## Dispersion tailoring of the quarter-wave Bragg reflection waveguide

Optics Express, Vol. 14, Issue 9, pp. 4073-4086 (2006)

http://dx.doi.org/10.1364/OE.14.004073

Acrobat PDF (250 KB)

### Abstract

We present analytical formulae for the polarization dependent first- and second-order dispersion of a quarter-wave Bragg reflection waveguide (QtW-BRW). Using these formulae, we develop several qualitative properties of the QtW-BRW. In particular, we show that the birefringence of these waveguides changes sign at the QtW wavelength. Regimes of total dispersion corresponding to predominantly material-dominated and waveguide-dominated dispersion are identified. Using this concept, it is shown that the QtW-BRW can be designed so as to provide anomalous group velocity dispersion of large magnitude, or very small GVD of either sign, simply by an appropriate chose of layer thicknesses. Implications on nonlinear optical devices in compound semiconductors are discussed.

© 2006 Optical Society of America

## 1. Introduction

2. A. S. Helmy, “Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications,” Opt. Express **14**, 1243–1252 (2006) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-3-1243. [CrossRef] [PubMed]

3. Y. Sakurai and F. Koyama, “Proposal of tunable hollow waveguide distributed Bragg reflectors,” Jap. J. Appl. Phys. **43**, L631–L633 (2004). [CrossRef]

04. E. Simova and I. Golub, “Polarization splitter/combiner in high index contrast Bragg reflector waveguides,” Opt. Express **11**, 3425–3430 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-25-3425. [CrossRef] [PubMed]

5. A. Mizrahi and L. Schächter, “Optical Bragg accelerators,” Phys. Rev. E. **70**, Art. 016505(2) (2004). [CrossRef]

6. S. Nakamura and K. Tajima, “Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch,” Jap. J. Appl. Phys. **35**, L1426–L1429 (1996). [CrossRef]

9. U. Peschel, T. Peschel, and F. Lederer, “A compact device for highly efficient dispersion compensation in fiber transmission,” Appl. Phys. Lett. **67**, 2111–2113 (1995). [CrossRef]

10. Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, “Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides,” J. Appl. Phys. **98**, 113102 (2005). [CrossRef]

11. M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express **13**, 6848–6855 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-6848. [CrossRef] [PubMed]

12. E. Valentinuzzi, “Dispersive properties of Kerr-like nonlinear optical structures,” J. Lightwave Technol. **16**, 152–155 (1998). [CrossRef]

13. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express **13**, 8452–8459 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-21-8452. [CrossRef] [PubMed]

14. T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, “Dispersion tailoring and compensation by modal interactions in OmniGuide fibers,” Opt. Express **11**, 1175–1196 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1175. [CrossRef] [PubMed]

15. A. Mizrahi and L. Schächter, “Bragg reflection waveguides with a matching layer,” Opt. Express **12**, 3156–3170 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3156. [CrossRef] [PubMed]

16. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E **69**, 016617 (2004). [CrossRef]

## 2. The quarter-wave Bragg reflection waveguide

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

19. P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media: I. General theory,” J. Opt. Soc. Am. **67**, 428–438 (1977). [CrossRef]

*n*

_{1}and

*n*

_{2}, with thickness a and b, respectively, where

*n*

_{1}>

*n*

_{2}. The cladding period is denoted as ʌ =

*a*+

*b*. The core has index

*n*

_{co}and thickness

*t*

_{co}. For this analysis it is assumed that the cladding consists of an infinite number of periods, although the results are valid for structures with a reasonably large number of periods [20]. In addition, it is assumed that the waveguide is symmetric about the core. This is not a requirement of BRWs, but calculation of dispersion becomes considerably more complex in the presence of waveguide asymmetry. Hence we shall consider symmetric waveguides here for simplicity, although the methodology is applicable to asymmetric structures as well.

*i*= 1, 2, co), where

*ω*is the angular frequency of the guided radiation and

*c*is the speed of light. The guided modes are determined by a mode dispersion equation, derived in [20] using a field transfer matrix description of the periodic cladding:

*iK*

_{TE(TM)}ʌ) is the one physically realizable eigenvalue of the field transfer matrix

*M*

^{FT}, can be expressed as

*ρ*< 1) and “odd” (

*ρ*> 1) TM modes in reference to their symmetry, as discussed in [20].

*λ*/4 with respect to the transverse propagation vector:

*k*

_{i}in the middle of the stop-band, ensuring strong reflection and hence optimal confinement in the core. As derived in [20], the effective index in this case is independent of polarization for slab waveguides and is equal to

*ω*

_{0}signifies the nominal angular frequency at which the quarter-wave condition is satisfied, as opposed to the variable

*ω*that will be used later in this paper. In addition, the matrix elements and eigenvalue reduce to

## 3. Analytical determination of dispersion

*k*to each transverse propagation vector and examine the resulting change in Eq. (2). As the polarization degeneracy is lifted when the waveguide is detuned from the quarter-wave condition, the

_{i}*k*

_{i}become polarization dependent by Eq. (1). Polarization subscripts on the

*k*

_{i}will be omitted for brevity; however, those on the matrix elements will remain in order to indicate that their defining equations are unique to a specific polarization, as shown in Eq. (3).

*iK*

_{TE}ʌ) -

*A*

_{TE}+

*B*

_{TE}= 0 at the quarter-wave condition. Next, we calculate the perturbed transfer matrix elements using Eq. (3),

_{TE}is a field transfer perturbation matrix defined analogously to Eq. (4) and

*u*

_{TE}is the normalized eigenvector of Eq. (4), equal to (1√2)[-1 1]

^{T}[20]. Finally, using Eq. (9) and Eqs. (13)–(15), Eq. (11) reduces to the TE mode detuning equation

*k*

_{i}to a change in effective index. We are now in a position to calculate first-order dispersion by expanding Eq. (1) to first order in Δ

*ω*about

*ω*

_{0},

*ω*at

*ω*=

*ω*

_{0}. A Taylor expansion of the square root, keeping terms up to first order in Δ

*ω*, results in

*∂n*

_{eff}/

*∂ω*arises entirely through the term

*α*, due to the polarization independence of

*n*

_{eff}under the quarter-wave condition. Furthermore, it is clear that the birefringence Δ

*n*

_{eff}of the QtW-BRW must change sign at the QtW wavelength;

*n*

_{i}′,

*n*

_{i}″… are properties of the material system studied, and can be expressed analytically using appropriate semi-empirical Sellmeier coefficients or Adachi [22

22. S. Adachi, “GaAs, AlAs, and Al_{x}Ga_{1-x}As material parameters for use in research and device applications,” J. Appl. Phys. **58**, R1–R29 (1985). [CrossRef]

23. M. A. Afromowitz, “Refractive index of Ga_{1-x}Al_{x}As,” Solid State Commun. **15**, 59–63 (1974). [CrossRef]

25. S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J Appl. Phys. **87**, 7825–7837 (2000). [CrossRef]

26. 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 superlattice waveguides using quantum-well intermixing,” IEEE J. Quantum Electron. **42**, 280–286 (2006). [CrossRef]

_{x}Ga

_{1-x}As QtW-BRW operating at a nominal wavelength of 1.55 μm. To provide a strong resonance in the cladding, we utilize a very large index difference, GaAs/Al

_{.75}Ga

_{.25}As, with material indices calculated using [25

25. S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J Appl. Phys. **87**, 7825–7837 (2000). [CrossRef]

*n*

_{1}> (

*n*

_{co},

*n*

_{2}) by definition, this requirement dictates that 0 <

*n*

_{eff}< min(

*n*

_{co},

*n*

^{2}) [20], resulting in the constraints

_{.90}Ga

_{.10}As and a core thickness of 350 nm. Material dispersion data is shown in Table 1. By Eq. (8), this 1-D waveguide has a polarization-independent effective index of 1.9235 at 1.55 μm.Cladding layer thicknesses (required only to model the BRW at wavelengths that do not correspond to the quarter-wave condition) are determined by Eqs. (1) and (7). Figure 2 shows the effective index curves for this waveguide, obtained by solving Eq. (2) at each wavelength in a region around 1.55 μm. Note the change of sign for the birefringence, as predicted. Normalized first-order dispersion

*ω*(

*∂n*

_{eff}/

*∂ω*ci) is shown in Fig. 3. The analytical solution at 1.55 μm obtained using Eq. (23) is plotted as well, showing complete agreement with the numerical curves.

*ω*

_{0},

*γ*

_{1}

*x*+

*γ*

_{2}

*x*

^{2})

^{1/2}≈ 1+

*γ*

_{2}

*x*/2+(

*γ*

_{2}/2-

*x*

^{2}, keeping terms up to second order in Δ

*ω*, gives

*G*

_{i}and

*H*

_{i}cancel out due to Eq. (23):

*α*terms, but also to the

*S*

_{i}terms, via

*J*

_{i}and

*P*

_{i}, which reflect the polarization dependence of first-order dispersion. In general, n

^{th}-order dispersion can be calculated in a similar iterative fashion by keeping terms up to order (Δ

*ω*)

^{n}in the expansion of Eq. (1) and applying the results of all lower orders, although the resulting accuracy diminishes due to the use of successive lower-order approximations. The analytical solutions are strictly valid at the nominal (quarter-wave) frequency. When calculating the effective index in a region around this frequency, the bandwidth over which these dispersion figures are valid is dependent upon the particular design ‒ it can be estimated by calculating the next-lowest order of dispersion. For this reason, choosing a nominal operating wavelength far from the material bandgap will result in an extended bandwidth. An example is given in [20] for first-order dispersion.

## 4. Discussion

22. S. Adachi, “GaAs, AlAs, and Al_{x}Ga_{1-x}As material parameters for use in research and device applications,” J. Appl. Phys. **58**, R1–R29 (1985). [CrossRef]

_{x}Ga

_{1-x}As at 1.55 μm using three of these models. Here and throughout this work, we describe second-order dispersion using the dispersion parameter

*β*

_{2}=

*∂*

^{2}

*β*/

*∂ω*

^{2}= (

*ω*/

*c*)

*∂*

^{2}

*n*/

*∂ω*

^{2}+ (2/

*c*)

*∂n*/

*∂ω*. While there is close agreement between the models of Afromowitz and Gehrsitz, the model of Adachi differs significantly, especially for low aluminum fraction. It should be noted that in this wavelength region, the latter model relies on an extrapolation of empirical data from photon energies closer to the bandgap for 0 ≤ x ≤0.38, and additional data from the same source as used by Afromowitz at x = 1 in the mid-IR; this is a possible source of the discrepancy. The uncertainty in material dispersion will lead to inaccuracies in calculating dispersion of the QtW-BRW; however, comparing the magnitudes of the quantities plotted in Figs. 3 and 4(b), it is clear that for this particular waveguide, material dispersion plays a relatively small role, which is more than one order of magnitude less than that of the waveguide dispersion.

*n*

_{co}. Cladding layer thicknesses are altered accordingly to maintain QtW operation by Eqs. (1) and (7). We define a normalized effective index

*B*, which bears some resemblance to the normalized propagation constant used in the analysis of total internal reflection waveguides; the variation of

*B*with

*t*

_{co}for waveguides of this composition is shown in Fig. 5:

*B*→ 1, the term

*α*→ 0, and thus

*∂n*

_{eff}/

*∂ω*approaches

*∂n*

_{co}/

*∂ω*and

*∂*

^{2}

*n*

_{eff}/

*∂ω*

^{2}→

*∂*

^{2}

*n*

_{co}/

*∂ω*

^{2}. This represents the situation where all power is confined within the core, with the physically unrealizable requirement of infinite core thickness. In this regime, the assumption of negligible material dispersion is invalid and will lead to relatively significant errors in estimating dispersion. An interesting result of this analysis of the limiting cases is that if the core material has normal GVD, there exists a waveguide design that will provide zero GVD, and a very wide range of core thicknesses that provide the low GVD desired for soliton formation.

*B*< 0.999, with and without the contribution of material dispersion. There is negligible difference for

*B*< 0.9, suggesting that accurate values of material dispersion are not required in this range. As expected, the normalized dispersion approaches

*ω*(

*∂n*

_{co}/

*∂ω*) ≈ .081 as B → 1 [see Fig. 4(b)]. Only the TE mode is considered here; the TM mode exhibits similar characteristics. The GVD parameter -

*β*

_{2}is plotted in Fig. 7 (using the negative dispersion parameter has no physical significance; it only allows the logarithmic scale to be used). The inset shows the region around

*B*≈ 1, where

*β*

_{2}approaches the core value of 0.23 fs

^{2}/μm [see Fig. 4(c)] As mentioned previously, there is a waveguide design for which

*β*

_{2}= 0, at

*B*≈ 0.996, or by Fig. 5,

*t*

_{co}≈ 3 μm. The small variation of

*B*with

*t*

_{co}in this range suggests that thickness tolerance is not an issue. Furthermore, the wide core need not be an impediment to single-mode operation, as guided modes in BRWs need to satisfy both the resonance condition in the core and the Bragg condition in the cladding [18

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

*n*

_{eff}<

*n*

_{i}it is clear that

*G*

_{i}and

*H*

_{i}are positive quantities and

*α*

_{TE},

*∂n*

_{eff}/

*∂μ*> 0 for all values of B.

*β*

_{2}and inversely proportional to the square of the pulse width [8]. This suggests an onset for soliton-like propagation within a BRW that is essentially controllable via the BRW parameters as seen in Fig. 7. This control over dispersion can take place also post waveguide growth. We have shown previously the capability of tuning the properties of BRWs using electro-optic as well as current injection effects [2

2. A. S. Helmy, “Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications,” Opt. Express **14**, 1243–1252 (2006) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-3-1243. [CrossRef] [PubMed]

*β*

_{2}. Note that this change in sign has been achieved solely through the waveguide dispersion control and does not include any excessive absorption due to a material resonance. This offers the capability of observing dark solitons in materials with self-focusing third-order nonlinearity using a low loss, tunable device [8], an attractive capability for optical switching and all-optical control devices.

## 5. Conclusion

*t*

_{co}) or by the waveguide form (for very narrow cores). In the latter regime, first-order and second-order dispersion asymptotically approach ∞ and -∞ respectively, as the effective index approaches zero. Between these extreme limits, there exists a waveguide design that shows zero GVD. Due to the low loss nature of BRWs, as they do not possess refractive index modulation in the direction of propagation, they offer an attractive building block for nonlinear optics in semiconductors.

## Acknowledgments

## References and links

1. | A. S. Helmy and B. R. West, “Phase matching using Bragg reflector waveguides,” in |

2. | A. S. Helmy, “Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications,” Opt. Express |

3. | Y. Sakurai and F. Koyama, “Proposal of tunable hollow waveguide distributed Bragg reflectors,” Jap. J. Appl. Phys. |

04. | E. Simova and I. Golub, “Polarization splitter/combiner in high index contrast Bragg reflector waveguides,” Opt. Express |

5. | A. Mizrahi and L. Schächter, “Optical Bragg accelerators,” Phys. Rev. E. |

6. | S. Nakamura and K. Tajima, “Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch,” Jap. J. Appl. Phys. |

7. | K. Cheng, ed., |

8. | G. P. Agrawal, |

9. | U. Peschel, T. Peschel, and F. Lederer, “A compact device for highly efficient dispersion compensation in fiber transmission,” Appl. Phys. Lett. |

10. | Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, “Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides,” J. Appl. Phys. |

11. | M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express |

12. | E. Valentinuzzi, “Dispersive properties of Kerr-like nonlinear optical structures,” J. Lightwave Technol. |

13. | G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express |

14. | T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, “Dispersion tailoring and compensation by modal interactions in OmniGuide fibers,” Opt. Express |

15. | A. Mizrahi and L. Schächter, “Bragg reflection waveguides with a matching layer,” Opt. Express |

16. | I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E |

17. | Y. Sakurai and F. Koyama, “Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors,” Jap. J. Appl. Phys. |

18. | P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. |

19. | P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media: I. General theory,” J. Opt. Soc. Am. |

20. | B. R. West and A. S. Helmy, “Properties of the quarter-wave Bragg reflection waveguide: Theory,” J. Opt. Soc. Am. B (to be published). |

21. | A. S. Deif, |

22. | S. Adachi, “GaAs, AlAs, and Al |

23. | M. A. Afromowitz, “Refractive index of Ga |

24. | A. N. Pikhtin and A. D. Yas’kov, “Dispersion of refractive-index of semiconductors with diamond and zincblende structures,” Sov. Phys. Semicond. |

25. | S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J Appl. Phys. |

26. | 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 superlattice waveguides using quantum-well intermixing,” IEEE J. Quantum Electron. |

27. | B. R. West and A. S. Helmy, “Analysis and design equations for phase matching using Bragg reflector waveguides,” IEEE J. Sel. Top. Quantum Electron. (to be published). |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.1480) Optical devices : Bragg reflectors

(230.7370) Optical devices : Waveguides

**ToC Category:**

Optical Devices

**History**

Original Manuscript: February 27, 2006

Revised Manuscript: April 17, 2006

Manuscript Accepted: April 18, 2006

Published: May 1, 2006

**Citation**

Brian R. West and A. S. Helmy, "Dispersion tailoring of the quarter-wave Bragg reflection waveguide," Opt. Express **14**, 4073-4086 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-4073

Sort: Year | Journal | Reset

### References

- A. S. Helmy and B. R. West, "Phase matching using Bragg reflector waveguides," in Proceedings of 18th Annual Meeting of the IEEE Lasers and Electro-Optics Society (Institute of Electrical and Electronics Engineers, Sydney, 2005), pp. 424-425.
- A. S. Helmy, "Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications," Opt. Express 14, 1243-1252 (2006) [CrossRef] [PubMed]
- Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. 43,L631-L633 (2004). [CrossRef]
- E. Simova and I. Golub, "Polarization splitter/combiner in high index contrast Bragg reflector waveguides," Opt. Express 11, 3425-3430 (2003), [CrossRef] [PubMed]
- A. Mizrahi and L. Schächter, "Optical Bragg accelerators," Phys. Rev. E. 70, 016505 (2004). [CrossRef]
- S. Nakamura, K. Tajima, "Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch," Jpn. J. Appl. Phys. 35, L1426-L1429 (1996). [CrossRef]
- K. Cheng, ed., Handbook of Optical Components and Engineering (Wiley Interscience, 2003).
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).
- U. Peschel, T. Peschel, and F. Lederer, "A compact device for highly efficient dispersion compensation in fiber transmission," Appl. Phys. Lett. 67, 2111-2113 (1995). [CrossRef]
- Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, "Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides," J. Appl. Phys. 98, 113102 (2005). [CrossRef]
- M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, "Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires," Opt. Express 13, 6848-6855 (2005), [CrossRef] [PubMed]
- E. Valentinuzzi, "Dispersive properties of Kerr-like nonlinear optical structures," J. Lightwave Technol. 16, 152-155 (1998). [CrossRef]
- G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005), [CrossRef] [PubMed]
- T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, "Dispersion tailoring and compensation by modal interactions in OmniGuide fibers," Opt. Express 11, 1175-1196 (2003) [CrossRef] [PubMed]
- A. Mizrahi and L. Schächter, "Bragg reflection waveguides with a matching layer," Opt. Express 12, 3156-3170 (2004). [CrossRef] [PubMed]
- I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004). [CrossRef]
- Y. Sakurai and F. Koyama, "Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors," Jpn. J. Appl. Phys. 43,5828-5831 (2004). [CrossRef]
- P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976). [CrossRef]
- P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media: I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977). [CrossRef]
- B. R. West and A. S. Helmy, "Properties of the quarter-wave Bragg reflection waveguide: Theory," J. Opt. Soc. Am. B (to be published).
- A. S. Deif, Advanced Matrix Theory for Scientists and Engineers (Routledge, 1987).
- S. Adachi, "GaAs, AlAs, and AlxGa1-xAs material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985). [CrossRef]
- M. A. Afromowitz, "Refractive index of Ga1-xAlxAs," Solid State Commun. 15, 59-63 (1974). [CrossRef]
- A. N. Pikhtin and A. D. Yas’kov, "Dispersion of refractive-index of semiconductors with diamond and zincblende structures," Sov. Phys. Semicond. 12, 622-626 (1978).
- S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000). [CrossRef]
- 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 superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006). [CrossRef]
- B. R. West and A. S. Helmy, "Analysis and design equations for phase matching using Bragg reflector waveguides," IEEE J. Sel. Top. Quantum Electron. (to be published).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.