J. Phys. Chem. C 2008, 112, 7953–7958
7953
Design of Ring Resonators Using Electro-Optic Polymer Waveguides† Byoung-Joon Seo,*,‡,⊥ Seongku Kim,‡ Harold Fetterman,‡ William Steier,§ Dan Jin,| and Raluca Dinu| Department of Electrical Engineering, UniVersity of California, Los Angeles, California 90095, Department of Electrical Engineering, UniVersity of Southern California, Los Angeles, California 90089, and Lumera Corporation, Bothell, Washington 98011 ReceiVed: December 19, 2007; ReVised Manuscript ReceiVed: January 28, 2008
New developments in optical nonlinear polymers have now enabled the implementation of novel new devices. The most important element of these new devices is usually a coupled ring resonator system that can be tuned with an electric field. In this study we have analyzed the design considerations, for these rings, and show how to determine and optimize all of the critical parameters. In addition, we demonstrate how to make the key characterization tests that confirm the component’s performance. All of this is structured so that it can be extended to the newer and higher electro-optic coefficient materials that are currently being fabricated. Future devices using this technology are expected to play a major role in the rapidly expanding field of optical signal processing. I. Introduction Recently Larry Dalton, in collaboration with Alex Jen, has revolutionized the field of nonlinear polymer materials for photonics. The nonlinear electro-optic coefficients now achieved have exceeded the ranges once considered feasible. This dramatic improvement has opened up new and important application areas. These include the conventional applications such as high-speed optical modulators and optical switches. Of even greater interest is the role of these polymers in developing high speed analog and digital signal processing devices. These devices really require the fabrication flexibility and high coefficients provided by these new materials. Operation at speeds greater than 150 GHz and half-wave voltages (Vπ) below 1/2 V are now possible. In this paper, we review the design and testing of the key element of the optical signal processing devices, the coupled Ring Resonator. This element we found is the main building block in this new technology and must be carefully configured and modeled. On the basis of this, we have already made first generation filters, real-time delays, arbitrary waveform generators and linearized modulators.1 We feel that this is just the first step, using the new materials, in a shift from high speed electronic systems to one which is largely photonic. II. Review of Ring Resonators The theory of ring resonators has been understood well in numerous studies.2–7 A simple schematic diagram of a ring resonator is shown in Figure 1. The rings we consider in this paper are basically racetracks since we must have reasonable length and also couple to optical transmission line structures. In next generation we will also couple rings to each other to obtain interesting cascade interference effects. Its optical transfer function, H, which is the complex ratio of input and output electric fields, can be summarized as * Corresponding author. † Part of the “Larry Dalton Festschrift” special issue. ‡ Department of Electrical Engineering, University of California, Los Angeles. § Department of Electrical Engineering, University of Southern California. | Lumera Corporation. ⊥ Current address: Jet Propulsion Laboratory, Pasadena, California 91109.
H ) Rce jτcω
t/ - Re jτωe jφ 1 - tRe jτωe jφ
(1)
where ω is optical angular frequency, τ (τc + τr) is the round trip time, R (RcRr) is the round trip loss factor, t is the transmission coefficient of the coupler, τc and τr are the transition time in the coupler and in the ring, and Rc and Rr are the optical loss factor in the coupler and ring, respectively. The optical loss factors represent the electric field attenuation on a linear scale and become unity in lossless waveguides. An additional optical phase change, φ, is due to the applied voltage to the electrode on the ring. The t can be also tuned by another applied voltage to the electrode on the coupler. It is a complex number in general since the optical phase as well as the amplitude of the propagating electric fields change due to applied voltages. III. Design Overview III.A. Considerations. Under the assumption that polymer materials are given, we begin designing a ring resonator by determining a waveguide structure. Two requirements are distinguished when selecting the type and dimensions of the waveguides. First, they should be highly confined waveguides because of the bending structure in the ring. Second, their confinement is low enough so that a reasonable amount of energy can be coupled. Note that these two factors are competing against each other. Furthermore, the waveguides also should be single moded to ensure no signal degradation occurs via modal dispersion.
Figure 1. A schematic diagram of a ring resonator. The rings we consider are basically racetracks since we must have reasonable length and also couple to optical transmission line structures.
10.1021/jp711914a CCC: $40.75 2008 American Chemical Society Published on Web 05/08/2008
7954 J. Phys. Chem. C, Vol. 112, No. 21, 2008
Seo et al. TABLE 1: Computed r, FSR, and Maximum Electrode Length over Ring (Le) with Given L and R when Propagation Losses for Round and Straight Waveguides Are Assumed as 5 and 2 dB/cm
Figure 2. Cross section of the designed optical waveguide. It is a single-mode waveguide whose confinement is high enough to support a 1-mm radius bending yet low enough to be coupled to an adjacent waveguide.
If we design the configurable coupler to control the coupling to the ring, there are additional considerations to fabrication. Since the optical coupler is sensitive to fabrication tolerance, the coupling effect is easily changed by fabrication error, the structure of the waveguides, and separation of two waveguides. Therefore, it becomes important to minimize fabrication errors and to design the optical waveguides and couplers such that each coupler is operating in the most tolerant region. III.B. Assumptions. It is well known that electro-optic effects are not suitable for the application that requires large refractive index change (e.g., n > 10-3). Having the results of electro-optic polymeric modulators fabricated previously as a reference,8–13 we set our design goal of a refractive index change less than 2 × 10-4 for the phase shifting or coupling operation. This amount of refractive index change corresponds to the voltage change about 20 V, assuming that the operating wavelength and r33 coefficient are 1.55 µm and 30 pm/V. Therefore, we assume the maximum voltage on the electrode is (20 V. This will significantly improve as we introduce the new materials developed by Professors Dalton and Jen. However we have focused this study on an existing, well-defined, available polymer. As the devices are developed with the new technology we expect that the operating voltages may be reduced to below 1 V. DH6/APC (Lumera Co.) is used as the initial demonstration electro-optic polymer core material. Single-layer films of DH6/ APC have shown an electro-optic coefficient r33 of 70 pm/V at 1.31 µm.14 For lower and upper cladding polymers, UV15LV (Master Bond Co.) and UFC170A (Uray Co.) are used. The index at 1.55 µm of the core is measured to be 1.61, and the indices of the lower and upper claddings have been measured to be 1.51 and 1.49, respectively.15 The channel-type or the rib-type waveguides can be considered when the electro-optic polymer material is used. In general, the channel-type waveguide has a more confined beam profile than the rib-type waveguide so that the optical propagation loss is smaller than rib-type waveguides when it is bent. Therefore, a number of applications which include high bending structures such as micro rings have used the channel waveguides.15 However, strong confinement of the channel waveguide makes coupling between waveguides difficult. According to our study, the waveguide separation between two single-mode channel waveguides must be smaller than 1 µm when fabricated with our polymer materials. We consider the inverted rib waveguide structure, which is similar to a usual rib waveguide as shown in Figure 2 and is known to be effective in minimizing the scattering loss from sidewall roughness.10
L [mm]
R [mm]
Le [mm]
R
FSR [GHz]
2 2 2 2 4 4 4 4
0.8 1.0 1.2 1.4 0.8 1.0 1.2 1.4
7.0 8.3 9.5 10.8 9.0 10.3 11.5 12.8
0.68 0.64 0.59 0.55 0.62 0.58 0.54 0.50
22 19 17 16 15 14 13 12
IV. Design of Ring Resonator Using Electro-Optic Polymer Waveguides Figures 1 and 2 show the ring resonator structure and the cross section of the waveguides at the coupler section. By designing ring resonators, we determine the physical dimensions: ring radius (R) and coupler length (L) in Figure 1 and rib thickness (t), rib height (h), rib width (w), and coupler separation (s) in Figure 2. IV.A. Determination of R and L. According to our earlier studies, the straight waveguide propagation loss in this initial demonstration material is around 2 dB/cm and the overall loss when the waveguides are bending (assume modest bending) is around 5 dB/cm. This information enables us to estimate design ranges of R and L since they are directly related to R and the free spectral range (FSR) as calculated in Table 1. Table 1 implies that the perimeter must be large enough to have an enough phase shift (φ) at the ring. On the other hand, the round trip loss will be too large if the perimeter is large. Therefore, we consider ranges of R and L combinations for our example design in section V. In parts C and D of section IV, we confirm that the designed radii are large enough that the bending loss is negligible and that the designed interaction lengths L are also large enough to permit reconfigurability of the coupler. IV.B. Determination of w, t, h, and s. In deterimnation of w, t, and h, we consider following four design factors. First, w, t, and h should be designed such that they satisfy the singlemode condition. The three lines in Figure 3 specify the boundaries of single mode conditions for t and h when w is 2, 3, and 4 µm, respectively. Figure 3 is obtained according to the analytical method investigation.16 Second, they satisfy the fabrication technology employed. If the conventional UV lithography is used, the conservative smallest limit for w is 2 µm. Furthermore, if we consider the conventional spin-coating method for the active core layer, the conservative maximum
Figure 3. Single-mode condition for the waveguide structure considered in Figure 2.
Design of Ring Resonators limit for t is 1 µm due to the dent of ribs.10 Third, for the bending loss point of view, we prefer larger w, thiner t, and higher h. Fourth, for coupler point of view, smaller w, thicker t and smaller h is needed. Note that the third and fourth criteria are exactly opposite. For our example design, we consider first three design factors first and then examine the fourth factor to confirm that our design solutions are valid. Consider three sets of test values of [w, t, h] (in units of µm): (1) [4,2,1], (2) [3,1.5,1], and (3) [2,1,1]. These sets are limits in terms of the third condition yet satisfy the first and second conditions. By use of our bending loss simulation method discussed in part C of section IV, we found only (3) has acceptable bending loss for the radii we already designed in part A of section IV. Therefore, we design the rib width, rib height, and slab thickness to be 2, 1, and 1 µm, respectively. Later in part D of section IV, this design set also satisfies the fourth design condition. We also use two different commercial software packages, Fimmwave (Photo Design Inc.) and BeamPROB (RSOFT Inc.) for the numerical simulations in this paper. Both numerical simulations and experiment show that this waveguide supports a single mode. Similar waveguide structures using similar polymer material are also used in ref 2, where bending waveguides with a 1-mm bending radius have a loss of approximately 4 dB/cm at 1.3µm. IV.C. Waveguide Bending. There are three different opticalloss mechanisms in optical waveguides: material loss, scattering loss, and bending loss. Material loss is power loss absorbed inside the material due to absorptions and imperfections in the bulk waveguiding material. Scattering loss is due to imperfect surface roughness at the interface of the core and cladding in both straight and curved waveguides. Bending loss is power leakage when the waveguide is bent and primarily determined by the waveguide confinement factor. We use the finite different method (FDM) and conformal transformation technique17 to find the optimal radius and the bending loss of the bending waveguide. The zero boundary condition is assumed for all the boundaries except the leaky side of the waveguide (where the energy is leaking). On the leaky side, we apply a plane-wave boundary condition.18 As a result, we calculate the bending loss as 0.02 dB/cm when the bending radius is 1 mm. This bending loss is much smaller than the propagation loss of the straight waveguide. Therefore we ignore the bending loss if the bending radius is larger than 1 mm. However, our experimental results (an example is discussed in section V) show overall propagating loss is ∼5-6 dB/cm for our bending radii. We believe that this is because the scattering loss becomes dominant for this regime, which is difficult to estimate theoretically due to lack of surface roughness statistics of our waveguides. Rather, our experimental measurement enables us estimate the roughness statistics. IV.D. Coupler Design. IV.D.1. Considerations. The cross section of the coupler is shown in Figure 2. Two symmetric waveguides are located a distance s apart. The energies carried by the two waveguides are coupled to each other. The driving electrode on top of only one waveguide applies an electric field to change the refractive index of the waveguide and to tune the amount of coupling (t). To optimize the operation of the coupler, the following should be considered. First, its interaction length should be optimized. Determining L has a tradeoff between the total round trip loss and tunability of the coupler. Second is that the coupler should be designed at the critical coupling state of the ring resonator. Since the ring resonator is most sensitive to the tuning of the
J. Phys. Chem. C, Vol. 112, No. 21, 2008 7955 coupler in its critical coupling state,3 it would be ideal for the transmission coefficient, t, to be matched to the round trip loss factor, R. Note that R is also coupled to L. Lastly, we need to find out the smallest separation length of two waveguides that can be utilized for configurable couplers. A coupler is configured by the electrode on top of the optical waveguide. To make the configurable coupler most sensitive to the applied voltage, the electric field generated by an applied voltage should be confined to only one waveguide. If the separation of two adjacent waveguides is too small, the amount of configurable coupling will decrease. We discuss these issues in this section. IV.D.2. ReWiew of Coupler Theory. According to the coupledmode theory, the coupling efficiency, η(|t|2), of a codirectional coupler, which is defined as the power ratio from one waveguide coupled to the other, can be denoted with theoretical variables such as coupling coefficients and the phase mismatches as
η)
|κab|2 β2c
sin 2 ( βcL)
(2)
where
βc ) √κabκba + δ2 δ)
βb-βa + κaa - κbb 2
κVu ) [C-1κˆ Vu]Vu cVu )
∫-∞∞ ∫-∞∞ (EV* × Hu × Eu × HV*)zˆdxdy ) cuV* ∞ ∞ κˆ Vu ) ω∫-∞ ∫-∞ EV*∆uEudxdy
It is evident that the theoretical variables used such as βc, δ, and κ are functions of physical parameters such as w, t, h, and s in Figure 2. It is generally difficult to find the theoretical variables from the physical parameters since all physical parameters are coupled to the theoretical variables as shown in eq 2. Instead of solving the exact solution, we can find an approximated solution with assumptions of: (1) κab ) κba ) κ > 0, and real: Lossless Symmetric Assumption. (2) κaa ) κbb ) 0: No Self-Coupling. (3) κ is only a function of the separation between two waveguides. (4) δ is only a function of the physical parameters which is related to the cross section of the coupler except the separation between two waveguides. These assumptions decouple the relation between the theoretical variables and the physical parameters. Then eq 2 becomes
η(L, κ, δ) )
κ2 sin 2(√κ2 + δ2L) κ2 + δ2
(3)
where κ is a coupling coefficient and δ is a phase difference due to applied electric field. Using Fimmwave, we find κ as a function of s for our designed waveguide structure as shown Figure 4a. By assumption that we apply (20 V to the electrode, that r33 is 30 pm/V, and that wavelength is 1.55 µm, we find configurable range of t with respect to L and s using eq 3 as listed in Table 2. The calculated values are in good agreement of our numerical simulation results using both Fimmwave and BeamPROB as shown in parts b-d of Figure 4, where the transmission coefficient t is plotted as a function of s when L is 2, 4, and 6
7956 J. Phys. Chem. C, Vol. 112, No. 21, 2008
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Figure 5. (a) A simulated result of the voltage potential when a DC voltage is applied to the microstrip electrode. (b) Electric field intensities at the core polymer region. The field is decaying exponentially and the electric field right underneath the electrode is approximately 70% larger than 4 µm away from the electrode. Therefore, we design configurable couplers such that the separation of two adjacent waveguides is larger than 4 µm.
Figure 4. Numerical simulation results for the couplers. Different lines correspond to different applied voltages. The coupler is designed such that its transmission coefficient is matched to the value of R since the ring resonator is most sensitive to tuning of the coupler in its critical coupling state.
TABLE 2: Configurable Range of t Calculated from Eq 3 L [mm]
S [µm]
t
2 2 2 4 4 4 6 6 6
4.5 4.6 4.7 4.3 4.6 5 4.7 5.1 5.5
0.64–0.7 0.69–0.74 0.74–0.78 0.5–0.9 0.0–0.85 0.5–0.8 0.6–1.0 0.0–1.0 0.5–9.5
mm, respectively. Different lines in the figures correspond to different applied voltages. Table 2 implies that the coupler length L should be large to have large tunability of t. Note that this competes with the requirement we discussed in part A of section IV. For our example device, we choose the coupler length to be 2 mm, and we design the coupler separation around to be 4.5 µm since the transmission coefficient should be around 0.6-0.7. IV.D.3. Smallest Separation for Configurable Couplers. Figure 5 shows a simulated result of the voltage potential when a DC voltage is applied to the electrode. The simulation is done with the finite element method (FEM) using Matlab, which finds an electromagnetic solution for the Poisson equation with proper boundary conditions. The electrode width and thickness are assumed to be 17 µm and negligible, respectively. The refractive index for the polymer materials are assumed to be 1.5 for the applied electric field. The separation between driving and ground electrode is 7 µm. The inverted rib optical waveguides at the coupler section is also shown in Figure 5a. Individual waveguides are the same as shown in Figure 2, and the separation of the two waveguides is assumed to be 4 µm. Since the gradient of the potential is the electric field, we find the electric field along y direction is strong right underneath the electrode and becomes weaker at the two electrode edges. Figure 5b shows electric field intensities at the core polymer region (y ) 3.0 µm and y ) 4.0 µm) along the vertical direction (y direction) as a function of horizontal location (x direction).
Figure 6. Experimental setup and spectral response of the ring resonator. The free spectral range (FSR), extinction ratio, finesse, and Q factor are measured as 0.12 nm (15.5 GHz), 18 dB, 3.36, and 4.34 × 104, respectively.
The field is decaying exponentially, and the electric field right underneath the electrode is approximately 70% larger than 4 µm away from the electrode. Therefore, we design configurable couplers such that the separation of two adjacent waveguides is larger than 4 µm. V. Verification of Ring Resonator First, we characterize the spectral response of the ring resonator. Figure 6 shows the spectral response of the ring resonator and its experimental measurement setup. The measured ring resonator has the design parameters [R, L, w, h, t, s] of [1.2 mm, 2 mm, 2 µm, 1 µm, 1 µm, 4.5 µm] . An AQ4321D (Ando) is used for the tunable laser source at 1.55 µm with TM mode polarization control. As seen in Figure 6b, the measurement is done in a 0.6-nm wavelength span, and the FSR, extinction ratio, finesse, and Q factor (loaded) are measured as 0.12 nm (15.5 GHz), 18 dB, 3.36 and 4.34 × 104, respectively. By use of these values, the effective group refractive index, R value, t value, total round trip optical loss, and the optical loss inside the ring are calculated as 1.66, 0.608, 0.535, 4.4, and 3.85 dB/cm, respectively. Figure 6b also shows the simulated spectral
Design of Ring Resonators
Figure 7. Spectral response of the ring resonator when (20 V is applied to the coupler. By fitting the measured response, we find that t becomes 0.535 ( 0.25 with an applied voltage to the coupler of (20 V.
J. Phys. Chem. C, Vol. 112, No. 21, 2008 7957
Figure 9. Measured t for the various couplers, which have s of 4.5, 4.6, and 4.5 µm and L of 2 mm.
inside the couplers are mismatched already due to imperfect fabrication, which has also been found in conventional electrooptic Mach-Zehnder devices.19 On the basis of the measurements, we summarize the parameters and their configurable range as follows: The FSR is 0.12 nm (15.5 GHz), and the R value is 0.608. The transmission coefficient, t, varies between 0.535-0.25 and 0.535 + 0.25 with (20 V applied voltage to the coupler electrode. The phase, φ, can be fully configurable (0 to 2π) with (30 V applied voltage to the ring electrode. VI. Conclusion Figure 8. Intensity response of the ring resonator when (20 V peakto-peak triangular signal is applied to the phase shifter. The measurement shows that the half-wave voltage (Vπ) of the φ phase shifter is 18.3 V. The corresponding r33 coefficient is also calculated as 23 pm/V.
response using these values. Since the propagation loss of the straight waveguide is measured around 2 dB/cm, the excess loss inside the ring is from scattering loss due to the roughness at the interface between core and cladding materials. We applied voltages of (20 V on the t electrode to verify the operation of the coupler. Figure 7 shows the spectral response while varying the driving voltages. The spectral response when no voltage is applied is shown as well for purpose of comparison. As seen in Figure 7, the response (extinction ratio) is changed depending on the voltage. By fitting the measured response, we find that t becomes 0.535 ( 0.25 with an applied voltage to the coupler of (20 V. We also find that the local minima shift with applied voltage. For verification of the φ phase shifter, we fix the input optical wavelength at 1.55 µm and apply (20 V peak-to-peak triangular signal to the ring electrode. Its response is shown in Figure 8. The measurement shows that the half-wave voltage (Vπ) of the phase shifter is 18.3 V. The corresponding r33 coefficient is also calculated as 23 pm/V. We also fabricate and test various couplers, which have s values of 405, 4.6, and, 4.5 µm and an L value of 2 mm. From the measured responses, we calculate the transmission coefficients t for the couplers, which are shown in Figure 9. Comparing to the simulated coupler in section IV, the measurements show that the measured transmission coefficients are different from the simulated values due to imperfect device fabrication. Furthermore, the different polarity of voltages leads to different output response even though output responses should be even functions with respect to applied voltages since the waveguides are symmetric. This implies that two waveguides
We have examined considerations for designing a ring resonator using electro-optic polymer waveguides. We have also shown that practical implementation can be made using the current state of polymer technology. The next step is to extend this technology to the newer materials that have been developed. This should be a straightforward process using relatively similar dimensions and producing significant reductions in operating voltages and optical losses. Finally, in a further enhancement the operational lengths will be shortened to obtain very high frequency operation. References and Notes (1) Seo, B-J. Optical signal processor using electro-optic polymer waveguides, Ph.D. dissertation; University of California: Los Angeles, 2007. (2) Tazawa, H.; Kuo, Y.-H; Dunayevskiy, I.; Luo; J; Jen, A. K.-Y; Fetterman, H. R.; Steier, W. H. Ring resonator-based electrooptic polymer traveling-wave modulator. IEEE J. LightwaVe Technol. 2006, 24 (9), 3514– 3519. (3) Choi, J. M. Ring fiber resonators based on fused-fiber grating adddrop filters: Application to resonator coupling. Opt. Lett. 2002, 27, 18. (4) Yariv, A. Critical coupling and its control in optical waveguidering resonator systems. IEEE Photonics Technol. Lett. 2002, 14 (4), 483– 485. (5) Jinguji, K. Synthesis of coherent two-port optical delay-line circuit with ring waveguides. IEEE J. LightwaVe Technol. 1996, 14, 8. (6) Kaalund, C. J.; Peng, G.-D. Pole-zero diagram approach to the design of ring resonator-based filters for photonic applications. IEEE J. LightwaVe Technol. 2004, 22 (6), 1548–1559. (7) Seo, B.-J.; Fetterman, H. R. True time delay element in lossy environment using eo waveguides. IEEE Photonics Technol. Lett. 2006, 18 (1), 10–12. (8) Zhang, H.; Oh, M-.C; Szep, A.; Steier, W. H; Zhang, C.; Dalton, L. R; Chang, D. H.; Fetterman, H. R. Push-pull electro-optic polymer modulators with low half-wave voltage and low loss at both 1310 and 1550 nm. Appl. Phys. Lett. 2001, 78 (20), 3136–3138. (9) Chang, D. H.; Erlig, H.; Oh, M; Zhang, C; Steier, W. H.; Dalton, L. R.; Fetterman, H. R. Time stretching of 102-ghz millimeter waves using novel 1.55mu m polymer electrooptic modulator. IEEE Photonics Technol. Lett. 2000, 12 (5), 537–539. (10) Kim, S.-K; Zhang, H.; Chang, D. H; Zhang, C.; Wang, C; Steier, W. H.; Fetterman, H. R. Electrooptic polymer modulators with an invertedrib waveguide structure. IEEE Photonics Technol. Lett. 2003, 15, 2.
7958 J. Phys. Chem. C, Vol. 112, No. 21, 2008 (11) Chen, D; Fetterman, H. R.; Chen, A.; Steier, W. H; Dalton, L. R; Wang, W.; Shi, Y. Demonstration of 110 ghz electro-optic polymer modulators. Appl. Phys. Lett. 1997, 70 (25), 3335–3337. (12) Oh, M. C.; Zhang, H.; Szep, A.; Chuyanov, V.; Steier, W. H; Zhang, C.; Dalton, L. R.; Erlig, H.; Tsap, B.; Fetterman, H. R. Electro-optic polymer modulators for 1.55 µm wavelength using phenyltetraene bridged chromophore in polycarbonate. Appl. Phys. Lett. 2000, 76, 3525–3527. (13) Oh, M. C.; Zhang, H; Zhang, C.; Erlig, H.; Chang, Y.; Tsap, B.; Chang, D.; Szep, A.; Steier, W. H; Fetterman, H. R.; Dalton, L. R. Recent advances in electrooptic polymer modulators incorporating highly nonlinear chromophore. IEEE J. Selected Top. Quantum Electron. 2001, 7, 5. (14) Tengi, C. C.; Man, H. T. A simple reflection technique for measuring the electro-optic coefficient of poled polymers. Appl. Phys. Lett. 1990, 56, 1734–1736.
Seo et al. (15) Rabiei, P.; Steier, W. H; Zhang, C.; Dalton, L. R. Polymer microring filters and modulators. IEEE J. LightwaVe Technol. 2002, 20 (11), 1968– 1975. (16) Soref, R.; Schimdtchen, J.; Peterman, K. Large single-mode rib waveguides in gesi-si and si-on-sio. IEEE J. Quantum Electron. 1991, 27 (8), 1971–1974. (17) Heiblum, M. Analysis of curved optical waveguides by conformal transformation. IEEE J. Quantum Electron. 1975, 11 (2), 75–83. (18) Nesterov, A.; Troppenz, U. Plane-wave boundary method for analysis of bent optical waveguides. IEEE J. LightwaVe Technol. 2003, 21 (10), 1.4. (19) Geary, K.; Kim, S.-K; Seo, B.-J.; Fetterman, H. R. Mach-zehnder modulator arm length mismatch measurement technique. IEEE J. LightwaVe Technol. 2005, 23, 1273–1277.
JP711914A
