## Bridging pole and coupled wave formalisms for grating waveguide resonance analysis and design synthesis

Optics Express, Vol. 15, Issue 15, pp. 9831-9842 (2007)

http://dx.doi.org/10.1364/OE.15.009831

Acrobat PDF (620 KB)

### Abstract

The algebraic polar expression of resonant reflection from a grating waveguide excited by a free space wave is formulated in terms of the physically meaningful phenomenological parameters of the coupled wave formalism. The reflection coefficient is simply represented as a circle in the complex plane which sheds light on the behaviour of the modulus and phase of anomalous reflection. Analytical expressions are derived for the phenomenological parameters that can now be calculated from optogeometrical quantities which are simple to measure. The relevance and usefulness of bridging the two formalisms is shown in the example of the design of an evanescent wave biosensor.

© 2007 Optical Society of America

## 1. Introduction

2. J. Marcou, N. Gremillet, and G. Tomin, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides. I - Theoretical study. II - Experimental study,” Opt. Commun. **32**, 63–71 (1980). [CrossRef]

3. J. Van Roey and P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin films gratings,” Appl. Opt. **20**, 423–429 (1981). [CrossRef] [PubMed]

4. L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. **55**, 377–380 (1985). [CrossRef]

5. G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. **15**, 886–887 (1985). [CrossRef]

6. I. A. Avrutsky and V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. **36**, 1527–1539 (1989). [CrossRef]

7. S. S. Wang, R. Magnusson, J. S. Bagby, and M. G. Moharam, “Guided-mode resonance in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A **7**, 1470–1474 (1990). [CrossRef]

8. N. M. Lyndin, V. A. Sychugov, A. V. Tishchenko, O. Parriaux, N. Athanassopoulou, P. Edwards, C. Maule, and J. Molloy, “Enhanced visibility grating waveguide biosensor,” in *Proceedings of 5 ^{th} European Conference on Optical Chemical Sensors and Biosensors EUROPT®ODE V*, Lyon and France, ed. (Elsevier, 2000), p. 63.

9. B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B **81**, 316–328 (2002). [CrossRef]

10. Y. Fang, A. Ferries, N. Fontaine, J. Mauro, and J. Balakrishnan, “Resonant waveguide biosensor for living cell sensing,” Biophys. J. **91**, 1925–1940 (2006). [CrossRef] [PubMed]

11. T. Clausnitzer, A. V. Tishchenko, E.-B. Kley, H.-J. Fuchs, D. Schelle, O. Parriaux, and U. Kroll, “Narrowband, polarization-independent free-space wave notch filter,” J. Opt. Soc. Am. A **22**, 2799–2803 (2005). [CrossRef]

12. J.-F. Bisson, O. Parriaux, J. C. Pommier, S. Tonchev, and K. Ueda, “A polarization-stabilized microchip laser using a resonant mirror,” Appl. Phys. B **85**, 519–524 (2006). [CrossRef]

13. N. Destouches, J. C. Pommier, O. Parriaux, T. Clausnitzer, N. Lyndin, and S. Tonchev, “Narrow band resonant grating of 100% reflection under normal incidence,” Opt. Express **14**, 12613–12622 (2006). [CrossRef] [PubMed]

14. T. Katchalski, G. Levy-Yurista, A. Friesem, G. Martin, R. Hierle, and J. Zyss, “Light modulation with electro-optic polymer-based resonant grating waveguide structures,” Opt. Express **13**, 4645–4650 (2005). [CrossRef] [PubMed]

15. N. Destouches, A. Tishchenko, J. Pommier, S. Reynaud, O. Parriaux, S. Tonchev, and M. Ahmed, “99% efficiency measured in the -1st order of a resonant grating,” Opt. Express **13**, 3230–3235 (2005). [CrossRef] [PubMed]

16. I. A. Avrutsky, Y. Zhao, and V. Kochrgin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated metal film,” Opt. Lett. **25**, 595–597 (2000). [CrossRef]

17. Y. Jourlin, E. Gamet, S. Tonchev, A. V. Tishchenko, O. Parriaux, and A. Last, “Low loss polarizing beam splitter using the long range plasmon mode along a continuous metal film,” Proc. SPIE **6187**, 61870 H (2006). [CrossRef]

_{0}n

_{c}sinθi±K

_{g}where k

_{0}=2π/λ is the free space wavenumber at vacuum wavelength λ, n

_{c}is the incidence medium refractive index and K

_{g}=2π/Λ is the spatial frequency of the grating of period Λ, θ

_{i}is the incidence angle in the cover medium.

## 2. The relationship between the pole and coupled wave representations

20. A. V. Tishchenko, M. Hamdoun, and O. Parriaux, “Two-dimensional coupled mode equation for grating waveguide excitation by a focused beam,” Opt. Quantum Electron. **35**, 475–491(2003). [CrossRef]

21. M. D. Salik and P. Chavel, “Resonant excitation analysis of waveguide grating couplers,” Opt. Commun. **193**, 127–131 (2001). [CrossRef]

22. M. Neviere, D. Maystre, and P. Vincent, “Application du calcul des modes de propagation a l’étude théorique des anomalies des réseaux recouverts de diélectriques,” J. Opt. **8**, 231–242 (1977). [CrossRef]

_{0}and a term r

_{g}resulting from the presence of a single pole in the k-vector space:

_{p}represents the complex pole coordinates, and a

_{p}is a complex constant coefficient which must somehow be determined. The reflection coefficient r(k) is a complex function of the real variable k. Since k describes the incidence conditions in the concrete case of waveguide grating coupling, it is real. Therefore, the scan of k will be experimentally made on the real axis by varying the incidence angle θ

_{i}or the period Λ or the wavelength λ:

_{p}and k

_{p}will first be connected to the phenomenological parameters of the coupled wave formalism.

_{g}(x) of the field of a guided mode propagating along x in the slab waveguide simply states that the longitudinal variation of the modal field is the sum of two terms [23

23. M. Nevière, “The homogeneous problem,” in *Electromagnetic Theory of Gratings*, R. Petit, ed., (Springer Verlag Berlin, 1980), pp. 123–157. [CrossRef]

_{g}(x) with the complex propagation constant of the modal field as a coefficient

_{i}- K

_{g}, is the detuning factor expressing how far the coupling process is from perfect resonance synchronism (k=β). In k, k

_{i}is taken with the + sign to express contradirectional first order coupling. Under these hypotheses the steady state solution a

_{g}of Eq. (3) is simply:

_{g}of the reflection. As resonant reflection relies upon waveguide mode excitation it is natural to assume that r

_{g}is proportional to a

_{g}, thus, by using (4):

_{c}is the proportionality coefficient whose meaning is that of a coupling diffraction efficiency of the guided mode towards the cover.

_{p}is the product of the coefficient κ describing the guided mode field feeding by the incident wave by the coefficient η

_{c}describing the contribution of the modal field to the resonant reflection. This doesn’t indicate how to easily quantify κ and η

_{c}. However, the main interest of the present paper is in the reflection as expressed in (5) where the κ and η

_{c}parameters are involved in the form of their product. The separation of the roles of κ and ηc is only needed when discussing the modal field which then has to be normalized. This is beyond the scope of the present paper; the interested reader will find in ref. [19

19. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta **33**, 607–619 (1986). [CrossRef]

24. N. M. Lyndin, O. Parriaux, and V. A. Sychugov, “Waveguide excitation by a Gaussian beam and a finite size grating,” Sens. Actuators B **41**, 23–29 (1997). [CrossRef]

## 3. Modulus and phase of the reflection coefficient

_{g}(k) and identify all meaningful k-values.

### 3.1 Complex circle representation of r(k)

_{g}(k) is a complex function having a first order pole. The locus of r

_{g}(k) in the complex plane upon the scan of k on the real axis is a circle as it is known from the theory of analytical functions. This is shown by finding out that there exists a complex point z

_{0}relative to which the modulus of (r(k)-z

_{0}) is constant for all k. The expression of the circle in the complex plane is given by substituting the identities (6) into expression (5):

_{g}(k) is centered at the complex point z

_{0}=ja

_{p}/2α and its radius is |a

_{p}|/2α. The location of the running point on the r

_{g}(k) circle for every value of the real parameter k is determined by the argument (9) of the complex phasor r

_{g}(k)-z

_{0}:

_{p}is the phase of the complex constant a

_{p}. Upon a scan along k from -∞ to +∞, the argument varies from π to -π. This means that the r

_{g}apex describes only one turn of the circle, and in the clockwise direction.

_{g}(k), tends to zero. Thus the rg(k) circle passes through the origin of the complex plane and the origin is the start (k→-∞) and end point (k→+∞) of the scan.

_{0}and the resonant part r

_{g}(k) just amounts to translating the described circle by the complex quantity r

_{0}. The graphic location of the r(k) circle is easy: knowing that the reflection modulus is 1 at the condition of resonant reflection r

_{M}=r(k

_{M}), the r(k) circle is tangent at this point to the circle of radius 1 centered at the origin of the complex plane (k

_{M}is the value taken by k at the resonant reflection point). Therefore, the latter, the r(k) circle center and the complex unit reflection point rM are on the same straight line. On the same straight line is also the point of minimum reflection r

_{m}. The r(k) circle is thus centered at point:

### 3.2 Relationships between radiation coefficient, coupling and propagation constants, and reflection coefficients

_{p}/2α from (10) into (12) gives:

_{β}, C and r

_{0}are on the same straight line, therefore on a diameter of the circle, and, in modulus, that r

_{β}and r

_{0}are equidistant from C. This means that the Fresnel reflection and the reflection at resonance are diametrically opposed on the circle. Now extracting a

_{p}from expression (12) of r

_{β}yields:

_{p}is proportional to the product of the radiation coefficient by the difference r

_{β}-r

_{0}.

_{g}which is the condition for maximum modal field excitation:

_{g}at its maximum.

_{β}is not easy to measure as the propagation constant β of the grating waveguide mode is unknown. Writing the circle center coordinate as:

_{β}in terms of easily measurable quantities:

_{p}|<α, the origin is outside the circle and the phase of r(k) varies as an oscillation around the phase ϕ

_{0}of r

_{0}. The phase variation is smaller than 2 arctan

_{p}|>α, the origin is contained in the circle and the phase experiences a 2π variation. In a third case the origin is on the circle (|a

_{p}|=α) and the phase variation is π. This is not an exceptional situation; it is often desired as for instance in biosensors [8

8. N. M. Lyndin, V. A. Sychugov, A. V. Tishchenko, O. Parriaux, N. Athanassopoulou, P. Edwards, C. Maule, and J. Molloy, “Enhanced visibility grating waveguide biosensor,” in *Proceedings of 5 ^{th} European Conference on Optical Chemical Sensors and Biosensors EUROPT®ODE V*, Lyon and France, ed. (Elsevier, 2000), p. 63.

_{0}=0).

_{M}, but only on the relative phases ϕ

_{0}-ϕ

_{M}and ϕ

_{m}-ϕ

_{M}. Once the modulus |r

_{0}| is known, its relative phase ϕ

_{0}-ϕ

_{M}is easily determined by finding graphically the intersection of the circle of radius |r

_{0}| and center O with the part of the r(k) circle corresponding to k values off the resonance domain. Less easy however is the determination of φ

_{m}-φ

_{M}because of the unknown sign of φ

_{m}relative to φ

_{M}. This phase difference can be straightforwardly determined by a simple polarimetric arrangement. After the quantities |r

_{m}| and |r

_{M}| have been measured for the polarization exhibiting resonance, a further power measurement is made in the resonance domain of the reflection coefficient of the orthogonal polarization |r

_{p}|. |r

_{p}| is essentially constant in the resonance domain, its unknown phase also. Then, the measurement of |r

_{m}| and |r

_{M}| is resumed with the incident polarization at 45 degrees from the incidence plane. An analyzer at 45 degrees placed after the reflection point projects the two polarizations in the same direction where they interfere. Let us first consider the interference product behind the analyser in the situation of resonant reflection r

_{M}: regardless of the origin being in the circle or not, the interference product is generally constructive when r

_{p}points in the same general direction as r

_{M}in the complex plane of Fig. 2, i.e. the measured power is larger than the quadratic mean of |r

_{M}| and |r

_{p}| corresponding to the case when rp is orthogonal to r

_{M}; it is generally destructive when r

_{p}points in the opposite general direction and its value is smaller than the quadratic mean. Considering now the interference product with the origin outside the circle with the parameter k set at k=k

_{m}, i.e. at r

_{m}, |r

_{m}| increases when r

_{p}generally points towards r

_{M}and decreases when r

_{p}points opposite. When the origin is contained in the circle, the converse applies.

_{p}in the direction of the analyser is for both r

_{m}and r

_{M}constructive or destructive, the origin is not in the circle. If it is constructive for r

_{m}and destructive for r

_{M}, or conversely, the circle contains the origin and the phase change across resonance is 2π. Whether the phase of the reflection undergoes an oscillation or a jump across resonance is a critical issue for all applications dealing with femtosecond pulse temporal control as analysed in ref [26].

_{β}r

_{M}r

_{0}and using the property r

_{0}+r

_{β}=rm+r

_{M}(since r

_{0}and r

_{β}are diametrically opposed) in the complex plane lead to the important relationships between reflection moduli:

_{M}=ϕ

_{m}if the origin O is outside the circle, and ϕ

_{M}=ϕ

_{m}+π if O is contained in the circle. The above relationships permit a concise expression for the modulus of r(k):

_{M}and k

_{m}for which the resonant reflection r

_{M}and the minimum reflection r

_{m}take place. This yields:

_{M}| is always 1 under plane wave excitation, therefore

*ρ*is always larger than 1. From expressions (20) α (therefore a

_{p}from (14)) can be expressed from easily measurable reflection coefficients and incidence parameter values k:

_{β}| and check on with the first expression of (17). Now knowing β, r

_{0}, α and r

_{β}the coupling constant a

_{p}can be retrieved using (14). All the phenomenological parameters are then known and the polar approximated function describing the resonant reflection can be calculated.

_{0}|, the resonant reflection |r

_{M}| and the minimum reflection |r

_{m}|, and their locations k

_{M}and k

_{m}. The scan of k can be performed by varying the incidence angle, the grating pitch, or the wavelength. These results are valid regardless of the polarization and of the side(s) at which the corrugation or index modulation is made. Furthermore, although resonant reflection can in a lossless structure always be 100% (|r

_{M}|=1) by a proper choice of the beam and grating parameters, the results of the present analysis are also valid when |r

_{M}| < 1, in the presence of scattering and absorption losses for instance; the retrieved α in such case is the sum of the radiation coefficient and of a loss coefficient. More generally, the above analysis also brings a clarification of the complex behaviour of the phase of reflection of a grating slab waveguide in the resonance domain.

## 4. Exact numerical simulation

27. K. Tiefenthaler and W. Lukosz, “Sensitivity of grating couplers as integrated-optical chemical sensors,” J. Opt. Soc. Am. B **6**, 209–220 (1989). [CrossRef]

28. C. Fattinger, H. Koller, D. Schlatter, and P. Wehrli, “The difference interferometer: a highly sensitive optical probe for quantification of molecular surface concentration,” Biosens. Bioelectron. **8**, 99–107 (1993). [CrossRef]

8. N. M. Lyndin, V. A. Sychugov, A. V. Tishchenko, O. Parriaux, N. Athanassopoulou, P. Edwards, C. Maule, and J. Molloy, “Enhanced visibility grating waveguide biosensor,” in *Proceedings of 5 ^{th} European Conference on Optical Chemical Sensors and Biosensors EUROPT®ODE V*, Lyon and France, ed. (Elsevier, 2000), p. 63.

9. B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B **81**, 316–328 (2002). [CrossRef]

10. Y. Fang, A. Ferries, N. Fontaine, J. Mauro, and J. Balakrishnan, “Resonant waveguide biosensor for living cell sensing,” Biophys. J. **91**, 1925–1940 (2006). [CrossRef] [PubMed]

_{2}surface in a water based solution are found to be obtained with a waveguide thickness h

_{opt}=70 nm [29].

_{2}layer is 56 nm and the depth of the binary HfO

_{2}corrugation is 30 nm so that the effective thickness of the resonant grating is close to the optimal thickness h

_{opt}needed for maximum surface sensing sensitivity. Such grating depth was chosen to satisfy the condition for high, close to 100% resonant reflection wα≫1 [30

30. M. S. Klimov, V. A. Sychugov, A. V. Tishchenko, and O. Parriaux, “Optimization of optical waveguide grating couplers,” Fiber Integ. Opt. **11**, 85–90 (1992). [CrossRef]

_{i}=30 degrees which implies a grating period of Λ=339 nm to excite the TE

_{0}mode. The scan of the k-parameter is made by varying the incidence angle θi in the substrate using the exact grating code of Lyndin [31] based on the modal method. As sketched in Fig. 1, the mode excitation is contradirectional. The reflection modulus is 1 at k

_{M}=125300 cm

^{-1}and |r

_{m}|=0.3616 at k

_{m}=124818 cm

^{-1}. The large number of digits retained here is irrelevant practically; it is only intended to permit a comparison with the polar model and check on the self-consistency of the exact and polar representations of r(k). Figure 3 represents the reflection coefficient modulus and phase versus θ

_{i}in the resonance domain. The phase of the maximum and minimum reflection coefficients is ϕ

_{M}=72 degrees, and ϕ

_{m}=-108 degrees respectively; this by the way means that the origin is contained in the circle. Representing the numerical results of Fig. 3 in the complex plane with k as a parameter gives the solid curve of Fig. 4 which is the exact representation of the complex function r(k).

^{-1}and β=125272 cm

^{-1}(such value for α implies that a beam of about 0.5 mm cross-section and above exhibits 100% resonant reflection). Returning to the modelling of the reflection coefficient by a polar function, the propagation constant can be calculated from expression (23) and the radiation coefficient from expression (22) by borrowing |r

_{0}|, |r

_{m}|, k

_{m}and k

_{M}from the exact modelling. The resulting phenomenological parameters are β=125275cm

^{-1}, and α=105 cm

^{-1}, β corresponding to an effective index of 1.5552 at wavelength λ=780 nm for the TE

_{0}mode excited by the -1

^{st}diffraction order. These values are very close to the numerical ones. The remaining discrepancy is small in the present structure and coupling conditions, but it can be larger. The main hypothesis in the polar representation (11) in terms of the phenomenological parameters is the constancy of the latter over the scanning range across resonance. This is only approximately true. For instance, scanning the parameter k by varying the grating period or the incidence angle obviously modifies slightly the interference conditions in the adjacent media between radiated waves, therefore α. Similarly, scanning k by varying the wavelength also slightly changes α(the modal field size changes) and the effective index.

_{β}can be retrieved exactly: its amplitude is 0.9754 and its phase 54 degrees. Using k

_{M}, k

_{m}, as well as |r

_{m}| and the phases φ

_{M}and φm given by the exact code permits to determine rβ - r

_{0}by using (17), therefore ap can be calculated from expression (14): its amplitude is 142 cm

^{-1}and its phase -45 degrees. Using the polar expression (1) with the calculated parameters α, β and a

_{p}one can represent the r(k) circle as shown by the crosses in Fig. 4. In Figs. 3 and 4 the exact numerical results closely coincide with the polar function calculated by using the retrieved phenomenological parameters; the density of crosses along the solid lines of Figs. 3 and 4 expresses the rate of change of r(k) upon an incremental variation Δθ

_{i}of the incidence angle in the substrate with Δθ

_{i}=0.02 degree. This confirms that the polar algebraic model of resonant reflection closely describes the features of this useful electromagnetic effect and can be used to find out the phenomenological parameters of a given structure as well as to design novel resonant devices. Such statement remains true as long as the coupled mode theory properly accounts for the involved coupling mechanism.

## 5. Conclusion

## Acknowledgments

## References and links

1. | V. A. Sychugov, A. V. Tishchenko, and A. A. Khakimov, “Resonant wave conversion in a corrugated dielectric waveguide,” Sov. Tech. Phys. Lett. |

2. | J. Marcou, N. Gremillet, and G. Tomin, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides. I - Theoretical study. II - Experimental study,” Opt. Commun. |

3. | J. Van Roey and P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin films gratings,” Appl. Opt. |

4. | L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. |

5. | G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. |

6. | I. A. Avrutsky and V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. |

7. | S. S. Wang, R. Magnusson, J. S. Bagby, and M. G. Moharam, “Guided-mode resonance in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A |

8. | N. M. Lyndin, V. A. Sychugov, A. V. Tishchenko, O. Parriaux, N. Athanassopoulou, P. Edwards, C. Maule, and J. Molloy, “Enhanced visibility grating waveguide biosensor,” in |

9. | B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B |

10. | Y. Fang, A. Ferries, N. Fontaine, J. Mauro, and J. Balakrishnan, “Resonant waveguide biosensor for living cell sensing,” Biophys. J. |

11. | T. Clausnitzer, A. V. Tishchenko, E.-B. Kley, H.-J. Fuchs, D. Schelle, O. Parriaux, and U. Kroll, “Narrowband, polarization-independent free-space wave notch filter,” J. Opt. Soc. Am. A |

12. | J.-F. Bisson, O. Parriaux, J. C. Pommier, S. Tonchev, and K. Ueda, “A polarization-stabilized microchip laser using a resonant mirror,” Appl. Phys. B |

13. | N. Destouches, J. C. Pommier, O. Parriaux, T. Clausnitzer, N. Lyndin, and S. Tonchev, “Narrow band resonant grating of 100% reflection under normal incidence,” Opt. Express |

14. | T. Katchalski, G. Levy-Yurista, A. Friesem, G. Martin, R. Hierle, and J. Zyss, “Light modulation with electro-optic polymer-based resonant grating waveguide structures,” Opt. Express |

15. | N. Destouches, A. Tishchenko, J. Pommier, S. Reynaud, O. Parriaux, S. Tonchev, and M. Ahmed, “99% efficiency measured in the -1st order of a resonant grating,” Opt. Express |

16. | I. A. Avrutsky, Y. Zhao, and V. Kochrgin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated metal film,” Opt. Lett. |

17. | Y. Jourlin, E. Gamet, S. Tonchev, A. V. Tishchenko, O. Parriaux, and A. Last, “Low loss polarizing beam splitter using the long range plasmon mode along a continuous metal film,” Proc. SPIE |

18. | V. A. Sychugov and A. V. Tishchenko, “Ray optics philosophy in the problem of corrugated-waveguide-excitation with an external lightbeam,” Photonics Optoelectron. |

19. | E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta |

20. | A. V. Tishchenko, M. Hamdoun, and O. Parriaux, “Two-dimensional coupled mode equation for grating waveguide excitation by a focused beam,” Opt. Quantum Electron. |

21. | M. D. Salik and P. Chavel, “Resonant excitation analysis of waveguide grating couplers,” Opt. Commun. |

22. | M. Neviere, D. Maystre, and P. Vincent, “Application du calcul des modes de propagation a l’étude théorique des anomalies des réseaux recouverts de diélectriques,” J. Opt. |

23. | M. Nevière, “The homogeneous problem,” in |

24. | N. M. Lyndin, O. Parriaux, and V. A. Sychugov, “Waveguide excitation by a Gaussian beam and a finite size grating,” Sens. Actuators B |

25. | A. Sychugov et al., “Corrugated waveguide structures in integrated and fibre optics,” in |

26. | D. Pietroy, A. V. Tishchenko, M. Flury, R. Stoian, and O. Parriaux, “Waveguide grating spectral phase-shifter for temporal femtosecond pulse splitting,” in |

27. | K. Tiefenthaler and W. Lukosz, “Sensitivity of grating couplers as integrated-optical chemical sensors,” J. Opt. Soc. Am. B |

28. | C. Fattinger, H. Koller, D. Schlatter, and P. Wehrli, “The difference interferometer: a highly sensitive optical probe for quantification of molecular surface concentration,” Biosens. Bioelectron. |

29. | O. Parriaux and G. J. Veldhuis, “Normalized analysis for the sensitivity optimization of integrated optical evanescent-wave sensors,” J. Lightwave Technol. |

30. | M. S. Klimov, V. A. Sychugov, A. V. Tishchenko, and O. Parriaux, “Optimization of optical waveguide grating couplers,” Fiber Integ. Opt. |

31. | A. V. Tishchenko and N. Lyndin, “The true modal method solves intractable problems: TM incidence on fine metal slits (but the C method also !),” Workshop on Grating Theory, Clermont-Ferrand, France, June 2004. |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(130.2790) Integrated optics : Guided waves

(230.7400) Optical devices : Waveguides, slab

(260.5740) Physical optics : Resonance

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 8, 2007

Revised Manuscript: May 14, 2007

Manuscript Accepted: May 16, 2007

Published: July 20, 2007

**Virtual Issues**

Vol. 2, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

David Pietroy, Alexandre V. Tishchenko, Manuel Flury, and Olivier Parriaux, "Bridging pole and coupled wave formalisms for grating waveguide resonance analysis and design synthesis," Opt. Express **15**, 9831-9842 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-15-9831

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### References

- V. A. Sychugov, A. V. Tishchenko, and A. A. Khakimov, "Resonant wave conversion in a corrugated dielectric waveguide," Sov. Tech. Phys. Lett. 5, 389-390 (1979).
- J. Marcou, N. Gremillet, and G. Tomin, "Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides. I - Theoretical study. II - Experimental study," Opt. Commun. 32, 63-71 (1980). [CrossRef]
- J. Van Roey, and P. E. Lagasse, "Coupled wave analysis of obliquely incident waves in thin films gratings," Appl. Opt. 20, 423-429 (1981). [CrossRef] [PubMed]
- L. Mashev, and E. Popov, "Zero order anomaly of dielectric coated gratings," Opt. Commun. 55, 377-380 (1985). [CrossRef]
- G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, "Total reflection of light from a corrugated surface of a dielectric waveguide," Sov. J. Quantum Electron. 15, 886-887 (1985). [CrossRef]
- I. A. Avrutsky and V. A. Sychugov, "Reflection of a beam of finite size from a corrugated waveguide," J. Mod. Opt. 36, 1527-1539 (1989). [CrossRef]
- S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, "Guided-mode resonance in planar dielectric-layer diffraction gratings," J. Opt. Soc. Am. A 7, 1470-1474 (1990). [CrossRef]
- N. M. Lyndin, V. A. Sychugov, A. V. Tishchenko, O. Parriaux, N. Athanassopoulou, P. Edwards, C. Maule, and J. Molloy, "Enhanced visibility grating waveguide biosensor," in Proceedings of 5th European Conference on Optical Chemical Sensors and Biosensors EUROPT®ODE V, Lyon, France, ed. (Elsevier, 2000), p. 63.
- B. T. Cunningham, P. Li, B. Lin, and J. Pepper, "Colorimetric resonant reflection as a direct biochemical assay technique," Sens. Actuators B 81, 316-328 (2002). [CrossRef]
- Y. Fang, A. Ferries, N. Fontaine, J. Mauro, and J. Balakrishnan, "Resonant waveguide biosensor for living cell sensing," Biophys. J. 91, 1925-1940 (2006). [CrossRef] [PubMed]
- T. Clausnitzer, A. V. Tishchenko, E.-B. Kley, H.-J. Fuchs, D. Schelle, O. Parriaux, and U. Kroll, "Narrowband, polarization-independent free-space wave notch filter," J. Opt. Soc. Am. A 22, 2799-2803 (2005). [CrossRef]
- J.-F. Bisson, O. Parriaux, J. C. Pommier, S. Tonchev, and K. Ueda, "A polarization-stabilized microchip laser using a resonant mirror," Appl. Phys. B 85, 519-524 (2006). [CrossRef]
- N. Destouches, J. C. Pommier, O. Parriaux, T. Clausnitzer, N. Lyndin, and S. Tonchev, "Narrow band resonant grating of 100% reflection under normal incidence," Opt. Express 14, 12613-12622 (2006). [CrossRef] [PubMed]
- T. Katchalski, G. Levy-Yurista, A. Friesem, G. Martin, R. Hierle, and J. Zyss, "Light modulation with electro-optic polymer-based resonant grating waveguide structures," Opt. Express 13, 4645-4650 (2005). [CrossRef] [PubMed]
- N. Destouches, A. Tishchenko, J. Pommier, S. Reynaud, O. Parriaux, S. Tonchev, and M. Ahmed, "99% efficiency measured in the -1st order of a resonant grating," Opt. Express 13, 3230-3235 (2005). [CrossRef] [PubMed]
- I. A. Avrutsky, Y. Zhao, and V. Kochrgin, "Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated metal film," Opt. Lett. 25, 595-597 (2000). [CrossRef]
- Y. Jourlin, E. Gamet, S. Tonchev, A. V. Tishchenko, and O. Parriaux, A. Last, "Low loss polarizing beam splitter using the long range plasmon mode along a continuous metal film," Proc. SPIE 6187, 61870 H (2006). [CrossRef]
- V. A. Sychugov and A. V. Tishchenko, "Ray optics philosophy in the problem of corrugated-waveguide-excitation with an external lightbeam," Photonics Optoelectron. 1, 79-89 (1993).
- E. Popov, L. Mashev, and D. Maystre, "Theoretical study of anomalies of coated dielectric gratings," Opt. Acta 33, 607-619 (1986). [CrossRef]
- A. V. Tishchenko, M. Hamdoun, and O. Parriaux, "Two-dimensional coupled mode equation for grating waveguide excitation by a focused beam," Opt. Quantum Electron. 35, 475-491(2003). [CrossRef]
- M. D. Salik and P. Chavel, "Resonant excitation analysis of waveguide grating couplers," Opt. Commun. 193, 127-131 (2001). [CrossRef]
- M. Neviere, D. Maystre, and P. Vincent, "Application du calcul des modes de propagation a l'étude théorique des anomalies des réseaux recouverts de diélectriques," J. Opt. 8, 231-242 (1977). [CrossRef]
- M. Nevière, "The homogeneous problem," in Electromagnetic Theory of Gratings, R. Petit, ed., (Springer Verlag Berlin, 1980), pp. 123-157. [CrossRef]
- N. M. Lyndin, O. Parriaux, and V. A. Sychugov, "Waveguide excitation by a Gaussian beam and a finite size grating," Sens. Actuators B 41, 23-29 (1997). [CrossRef]
- A. Sychugov et al., "Corrugated waveguide structures in integrated and fibre optics," in Proceedings of the General Physics Institute of the Academy of Sciences of the USSR34, 1991 (in Russian).
- D. Pietroy, A. V. Tishchenko, M. Flury, R. Stoian, and O. Parriaux, "Waveguide grating spectral phase-shifter for temporal femtosecond pulse splitting," in Proceedings of the 13th European Conference on Integrated Optics, 25-27 April 2007, Copenhagen, Denmark.
- K. Tiefenthaler and W. Lukosz, "Sensitivity of grating couplers as integrated-optical chemical sensors," J. Opt. Soc. Am. B 6, 209-220 (1989). [CrossRef]
- C. Fattinger, H. Koller, D. Schlatter, and P. Wehrli, "The difference interferometer: a highly sensitive optical probe for quantification of molecular surface concentration," Biosens. Bioelectron. 8, 99-107 (1993). [CrossRef]
- O. Parriaux and G. J. Veldhuis, "Normalized analysis for the sensitivity optimization of integrated optical evanescent-wave sensors," J. Lightwave Technol. 16, 573-582 (1998). [CrossRef]
- M. S. Klimov, V. A. Sychugov, A. V. Tishchenko, and O. Parriaux, "Optimization of optical waveguide grating couplers," Fiber Integ. Opt. 11, 85-90 (1992). [CrossRef]
- A. V. Tishchenko and N. Lyndin, "The true modal method solves intractable problems: TM incidence on fine metal slits (but the C method also !)," Workshop on Grating Theory, Clermont-Ferrand, France, June 2004.

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