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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 8 — Aug. 10, 2007
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Bridging pole and coupled wave formalisms for grating waveguide resonance analysis and design synthesis

David Pietroy, Alexandre V. Tishchenko, Manuel Flury, and Olivier Parriaux  »View Author Affiliations


Optics Express, Vol. 15, Issue 15, pp. 9831-9842 (2007)
http://dx.doi.org/10.1364/OE.15.009831


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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

A dielectric slab layer with grating coupler propagating at least one guided mode is the simplest form of resonant grating. A free space incident wave satisfying the k-vector synchronism condition between the in-plane projection ki of the incident k-vector, the grating Kg-vector and the propagation constant β of a guided mode couples to the said mode and experiences a quite complex redistribution of field in the zeroth transmitted and reflected diffraction orders, the coupling being usually made via the + or -1st order of the grating so as to avoid power losses via higher diffraction orders.

The condensed set of three phenomenological parameters of the coupled wave representation is very useful for all problems involving grating coupling. They not only give a physical understanding of the coupling and resonance effects involved, they also and most importantly enable the designer to solve the inverse problem quasi-mentally. One important remaining problem which the present paper addresses is the experimental access to these parameters in a given structure. α, β and κ are difficult to measure directly. Therefore, access to them will be found first by bridging the coupled wave vision and the pole representation of the reflection coefficient r(k), then by analysing the modulus and phase of r(k), and identifying the experimental situation giving an easy and accurate access to the needed parameters. The proposed methodology provides a powerful and physically intelligible tool for characterizing and also for designing resonant diffractive phenomena.

2. The relationship between the pole and coupled wave representations

A clarification can however be made by considering the so-called inhomogeneous problem where the occurrence of grating mode excitation is expressed as a pole of the reflection function. Numerical and curve fitting approaches were presented for the retrieval of phenomenological parameters [20

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

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

] from such polar function. We write here the polar function in terms of observable physical quantities, provide a clear picture of the resonant reflection phenomenon and give analytical relationships between phenomenological parameters.

Fig. 1. Contradirectional grating mode coupling of a plane wave into a slab waveguide

As a result of the theory of analytical functions, the reflection at and around the resonance can be expressed from the pole-zero model of Ref. [22

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]

] as the sum of two terms: an essentially constant regular term r0 and a term rg resulting from the presence of a single pole in the k-vector space:

r(k)=r0+apkkp
(1)

where kp represents the complex pole coordinates, and ap 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 θi or the period Λ or the wavelength λ:

k=2πncsinθiλ2πΛ
(2)

The algebraic expression (1) will now be given a physical meaning. The parameters ap and kp will first be connected to the phenomenological parameters of the coupled wave formalism.

ag∂x=κq(x)exp[j(kβ)x]αag
(3)

where α is the radiation coefficient, β the propagation constant of the mode, κ the coupling coefficient, and q(x) the profile of the incident beam taken equal to 1 here as we are restricting our interest to an incident wave and grating of infinite extent. (k-β), with k=ki - Kg, is the detuning factor expressing how far the coupling process is from perfect resonance synchronism (k=β). In k, ki is taken with the + sign to express contradirectional first order coupling. Under these hypotheses the steady state solution ag of Eq. (3) is simply:

ag=jκk(β+jα)
(4)

It is worth remarking at this stage that k=β is the condition for maximum field in the slab waveguide.

rg(k)=ηcag=jκηck(β+jα)
(5)

where ηc is the proportionality coefficient whose meaning is that of a coupling diffraction efficiency of the guided mode towards the cover.

Comparing expressions (1) and (5) establishes the relationship between the pole coefficients and the phenomenological coefficients of the coupled wave representation:

ap=jκηcandkp=β+jα
(6)

3. Modulus and phase of the reflection coefficient

A more general and synthetic modelling of the reflection across a waveguide mode resonance will now be given by resorting to a graphic representation. Now as expression (1) for the reflection of a plane wave from a grating waveguide has received a physical content, we will proceed to the scan of the free real parameter k across resonance and give a representation of the effect of the scan on rg(k) and identify all meaningful k-values.

3.1 Complex circle representation of r(k)

rg(k) is a complex function having a first order pole. The locus of rg(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 z0 relative to which the modulus of (r(k)-z0) is constant for all k. The expression of the circle in the complex plane is given by substituting the identities (6) into expression (5):

rg(k)=ap(kβ)jα
(7)

Removing the (k-β) parameter from the two real equations which the complex expression (7) corresponds to gives the equation of a circle:

[Re(rg)+Im(ap)2α]2+[Im(rg)Re(ap)2α]2=(ap2α)2
(8)
Fig. 2. Representation of the reflection coefficient r(k) in the complex plane.

As illustrated in Fig. 2, the circle rg(k) is centered at the complex point z0=jap/2α and its radius is |ap|/2α. The location of the running point on the rg(k) circle for every value of the real parameter k is determined by the argument (9) of the complex phasor rg(k)-z0 :

Arg[rg(k)z0]=ϕpπ22arctan(kβα)
(9)

where ϕp is the phase of the complex constant ap. Upon a scan along k from -∞ to +∞, the argument varies from π to -π. This means that the rg apex describes only one turn of the circle, and in the clockwise direction.

It is noteworthy that well outside resonance (k→±∞) there is no mode excitation, therefore the resonant part of reflection, rg(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.

Stating now that the reflection coefficient r(k) of the resonant grating is the sum of the regular part r0 and the resonant part rg(k) just amounts to translating the described circle by the complex quantity r0. The graphic location of the r(k) circle is easy: knowing that the reflection modulus is 1 at the condition of resonant reflection rM=r(kM), the r(k) circle is tangent at this point to the circle of radius 1 centered at the origin of the complex plane (kM 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 rm. The r(k) circle is thus centered at point:

C=r0+jap2α
(10)

and its radius is still R=ap2α. Expression (11) hereunder is the polar expression (1) for r(k) with all phenomenological parameters substituted:

r(k)=r0jκηc(kβ)jα
(11)

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

Let us write r(k) at resonant waveguide mode excitation, i.e. at k=β where r(β) is denoted rβ. Setting k=β in (11) yields:

rβ=r(β)=r0+japα
(12)

Substituting jap/2α from (10) into (12) gives:

rβ=2Cr0
orrβC=Cr0
(13)

Complex equality (13) states in phase that rβ, C and r0 are on the same straight line, therefore on a diameter of the circle, and, in modulus, that rβ and r0 are equidistant from C. This means that the Fresnel reflection and the reflection at resonance are diametrically opposed on the circle. Now extracting ap from expression (12) of rβ yields:

ap=jα(rβr0)
(14)

This states that the coefficient ap is proportional to the product of the radiation coefficient by the difference rβ-r0.

κ=αag(β)
(15)

which states that the coupling coefficient is equal to the product of the radiation coefficient α by the amplitude of the guided wave field ag at its maximum.

From an experimental point of view, rβ is not easy to measure as the propagation constant β of the grating waveguide mode is unknown. Writing the circle center coordinate as:

C=r0+rβ2=rM+rm2
(16)

permits to express rβ in terms of easily measurable quantities:

rβ=rM+rmr0
(17)

Three cases must be considered in the phase dependence of r(k) depending on the position of the circle relative to the origin. In the first case |ap|<α, the origin is outside the circle and the phase of r(k) varies as an oscillation around the phase ϕ0 of r0. The phase variation is smaller than 2 arctan (rMrmrM+rm), i.e. smaller or much smaller than π. In the second case |ap|>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 (|ap|=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 5th European Conference on Optical Chemical Sensors and Biosensors EUROPT®ODE V, Lyon and France, ed. (Elsevier, 2000), p. 63.

] to have zero reflection outside resonance (r0=0).

Without measuring the phase of the reflection coefficients it is difficult to find out whether the origin is contained in the r(k) circle or not. But the absolute phase of a reflection coefficient is difficult to measure. We can bypass this difficulty by noting that α does not depend on ϕM, but only on the relative phases ϕ0M and ϕmM. Once the modulus |r0| is known, its relative phase ϕ0M is easily determined by finding graphically the intersection of the circle of radius |r0| 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 φmM because of the unknown sign of φm relative to φM. This phase difference can be straightforwardly determined by a simple polarimetric arrangement. After the quantities |rm| and |rM| 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 |rp|. |rp| is essentially constant in the resonance domain, its unknown phase also. Then, the measurement of |rm| and |rM| 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 rM: regardless of the origin being in the circle or not, the interference product is generally constructive when rp points in the same general direction as rM in the complex plane of Fig. 2, i.e. the measured power is larger than the quadratic mean of |rM| and |rp| corresponding to the case when rp is orthogonal to rM ; it is generally destructive when rp 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=km, i.e. at rm, |rm| increases when rp generally points towards rM and decreases when rp points opposite. When the origin is contained in the circle, the converse applies.

To summarize, if the interferential contribution of rp in the direction of the analyser is for both rm and rM constructive or destructive, the origin is not in the circle. If it is constructive for rm and destructive for rM, 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

26. 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.

].

The mode propagation constant β is also a phenomenological parameter which is difficult to determine in a grating waveguide. Applying the Pythagoras theorem in triangle rβ rM r0 and using the property r0+rβ=rm+rM (since r0 and rβ are diametrically opposed) in the complex plane lead to the important relationships between reflection moduli:

r02+rβ2=rM2+rm2
rβr0cos(ϕβϕ0)=rMrmcos(ϕMϕm)
(18)

ϕMm if the origin O is outside the circle, and ϕMm+π if O is contained in the circle. The above relationships permit a concise expression for the modulus of r(k):

r(k)2=(r02(kβ)2+rβ2α2+2α(kβ)(rM2r02)(r02rm2))((kβ)2+α2)
(19)

By setting the first derivative of |r(k)| with respect to the parameter k to zero it is possible to obtain an expression the k-values kM and km for which the resonant reflection rM and the minimum reflection rm take place. This yields:

kMβ=αρ
kmβ=αρ
(20)

where ρ=rM2r02r02rm2

Theoretically |rM| is always 1 under plane wave excitation, therefore ρ is always larger than 1. From expressions (20) α (therefore ap from (14)) can be expressed from easily measurable reflection coefficients and incidence parameter values k:

α=ρkMkm1+ρ2
(21)

Expressions (20) also reveal a very interesting property of the resonance domain which can be expressed as:

(kMβ)(βkm)=α2
(22)

Therefore, the mode propagation constant β can be expressed as:

β=kM+km2+(kMkm)(ρ21)2(ρ2+1)
(23)

Expression (23) is a result in itself in that it gives the possibility to measure the propagation constant, or effective index, of a corrugated waveguide mode without ambiguity. As a matter of fact the effective index of a mode is usually measured by finding the so-called “m-line” observed in the reflected beam; the internal features of the m-line is generally overlooked which either leads to a low accuracy estimate of the effective index or to an error because there has so far been no criterion to identify what feature of the resonant reflection domain bears the stamp of the coupling synchronism condition: the reflection maximum, or the minimum, or half-way between? Expression (23) can also be used to measure |rβ| and check on with the first expression of (17). Now knowing β, r0, α and rβ the coupling constant ap can be retrieved using (14). All the phenomenological parameters are then known and the polar approximated function describing the resonant reflection can be calculated.

4. Exact numerical simulation

The described methodology of parameter retrieval will now be tested against the exact modelling of a grating coupled structure. The chosen structure is that of an evanescent wave biosensor for non-labelled species where the bioreaction takes place at the surface of a waveguide under the monitoring of the evanescent field tail of a guided mode. A number of grating excitation/readout schemes have been proposed [27

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

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]

]. The scheme which best illustrates the usefulness of the present phenomenological approach is that which makes use of resonant reflection from a grating coupled slab waveguide with biospecies immobilized at its surface and excited from the substrate [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 5th European Conference on Optical Chemical Sensors and Biosensors EUROPT®ODE V, Lyon and France, ed. (Elsevier, 2000), p. 63.

, 9

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

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]

].

Fig. 3. (a). Amplitude and b) phase variations of r(k) across resonance. Solid line: exact numerical modelling. Crosses: polar function.

The structure is made of a pyrex glass substrate coated with an ion plated hafnium oxide layer of 2.12 refractive index and excited with a 780 nm laser beam of TE polarization. The optimum sensitivity conditions for the detection of nanometer sized biospecies at the corrugated HfO2 surface in a water based solution are found to be obtained with a waveguide thickness hopt=70 nm [29].

Fig. 4. Polar representation of the reflection coefficient of a resonant grating. Solid line: exact numerical simulations. Crosses: polar function.

From the knowledge of β, rβ can be retrieved exactly: its amplitude is 0.9754 and its phase 54 degrees. Using kM, km, as well as |rm| and the phases φM and φm given by the exact code permits to determine rβ - r0 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 ap 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 Δθi of the incidence angle in the substrate with Δθi=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.

The retrieval of the phenomenological parameters could actually be made directly from an experimental scan of the parameter k and expression (11) by numerical optimisation without analytical development. However, the present development gives much physical insight into this resonant coupling phenomenon as well as meaningful and useful quantitative expressions for β and α.

5. Conclusion

The outcome of the present analysis has direct implications in resonant structure characterization and design. Beyond this, the synthesis accomplished in the present work is a basis for the exploration of novel applications of this still puzzling effect of anomalous reflection. This effect can be associated with mirrors, filters to give rise to optical functions using its adjustable spectral phase dependence and quasi 100% modulus. One promising field of application is femtosecond laser pulse processing [26

26. 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.

].

Acknowledgments

This work is a contribution of the TSI laboratory (now Laboratoire Hubert Curien) on waveguide characterization and resonant gratings in the Network of Excellence of the European Community on Microoptics NEMO.

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. 5, 389–390 (1979).

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 5th 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]

18.

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).

19.

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

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]

23.

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

25.

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).

26.

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.

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]

29.

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]

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]

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

  1. 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).
  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]
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