## Interpretation of Fano lineshape reversal in the reflectivity spectra of photonic crystal slabs |

Optics Express, Vol. 18, Issue 25, pp. 26569-26582 (2010)

http://dx.doi.org/10.1364/OE.18.026569

Acrobat PDF (1535 KB)

### Abstract

Resonant coupling of light to leaky modes of a photonic crystal slab leads to asymmetric Fano lineshapes in the reflectivity spectra. The generally accepted picture, for a lossless system, is that the sign of the real-valued parameter *q* controls the asymmetry of a Fano resonance. For the reflectivity of a symmetric slab this situation occurs if the amplitude reflection coefficient of the slab goes through zero. In this article, we show that it is also possible to change the asymmetry of a resonance by angle tuning without reaching a condition of zero amplitude. Moreover, we show that the picture of a real-valued parameter *q* that controls the asymmetry is incomplete.

© 2010 Optical Society of America

## 1. Introduction

2. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. **82**, 2257 (2010) [CrossRef]

3. R. K. Adair, C. K. Bockelman, and R. E. Peterson, “Experimental corroboration of the theory of neutron resonance scattering,” Phys. Rev. **76**, 308 (1949). [CrossRef]

4. K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. Iye, “Tuning of the fano effect through a quantum dot in an aharonov-bohm interferometer,” Phys. Rev. Lett. **88**, 256806 (2002). [CrossRef] [PubMed]

5. M. Mendoza, P. A. Schulz, R. O. Vallejos, and C. H. Lewenkopf, “Fano resonances in the conductance of quantum dots with mixed dynamics,” Phys. Rev. B **77**, 155307 (2008). [CrossRef]

6. C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Commun. **225**, 331 (2003). [CrossRef]

7. M. J. A. de Dood, E. F. C. Driessen, D. Stolwijk, and M. P. van Exter, “Observation of coupling between surface plasmons in index-matched hole arrays,” Phys. Rev. B **77**, 115437 (2008). [CrossRef]

8. A. Bärnthaler, S. Rotter, F. Libisch, J. Burgdörfer, S. Gehler, U. Kuhl, and H.-J. Stöckmann, “Probing decoherence through fano resonances,” Phys. Rev. Lett. **105**, 056801 (2010). [CrossRef] [PubMed]

9. M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. **70**, 1438 (1997). [CrossRef]

11. M. Galli, M. Agio, L. C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, and P. Bellutti, “Spectroscopy of photonic bands in macroporous silicon photonic crystals,” Phys. Rev. B **65**, 113111 (2002). [CrossRef]

*q*parameter, contains information on the decoherence and dephasing of the underlying quantum system [8

8. A. Bärnthaler, S. Rotter, F. Libisch, J. Burgdörfer, S. Gehler, U. Kuhl, and H.-J. Stöckmann, “Probing decoherence through fano resonances,” Phys. Rev. Lett. **105**, 056801 (2010). [CrossRef] [PubMed]

12. A. A. Clerk, X. Waintal, and P. W. Brouwer, “Fano resonances as a probe of phase coherence in quantum dots,” Phys. Rev. Lett. **86**, 4636 (2001). [CrossRef] [PubMed]

4. K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. Iye, “Tuning of the fano effect through a quantum dot in an aharonov-bohm interferometer,” Phys. Rev. Lett. **88**, 256806 (2002). [CrossRef] [PubMed]

13. S. Klaiman, N. Moiseyev, and H. R. Sadeghpour, “Interpretation of the fano lineshape reversal in quantum waveguides,” Phys. Rev. B **75**, 113305 (2007). [CrossRef]

*ω*and in-plane wave vector

*k*

_{||}, which are above the light line defined by

*ω*=

*ck*

_{||}, with

*c*the speed of light in vacuum. These modes are either propagating (Fabry-Perot) modes of the slab, or leaky modes that couple incident light from the surrounding media to a guided mode of the slab via diffraction, picking up an additional crystal momentum equal to a reciprocal lattice vector. The interference of this resonant mode with the propagating Fabry-Perot mode leads to Fano resonances [14

14. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B **65**, 235112 (2002). [CrossRef]

9. M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. **70**, 1438 (1997). [CrossRef]

11. M. Galli, M. Agio, L. C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, and P. Bellutti, “Spectroscopy of photonic bands in macroporous silicon photonic crystals,” Phys. Rev. B **65**, 113111 (2002). [CrossRef]

*q*, can be interpreted as the amplitude of the resonant contribution relative to the background. The sign of

*q*controls the asymmetry of the lineshape [1, 15

15. M. Galli, S. L. Portalupi, M. Belotti, L. C. Andreani, L. O’Faolain, and T. F. Krauss, “Light scattering and fano resonances in high-q photonic crystal nanocavities,” Applied Physics Letters **94**, 071101 (2009). [CrossRef]

16. S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. **80**, 908 (2002). [CrossRef]

*d/a*of slab thickness over lattice constant. This shifts the resonance frequency of a leaky mode over a zero in the direct reflection of the slab and reverses the asymmetry. Experimentally this reversal has recently been demonstrated in reflection spectra of

*p*-polarized light from a photonic crystal waveguide via angle tuning [17, 18

18. E. F. C. Driessen, D. Stolwijk, and M. J. A. de Dood , “Asymmetry reversal in the reflection from a two-dimensional photonic crystal,” Opt. Lett. **32**, 3137 (2007) [CrossRef] [PubMed]

*π*phase shift. It is this phase shift that is responsible for the reversal of the asymmetry.

*p*-polarized light goes through zero. In order to describe the reversal of the asymmetry a complex

*q*parameter is needed in a system that obeys both time reversal symmetry and energy conservation. We stress that the origin of the complex

*q*in our work is not due to dephasing or decoherence as reported in literature [8

8. A. Bärnthaler, S. Rotter, F. Libisch, J. Burgdörfer, S. Gehler, U. Kuhl, and H.-J. Stöckmann, “Probing decoherence through fano resonances,” Phys. Rev. Lett. **105**, 056801 (2010). [CrossRef] [PubMed]

12. A. A. Clerk, X. Waintal, and P. W. Brouwer, “Fano resonances as a probe of phase coherence in quantum dots,” Phys. Rev. Lett. **86**, 4636 (2001). [CrossRef] [PubMed]

*q*parameter, the inverse statement is not true.

## 2. Experiment

*a*= 820 nm were fabricated in a commercially grown AlGaAs layer structure [19

19. Philips MiPlaza - Cedova, http://www.cedova.com.

*μ*m thick Al rich Al

_{0.7}Ga

_{0.3}As layer, a 150 nm thick Ga rich Al

_{0.35}Ga

_{0.65}As layer, and a 100 nm thick GaAs capping layer. To create the hole pattern, a 150 nm silicon nitride layer is deposited on top of the structure and serves as a mask during the final reactive ion etching step. The lattice of holes is created by e-beam lithography in a ∼ 500 nm thick layer of positive tone e-beam resist, ZEP 520A [20

20. Z EON corporation, http://www.zeon.co.jp.

_{3}/Ar plasma. After removal of the e-beam resist in a low pressure O

_{2}plasma, the hole pattern is etched deep into the AlGaAs heterostructure in a BCl

_{2}/Cl

_{2}/N

_{2}reactive ion etch process at 100 W RF power, a pressure of ∼ 4.5

*μ*bar and flow rates of 15, 7.5 and 10 sccm respectively. The nitrogen flow in this process was optimized to create near vertical side walls of the holes. After etching the holes, the nitride mask is removed using the CHF

_{3}/Ar etching procedure described above.

_{2}O:buffered hydrofluoric acid (BHF) solution for 15 sec. The sample is then placed in a 3:1 citric acid:H

_{2}O

_{2}solution for 120 sec to selectively remove the GaAs capping layer. The freestanding membrane is created by etching the sacrificial Al

_{0.7}Ga

_{0.3}As layer in a concentrated 1:4 HF (40%):H

_{2}O solution for 60 sec followed by a rinsing step in pure water and critical point drying. The resulting freestanding membrane covers an area of ∼ 300 × 300

*μ*m

^{2}and is used to measure specular reflectivity spectra at oblique angles of incidence. Afterwards, the membrane is transferred to a transparent gel layer [21

21. Gel-Pak, http://www.gelpak.com.

*μ*m multimode fiber. The output of this fiber is imaged onto the sample with a 1.5 times magnification to create a 75

*μ*m spot on the sample. The reflected light is collected into a 400

*μ*m fiber and then sent to a fiber-coupled grating spectrometer with an InGaAs array (Ocean Optics NIR512) with a ≈ 3 nm spectral resolution. Apertures in the beam limit the numerical aperture of the input beam to NA < 0.025. A Glan-Thompson polarizing beamsplitter cube is placed in a parallel part of the beam and is used to measure both the

*s*- and

*p*-polarized reflectivity as a function of wavelength and angle of incidence.

## 3. Results

*p-*polarized with the parallel wave vector

**k**

_{||}oriented along the ΓX symmetry direction of the photonic lattice. The numerical calculations assume an ideal two-dimensional photonic crystal slab with air on both sides. The calculations use the tabulated value for the refractive index of the Al

_{0.35}Ga

_{0.65}As at a wavelength of 1.5

*μ*m;

*n*

_{2}= 3.1975. The thickness of the slab and the value of

*r/a*are identical to those of the experimental structure.

*R*(

*ω*) given by The first term represents the direct contribution with an amplitude

*r*, while the second term represents the resonant contribution with an amplitude

_{D}*r*. The resonance is characterized by a frequency

_{R}*ω*

_{0}, and a linewidth Γ

_{0}. The phase Δ

*ξ*represents the phase difference between the resonant and the non-resonant contribution at the resonance frequency and controls the asymmetry of the resonance. We assume that the amplitude of the slowly varying non-resonant contribution as a function of frequency

*ω*,

*r*(

_{D}*ω*), can be approximated well with where

*r*

_{0},

*r*

_{1}and

*r*

_{2}are fit parameters.

*p-*polarized (−1,±1) mode is reversed by tuning the angle of incidence, creating a nearly symmetric lineshape at an angle of incidence of 70°. Based on the asymmetry of the lineshape in reflectivity measurements for every 5° (not shown) we estimate that the asymmetry reversal occurs at an angle of incidence of 71 ± 1°. This corresponds to a Brewster’s angle

*θ*for a uniform dielectric slab in air with an effective refractive index of tan(

_{B}*θ*) equal to

_{B}*n*

_{eff}= 2.9 ± 0.2. We expect this value to be comparable to the effective refractive index estimated from the direct (non-resonant) reflectivity of the slab at the resonance frequency. This contribution is modeled by the Fresnel reflection coefficients of the dielectric slab with an effective refractive index that represents the average effect of the holes. Since this background is close to zero for all frequencies for angles of incidence close to Brewster’s angle we analyze calculated spectra over a broad frequency range for angles of incidence of 60° and 80°. From these fits, we obtain values of the effective refractive index of 3.02 and 2.89, consistent with the effective refractive index found from analyzing the asymmetry of the resonant contribution.

*ω*

_{0}) and the corresponding linewidth (Γ

_{0}). The phase difference Δ

*ξ*changes sign as the resonant Fano lineshape changes the asymmetry. The relatively large error bars for Δ

*ξ*are representative for the variation in the fitted value of Δ

*ξ*for different choices of the background (direct) contribution; e.g. by setting both

*r*

_{1}and

*r*

_{2}equal to zero in Eq. (2). Close to Brewster’s angle for the symmetric slab, the background (direct) contribution reaches zero amplitude and the phase difference Δ

*ξ*becomes undefined. The comparison between the experimental and calculated spectra presented above confirms the picture that the asymmetry reversal occurs around the true zero in the direct reflectivity at Brewster’s angle.

*n*= 1.4. Figure 2 shows measured (symbols) and calculated (solid gray lines) reflection spectra for an asymmetric slab for angles of incidence of 75° (left), 78° (middle), and 83° (right).

_{gel}*p-*polarized (−1,±1) mode also occurs in the asymmetric case when there is no Brewster’s angle. In this case, the asymmetry reversal is observed at a significantly larger angle of incidence of 78±1°, compared to the symmetric structure. The parameters of the fitted Fano lineshape are summarized in Table 1.

## 4. Discussion

*q*of the Fano resonance. We then apply this truncated scattering matrix method to interpret our experimental data and compare the results to those obtained using a complete scattering matrix.

### 4.1. Scattering matrix formalism

23. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the fano resonance in optical resonators,” J. Opt. Soc. Am. A **20**, 569 (2003). [CrossRef]

*m*inputs and

*m*outputs that can be described by temporal coupled-mode theory featuring a

*m*×

*m*scattering matrix [23

23. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the fano resonance in optical resonators,” J. Opt. Soc. Am. A **20**, 569 (2003). [CrossRef]

9. M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. **70**, 1438 (1997). [CrossRef]

11. M. Galli, M. Agio, L. C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, and P. Bellutti, “Spectroscopy of photonic bands in macroporous silicon photonic crystals,” Phys. Rev. B **65**, 113111 (2002). [CrossRef]

*S*needs to be unitary and symmetric [23

23. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the fano resonance in optical resonators,” J. Opt. Soc. Am. A **20**, 569 (2003). [CrossRef]

### 4.2. Example: 2-port asymmetric slab

*C*, Γ and

**d**can be made explicit and insight can be gained by studying this case in more detail. Therefore we will restrict ourselves to the simplest asymmetric photonic crystal structures where no higher order diffraction occurs. In that case the system has only two inputs and two outputs and the scattering matrix reduces to a 2 × 2 matrix. The inputs and outputs of this system correspond to the reflected and transmitted modes. Compared to the earlier work reported in Ref. [23

**20**, 569 (2003). [CrossRef]

*n*

_{2}and is sandwiched between a substrate with a refractive index

*n*

_{3}and a superstrate with a refractive index

*n*

_{1}.

*R*(

*ω*) = |

*S*

_{11}(

*ω*)|

^{2}can be written in a form that is identical to the original result of Fano [1]: Here

*ε*is the normalized detuning in units of the linewidth

*ε*= (

*ω*–

*ω*

_{0})/Γ

_{0}, and the asymmetry parameter

*q*is given by A similar expression can be derived for the transmission with an asymmetry parameter

*q̃*= −

*q*

^{−1}. For a symmetric slab

*q*reduces to a real-valued parameter since the Fresnel transmission coefficient has a

*π*phase difference with the reflection coefficient. This is no longer true for the more general case of an asymmetric slab for which the phase difference between

*t*′ and

*r*′ varies as a function of angle of incidence and the resulting

*q*parameter is complex.

*p*-polarized light for the fundamental TE mode supported by a waveguide layer with an effective refractive index

*n*

_{eff}equal to

*S*

_{11}in Eq. (5) at the resonance frequency and we used complex Fresnel coefficients

*r*and

*t*defined as Here the coefficients

*r*and

_{ij}*t*refer to the Fresnel coefficients of a single interface between layer

_{ij}*i*and

*j*. The figure shows the phase difference as a function of angle of incidence and dimensionless quantity

*n*

_{eff}

*d/λ*. Figure 3(a) shows the phase difference for a symmetric structure with air on both sides, while Fig. 3(b) shows the phase difference for the same slab with the air on one side replaced by a transparent gel with refractive index

*n*

_{3}= 1.4. This figure clearly shows large, abrupt jumps in the phase difference for both symmetric and asymmetric slab structures. The phase jumps in the symmetric case are easily understood as points where the direct reflectivity reaches zero. This occurs at Brewster’s angle (vertical dashed line in (a)) and whenever the reflectivity becomes zero due to interference in the film (horizontal solid lines). This Fabry-Perot condition is satisfied whenever the optical path length of the film defined as

*n*

_{eff}

*d*cos

*θ*is equal to

*mλ/*2, with

*m*integer. This situation changes for the asymmetric structure (b). The Fabry-Perot condition produces a minimum in the reflectivity, but the reflectivity does not reach zero. As can be seen in the figure, for angles smaller than arctan(

*n*

_{3}

*/n*

_{1}) the phase jumps at the minimum in reflectivity. To understand the phase jump for larger angles of incidence we write the reflectivity of the asymmetric slab as [26]: with

*β*=

*k*

_{2}

*.*

_{z}d*n*

_{3}

*/n*

_{1}) between the superstrate and substrate [26]. The second angle depends on the refractive index of the slab, superstrate and substrate material. For a slab with

*d/a*< 0.5 this angle is always larger than Brewster’s angle between the superstrate and the slab material. From the figure, it can be seen that the phase difference jumps for angles close to this larger angle for dimensionless quantities

*n*

_{eff}

*d/λ*that are not close to

*m/*2, with

*m*being integer. The value of the refractive index of the substrate plays an important role in determining the two angles and also controls the apparent repulsion between the phase jumps due to the Fabry-Perot effect and due to angle tuning.

*q*as a function of angle for a value of

*n*

_{eff}*d/λ*= 0.3. This value corresponds to the experimentally measured frequency of the Fano resonance of the symmetric structure at Brewster’s angle. The real part of

*q*is represented by the solid blue line, while the complex part of

*q*is shown by the red dash-dot line. For the symmetric structure the

*q*is real-valued, while the

*q*parameter for the asymmetric structure in Fig. 3(d) is clearly complex with a non-trivial phase that depends on the angle of incidence.

### 4.3. Asymmetry reversal with non-zero background

*a*= 2

*μ*m and use the same radius of holes

*r*= 160 nm and slab thickness

*d*= 122 nm as in the experimental structure. The refractive index of the slab material is taken to be

*n*= 3.157, equal to the infrared refractive index of Al

_{slab}_{0.35}Ga

_{0.65}As at a wavelength of 2.5

*μ*m.

*ξ*for the symmetric structure. These data should be compared to the data in Figs. 6 and 7 obtained for the asymmetric structure.

*θ*= 72°), the direct reflectivity reaches zero, and the Fano lineshape reduces to the symmetric Lorentzian lineshape of the resonant contribution.

*ξ*between the resonant and the non-resonant contribution, for the

*p*-polarized (−1,±1) mode over a large range of angles. This phase difference is obtained at the resonance frequency, by fitting the Fano model of Eq. (1) to the calculated reflection spectra. The phase difference Δ

*ξ*influences the interference between the direct and resonant reflectivity and controls the asymmetry of the Fano lineshape. Whenever this phase difference is an integer multiple of

*π*radians, the asymmetry reversal of the Fano lineshape occurs. As can be seen in Fig. 5, the phase difference goes through zero for an angle of incidence of 72°. This angle is exactly equal to Brewster’s angle for the symmetric structure and the direct reflectivity is zero (see Fig. 4).

**G**. In dimensionless units this diffraction condition is given by where

**k**is the wave vector in the substrate/superstrate and

_{s}**k**

_{||}is the parallel component of the incident wave vector,

*ω*

_{0}is the dimensionless frequency of the resonance and

*n*is the refractive index of the substrate/superstrate. The inset of Fig. 5 shows the dispersion

_{s}*ω*

_{0}(

*k*

_{||}) of the

*p*-polarized (−1,±1) leaky mode (solid line) obtained from the calculated spectra together with the folded light line in air (dashed line). The crossing of these lines corresponds to the condition for diffraction in air, which occurs for angles larger than 12°.

*ξ*between the direct and the resonant contribution is shown by the solid line in Fig. 7. The parameters of the structure are identical to those of the symmetric structure of Fig. 5, with the only difference that the air on one side has been replaced by a dielectric with a refractive index of 1.4. As can be seen, the phase difference goes through zero for an angle of incidence of 77.2°, which is larger than Brewster’s angle for the symmetric structure. Similar to the symmetric structure we observe abrupt changes in the phase difference due to higher order diffraction in the air and in the substrate, as indicated by the vertical arrows. The inset shows the dispersion relation of the leaky mode (solid line) together with folded light lines in air (dashed line) and in the substrate (dash-dot lines). From the crossings of these lines with the dispersion of the leaky mode we conclude that there are four higher diffraction orders. The diffraction order into the substrate that uses the (−1,0) reciprocal lattice vector is present for all angles of incidence.

## 5. Conclusions

*p*-polarized light for a photonic crystal slab on a gel substrate, show several Fano lineshapes on top of a slowly varying background. The asymmetry of the Fano lineshape can be reversed by tuning the angle of incidence. For symmetric slabs, with air on both sides, the angle at which the asymmetry reverses is equal to Brewster’s angle, and the direct contribution reaches zero. For asymmetric slabs this is no longer true: the reversal is observed for an angle of incidence beyond Brewster’s angle, and the direct reflectivity no longer reaches zero. A truncated two-port coupled-mode theory can be applied to both the symmetric and the asymmetric structures, which reveals the underlying mechanism of the asymmetry reversal. For asymmetric photonic crystal slabs with

*d/a*< 0.5, the reversal of the asymmetry occurs for angles larger then Brewster’s angle. We show that the resonances in reflection spectra of a lossless photonic crystal waveguide should be described by a complex

*q*parameter in the Fano model. Only for a symmetric structure the

*q*parameter, which gives the ratio between resonant and direct contribution, can be taken as real. The reversal of the asymmetry occurs whenever the phase difference between the resonant and the non-resonant contribution is an integer multiple of

*π*, which does not necessarily coincide with a minimum in the direct reflectivity.

## Acknowledgments

## References and links

1. | U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. |

2. | A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. |

3. | R. K. Adair, C. K. Bockelman, and R. E. Peterson, “Experimental corroboration of the theory of neutron resonance scattering,” Phys. Rev. |

4. | K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. Iye, “Tuning of the fano effect through a quantum dot in an aharonov-bohm interferometer,” Phys. Rev. Lett. |

5. | M. Mendoza, P. A. Schulz, R. O. Vallejos, and C. H. Lewenkopf, “Fano resonances in the conductance of quantum dots with mixed dynamics,” Phys. Rev. B |

6. | C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Commun. |

7. | M. J. A. de Dood, E. F. C. Driessen, D. Stolwijk, and M. P. van Exter, “Observation of coupling between surface plasmons in index-matched hole arrays,” Phys. Rev. B |

8. | A. Bärnthaler, S. Rotter, F. Libisch, J. Burgdörfer, S. Gehler, U. Kuhl, and H.-J. Stöckmann, “Probing decoherence through fano resonances,” Phys. Rev. Lett. |

9. | M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. MacKenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. |

10. | V. N. Astratov, D. M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De la Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B |

11. | M. Galli, M. Agio, L. C. Andreani, M. Belotti, G. Guizzetti, F. Marabelli, M. Patrini, P. Bettotti, L. Dal Negro, Z. Gaburro, L. Pavesi, A. Lui, and P. Bellutti, “Spectroscopy of photonic bands in macroporous silicon photonic crystals,” Phys. Rev. B |

12. | A. A. Clerk, X. Waintal, and P. W. Brouwer, “Fano resonances as a probe of phase coherence in quantum dots,” Phys. Rev. Lett. |

13. | S. Klaiman, N. Moiseyev, and H. R. Sadeghpour, “Interpretation of the fano lineshape reversal in quantum waveguides,” Phys. Rev. B |

14. | S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B |

15. | M. Galli, S. L. Portalupi, M. Belotti, L. C. Andreani, L. O’Faolain, and T. F. Krauss, “Light scattering and fano resonances in high-q photonic crystal nanocavities,” Applied Physics Letters |

16. | S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. |

17. | E. Flück, “Local interaction of light with periodic photonic structures,” Ph.D. thesis, University of Twente (2003). |

18. | E. F. C. Driessen, D. Stolwijk, and M. J. A. de Dood , “Asymmetry reversal in the reflection from a two-dimensional photonic crystal,” Opt. Lett. |

19. | Philips MiPlaza - Cedova, http://www.cedova.com. |

20. | Z EON corporation, http://www.zeon.co.jp. |

21. | Gel-Pak, http://www.gelpak.com. |

22. | Lj. Babić, R. Leijssen, E. Driessen, and M. J. A. de Dood, in preparation. |

23. | S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the fano resonance in optical resonators,” J. Opt. Soc. Am. A |

24. | H. A. Haus, |

25. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

26. | M. Born and E. Wolf, |

**OCIS Codes**

(020.3690) Atomic and molecular physics : Line shapes and shifts

(260.5740) Physical optics : Resonance

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: September 1, 2010

Revised Manuscript: November 12, 2010

Manuscript Accepted: November 22, 2010

Published: December 3, 2010

**Citation**

Lj. Babić and M. J. A. de Dood, "Interpretation of Fano lineshape reversal in the reflectivity spectra of photonic
crystal slabs," Opt. Express **18**, 26569-26582 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-26569

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### References

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