Broadened phase-matching bandwidth in waveguide-frequency-doubling devices

Transcription

1 Broadened phase-matching bandwidth in waveguide-frequency-doubling devices Rabi Rabady Electrical Engineering Department, Jordan University of Science and Technology, P.O. Box 3030, Irbid 110, Jordan Received 6 July 009 revised 1 October 009 accepted 6 October 009 posted 7 October 009 (Doc. ID ) published 10 November 009 Second harmonic generation in optical planar waveguides is the most promising mechanism for frequency doubling of laser emission since light can be highly confined to the nonlinear waveguide medium. However, this advantage is achievable only by precise phase matching between the fundamental wave and the doubled frequency wave, which is hard to control at the fabrication stage. Two tasks are addressed: a basic design for the layer thicknesses of a two-layer waveguide to achieve phase matching, and second, a multi-layer-waveguide design to achieve broadened phase matching bandwidth. 009 Optical Society of America OCIS codes: , , , Introduction Naturally short wavelength coherent sources from a compact semiconductor and solid state lasers are not available. Nevertheless, second harmonic generation in a nonlinear medium can be utilized to achieve frequency doubling of light, and consequently to realize coherent sources of light at blue and ultraviolet wavelength, which is useful for many important applications as in high-density optical storage and laser printers. The phase matching (PM) between the fundamental wave and the doubled-frequency wave represents a major concern when it is desired to achieve effective conversion efficiency, in addition to other factors, such as the nonlinearity coefficient of the waveguide layers media, the power of the fundamental wave, the dimensions (length and cross-sectional area) of the waves interacting region, and the overlap integral of the fundamental and second harmonic guided fields [1,]. Normally, the PM bandwidth is narrow, which makes it highly sensitive to the material constants uncertainty and fabrication errors. Broadening the PM bandwidth should contribute directly to higher second harmonic generation (SHG) /09/ $15.00/0 009 Optical Society of America efficiency by allowing more power flow from the fundamental wave to the second harmonic (SH) wave during the conversion process. Additionally, the conversion efficiency can be sensitive to the device temperature variation that alters the materials optical constants. The design of a frequency-doubling device that tolerates the unavoidable phase mismatch should lead to a stable and useful amount of the doubled-frequency output light. A common technique is the Cherenkov second harmonic generation scheme [3], which allows the second harmonic wave to propagate in the substrate with a large and dense number of propagating modes, thus increasing the chances to satisfy the phased matching condition with a propagating mode of the fundamental wave to the practical level. Moreover, one could enhance the phase matching tolerance by increasing the PM bandwidth, which depends on a diffractive grating that is integrated within the device and has a randomized or chirped period to broaden the allowed values for the waveguide effective refractive indices of the fundamental and second harmonic waves and, hence, increases their overlap and effectively broadens the PM bandwidth [4]. The main focus of this paper is to enhance the tolerance of the frequency-doubling device against the narrow PM bandwidth by allowing the waveguide effective refractive index of the SH wave to be tightly 0 November 009 / Vol. 48, No. 33 / APPLIED OPTICS 6417

2 sandwiched between two (or more) waveguide effective refractive indices of the fundamental wave. Generally, the simple frequency-doubling device can be configured as shown in Fig. 1 by using a two-layer waveguide, where the maximum conversion factor is obtained when the effective refractive indices of the two waves are equal. The first proposed approach is configured as in Fig. (a), which depends on designing the thicknesses of a three-layer waveguide such that two refractive indices of the fundamental wave tightly sandwich the effective refractive index of the SH wave consequently, the effective PM bandwidth is approximately doubled as illustrated in Fig. (b), where the dashed bold curve represents the effective broadened-bandwidth conversion factor profile. The second proposed approach is configured as in Fig. 3(a), which depends on designing the thicknesses of a four-layer waveguide such that the effective refractive index of the SH wave is aligned with one of the effective refractive indices of the fundamental wave and is tightly sandwiched between the other two effective refractive indices consequently, the effective PM bandwidth is approximately tripled as illustrated in Fig. 3(b), where the dashed bold curve represents the effective broadened conversion factor profile. Generally, the principle of modal sandwiching can be applied to a larger number of guided modes to realize further bandwidth broadening that satisfies the needs. Obviously, the proposed approaches cannot be realized by a single-layer thin film waveguide since the number of constraints should be equal to or less than the degrees of freedom of the design, which are chosen as the waveguide layers thicknesses. Moreover, the number of required layers can be increased further to account for additional constraints, such Fig.. (Color online) (a) Three-layer waveguide assisted by top diffractive gratings for coupling the fundamental wave. A proposed scheme that enables the design of layer thicknesses to achieve a broadened conversion factor profile. (b) Modal sandwiching of the second harmonic guided mode by two fundamental guided modes. A proposed scheme that enables achieving a broadened conversion factor profile. Fig. 1. (Color online) Two-layer waveguide assisted by top diffractive gratings for coupling the fundamental wave, a scheme that enables the design of layer thicknesses to satisfy phase matching condition. as achieving the maximum integral overlap between the fundamental and the SH guided fields. Nevertheless, the focus of this work is to present a new method for broadening the PM bandwidth by modal sandwiching. The recent advancements in electro-optical polymers [5], which show higher electro-optical coefficient than inorganic crystals, made them attractive to be used in frequency-doubling devices. Therefore, the thin films are assumed in this paper to be electrooptical polymers, which not only eliminates the challenge of achieving lattice constant matching between neighboring layers but also simplifies the design since there should be only a single type of ray and a single refractive index for each layer instead of 6418 APPLIED OPTICS / Vol. 48, No. 33 / 0 November 009

3 objective of achieving modal sandwiching. All of the proposed tasks are based on designing the waveguide layer thicknesses to control the multilayer waveguide effective refractive indices, which is governed by the modal equation [6] f ðn ÞM 11 ξ 0 M 1 ξ pþ1 M 1 þ ξ 0 ξ pþ1 M 1 0 ð1þ where M ij is the ði jþ component in a M matrix that is defined as M Yp M11 M 1 m l M 1 M cosðql iξ 1 l iξ l sinðq l sinðq l m l cosðq l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q l k n l ðn Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξ l n l ðn Þ ðþ Fig. 3. (Color online) Four-layer waveguide assisted by top diffractive gratings for coupling the fundamental wave. An advanced proposed scheme that enables the design of layer thicknesses to achieve a broadened conversion factor profile. (b) Modal sandwiching of the second harmonic guided mode by three fundamental guided modes. A proposed scheme that enables achieving a broadened conversion factor profile. two refractive indices (ordinary and extraordinary) as in organic crystalline thin films. In the following, the theory and numerical simulation that supports and clarifies the proposed designs are presented.. Theory The main objective here is to design for the layer thicknesses of multi-layer waveguide with specific refractive indices, as shown in Figs. 1, (a), and 3(a), to satisfy certain conditions that are decided by the where p is the number of waveguide layers, n is the waveguide effective refractive index, and k is the propagation constant in free space. The layers refractive indices are assumed to be defined, whereas, the layer thicknesses are to be determined. Practically, it would be more reliable to control the thickness than the optical constants of the thin film at the fabrication stage. The fundamental wave is coupled to the frequency-doubling device by a shallow diffractive gratings coupler as shown in Figs. 1, (a), and 3(a). The first case of design, shown in Fig. 1, is to phase match the fundamental wave with the SH wave, which requires determining the first and second layer thicknesses ðt 1 t Þ of a multilayer waveguide that is made of two nonlinear thin films with defined refractive indices for the first and second layers as ðn 1f n f n 1s n s Þ, where the subscript letters ðf sþ refer to the fundamental and SH wavelengths, respectively. The effective refractive index of the two-layer waveguide should be chosen greater than the maximum of ðn 3f n 3s Þ and smaller than the minimum of ðn 1f n f n 1s n s Þ, where n 3f and n 3s are the substrate refractive index at the fundamental and SH wavelengths, respectively. Practically, the refractive index of the waveguiding layer increases as the layer becomes closer to the substrate, and the refractive index for the same layer is higher at the SH wavelength. Therefore, the waveguide effective refractive index should be chosen between n 3s and n f. Once the waveguide effective refractive index is decided, Eq. (1) can be employed twice, for the fundamental wave and the SH wave, with p and using the same waveguide effective refractive index to impose the PM condition. This should lead to two nonlinear equations for ðt 1 t Þ as follows: 0 November 009 / Vol. 48, No. 33 / APPLIED OPTICS 6419

4 f 1 ðn ÞM 11f ξ 0 M 1f ξ 3f M 1f þ ξ 0 ξ 3f M 1f 0 f ðn ÞM 11s ξ 0 M 1s ξ 3s M 1s þ ξ 0 ξ 3s M 1s 0 where M f Y m lf M 1f M 1f M f ð3þ iξ 1 lf sinðq lf m lf iξ lf sinðq lf cosðq lf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q lf k n lf ðn Þ ξ lf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n lf ðn Þ ð4þ M s Y m ls M11s M 1s M 1s M s cosðqls iξ 1 ls sinðq ls m ls iξ ls sinðq ls cosðq ls qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ls k n ls ðn Þ ξ ls qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ls ðn Þ : ð5þ The set of the two nonlinear equations, Eqs. (3), can be solved numerically, as clarified by example 1 in Section 3. A similar procedure can be repeated for the second case of design that depends on designing the thicknesses of a three-layer waveguide shown in Fig. (a) to facilitate a PM bandwidth broadening by a tight sandwiching of the waveguide effective refractive index n of the SH wave between two waveguide effective refractive indexes ðn 1 n 3Þ of the fundamental wave such that n 1<n <n 3 and Δn ðn 3 n 1Þ, as shown in Fig. (b). Therefore, Eq. (1) can be employed three times and yields f 1 ðn 1ÞM 11f 1 ξ 0f 1 M 1f 1 ξ 4f 1 M 1f 1 þ ξ 0f 1 ξ 4f 1 M 1f 1 0 f ðn 3ÞM 11f ξ 0f M 1f ξ 4f M 1f þ ξ 0f ξ 4f M 1f 0 f 3 ðn ÞM 11s ξ 0 M 1s ξ 4s M 1s þ ξ 0 ξ 4s M 1s 0 where ð6þ 1 M 1f 1 M 1f 1 M f 1 1 iξ 1 lf 1 sinðq lf 1 iξ lf 1 sinðq lf 1 cosðq lf 1 k n lf ðn 1Þ n lf ðn 1Þ ð7þ M f 1 Y3 m lf 1 m lf 1 q lf 1 ξ lf 1 M 1f M 1f M f iξ 1 lf sinðq lf iξ lf sinðq lf cosðq lf k n lf ðn 3Þ n lf ðn 3Þ ð8þ M f Y3 m lf m lf q lf ξ lf M s Y3 m ls M11s M 1s M 1s M s cosðqls iξ 1 ls sinðq ls m ls iξ ls sinðq ls cosðq ls q ls k n ls ðn Þ ξ ls n ls ðn Þ : ð9þ Again, a similar procedure can be used for the last case of design, which depends on designing the thicknesses of the four-layer waveguide shown in Fig. 3(a) to facilitate a PM bandwidth broadening by a tight sandwiching of the waveguide effective refractive index n 4 of the SH wave between the waveguide effective refractive indices ðn 1 n n 3Þ of the fundamental wave such that n 1 < n < n 3, n 4 n, and Δn ðn 3 n 1Þ, as shown in Fig. 3(b). Therefore, Eq. (1) can be employed four times and yields f 1 ðn 1ÞM 11f 1 ξ 0f 1 M 1f 1 ξ 5f 1 M 1f 1 þ ξ 0f 1 ξ 5f 1 M 1f 1 0 f ðn ÞM 11f ξ 0f M 1f ξ 5f M 1f þ ξ 0f ξ 5f M 1f 0 f 3 ðn 3ÞM 11f 3 ξ 0f 3 M 1f 3 ξ 5f 3 M 1f 3 þ ξ 0f 3 ξ 5f 3 M 1f 3 0 f 4 ðn ÞM 11s ξ 0 M 1s ξ 5s M 1s þ ξ 0 ξ 5s M 1s 0 where ð10þ 640 APPLIED OPTICS / Vol. 48, No. 33 / 0 November 009

5 M f 1 Y4 m lf 1 1 M 1f 1 M 1f 1 M f 1 m lf 1 1 iξ 1 lf 1 sinðq lf 1 iξ lf 1 sinðq lf 1 cosðq lf 1 q lf 1 k n lf ðn 1Þ ξ lf 1 n lf ðn 1Þ ð11þ M 1f M 1f M f iξ 1 lf sinðq lf iξ lf sinðq lf cosðq lf k n lf ðn Þ n lf ðn Þ ð1þ M f Y4 m lf m lf q lf ξ lf 3 M 1f 3 M 1f 3 M f 3 3 iξ 1 lf 3 sinðq lf 3 iξ lf 3 sinðq lf 3 cosðq lf 3 k n lf ðn 3Þ n lf ðn 3Þ ð13þ M f 3 Y4 m lf 3 m lf 3 q lf 3 ξ lf 3 M s Y4 m ls M11s M 1s M 1s M s cosðqls iξ 1 ls sinðq ls m ls iξ ls sinðq ls cosðq ls q ls k n ls ðn Þ ξ ls n ls ðn Þ : ð14þ Obviously, the three sets of nonlinear Eqs. (3), (6), and (10) do not have a straightforward analytical solution. Numerical root finding methods [7,8], however, can assist to find a solution for the three sets and facilitate an enhanced design for such an important applied technological problem. To quantify the broadening of the phase matching using the modal sandwiching method, we start with the conversion factor versus the detuning from PM condition relation, which is given by [1] FðxÞ sin ðxþ x x klðδn Þ ð15þ where x is the detuning from the PM condition, k is the propagation vector, l is the length of the device, and δn is the effective refractive index mismatch between the fundamental and SH guided modes. And for the case of modal sandwiching of the SH mode by two fundamental modes as in the first scheme, one can write FðxÞ sin ðxþ x þ sin ðx ΔxÞ ðx ΔxÞ x klðδn Þ Δx klðδn Þ ð16þ where Δn is the effective refractive indices between the two fundamental modes [i.e., Δn ðn 3 n 1Þ from Fig. (b)]. One important point that needs to be addressed here is the maximum value of the allowed Δx, and therefore the values of n 3 and n 1, to control the ripple of FðxÞ that is given in Eq. (16), which is associated to the bandwidth broadening by modal sandwiching. Figure 4 shows the conversion factor versus the detuning from PM of the two-mode sandwiching scheme that is described by Eq. (16) the solid curve represents the behavior without modal sandwiching that is given in Eq. (15), the dashed curve represents the behavior given in Eq. (16) when 5% ripple is allowed, and the dotted curve represents the behavior given in Eq. (16) when 0% ripple was allowed. In these, the ripple was considered as the ratio of the difference between the maximum and minimum of FðxÞ within the broadened bandwidth to the maximum value. Obviously, a trade-off between 3 db bandwidth and the allowed ripple is observed when considering two allowed values for the ripple as depicted from Fig. 4. A similar procedure can be followed to quantitate the second scheme of PM bandwidth broadening shown in Fig. 3(b) using three sandwiching fundamental modes of the SH mode with the following conversion factor: FðxÞ sin ðxþ x þ sin ðx Δx 1 Þ ðx Δx 1 Þ þ sin ðx Δx Þ ðx Δx Þ x klðδn Þ Δx 1 klðn n 1 Þ Δx klðn 3 n Þ ð17þ where all parameters are defined as before but with three fundamental modes considerations. It is important to note here that in both schemes the device length plays a major roll in the performance of the conversion process. The more device length that is needed to get more fundamental and SH powers, the less tolerance there is in the detuning from PM condition therefore, less effective broadened PM bandwidth can be realized from both schemes. These trade-offs and other optimization aspects, such as the overlap integrals between the interacting fields [9] can be treated by introducing more degrees of 0 November 009 / Vol. 48, No. 33 / APPLIED OPTICS 641

6 Fig. 4. (Color online) Conversion factor versus the detuning from phase matching: the solid curve represents the behavior without modal sandwiching, and the dashed curve represents the conversion factor profile of the first scheme with modal sandwiching when a 5% ripple was allowed, whereas, the dotted curve represents the conversion factor profile when a 0% ripple was allowed. freedom to the device s design, such as additional layers, which can be a future comprehensive work. In Section 3 some numerical examples are considered to clarify the above schemes. Fig. 6. (Color online) (a) Evolution of layers thicknesses of the two-layer-waveguide design when using genetically assisted random search method to solve Eqs. (3). (b) Error evolution of the twolayer-waveguide design when using genetically assisted random search method to solve Eqs. (3). Fig. 5. (Color online) (a) Evolution of layers thicknesses of the two-layer waveguide design when using the Newton method to solve Eqs. (3). (b) Error evolution of the two-layer waveguide design when using the Newton method to solve Eqs. (3). 3. Simulations and Numerical Examples As stated above, Eqs. (3), (6), and (10) need to be solved numerically, but an appropriate choice of the numerical method is important to obtain reasonable convergence. For the sake of comparison, to be able to adopt an effective method, the Newton method and the genetically assisted random search method were applied for the first case of achieving the basic PM condition in the two-layer waveguide. The Newton method proved to be relatively impractical for solving this kind of nonlinear set of equations since it suffers frequent divergence and demands finding the derivative of the functions set with respect to layer thicknesses, which is highly complicated especially for the higher dimension problems. Additionally, the convergence of the Newton method is sensitive to the initial conditions, which is hard to predict successfully for this type of function. Because of the oscillatory behavior and the complexity of the functions given in Eqs. (3), (6), and (10), the simple random search method was employed and assisted by a primitive genetic method that is based on the survival of the best set of parameters. After comparing the performance of the two methods, the genetically assisted random search method was adopted solely for the second case of three-layer-waveguide 64 APPLIED OPTICS / Vol. 48, No. 33 / 0 November 009

7 Fig. 7. (Color online) (a) Evolution of layers thicknesses of the three-layer-waveguide design when using genetically assisted random search method to solve Eqs. (6). (b) Error evolution of the three-layer-waveguide design when using genetically assisted random search method to solve Eqs. (6). (c) The electric field intensity profile of the fundamental modes and the SH mode across the three waveguide layers for the three-layer waveguide that was proposed in example. design and the third case of four-layer-waveguide design that were dedicated to realize a PM broadened bandwidth. The genetically assisted random search method proved to be more effective compared with other methods that increase their complexity dramatically with the dimension of the problem the higher dimension of the problem was handled by the genetically assisted random search method simply by increasing the number of iterations. Let us demonstrate this by numerical examples. Example 1 The following parameters for the twolayer waveguide shown in Fig. (a) were chosen: n 0 1 n 1f 1:6 n f 1:7 n 3f 1:5 n 1s 1:65 n s 1:73 n 3s 1:5 n 1:59 λ 1000 nm: The subscript number refers to the layer index and the subscript letter ðf sþ refers to the fundamental and SH wavelengths, respectively. The Newton method was employed first to solve Eqs. (3). Figure 5(a) shows the convergence toward a good solution (t 1 551:69 nm and t 351:6 nm) after several failed initial starts. Figure 5(b) shows the evolution of relative error that was taken as the maximum of all thicknesses estimated relative error, and each thickness relative error was estimated as the absolute thicknesses differences between two successive iterations normalized by the corresponding current thicknesses. Afterward, the genetically assisted random search method was employed to solve Eqs. (3) again. Figure 6(a) shows the evolution of the best achieved solutions (t 1 553:77 nm and t 350:39 nm) after performing the random search for 150,000 iterations. Figure 6(b) shows the evolution of the error of the best achieved solutions, which was considered as the maximum value for absolutes of f 1 and f that are given in Eqs. (5). In both methods, the error measurement was chosen to fit the nature of the numerical method. Clearly, the two methods converged to close values, but with two major advantages for the genetically assisted random search: the simple employment and the certainty of convergence. Example II The following parameters for the three-layer waveguide shown in Fig. (a) were chosen: n 0 1 n 1f 1:6 n f 1:7 n 3f 1:75 n 4f 1:5 n 1s 1:6 n s 1:73 n 3s 1:77 n 3s 1:515 n 1 1:55 n 1:5515 n 3 1:553 λ 1000 nm: The genetically assisted random search method was employed to solve Eqs. (6) again. Figure 7(a) shows 0 November 009 / Vol. 48, No. 33 / APPLIED OPTICS 643

8 Fig. 8. (Color online) (a) Evolution of layers thicknesses of the four-layer-waveguide design when using genetically assisted random search method to solve Eqs. (10). (b) Error evolution of the four-layer-waveguide design when using genetically assisted random search method to solve Eqs. (10). (c) The electric field intensity profile of the fundamental modes and the SH mode across the four waveguide layers for the four-layer waveguide that was proposed in example (3). the evolution of the best achieved solutions (t : nm, t 1045:6 nm, and t 3 635:65 nm) after performing the random search for 00,000 iterations. Figure 7(b) shows the evolution of the error of the best achieved solutions, which was considered as the maximum value for absolutes of f 1, f, and f 3 that are given in Eqs. (6). Figure 7(c) shows the electric field intensity of the fundamental modes and the SH mode across the three waveguide layers for the adopted solution such illustration is rather important to check the overlap integrals between the fundamental modes and SH modes for all achieved good solutions and, therefore, to choose that one which achieves the optimal modal sandwiching and overlap integrals. Example III The following parameters for the four-layer waveguide shown in Fig. 3(a) were chosen: n 0 1 n 1f 1:6 n f 1:7 n 3f 1:75 n 4f 1:78 n 5f 1:5 n 1s 1:6 n s 1:73 n 3s 1:77 n 4s 1:80 n 5s 1:515 n 1 1:55 n 1:5515 n 3 1:553 λ 1000 nm: The genetically assisted random search method was employed to solve Eqs. (10) again. Figure 8(a) shows the evolution of the best achieved solutions (t 1 64:61 nm, t 67:43 nm, t 3 594:0 nm, and t 4 10:96 nm) after performing the random search for 600,000 iterations. Figure 8(b) shows the evolution of the error of the best achieved solutions, which was considered as the maximum value for absolutes of f 1, f, f 3, and f 4 that are given in Eqs. (10). Figures 7(a) and 7(b) and Figs. 8(a) and 8(b) show smoothness for the error curves but not the thicknesses curves since the method depends on random guesses in each case and only that random guess of thicknesses that satisfies the corresponding equations set with error that is less than all previously achieved solutions is reserved and recorded Therefore the error curve was smooth and decreasing, whereas, the thicknesses curves were not since the guesses were random. Moreover, it is for the same reason that the final best achieved solution of the thicknesses was considered as the outcome of the method since it has the lowest error. Figure 8(c) shows the electric field intensity of the fundamental modes and the SH mode across the three waveguide layers for the adopted solution. Clearly, the frequency of the field changing direction, which increases with the total number of modes the 644 APPLIED OPTICS / Vol. 48, No. 33 / 0 November 009

9 waveguide have, decreased as compared to the previous example. This is attributed to the fact that the waveguide total thickness in the previous example is larger and, therefore, should carry a higher number of modes, which is well known from the optical waveguide theory. 4. Summary A multilayer electro-optic-polymer multilayer waveguide was proposed and theoretically modeled to achieve a broadened PM bandwidth by designing for the layers thicknesses to allow a tight modal sandwiching of the fundamental guided modes for a SH guided mode. Depending on this principle, two schemes were considered and each produced a number of constrains that dictated the minimum number of layers for the multilayer waveguide. The genetically assisted random search method was employed and proved to be simple and effective for solving the set of modal equations that were generated by imposing the constraints associated with each case. Broadening the PM bandwidth should contributes to higher conversion efficiency of frequencydoubling devices, which can be key a component in many technological applications. The author would like to thank Professor Omar Alasfar for his valuable advice. References 1. A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart and Winston, 1985).. M. Fujimura, T. Suhara, and H. Nishihara, Theoretical analysis of resonant waveguide optical second harmonic generation devices, J. Lightwave Technol. 14, (1996). 3. G. Leo, R. Drenten, and M. Jongerius, Cherenkov secondharmonic generation in multilayer waveguide structures, IEEE J. Quantum Electron. 8, (199). 4. K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation, IEEE J. Quantum Electron. 30, (1994). 5. M. C. Oh, H. Zhang, C. Zhang, H. Erlig, C. Yian, B. Tsap, D. Chang, A. Szep, W. Steier, H. Fetterman, and L. Dalton, Recent advances in electroptic polymer modulators incorporating highly nonlinear chromophore, IEEE J. Sel. Top. Quantum Electron. 7, (001). 6. A. Stratonnikov, A. Bogatov, A. Drakin, and F. Kzamenets, A semianalytical method of mode determination for a multilayer planar optical waveguide, J. Opt. A Pure Appl. Opt. 4, (00). 7. W. Press, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1997). 8. S. Chapra and R. Canale, Numerical Methods for Engineers, 3rd ed. (McGraw-Hill, 006). 9. C. Flueraru and C. P. Grover, Overlap integral analysis for second-harmonic generation within inverted waveguide using mode dispersion phase match, IEEE Photonics Technol. Lett. 15, (003). 0 November 009 / Vol. 48, No. 33 / APPLIED OPTICS 645