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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 3 — Feb. 6, 2006
  • pp: 1243–1252
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Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications

A. S. Helmy  »View Author Affiliations


Optics Express, Vol. 14, Issue 3, pp. 1243-1252 (2006)
http://dx.doi.org/10.1364/OE.14.001243


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Abstract

A novel design to achieve phase matching between modes of a vertical distributed Bragg reflector waveguide and those of a conventional total internal reflection waveguide is reported for the first time. The device design and structure lend themselves to monolithic integration with active devices using well developed photonic fabrication technologies. Due to the lack of any modulation of the optical properties in the direction of propagation, the device promises very low insertion loss. This property together with the large overlap integral between the interacting fields dramatically enhances the conversion efficiency. The phase matching bandwidth, tunability and dimensions of these structures make them excellent contenders to harness optical nonlinearities in compact, low insertion loss monolithically integrable devices.

© 2006 Optical Society of America

1. Introduction

Optical parametric oscillators (OPOs) have become indispensable coherent sources for the mid infra red. Their operating wavelength span is limited by the transparency window of lithium niobate however, because periodically poled lithium niobate (PPLN) is the most commonly used nonlinear element in OPOs. Compound semiconductors such as GaAs, in contrast, exhibit higher nonlinear coefficients near the material resonances in comparison to PPLN, and have a large transparency window. In the case of GaAs the transparency window spans the spectral range 1–17 μm. GaAs also has high optical damage threshold and a mature fabrication technology for making waveguides in comparison to PPLN. Therefore, achieving parametric oscillation in semiconductors monolithically can vastly improve the efficiency of this class of coherent sources [1

01. C. B. Ebert, L. A. Eyres, M. M. Fejer, and J. H. Harris, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227, 183–192 (1999).

]. More importantly, the potential for monolithically integrating these nonlinear elements with active sources to form an OPO chip is an attractive option for numerous applications. A coherent source in this form factor is bound to redefine how coherent sources are used due to its versatility, ruggedness, and compactness [2

02. J. B. Khurgin, E. Rosencher, and Y. J. Ding, “Analysis of all-semiconductor intracavity optical parametric oscillators,” J. Opt. Soc. Am. B 15, 1726–1734 (1998). [CrossRef]

]. It is clear, however, that achieving efficient, low loss and tunable phase matching in a semiconductor material is pivotal for the realization of the monolithic OPOs discussed.

Although semiconductors possess large nonlinearities, they also have large dispersion, particularly at wavelengths close to their bandgap. This makes the problems of phase matching challenging in such material systems. In order to phase match the two interacting waves, their wave vectors must obey the relation k 2ω = 2kω which leads to the condition that requires the effective indices of both waves to be equal in order for perfect phase matching to occur, namely n = nω. The difficulty of achieving this in semiconductors is usually most severe while operating near the bandgap resonances where dispersion prevents such condition from naturally taking place. Various means have been devised to overcome this problem; form birefringence [3

03. A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, and E. Rosencher, “Second-harmonic generation at λ= 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2945 (1998). [CrossRef]

], quasi-phase matching [4

04. A. S. Helmy, D. C. Hutchings, T. C. Kleckner, J. H. Marsh, A. C. Bryce, J. M. Arnold, C. R. Stanley, J. S. Aitchison, C. T. A. Brown, K. Moutzouris, and M. Ebrahimzadeh, “Quasi phase matching in GaAs-AlAs superlattice waveguides via bandgap tuning using quantum well intermixing,” Opt. Lett. 25, 1370–1373 (2000). [CrossRef]

], high Q resonant cavities [5

05. R. Haidar, N. Forget, and E. Rosencher, “Optical parametric oscillation in micro-cavities based on isotropic semiconductors: a theoretical study,” IEEE J. Quantum Electron. 39, 569–576 (2003). [CrossRef]

] and photonic bandgap structures were all studied [6

06. D. Faccio, F. Bragheri, and M. Cherchi, “Optical Bloch-mode-induced quasi phase matching of quadratic interactions in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 21, 296–301 (2004). [CrossRef]

]. On one hand, these solutions produce a route to phase matching; however, they provide devices that are difficult to integrate with other active and passive photonic components. One of these approaches, which uses quasi-phase matching, lends itself to monolithic integration. However, so far it provides imperfect quasi-phase matching which makes the attainable effective nonlinearity too low to be of practical use [4

04. A. S. Helmy, D. C. Hutchings, T. C. Kleckner, J. H. Marsh, A. C. Bryce, J. M. Arnold, C. R. Stanley, J. S. Aitchison, C. T. A. Brown, K. Moutzouris, and M. Ebrahimzadeh, “Quasi phase matching in GaAs-AlAs superlattice waveguides via bandgap tuning using quantum well intermixing,” Opt. Lett. 25, 1370–1373 (2000). [CrossRef]

]. Another very promising demonstration of quasi-phase matching in semiconductors relies on inverting the domain of the nonlinearity by growing the semiconductor with different orientations, and hence perfect quasi-phase matching has been reported [1

01. C. B. Ebert, L. A. Eyres, M. M. Fejer, and J. H. Harris, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227, 183–192 (1999).

]. Monolithic integration using this technique is yet to be demonstrated though, and is likely to prove challenging as the technique involves etch and re-growth steps to invert the domain of the nonlinearity in the semiconductor and hence adding AlGaAs is at best challenging. Operational table-top OPOs were demonstrated using these materials nonetheless [7

07. K. L. Vodopyanov, O. Levi, P.S. Kuo, T.J. Pinguet, J.S. Harris, M.M. Fejer, B. Gerard, L. Becouarn, and E. Lallier “Optical parametric oscillation in quasi-phase-matched GaAs,” Opt. Lett. 29, 1912–1914, (2004). [CrossRef] [PubMed]

]. From the aforementioned overview it is clear that a means of phase matching, which has low insertion loss, large nonlinear coefficient, and can be readily integrable with mainstream photonic devices using the available technologies is yet to be found.

In this work we present a simple means to achieve phase matching in compound semiconductor heterostructures through the use of novel waveguide design [8

08. A. S. Helmy and Brian R. West “Phase Matching using Bragg Reflector Waveguides,” IEEE LEOS Annual Meeting, Sydney , (2005). [CrossRef]

]. In this work phase matching is theoretically demonstrated through second harmonic generation (SHG) using a TE-polarized pump at 1550 nm wavelength which produces a TM-polarized SH radiation at 775 nm. Parametric conversion can also be demonstrated using the same structure by launching a TM-polarized SH at 775 nm; TE-polarized parametric florescence can then be obtained at 1550 nm.

The paper is organized as follows; in Section 2 the properties of Bragg reflection waveguides (BRWs) will first be discussed, then a demonstration of how these waveguides provide a propagation constant lower than that of the constituent waveguide materials will be presented. In Section 3, the mathematical formulation that describes the wave propagation in BRWs will be given, then in Section 4 the waveguiding condition for a BRW will be solved simultaneously with that describing waveguiding in conventional total internal reflection (TIR) waveguides, to obtain a structure that satisfies the condition n = nω. A discussion of this technique will be presented in Section 5 followed by a study of the tuning behavior in Section 6.

2. Features of Bragg reflection waveguides

The BRWs operates by providing reflection for one of the guided waves involved in the SHG process described above using stacks of periodic or quasi periodic layers on both sides of the core, as can be seen in Fig. 1. These waveguides have attracted substantial interest since their initial analysis [9

09. P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976). [CrossRef]

]. Their interesting birefringence properties [10

10. P. Yeh, A. Yariv, and C Hong “Electromagnetic propagation in periodic stratified media: I. General Theory,” J. Appl. Phys. , 67, 423–438 (1977).

,11

11. S. R. A. Dods, “Bragg reflection waveguides,” J. Opt. Soc. Am. A , 6, 1465–1475 (1989). [CrossRef]

] were utilized to produce devices such as polarization splitters/combiners [12

12. E. Simova and I. Golub, “Polarization splitter/combiner in high index contrast reflector waveguides,” Opt. Express 11, 3425–3430 (2003). [CrossRef] [PubMed]

] while their versatile waveguiding properties were used to tailor the profile of their guided modes [13

13. A. Mizrahi and L. Schächter, “Bragg reflection waveguides with a matching layer,” Opt. Express 12, 3156–3170 (2004). [CrossRef] [PubMed]

]. BRWs are also attractive for nonlinear propagation, where spatial optical solitons have been studied [14

14. C. Wätcher, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection waveguide,” J. Appl. Phys. 71, 3688–3692 (1992). [CrossRef]

], and nonlinear optical modes have been found to propagate at higher optical powers in waveguides that have no bound modes in the linear regime [15

15. P. M. Lambkin and K. A. Shore, “Nonlinear semiconductor Bragg reflection waveguide structures,” IEEE J. Quantum Eletron. 27, 824–828 (1991). [CrossRef]

].

Fig. 1. A schematic diagram of a BRW with the propagation direction orthogonal to the Bragg stack.
Fig. 2. Plot of the refractive index dispersion of a BRW similar to that shown in Fig. 1. The waveguide structure that resulted in this dispersion curve is a 200 nm core (nc) of Al0.24G0.76aAs, sandwiched in a Bragg stack made of alternating layers of Al0.3G0.7aAs and Al0.5G0.5aAs (n1 and n2).

3. Waveguiding condition for Bragg reflection waveguides

Assuming invariance in the y direction, the electric field propagating in the waveguide seen in Fig. 1 can be written as,

E(x,z,t)={EK(xdc2)eiK(xdc2)ei(ωtβz)x>dc2[C1cos(kcx)+C2sin(kcx)]ei(ωtβz)dc2<xdc2EK(xdc2)eiK(x+dc2)ei(ωtβz)xdc2
(1)

kc2=(ω.ncc)2=β2+kx2
(2)

Where kx is the wave vector in the waveguide transverse direction.

kixdi=π2,i=1,2
(3)
kix2=(ω.nic)2β2,i=1,2
(4)

where k1x, k2x are the transverse wave vectors in both stack layers. In the special case of quarter-wavelength stack, the waveguiding condition then becomes [9

09. P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976). [CrossRef]

,10

10. P. Yeh, A. Yariv, and C Hong “Electromagnetic propagation in periodic stratified media: I. General Theory,” J. Appl. Phys. , 67, 423–438 (1977).

],

ik1xeiKΛ+k1xk2xeiKΛ+k2xk1x={kctan(kcdc)forevenTEmodeskccot(kcdc)foroddTEmodes
ik1xnc2eiKΛ+n22k1xn12k2xeiKΛ+n12k2xn22k1x={kcn12tan(kcdc)forevenTEmodeskcn12cot(kcdc)foroddTEmodes
(5)

This condition is obtained through matching the field and its derivative (/∂x) at the interfaces of the core with both claddings to satisfy the necessary guiding condition for a bound mode. By designing the structure using quarter-wavelength stacks in the claddings we ensure that we operate right at the center of the stop band of the Bragg stack. This Bloch wave vector K is in the form,

K=Λ±iκi,m=1,2..
(6)

The imaginary component κi determines the exponential decay/growth of the BR mode envelope into the claddings, and will be at a maximum in the center of the forbidden gap with a sign appropriate for field decay, which is described as follows for the quarter-wavelength claddings [17

17. P. Yeh, Optical Waves in layered media, (Wiley, 1988).

],

Λκi{ln(k2xk1x)forTEmodesln(n12k2xn22k1x)forTEmode
(7)

A question may arise about how we define the thicknesses of the cladding layers d1 and d2 prior to the design. In our case of quarter-wavelength claddings both thicknesses are defined as per Eq. (3), and hence can be expressed as a function of the propagation constant and the wave vector of their respective layer. This enables their determination after solving for the propagation constantβ. Therefore there is no need for prior knowledge of d1 and d2 when using the method described here since we inherently set their contribution (kidi) to be π/2. In the section that follows we shall simultaneously solve the waveguiding condition of both the TIR and BRW modes for a given waveguides in order to fulfill the condition n = nω.

4. Simultaneously solving for TIR and BRW waveguiding conditions

The BRW is used to compensate for the large difference in the propagation constant of the fundamental and SH waves (@ 775 nm) due to material dispersion by providing a guided mode at the SH. Henceforth a TIR waveguide for the fundamental wave (@ 1550 nm) needs to be designed in the structure. For this purpose we use a core of higher refractive index to provide TIR waveguiding @ 1550 nm. Owing to the one dimensional nature of the problem at hand and the stratified periodic medium, the fundamental mode can therefore be easily analysed using the field transfer matrix method [18

18. J. Chilwell and I. Hodgkinson, “Thin-film field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984) [CrossRef]

]. The technique is very well documented else where and is based on constructing a transfer matrix for the layer stack, which accounts for the phase accumulated in every layer [18

18. J. Chilwell and I. Hodgkinson, “Thin-film field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984) [CrossRef]

],

Mstructure=(m11m12m21m22)=j=1lMj,totalnumberoflayers=l
(8)

The initial layer matrices Mj are defined as,

Mj=(cosΦjiγjsinΦjiγjsinΦjcosΦj)
(9)
Φj=kjxdj
(10)
Fig. 3. A graphical example of the solution of both the TIR and BR modes. The waveguide structure that resulted in this dispersion curve is a 310 nm core of 30% AlGaAs, sandwiched in a Bragg stack made of alternating quarter wave layers of 20% and 40% AlGaAs.
γj={nj2neff2cμoforTEmodesnj2neff2cεoforTMmodes
(11)

χM(neff)=γcm11+γcγsm12+m21+γsm22,
(12)

The core is made of Al0.3G0.7aAs layer, while the quarter-wavelength Bragg stacks on either side are made of alternating Al0 2G0.8aAs and Al0.4G0.6aAs. For demonstration purposes both Eqs. (5) and (13) are solved and presented here graphically. As can be seen in Fig. 3, the propagation constant of both modes match at a core thickness of 310 nm. The respective modes are then calculated and plotted, as shown in Fig. 4, where the 45% of the TE mode of the fundamental at 1550 nm is spatially centered in the core of the structure overlapping with 23 % of the SH field which is the TM BR mode at 775 nm. The effective index of these guides coincided at a value of 3.2236 for this structure.

Fig. 4. Field profiles of both the TE-polarized TIR mode @ 1550 nm and the TM-polarized BRW mode at 775nm.

5. Discussion

Fig. 5. Modal index of both the TIR and BR modes due to the change of the core bandgap, and hence refractive index.

Therefore the effect of such modes, if they exist, on the efficiency of second harmonic generation is thought to be negligible.

6. Tuning

The bandwidth of the BR mode depends on that of the bandgap of the cladding Bragg stack, which for a quarter-wavelength structure has a bandwidth [17

17. P. Yeh, Optical Waves in layered media, (Wiley, 1988).

],

Δωgapωo2π(k2xk1xk2x),
(13)

The design offers distinct advantages over the techniques which have been developed previously;

  • No patterning along the direction of propagation and hence have the potential to possess low optical losses in comparison with other quasi-phase matching designs. However it must be noted that Fresnel and modal phase matching share this advantage with our technique.
  • Gain can be provided in structures through carrier injection via electrical pumping. The electrical pumping is readily available in such structures due to the work which has been carried out developing vertical cavity surface emitting lasers.
  • The structure lends itself to be grown along side active and passive photonic devices for monolithic integration.

7. Conclusions

A design which utilizes Bragg reflector waveguides to provide phase matching for second harmonic generation at a wavelength of 775 nm is reported for the first time. The structure offers an attractive alternative to the techniques investigated to date as it involves no patterning along the propagation direction and allows more control over the overlap between the interacting waves and hence lends itself to more efficient nonlinear interactions. However the bandwidth efficiency and tuning capabilities of this phase matching technique need to be experimentally verified.

Acknowledgments

The Author would like to thank I. Golub and Brian R. West for stimulating discussion. This work has been supported by the department of ECE at the University of Toronto and NSERC.

References and links

01.

C. B. Ebert, L. A. Eyres, M. M. Fejer, and J. H. Harris, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227, 183–192 (1999).

02.

J. B. Khurgin, E. Rosencher, and Y. J. Ding, “Analysis of all-semiconductor intracavity optical parametric oscillators,” J. Opt. Soc. Am. B 15, 1726–1734 (1998). [CrossRef]

03.

A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, and E. Rosencher, “Second-harmonic generation at λ= 1.6 μm in AlGaAs/Al2O3 waveguides using birefringence phase matching,” Appl. Phys. Lett. 72, 2942–2945 (1998). [CrossRef]

04.

A. S. Helmy, D. C. Hutchings, T. C. Kleckner, J. H. Marsh, A. C. Bryce, J. M. Arnold, C. R. Stanley, J. S. Aitchison, C. T. A. Brown, K. Moutzouris, and M. Ebrahimzadeh, “Quasi phase matching in GaAs-AlAs superlattice waveguides via bandgap tuning using quantum well intermixing,” Opt. Lett. 25, 1370–1373 (2000). [CrossRef]

05.

R. Haidar, N. Forget, and E. Rosencher, “Optical parametric oscillation in micro-cavities based on isotropic semiconductors: a theoretical study,” IEEE J. Quantum Electron. 39, 569–576 (2003). [CrossRef]

06.

D. Faccio, F. Bragheri, and M. Cherchi, “Optical Bloch-mode-induced quasi phase matching of quadratic interactions in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 21, 296–301 (2004). [CrossRef]

07.

K. L. Vodopyanov, O. Levi, P.S. Kuo, T.J. Pinguet, J.S. Harris, M.M. Fejer, B. Gerard, L. Becouarn, and E. Lallier “Optical parametric oscillation in quasi-phase-matched GaAs,” Opt. Lett. 29, 1912–1914, (2004). [CrossRef] [PubMed]

08.

A. S. Helmy and Brian R. West “Phase Matching using Bragg Reflector Waveguides,” IEEE LEOS Annual Meeting, Sydney , (2005). [CrossRef]

09.

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976). [CrossRef]

10.

P. Yeh, A. Yariv, and C Hong “Electromagnetic propagation in periodic stratified media: I. General Theory,” J. Appl. Phys. , 67, 423–438 (1977).

11.

S. R. A. Dods, “Bragg reflection waveguides,” J. Opt. Soc. Am. A , 6, 1465–1475 (1989). [CrossRef]

12.

E. Simova and I. Golub, “Polarization splitter/combiner in high index contrast reflector waveguides,” Opt. Express 11, 3425–3430 (2003). [CrossRef] [PubMed]

13.

A. Mizrahi and L. Schächter, “Bragg reflection waveguides with a matching layer,” Opt. Express 12, 3156–3170 (2004). [CrossRef] [PubMed]

14.

C. Wätcher, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection waveguide,” J. Appl. Phys. 71, 3688–3692 (1992). [CrossRef]

15.

P. M. Lambkin and K. A. Shore, “Nonlinear semiconductor Bragg reflection waveguide structures,” IEEE J. Quantum Eletron. 27, 824–828 (1991). [CrossRef]

16.

T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, “Dispersion and Modulation of the Linear Optical Properties of GaAs/AlAs Superlattices Waveguides using Quantum Well Intermixing,” IEEE J. Quantum Eletron. (Accepted).

17.

P. Yeh, Optical Waves in layered media, (Wiley, 1988).

18.

J. Chilwell and I. Hodgkinson, “Thin-film field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984) [CrossRef]

19.

J. Khurgin, “Improvement of frequency-conversion efficiency in waveguides with rotationally twinned layers,” Opt. Lett , 13, 603–605 (1988). [CrossRef] [PubMed]

20.

S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, and M. Calligaro, “Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides,” Appl. Phys. Lett. 84, 2974–2976 (2004). [CrossRef]

21.

P. Dong and A. G. Kirk, “Nonlinear frequency conversion in waveguide directional couplers,” Phys. Rev. Lett. 93, 133901 (2004). [CrossRef] [PubMed]

22.

N. Yokouchi, A. J. Danner, and K. D. Choquette, “Two-dimensional photonic crystal confined vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. , 9, 1439–1447 (2003). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(230.1480) Optical devices : Bragg reflectors
(230.7370) Optical devices : Waveguides

ToC Category:
Optical Devices

History
Original Manuscript: December 21, 2005
Revised Manuscript: January 30, 2006
Manuscript Accepted: January 31, 2006

Citation
A. S. Helmy, "Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications," Opt. Express 14, 1243-1252 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1243


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References

  1. C. B. Ebert, L. A. Eyres, M. M. Fejer, J. H. Harris, "GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices," J. Cryst. Growth 227, 183-192 (1999).
  2. J. B. Khurgin, E. Rosencher, Y. J. Ding, "Analysis of all-semiconductor intracavity optical parametric oscillators," J. Opt. Soc. Am. B 15, 1726-1734 (1998). [CrossRef]
  3. A. Fiore, S. Janz, L. Delobel, P. van der Meer, P. Bravetti, V. Berger, and E. Rosencher, "Second-harmonic generation at λ= 1.6 µm in AlGaAs/Al2O3 waveguides using birefringence phase matching," Appl. Phys. Lett. 72, 2942-2945 (1998). [CrossRef]
  4. A. S. Helmy, D. C. Hutchings, T. C. Kleckner, J. H. Marsh, A. C. Bryce, J. M. Arnold, C. R. Stanley, and J. S. Aitchison, C. T. A. Brown, K. Moutzouris and M. Ebrahimzadeh, "Quasi phase matching in GaAs-AlAs superlattice waveguides via bandgap tuning using quantum well intermixing," Opt. Lett. 25, 1370-1373 (2000). [CrossRef]
  5. R. Haidar, N. Forget, E. Rosencher, "Optical parametric oscillation in micro-cavities based on isotropic semiconductors: a theoretical study," IEEE J. Quantum Electron. 39, 569-576 (2003). [CrossRef]
  6. D. Faccio, F. Bragheri, M. Cherchi, "Optical Bloch-mode-induced quasi phase matching of quadratic interactions in one-dimensional photonic crystals," J. Opt. Soc. Am. B 21, 296-301 (2004). [CrossRef]
  7. K. L. Vodopyanov, O. Levi, P.S. Kuo, T.J. Pinguet, J.S. Harris, M.M. Fejer, B. Gerard, L. Becouarn, E. Lallier "Optical parametric oscillation in quasi-phase-matched GaAs," Opt. Lett. 29, 1912-1914, (2004). [CrossRef] [PubMed]
  8. A. S. Helmy, BrianR.  West "Phase Matching using Bragg Reflector Waveguides," IEEE LEOS Annual Meeting, Sydney, (2005). [CrossRef]
  9. P. Yeh, A. Yariv, " Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976). [CrossRef]
  10. P. Yeh, A. Yariv, C Hong "Electromagnetic propagation in periodic stratified media: I. General Theory," J. Appl. Phys.,  67, 423-438 (1977).
  11. S. R. A. Dods, "Bragg reflection waveguides," J. Opt. Soc. Am. A,  6, 1465-1475 (1989). [CrossRef]
  12. E. Simova, I. Golub, "Polarization splitter/combiner in high index contrast reflector waveguides," Opt. Express 11, 3425-3430 (2003). [CrossRef] [PubMed]
  13. A. Mizrahi, L. Schächter, "Bragg reflection waveguides with a matching layer," Opt. Express 12, 3156-3170 (2004). [CrossRef] [PubMed]
  14. C. Wätcher, F. Lederer, L. Leine, U. Trutschel, M. Mann, " Nonlinear Bragg reflection waveguide," J. Appl. Phys. 71, 3688-3692 (1992). [CrossRef]
  15. P. M. Lambkin, K. A. Shore, "Nonlinear semiconductor Bragg reflection waveguide structures," IEEE J. Quantum Eletron. 27, 824-828 (1991). [CrossRef]
  16. T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, J. S. Aitchison, "Dispersion and Modulation of the Linear Optical Properties of GaAs/AlAs Superlattices Waveguides using Quantum Well Intermixing," IEEE J. Quantum Eletron. (Accepted).
  17. P. Yeh, Optical Waves in layered media, (Wiley, 1988).
  18. J. Chilwell, I. Hodgkinson, "Thin-film field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides," J. Opt. Soc. Am. A 1, 742-753 (1984) [CrossRef]
  19. J. Khurgin, "Improvement of frequency-conversion efficiency in waveguides with rotationally twinned layers," Opt. Lett,  13, 603-605 (1988). [CrossRef] [PubMed]
  20. S. Ducci, L. Lanco, V. Berger, A. De Rossi, V. Ortiz, M. Calligaro, "Continuous-wave second-harmonic generation in modal phase matched semiconductor waveguides," Appl. Phys. Lett. 84,2974-2976 (2004). [CrossRef]
  21. P. Dong, A. G. Kirk, "Nonlinear frequency conversion in waveguide directional couplers," Phys. Rev. Lett. 93,133901 (2004). [CrossRef] [PubMed]
  22. N. Yokouchi, A. J. Danner, K. D. Choquette, "Two-dimensional photonic crystal confined vertical-cavity surface-emitting lasers," IEEE J. Sel. Top. Quantum Electron.,  9, 1439-1447 (2003). [CrossRef]

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