## Band gaps and leaky-wave effects in resonant photonic-crystal waveguides

Optics Express, Vol. 15, Issue 2, pp. 680-694 (2007)

http://dx.doi.org/10.1364/OE.15.000680

Acrobat PDF (283 KB)

### Abstract

The paper addresses the leaky stop bands associated with resonant photonic crystal slabs and periodic waveguides. We apply a semianalytical model pertinent to the second band to compute the dispersion curves describing the leaky stop band and verify its correctness by rigorous band computations. This approximate model provides clear insights into the physical properties of the leaky stop band in terms of explicit analytical expressions found. In particular, it enables comparison of the structure of the bands computed in complex propagation constant, implying spatially decaying leaky modes, with the bands computed in complex frequency, implying temporally decaying modes. It is shown that coexisting Bragg-coupling and energy-leakage mechanisms perturb the bands in complex propagation constant whereas these mechanisms are decoupled in complex frequency. As a result, the bands in complex frequency are well defined exhibiting a clear gap. These conclusions are verified by numerical diffraction computations for both weak and strong grating modulations where the resonance peaks induced by external illumination are shown to closely track the band profile computed in complex frequency. Thus, in general, phase matching to a resonant leaky mode occurs via real propagation constant that is found by dispersion computations employing complex frequency.

© 2007 Optical Society of America

## 1. Introduction

4. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express **12**,5661–5674 (2004). [CrossRef] [PubMed]

5. D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, “Optical fiber endface biosensor based on resonances in dielectric waveguide gratings,” Proceedings of the SPIE **3911**,86–94 (2000). [CrossRef]

6. B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction,” Sens. Actuators B **85**,219–226 (2002). [CrossRef]

7. G. Purvinis, P. S. Priambodo, M. Pomerantz, M. Zhou, T. A. Maldonado, and R. Magnusson, “Second-harmonic generation in resonant waveguide gratings incorporating ionic self-assembled monolayer polymer films,” Opt. Lett. **29**,1108–1110 (2004). [CrossRef] [PubMed]

4. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express **12**,5661–5674 (2004). [CrossRef] [PubMed]

8. Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express **12**,1885–1891 (2004). [CrossRef] [PubMed]

9. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation loss,” IEEE J. Quantum Electron. **QE-21**,144–150 (1985). [CrossRef]

10. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. **33**,2038–2059 (1997). [CrossRef]

11. P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. **20**,345–351 (1979). [CrossRef]

8. Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express **12**,1885–1891 (2004). [CrossRef] [PubMed]

4. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express **12**,5661–5674 (2004). [CrossRef] [PubMed]

11. P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. **20**,345–351 (1979). [CrossRef]

*k*versus complex propagation coefficient

_{0}*β*=

*β*+

_{R}*jβ*that included the null in the imaginary part at the nonleaky edge. Noting that the

_{I}*k*-

_{0}*β*plot exhibited a forbidden gap that was not well defined, they turned to the complex frequency plane and demonstrated a clear band by numerical computations [11

_{R}11. P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. **20**,345–351 (1979). [CrossRef]

*k*-

_{0}*β*have since been reported in several papers [12

_{R}12. T. Tamir and S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A **14**,1607–1616 (1997). [CrossRef]

13. D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, and J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A **17**,1221–1230 (2000). [CrossRef]

**12**,5661–5674 (2004). [CrossRef] [PubMed]

**12**,5661–5674 (2004). [CrossRef] [PubMed]

13. D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, and J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A **17**,1221–1230 (2000). [CrossRef]

14. A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, “Resonant scattering and mode coupling in two-dimensional textured planar waveguides,” J. Opt. Soc. Am. A. **18**,1160–1170 (2001). [CrossRef]

15. D. Gerace and L. C. Andreani, ”Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs,” Phys. Rev. E **69**,056603 (2004). [CrossRef]

**20**,345–351 (1979). [CrossRef]

## 2. Dispersion characteristics of periodic waveguides

*TE*mode (

*TE*). Figure 2(a) shows the first Brillouin zone where we display the first three stop bands. The solid line denotes the leaky mode and the dashed curves provide the light lines corresponding to the substrate (

_{0}*k*/

_{0}n_{s}*K*) and cover (

*k*/

_{0}n_{c}*K*). At a particular frequency beyond the first stop band, a leaky wave with

*β*≠0 appears as shown in Fig. 2(b). When the frequency increases, a new leaky wave emerges and radiates into the cover or substrate when the Rayleigh conditions, expressed in the first zone,

_{I}*k*is the wave number in free space,

_{0}*m*is the index of the diffracted order,

*K*=

*2π*/

*Λ*, and

*β*is the propagation constant. These conditions are satisfied at the intersections of the folded light lines (dashed lines) and the

*β*curve (solid line) as shown in Fig. 2(a).

_{R}*β*curve as shown in Fig. 2(b).

_{I}*β*=

_{R}*qK*/2, where

*q*is an integer denoting the band. The stop band can be identified by its significantly high value of

*β*as shown in Fig. 2(b). As the figure shows, the coupling of two counter-propagating waveguide modes within a band is generally considerably stronger than the coupling between the waveguide mode and the radiating wave at moderate modulation levels. In the present example, the first stop band (

_{I}*q*=1) occurs around normalized frequency

*k*/

_{0}*K*=0.315; the second stop band (

*q*=2) is near frequency 0.571; and the third stop band (

*q*=3) is at frequency 0.816. Details of these stop bands are presented in Fig. 3. As the first band is located in the non-leaky regime, it is a non-leaky stop band. Higher bands, such as the second and third stop bands, are thus leaky stop bands. In the first stop band,

*β*is a constant across the band.

_{R}## 3. Second-band coupled-mode model

*Δε*, the dispersion properties at the second stop band can be approximated as [9

9. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation loss,” IEEE J. Quantum Electron. **QE-21**,144–150 (1985). [CrossRef]

*h*,

_{1}*h*and

_{2}*h*are given by [9

_{3}9. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation loss,” IEEE J. Quantum Electron. **QE-21**,144–150 (1985). [CrossRef]

*γ*,

_{0}*γ*and

_{1}*γ*are the

_{2}*0*,

^{th}*1*and

^{st}*2*Fourier coefficients of the relative permittivity modulation profile, respectively. For a binary profile,

^{nd}*γ*,

_{0}*γ*and

_{1}*γ*are constant throughout (i. e. not x-dependent) the grating layer.

_{2}*G(x,x́)*is the Green’s function for the diffracted field [10

10. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. **33**,2038–2059 (1997). [CrossRef]

*h*is constant, while

_{3}*h*and

_{1}*h*are proportional to

_{2}*Δε*and

^{2}*Δε*, respectively. These assumptions are appropriate for small modulation with the frequency under consideration varying across a limited range. At high modulation, these assumptions do not work well because of two factors. First, the stop-band width generally expands as modulation increases. Second, the band may shift when higher-order coupling becomes significant [20

20. D. L. Jaggard and C. Elachi, “Floquet and coupled-waves analysis of higher-order Bragg coupling in a periodic medium,” J. Opt. Soc. Am. **66**,674–682 (1976). [CrossRef]

## 4. Physical properties of the leaky stop band

*h*=0 and Eqs. (2) and (3) become

_{1}*Δkh*|<|

_{3}*h*|. Figure 5(b) is computed with Eq. (6) presenting the dispersion characteristics with real propagation constant; note that there exists no solution for frequencies Re(

_{2}*Δkh*)<|

_{3}*h*|. Outside the bands in Fig. 5, the imaginary part of either propagation constant or frequency is zero as there is no radiation leakage in this case. In addition, the real parts of the curves in Figs. 5(a) and 5(b) are exactly the same.

_{2}### 4.1. Band structure in complex propagation constant

*Δε*=0.34, the results obtained from Eqs. (2) or (3) agree with rigorous computations verifying that the coupled-mode model holds well around the leaky band.

*Δk*=

*h*/

_{2}*h*, we get

_{3}*Δβ*= 0 from Eq. 2, which defines the upper band edge in Fig. 6. At this edge,

*h*has no effect on the propagation constant as

_{1}*β*= 0 and the wave appears nonleaky. The lower edge is not as obvious. If we use the criterion that the band edge should appear at minimum

_{I}*∂k*/

*∂β*(which is analogous to group velocity), it can be shown that the lower edge locates at approximately Δ

_{R}*k*= -

*h*

_{2}/

*h*

_{3}in the case of small modulation. At this edge, by Eq. (2), we have

*h*is generally a complex number as indicated in Eq. 4). Thus, both the real and imaginary parts of the propagation constant are nonzero and proportional to both

_{1}*h*and

_{1}*h*. At frequencies far from the band edges with |Δ

_{2}*k*| ≫

*h*

_{2}/

*h*

_{3}, one has

*h*) away from the band as shown in Fig. 6. This indicates that the real part of

_{1}*h*represents the actual amount of energy leaked out of the waveguide as discussed by Rosenblatt et al. [10

_{1}10. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. **33**,2038–2059 (1997). [CrossRef]

### 4.2. Band structure in complex frequency

*β*= 0, the model gives two solutions Δ

*k*=

*h*

_{2}/

*h*

_{3}and Δ

*k*= -(

*h*

_{2}+ 2

*jh*

_{1})/

*h*

_{3}. The former represents the upper band edge as shown in Fig. 7. At this edge, Δ

*k*is real agreeing with the results from analysis obtained with real frequency; since

*k*=0, no leakage occurs. The latter expression represents the lower band edge as shown in the figure where Δ

_{0,I}*k*has an imaginary part equal to-2Re(

*h*

_{1})/

*h*

_{3}inducing energy leakage. At frequencies far from the band edges, one has Δ

*k*= (±Δ

*β*-

*jh*

_{1})/

*h*

_{3}with the imaginary part of Δ

*k*converging to -Re(

*h*

_{1})/

*h*

_{3}as shown in Fig. 7.

### 4.3 Summary of analytical results

*Δβ*=

*Δkh*. In the stop band and at the leaky edge, the results in real and complex frequency differ significantly in their relations to the coexisting coupling mechanisms (Bragg coupling and leaky wave coupling). For example, the representation with complex

_{3}*β*keeps track of the spatial decay associated with each mechanism whereas the representation in complex

*k*lumps the effects of these mechanisms together with the mode decay represented as temporal decay or radiation lifetime given by

*Im(k)*. In particular, as seen in the table, both

^{-1}*Δβ*and

_{R}*Δβ*in the band center and at the lower (leaky) edge, depend on both coefficients

_{I}*h*(leak) and

_{1}*h*(Bragg). In contrast, for complex

_{2}*k*,

*Δk*at the lower edge depends only on

_{R}*h*whereas

_{2}*Δk*depends only on

_{I}*h*. Thus, for the band in complex frequency, the two mechanisms are decoupled accounting for the well-defined band gap observed. Moreover, there are no solutions found in the band center consistent with no states existing there. Figure 8 shows a comparison of the band structure for these two cases computed with RCWA.

_{1}*h*. For the example structure treated (Fig. 1) whose results are presented in Figs. 6–8, the coefficient

_{2}*h*is real and positive since the modulation profile is symmetric. The fill factor F<0.5 and thus

_{2}*Δβ*is zero at the upper edge in this particular example [13

13. D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, and J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A **17**,1221–1230 (2000). [CrossRef]

8. Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express **12**,1885–1891 (2004). [CrossRef] [PubMed]

*Δβ*are proportional to

*Δε*since

^{1.5}*h*∝

_{1}*Δε*and

^{2}*h*∝

_{2}*Δε*. Again, the difference between the real parts of β in the two representations originates in the leakage of the energy from the waveguide grating. At moderate modulation, the real parts of the two sets converge quickly when the frequency is away from the band as shown in Fig. 8.

### 4.4 Extension to strong modulation

*h*∝

_{1}*Δε*, the difference between the real and complex frequency dispersion formulations will be amplified under strong modulation. This is illustrated in Fig. 9. From Figs. 9(a) and 9(b), we see that the dispersion plots in real frequency do not give a clearly defined band gap while the set in complex frequency does. The KH model provides qualitatively correct results even for this high level of grating modulation. Again, both representations obtain the same nonleaky edge located at approximately

^{2}*k*/

_{0}*K*=0.577 for this example. It is interesting to note that the nonleaky edge has shifted to the lower side of the band (compare with Figs. 6 and 7). This will be explained in a future publication; we can show that the transition point is at

*h*=

_{2}*Re(h*.

_{1})## 5. Guided-mode resonance excitation on periodic waveguides

*TE*; for example,

_{m,ν}*TE*is the resonance formed by the coupling between the second diffracted order and the

_{2,1}*TE*mode.

_{1}*TE*,

_{1,0}*TE*and

_{2,0}*TE*can be observed at normalized frequencies (

_{2,1}*k*/

_{0}*K*) 0.57, 1.06 and 1.23 respectively. Because GMR

*TE*resides in the zero-order diffraction regime, there is no energy lost to propagating higher-order diffracted waves and the resonance peak has 100% reflectance theoretically. The higher-order diffraction regime begins approximately at frequency 0.67, at which the ±1 diffraction orders emerge from the waveguide as propagating waves on the substrate side. Additional propagating orders will appear as the frequency increases.

_{1,0}21. T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. **14**,235–254 (1977). [CrossRef]

22. C. Peng and W. Challener, “Input-grating couplers for narrow Gaussian beam: influence of groove depth,” Opt. Express **12**,6481–6490 (2004). [CrossRef] [PubMed]

## 6. Association of dispersion and resonance

**12**,1885–1891 (2004). [CrossRef] [PubMed]

*Δε*. The width of the stop band is ~2

*h*/

_{2}*h*(see Table I), in both cases, is set primarily by

_{3}*h*which is proportional to the value of the second Fourier grating harmonic with amplitude

_{2}*γ*. At the resonant (leaky) edge, the difference in the β-values is due to the leaked wave as demonstrated in Sec. 4. The representation in complex frequency delivers the real propagation constant directly and matches the results from the diffraction computations as demonstrated in Fig 13. Figure 13(c) indicates phase matching to the deviated part of the curve. It implies that, as an example, for incident wave with θ=0.2°, a resonance should be found at the point marked by ‘*’ in Fig. 13(c) based on Eq. (8). However, there are no states in the band and this picture is not correct. The phase matching relation that pertains to the correct dispersion plot in Fig. 13(b) is

_{2}*β*is real and found by dispersion computations employing complex frequency.

_{ν}*h*(leak) and

_{1}*h*(Bragg). Thus, the leaked wave perturbs the real part of β on which phase-matching proceeds. Assignment of a complex propagation constant to the problem requires spatial Bragg- and radiation decay via Im(

_{2}*β*). In contrast, by casting the dispersion computation into the complex frequency domain, the spatial decay mechanisms associated with Bragg reflection and radiation leakage are converted into a single time decay coefficient as described in Sec. 4.3. This renders an unperturbed, real

_{ν}*β*for input-wave phase matching, avoiding explicit dependence on separate spatial decay mechanisms, and provides consistent results as shown in Fig. 13.

_{ν}**12**,5661–5674 (2004). [CrossRef] [PubMed]

## 6. Conclusion

**QE-21**,144–150 (1985). [CrossRef]

*β*that is found by dispersion computations employing complex frequency.

_{ν}## Acknowledgment

## References and links

1. | A. Yariv, |

2. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

3. | K. Sakoda, |

4. | Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express |

5. | D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, “Optical fiber endface biosensor based on resonances in dielectric waveguide gratings,” Proceedings of the SPIE |

6. | B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction,” Sens. Actuators B |

7. | G. Purvinis, P. S. Priambodo, M. Pomerantz, M. Zhou, T. A. Maldonado, and R. Magnusson, “Second-harmonic generation in resonant waveguide gratings incorporating ionic self-assembled monolayer polymer films,” Opt. Lett. |

8. | Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express |

9. | R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation loss,” IEEE J. Quantum Electron. |

10. | D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. |

11. | P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. |

12. | T. Tamir and S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A |

13. | D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, and J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A |

14. | A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, “Resonant scattering and mode coupling in two-dimensional textured planar waveguides,” J. Opt. Soc. Am. A. |

15. | D. Gerace and L. C. Andreani, ”Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs,” Phys. Rev. E |

16. | T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE |

17. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

18. | S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguide,” IEEE Trans. Microwave Theory Tech. |

19. | T. Tamir and S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. of Lightwave Technol. |

20. | D. L. Jaggard and C. Elachi, “Floquet and coupled-waves analysis of higher-order Bragg coupling in a periodic medium,” J. Opt. Soc. Am. |

21. | T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. |

22. | C. Peng and W. Challener, “Input-grating couplers for narrow Gaussian beam: influence of groove depth,” Opt. Express |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.1970) Diffraction and gratings : Diffractive optics

(310.0310) Thin films : Thin films

(310.2790) Thin films : Guided waves

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: December 1, 2006

Revised Manuscript: January 11, 2007

Manuscript Accepted: January 12, 2007

Published: January 22, 2007

**Citation**

Y. Ding and R. Magnusson, "Band gaps and leaky-wave effects in resonant photonic-crystal waveguides," Opt. Express **15**, 680-694 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-680

Sort: Year | Journal | Reset

### References

- A. Yariv, Optical Electronics in Modern Communications, 5th edition (Oxford University Press, New York, 1997).
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995).
- K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
- Y. Ding and R. Magnusson, "Resonant leaky-mode spectral-band engineering and device applications," Opt. Express 12,5661-5674 (2004). [CrossRef] [PubMed]
- D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, "Optical fiber endface biosensor based on resonances in dielectric waveguide gratings," Proceedings of the SPIE 3911, 86-94 (2000). [CrossRef]
- B. T. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, and B. Hugh, "A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interaction," Sens. Actuators B 85, 219-226 (2002). [CrossRef]
- G. Purvinis, P. S. Priambodo, M. Pomerantz, M. Zhou, T. A. Maldonado, and R. Magnusson, "Second-harmonic generation in resonant waveguide gratings incorporating ionic self-assembled monolayer polymer films," Opt. Lett. 29, 1108-1110 (2004). [CrossRef] [PubMed]
- Y. Ding and R. Magnusson, "Use of nondegenerate resonant leaky modes to fashion diverse optical spectra," Opt. Express 12, 1885-1891 (2004). [CrossRef] [PubMed]
- R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation loss," IEEE J. Quantum Electron. QE-21, 144-150 (1985). [CrossRef]
- D. Rosenblatt, A. Sharon, and A. A. Friesem, "Resonant grating waveguide structures," IEEE J. Quantum Electron. 33, 2038-2059 (1997). [CrossRef]
- P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979). [CrossRef]
- T. Tamir and S. Zhang, "Resonant scattering by multilayered dielectric gratings," J. Opt. Soc. Am. A 14, 1607-1616 (1997). [CrossRef]
- D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, and J. M. Bendickson, "Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence," J. Opt. Soc. Am. A 17, 1221-1230 (2000). [CrossRef]
- A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001). [CrossRef]
- D. Gerace and L. C. Andreani, "Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs," Phys. Rev. E 69, 056603 (2004). [CrossRef]
- T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985). [CrossRef]
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1086 (1995). [CrossRef]
- S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguide," IEEE Trans. Microwave Theory Tech. MTT-23, 123-133 (1975). [CrossRef]
- T. Tamir and S. Zhang, "Modal transmission-line theory of multilayered grating structures," J. of Lightwave Technol. 14, 914-927 (1996). [CrossRef]
- D. L. Jaggard and C. Elachi, "Floquet and coupled-waves analysis of higher-order Bragg coupling in a periodic medium," J. Opt. Soc. Am. 66, 674-682 (1976). [CrossRef]
- T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235-254 (1977). [CrossRef]
- C. Peng and W. Challener, "Input-grating couplers for narrow Gaussian beam: influence of groove depth," Opt. Express 12, 6481-6490 (2004). [CrossRef] [PubMed]

## 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.

### Figures

Fig. 1. |
Fig. 2. |
Fig. 3. |

Fig. 4. |
Fig. 5. |
Fig. 6. |

Fig. 7. |
Fig. 8. |
Fig. 9. |

Fig. 10. |
Fig. 11. |
Fig. 12. |

Fig. 13. |
Fig. 14. |
Fig. 15. |

« Previous Article | Next Article »

OSA is a member of CrossRef.