials.

Typical receiver power levels are on the order of 0.1 mW in this section (i.e., after attenuation due to typical insertion losses and fiber to chip coupling losses). Receiver noise generally cannot be ignored at these optical power levels, so that receiver noise needs to be added to Nex is Eq. (14). The development of ultra-low capacitance integrated photodetectors has enabled the design of receivers with record low noise floors [21

21. J. Witzens, G. Masini, S. Sahni, B. Analui, and C. Gunn, “10 Gbits/s transceiver on silicon”, Proc. SPIE 6996, 699610 1–10 (2008).

] which would allow the NF to be shot noise dominated even at power levels as low as a few hundred μW. However, for simplicity’s sake typical receive optical subassembly (ROSA) input referred noise currents of In,receiver = 35 pA/Hz (rms) are assumed (as can be obtained from off-the-shelf parts). The photodetector sensitivity is assumed to be 0.85 A/W.

5.4. Link architecture with homodyne amplification

In this section we address other architecture improvements targeted specifically at alleviating the power handling limitations of SWG based MZIs by using homodyne amplification of the signal inside the Tx sub-system.

It should also be noted that this scheme only makes sense when the power handling capabilities of the phase modulator and the passive optics are highly asymmetric as with polymer based SWGs, since it would otherwise be more optimum to split the power equally between two phase modulators in a push-pull scheme.

6. Conclusion

Acknowledgments

The authors would like to thank Gernot Pomrenke, of the Air Force Office of Scientific Research, for his support through an AFOSR Young Investigators Program Grant, and would like to acknowledge support from the NSF STC MDITR Center and the Washington Research Foundation.

References and links

1.

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007). [CrossRef] [PubMed]

2.

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010). [CrossRef]

3.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

4.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008). [CrossRef]

5.

L. T. Nichols, K. J. Williams, and R. D. Esman, “Optimizing the Ultrawide-Band Photonic Link,” IEEE Trans. Microw. Theory Tech. 45(8), 1384–1389 (1997). [CrossRef]

6.

R. Taylor and S. R. Forrest, “Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture,” IEEE Photon. Technol. Lett. 10(1), 144–146 (1998). [CrossRef]

7.

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000). [CrossRef] [PubMed]

8.

W. K. Burns, M. M. Howerton, R. P. Moeller, R. W. McElhanon, and A. S. Greenblatt, “Low Drive Voltage, Broad-Band LiNbO3 Modulators With and Without Etched Ridges,” J. Lightwave Technol. 17(12), 2551–2555 (1999). [CrossRef]

9.

F. Lucchi, D. Janner, M. Belmonte, S. Balsamo, M. Villa, S. Giurgiola, P. Vergani, and V. Pruneri, “Very low voltage single drive domain inverted LiNbO(3) integrated electro-optic modulator,” Opt. Express 15(17), 10739–10743 (2007). [CrossRef] [PubMed]

10.

PhotonicSystems, part number PSI-3600-MOD-D1.

11.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]

12.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]

13.

P. Dong, S. Liao, D. Feng, H. Liang, D. Zheng, R. Shafiiha, C.-C. Kung, W. Qian, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Low Vpp, ultralow-energy, compact, high-speed silicon electro-optic modulator,” Opt. Express 17(25), 22484–22490 (2009). [CrossRef] [PubMed]

14.

T. Pinguet, V. Sadagopan, A. Mekis, B. Analui, D. Kucharski, and S. Gloeckner, “A 1550 nm, 10 Gbps optical modulator with integrated driver in 130 nm CMOS”, Proc. IEEE conf. on Group IV Photonics, 1–3 (2007).

15.

R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]

16.

R. Ding, T. Baehr-Jones, Y. Liu, R. Bojko, J. Witzens, S. Huang, J. Luo, S. Benight, P. Sullivan, J.-M. Fedeli, M. Fournier, L. Dalton, A. Jen, M. Hochberg, “Demonstration of a low VπL modulator with GHz bandwidth based on electro-optic polymer-clad silicon slot waveguides,” submitted for publication.

17.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008). [CrossRef]

18.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997). [CrossRef]

19.

T. E. Darcie and P. F. Driessen, “Class-AB Techniques for High-Dynamic-Range Microwave-Photonic Links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006). [CrossRef]

20.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008). [CrossRef]

21.

J. Witzens, G. Masini, S. Sahni, B. Analui, and C. Gunn, “10 Gbits/s transceiver on silicon”, Proc. SPIE 6996, 699610 1–10 (2008).

22.

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003). [CrossRef]

OCIS Codes
(230.2090) Optical devices : Electro-optical devices
(130.4110) Integrated optics : Modulators

ToC Category:
Integrated Optics

History
Original Manuscript: April 12, 2010
Revised Manuscript: June 20, 2010
Manuscript Accepted: June 22, 2010
Published: July 26, 2010

Citation
Jeremy Witzens, Thomas Baehr-Jones, and Michael Hochberg, "Design of transmission line driven slot waveguide
Mach-Zehnder interferometers 
and application to analog optical links," Opt. Express 18, 16902-16928 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16902


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References

  1. J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007). [CrossRef] [PubMed]
  2. D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010). [CrossRef]
  3. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]
  4. T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008). [CrossRef]
  5. L. T. Nichols, K. J. Williams, and R. D. Esman, “Optimizing the Ultrawide-Band Photonic Link,” IEEE Trans. Microw. Theory Tech. 45(8), 1384–1389 (1997). [CrossRef]
  6. R. Taylor and S. R. Forrest, “Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture,” IEEE Photon. Technol. Lett. 10(1), 144–146 (1998). [CrossRef]
  7. Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000). [CrossRef] [PubMed]
  8. W. K. Burns, M. M. Howerton, R. P. Moeller, R. W. McElhanon, and A. S. Greenblatt, “Low Drive Voltage, Broad-Band LiNbO3 Modulators With and Without Etched Ridges,” J. Lightwave Technol. 17(12), 2551–2555 (1999). [CrossRef]
  9. F. Lucchi, D. Janner, M. Belmonte, S. Balsamo, M. Villa, S. Giurgiola, P. Vergani, and V. Pruneri, “Very low voltage single drive domain inverted LiNbO(3) integrated electro-optic modulator,” Opt. Express 15(17), 10739–10743 (2007). [CrossRef] [PubMed]
  10. PhotonicSystems, part number PSI-3600-MOD-D1.
  11. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004). [CrossRef] [PubMed]
  12. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef] [PubMed]
  13. P. Dong, S. Liao, D. Feng, H. Liang, D. Zheng, R. Shafiiha, C.-C. Kung, W. Qian, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Low Vpp, ultralow-energy, compact, high-speed silicon electro-optic modulator,” Opt. Express 17(25), 22484–22490 (2009). [CrossRef] [PubMed]
  14. T. Pinguet, V. Sadagopan, A. Mekis, B. Analui, D. Kucharski, and S. Gloeckner, “A 1550 nm, 10 Gbps optical modulator with integrated driver in 130 nm CMOS”, Proc. IEEE conf. on Group IV Photonics, 1–3 (2007).
  15. R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
  16. R. Ding, T. Baehr-Jones, Y. Liu, R. Bojko, J. Witzens, S. Huang, J. Luo, S. Benight, P. Sullivan, J.-M. Fedeli, M. Fournier, L. Dalton, A. Jen, M. Hochberg, “Demonstration of a low VπL modulator with GHz bandwidth based on electro-optic polymer-clad silicon slot waveguides,” submitted for publication.
  17. Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008). [CrossRef]
  18. C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997). [CrossRef]
  19. T. E. Darcie and P. F. Driessen, “Class-AB Techniques for High-Dynamic-Range Microwave-Photonic Links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006). [CrossRef]
  20. E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008). [CrossRef]
  21. J. Witzens, G. Masini, S. Sahni, B. Analui, and C. Gunn, “10 Gbits/s transceiver on silicon”, Proc. SPIE 6996, 699610 1–10 (2008).
  22. J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003). [CrossRef]

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