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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 24 — Nov. 26, 2007
  • pp: 15734–15740
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Narrow-band filtering with whispering modes in gratings made of fibers

Anne-Laure Fehrembach, Evgeny Popov, Gérard Tayeb, and Daniel Maystre  »View Author Affiliations


Optics Express, Vol. 15, Issue 24, pp. 15734-15740 (2007)
http://dx.doi.org/10.1364/OE.15.015734


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Abstract

We present a numerical study of whispering modes in gratings made of fibers. Due to the strong localization of the modes inside each fiber, it is possible to obtain narrow-band filters with very broad angular tolerance.

© 2007 Optical Society of America

1. Introduction

Grating anomalies attract attention since the famous observation by R. Wood that grating efficiency can vary several orders of magnitude within a very short spectral interval [1

1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag. 4, 396–402 (1902).

]. While very boring for grating manufacturers, these anomalies, due to excitation of eigenmodes in the grating structure, have found practical application in detection and filtering [2

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

5

5. E. Popov, L. Mashev, and D. Maystre, “Theoretical Study of the Anomalies of Coated Dielectric Gratings,” Opt. Acta 33, 607–619 (1986). [CrossRef]

]. The resonances can be due to surface waves (surface plasmons) along dielectric-metallic interfaces [6

6. D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 17, 196–200 (1976). [CrossRef]

], guided modes of dielectric waveguides [5

5. E. Popov, L. Mashev, and D. Maystre, “Theoretical Study of the Anomalies of Coated Dielectric Gratings,” Opt. Acta 33, 607–619 (1986). [CrossRef]

], cavity [7

7. E. Popov, S. Enoch, G. Tayeb, M. Nevière, B. Gralak, and N. Bonod, “Enhanced transmission due to non-plasmon resonances in one and two dimensional gratings,” Appl. Opt. 43, 999–1008 (2004). [CrossRef] [PubMed]

] or Fabry-Perot resonances [8

8. P. Lalanne, J.-P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: A coupled Bloch-mode insight,” J. Lightwave Technol. 24, 2442–2449 (2006). [CrossRef]

].

The resonant anomalies can be characterized by sharp maxima or minima whose width depends on the resonance finesse and losses. For lossless dielectric waveguides with shallow corrugation, the width of the peak can be reduced to less than several angstroms. However, excitation of guided modes, which leads to very narrow spectral lines, is characterized by strong sensitivity with respect to the angle of incidence and thus requiring very tight tolerances with respect to the collimation of the light beams [9

9. E. Popov and B. Bozhkov, “Corrugated waveguides as resonance optical filters — advantages and limitations,” J. Opt. Soc. Am. A 18, 1758–1764 (2001). [CrossRef]

]. Several approaches are proposed to decrease the angular constraints of such devices, keeping the spectral lines as narrow as possible. The first approach is to flatten the mode dispersion curve by using the Bragg interaction between counter-propagating modes [10

10. F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998). [CrossRef]

]. A second approach is to juxtapose the flatten dispersion curves of two modes, which is possible with deep gratings [11

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

]. The third approach is based on the use of cavity resonances, which are generally characterized by strong spectral variation and much weaker angular dependence. This difference can be understood by taking into account that resonances localized in the direct space have larger support in the inverse space. The problem is that the creation of cavity resonances with large finesse requires use of metallic walls, because the only-dielectric grating provides strong coupling between the consecutive periods and thus reducing the finesse. Unfortunately, in the visible, metals have losses, strongly enhanced when resonances are excited, which significantly reduces the application performances. On the other hand, since Lord Rayleigh [12

12. Lord Rayleigh, “Whispering gallery modes,” Phil. Mag.20, 1001–1004 (1910); idem “The problem of the Whispering Gallery,” Scientific Papers5, 617–620, Cambridge University, Cambridge (1912).

], it is known that even purely dielectric systems can support well-localized modes, called whispering (gallery) modes, because they were discovered in acoustics. These modes can exist in optical fibers and in dielectric spheres, provided the optical dimensions are large enough. They are characterized by strong field maxima localized inside but close to the surface of the object and can be considered from a geometrical point of view as due to total reflection of the beam propagating inside the fiber or the sphere.

2. Narrow-band and angularly tolerant resonances in gratings made of fibers

To simplify the understanding of the phenomena, we use step-profile fibers without cladding, suspended in air. There are several complications that should appear and must be solved for any practical application. First, the whispering modes are very sensitive to fiber dimensions, thus identical fibers must be used. Second, suspending fibers in air is technically almost impossible, thus a dielectric matrix has to be used. Third, in order to maintain the required distance between the fibers during their assembly in the grating, there are two possibilities: either to use fibers with low-index cover, with total diameter equal to the grating period, or to use some self-assembling technique, such as putting the fibers on a Si surface with etched equidistant grooves. However, such technical analysis lies outside the scopes of this paper. Figure 1 represents schematically the rod grating under study. The working polarization is TE, electric field vector parallel to the fibers. All along the paper, the plane of incidence is perpendicular to the fibers axis (e.g. the (Oxy) plane). The wavelength is close to 1.55 µm, the fibers are made of Si with refractive index n 2=3.45, and suspended in air (refractive index n 1=1). The period d is equal to 1.45 µm. Consequently, near normal incidence, the only propagating orders are the reflected and transmitted zeroth orders.

Fig. 1. Rod grating under study.

As can be expected from general theoretical considerations [5

5. E. Popov, L. Mashev, and D. Maystre, “Theoretical Study of the Anomalies of Coated Dielectric Gratings,” Opt. Acta 33, 607–619 (1986). [CrossRef]

], in the vicinity of resonance excitation, one can expect the reflectivity to vary from 0 to 100%. Figure 2 presents the behavior of the reflectivity as a function of the fiber diameter Φ.

Fig. 2. Reflectivity of the rod grating of Fig. 1 versus the rod diameter (in µm). Period d=1.45 µm, wavelength λ=1.55 µm, TE polarization, normal incidence, rod refractive index equals to 3.45.

Fig. 3. Spectral dependence of the reflectivity of the rod grating presented in Fig. 1. Period d=1.45 µm. The rod diameters Φ correspond to the anomalies indicated with arrows in Fig. 2.

Let us thoroughly study the system composed of fibers with diameter 0.8947 µm, chosen among the others as having a spectral response most suitable for narrow-band reflection filtering. We plot on Fig. 4 the angular dependence of the reflectivity. It presents a flat top resonance expanding over a wide angular range. The reflectivity decreases quickly for angles greater than 4° because the first order of the grating becomes propagative above 4°.

Fig. 4. Angular dependence of the reflectivity of the rod grating with period 1.45 µm, rod diameter 0.8947 µm.

An angular tolerant resonance is especially interesting in practical filtering applications for two reasons. First, experimental incident beams have non-zero divergence, which may cause a loss of filtering efficiency for low angular tolerant resonances [26

26. G. Niederer, H. P. Herzig, J. Shamir, H. Thiele, M. Schnieper, and C. Zschokke, “Tunable, oblique incidence resonant grating filter for telecommunications,” Appl. Opt. 43, 1683–1694 (2004). [CrossRef] [PubMed]

]. Second, an angular tolerant filter will be efficient even for focused incident beam and devices having small dimensions. As an example, the grating composed of 0.8947 µm diameter fibers will keep its efficiency for incident beams with convergence angle up to 8°, which corresponds to a Gaussian beam with waist diameter of about 13 µm (for a wavelength of 1.55 µm). We plot on Fig. 5 (solid curve) the reflectivity versus the incident wavelength for a finite-size grating of 30 µm length, illuminated by a Gaussian beam (invariant along the z-axes) with 20 µm waist. The calculations were performed using the Scattering Matrix Method [27

27. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994). [CrossRef]

]. The curve is almost the same as that obtained for the infinite grating illuminated by a plane wave under normal incidence (dashed curve), except for the peak observed around 1.544 µm. This extra resonance is caused by the non-normal incident plane waves that compose the Gaussian beam and which can couple modes having anti-symmetric fields with respect to the normal to the grating, contrary to the normally incident plane wave. For confirmation, the extra resonance also appears when the infinite grating is illuminated by a plane wave with an angle of incidence of 2° with respect to the z-axes in the (Oxy) plane.

Fig. 5. Spectral dependence of the reflectivity of the finite rod grating illuminated by a Gaussian beam (solid curve), infinite grating illuminated by a plane wave under normal incidence (dashed curve) and under an incidence of 2° (dotted curve) (period 1.45 µm, rod diameter 0.8947µm).

3. Physical insight into the origin of the resonance properties

In order to find the physical origin of these anomalies, we plot on Fig. 6 the modulus of the electric field inside the structure at the resonance wavelength. One can observe that the field is particularly localized into the fibers.

Fig. 6. Modulus of the electric field for the finite rod grating (period 1.45 µm, rod diameter 0.8947 µm) illuminated by a Gaussian beam with 20 µm diameter at waist.

Fig. 7. Modulus of the electric field at the resonance wavelength for a single fiber, with diameter (a) 1.0686 µm, (b) 0.8947 µm, (c) 0.7069 µm, (d) 0.5162 µm, (e) 0.3306 µm. The modulus of the incident plane wave is normalized to 1.

The natural question that arises is to know to what extend the WsM resonances remain localized when the fibers are assembled into a grating, as there is an inevitable coupling between the modes in the adjacent fibers through the air gap, which decreases sharply with the increase of the diameter. Let us consider a system of two fibers, with diameter 0.8947 µm. The coupling leads to a splitting of otherwise degenerate mode of the single fiber into four modes, because the cylindrical rotation symmetry is broken into a reflection symmetry with respect to the x- and y-axes (see Fig. 1), thus permitting modes that are symmetrical or antisymmetrical with respect to these axes. The four poles (in the spectral range of interest) of the scattering matrix of the structure are reported in Tab. 1. They are actually distributed in the neighboring of the pole of one single fiber (1.5466+i 1.9 10-3), and are characterized by a smaller or larger imaginary part. Hence, the coupling leads to both stronger and weaker localized modes. Note that the mode which is symmetrical with respect to x- and y-axes has an imaginary part twice as large as the other modes. This can be explained by the fact that the field of this mode has the same symmetry as that of the plane waves radiated along the x- and y-axes, i.e., this mode is more easily radiated along both axes (including their positive and negative directions) than the other modes.

Table.1. Poles of the structure composed of two fibers with diameter 0.8947 µm, separated with a distance of 1.45 µm.

table-icon
View This Table

Last, it is expected that the coupling between the modes of the fibers depends on the distance between the fibers. Hence it may be possible to tune the spectral bandwidth of the resonance by changing the distance between the fibers. We show in Fig. 8 that a twice shorter (2 nm) bandwidth can be obtained with the same 0.8947 µm diameter fibers separated by a longer grating period of 1.49 µm. The angular tolerance of the resonance is limited, as for the grating with period 1.45 µm, by the first order of the grating which becomes propagative for angles greater than 2.3°.

Fig. 8. Spectral dependence of the reflectivity of the rod grating presented in Fig. 1. Period d=1.49 µm (solid curve) and d=1.45 µm (dashed curve), rod diameter Φ=0.8947 µm, illuminated by a plane wave in normal incidence.

4. Conclusion

References and links

1.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag. 4, 396–402 (1902).

2.

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

3.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tischenko, E. Popov, and L. Mashev, “Diffraction characteristics of Planar corrugated waveguides,” Opt. Quantum. Electron. 18, 123–128 (1986). [CrossRef]

4.

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

5.

E. Popov, L. Mashev, and D. Maystre, “Theoretical Study of the Anomalies of Coated Dielectric Gratings,” Opt. Acta 33, 607–619 (1986). [CrossRef]

6.

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 17, 196–200 (1976). [CrossRef]

7.

E. Popov, S. Enoch, G. Tayeb, M. Nevière, B. Gralak, and N. Bonod, “Enhanced transmission due to non-plasmon resonances in one and two dimensional gratings,” Appl. Opt. 43, 999–1008 (2004). [CrossRef] [PubMed]

8.

P. Lalanne, J.-P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: A coupled Bloch-mode insight,” J. Lightwave Technol. 24, 2442–2449 (2006). [CrossRef]

9.

E. Popov and B. Bozhkov, “Corrugated waveguides as resonance optical filters — advantages and limitations,” J. Opt. Soc. Am. A 18, 1758–1764 (2001). [CrossRef]

10.

F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998). [CrossRef]

11.

D. Brundrett, E. Glytsis, T. Gaylord, and J. 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.

Lord Rayleigh, “Whispering gallery modes,” Phil. Mag.20, 1001–1004 (1910); idem “The problem of the Whispering Gallery,” Scientific Papers5, 617–620, Cambridge University, Cambridge (1912).

13.

H. Mahlein, “Fiber-optic communication in the wavelength-division multiplex mode,” Fiber Integr. Opt. 4, 339–372 (1983). [CrossRef]

14.

C. M. de Blok and P. Mathiesse, “Core alignment procedure for single-mode-fiber jointing,” Electron. Lett. 20, 109–110 (1984). [CrossRef]

15.

R. Morgan, J. S. Barton, P. G. Harper, and J. D. C. Jones, “Wavelength dependence of bending loss in monomode optical fibers: effect of the fiber buffer coating,” Opt. Lett. 15, 947–949 (1990). [CrossRef] [PubMed]

16.

F. M. Haran, J. S. Barton, and J. D. C. Jones, “Determination of monomode fiber buffer properties from bend-loss measurements,” Opt. Lett. 18, 1618–1620 (1993). [CrossRef] [PubMed]

17.

R. Morgan, J. D. C. Jones, J. S. Barton, and P. G. Harper, “Determination of monomode fiber buffer properties,” IEEE J. Lightwave Technol. 12, 1355 (1994). [CrossRef]

18.

F. M. Haran, J. S. Barton, S. R. Kidd, and J. D. C. Jones, “Optical-fiber interferometric sensors using buffer guided light,” Meas. Sci. Technol. 5, 526–530 (1994). [CrossRef]

19.

F. M. Haran, J. S. Barton, and J. D. C. Jones, “Bend loss in buffered over-moded optical-fiber - LP11 mode and whispering-gallery mode interaction,” Electron. Lett. 30, 1433–1434 (1994). [CrossRef]

20.

Y. Powell-Friend, L. Phillips, T. George, and A. Sharma, “A simple technique for investigating whispering gallery modes in optical fibers,” Rev. Sci; Instrum. 69, 2868–2870 (1998). [CrossRef] [PubMed]

21.

S. Oliver, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Oesterle, “Miniband transmission in a photonic crystal coupled-resonator optical waveguide,” Opt. Lett. 26, 1019–1021 (2001). [CrossRef]

22.

T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4–6 (2003). [CrossRef]

23.

A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Groucher, and G. A. Gehring, “Defect states and commensurability in dual-period AlxGa1-xAs photonic crystal waveguides,” Phys. Rev. B 68, 033303 (2003). [CrossRef]

24.

S Deng, W. Cai, and V. Asratov, “Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides,” Opt. Express 12, 6468–6480 (2004). [CrossRef] [PubMed]

25.

D. Maystre and R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Chap. 3.

26.

G. Niederer, H. P. Herzig, J. Shamir, H. Thiele, M. Schnieper, and C. Zschokke, “Tunable, oblique incidence resonant grating filter for telecommunications,” Appl. Opt. 43, 1683–1694 (2004). [CrossRef] [PubMed]

27.

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994). [CrossRef]

28.

G. Tayeb and D. Maystre, “Rigorous theoretical study of finite size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14, 3323–3332 (1997). [CrossRef]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(260.5740) Physical optics : Resonance
(230.7408) Optical devices : Wavelength filtering devices

ToC Category:
Diffraction and Gratings

History
Original Manuscript: September 17, 2007
Revised Manuscript: November 6, 2007
Manuscript Accepted: November 6, 2007
Published: November 12, 2007

Citation
Anne-Laure Fehrembach, Evgeny Popov, Gérard Tayeb, and Daniel Maystre, "Narrow-band filtering with whispering modes in gratings made of fibers," Opt. Express 15, 15734-15740 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15734


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References

  1. R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Phylos. Mag. 4, 396-402 (1902).
  2. L. Mashev and E. Popov, "Zero order anomaly of dielectric coated grating," Opt. Commun. 55, 377-380 (1985). [CrossRef]
  3. G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tischenko, E. Popov, and L. Mashev, "Diffraction characteristics of Planar corrugated waveguides," Opt. Quantum. Electron. 18, 123-128 (1986). [CrossRef]
  4. S. Wang, R. Magnusson, J. Bagby, and M. Moharam, "Guided-mode resonances in planar dielectric-layer diffraction gratings," J. Opt. Soc. A 7, 1470-1474 (1990). [CrossRef]
  5. E. Popov, L. Mashev, and D. Maystre, "Theoretical Study of the Anomalies of Coated Dielectric Gratings," Opt. Acta 33, 607-619 (1986). [CrossRef]
  6. D. Maystre and R. Petit, "Brewster incidence for metallic gratings," Opt. Commun. 17, 196-200 (1976). [CrossRef]
  7. E. Popov, S. Enoch, G. Tayeb, M. Nevière, B. Gralak, and N. Bonod, "Enhanced transmission due to non-plasmon resonances in one and two dimensional gratings," Appl. Opt. 43, 999-1008 (2004). [CrossRef] [PubMed]
  8. P. Lalanne, J.-P. Hugonin, and P. Chavel, "Optical properties of deep lamellar gratings: A coupled Bloch-mode insight," J. Lightwave Technol. 24, 2442-2449 (2006). [CrossRef]
  9. E. Popov and B. Bozhkov, "Corrugated waveguides as resonance optical filters - advantages and limitations," J. Opt. Soc. Am. A 18, 1758-1764 (2001). [CrossRef]
  10. F. Lemarchand, A. Sentenac, and H. Giovannini, "Increasing the angular tolerance of resonant grating filters with doubly periodic structures," Opt. Lett. 23, 1149-1151 (1998). [CrossRef]
  11. D. Brundrett, E. Glytsis, T. Gaylord, and J. 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. Lord Rayleigh, "Whispering gallery modes," Phil. Mag. 20, 1001-1004 (1910); idem "The problem of the Whispering Gallery," Scientific Papers 5, 617-620, Cambridge University, Cambridge (1912).
  13. H. Mahlein, "Fiber-optic communication in the wavelength-division multiplex mode," Fiber Integr. Opt. 4,339-372 (1983). [CrossRef]
  14. C. M. de Blok and P. Mathiesse, "Core alignment procedure for single-mode-fiber jointing," Electron. Lett. 20,109-110 (1984). [CrossRef]
  15. R. Morgan, J. S. Barton, P. G. Harper, and J. D. C. Jones, "Wavelength dependence of bending loss in monomode optical fibers: effect of the fiber buffer coating," Opt. Lett. 15,947-949 (1990). [CrossRef] [PubMed]
  16. F. M. Haran, J. S. Barton, and J. D. C. Jones, "Determination of monomode fiber buffer properties from bend-loss measurements," Opt. Lett. 18,1618-1620 (1993). [CrossRef] [PubMed]
  17. R. Morgan, J. D. C. Jones, J. S. Barton, and P. G. Harper, "Determination of monomode fiber buffer properties," IEEE J. Lightwave Technol. 12,1355 (1994). [CrossRef]
  18. F. M. Haran, J. S. Barton, S. R. Kidd, and J. D. C. Jones, "Optical-fiber interferometric sensors using buffer guided light," Meas. Sci. Technol. 5,526-530 (1994). [CrossRef]
  19. F. M. Haran, J. S. Barton, and J. D. C. Jones, "Bend loss in buffered over-moded optical-fiber - LP11 mode and whispering-gallery mode interaction," Electron. Lett. 30,1433-1434 (1994). [CrossRef]
  20. Y. Powell-Friend, L. Phillips, T. George, and A. Sharma, "A simple technique for investigating whispering gallery modes in optical fibers," Rev. Sci; Instrum. 69, 2868-2870 (1998). [CrossRef] [PubMed]
  21. S. Oliver, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Oesterle, "Miniband transmission in a photonic crystal coupled-resonator optical waveguide," Opt. Lett. 26, 1019-1021 (2001). [CrossRef]
  22. T. D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, and L. Goldstein, "Two-dimensional photonic crystal coupled-defect laser diode," Appl. Phys. Lett. 82, 4-6 (2003). [CrossRef]
  23. A. D. Bristow, D. M. Whittaker, V. N. Astratov, M. S. Skolnick, A. Tahraoui, T. F. Krauss, M. Hopkinson, M. P. Groucher, and G. A. Gehring, "Defect states and commensurability in dual-period AlxGa1-xAs photonic crystal waveguides," Phys. Rev. B 68, 033303 (2003). [CrossRef]
  24. S Deng, W. Cai, and V. Asratov, "Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides," Opt. Express 12, 6468-6480 (2004). [CrossRef] [PubMed]
  25. D. Maystre and R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Chap. 3.
  26. G. Niederer, H. P. Herzig, J. Shamir, H. Thiele, M. Schnieper, and C. Zschokke, "Tunable, oblique incidence resonant grating filter for telecommunications," Appl. Opt. 43, 1683-1694 (2004). [CrossRef] [PubMed]
  27. D. Felbacq, G. Tayeb, and D. Maystre, "Scattering by a random set of parallel cylinders," J. Opt. Soc. Am. A 11, 2526-2538 (1994). [CrossRef]
  28. G. Tayeb and D. Maystre, "Rigorous theoretical study of finite size two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A 14, 3323-3332 (1997). [CrossRef]

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