## Tuning the transmission lineshape of a photonic crystal slab guided-resonance mode by polarization control |

Optics Express, Vol. 21, Issue 18, pp. 20675-20682 (2013)

http://dx.doi.org/10.1364/OE.21.020675

Acrobat PDF (2799 KB)

### Abstract

We demonstrate a system consisting of a two-dimensional photonic crystal slab and two polarizers which has a tunable transmission lineshape. The lineshape can be tuned from a symmetric Lorentzian to a highly asymmetric Fano lineshape by rotating the output polarizer. We use temporal coupled mode theory to explain the measurement results. The theory also predicts tunable phase shift and group delay.

© 2013 OSA

## 1. Introduction

1. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Fano resonances in nanoscale structures, Rev. Mod. Phys. **82**, 2257–2298 (2010). [CrossRef]

2. 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, Appl. Phys. Lett. **94**, 071101 (2009). [CrossRef]

3. P. T. Valentim, J. P. Vasco, I. J. Luxmoore, D. Szymanski, H. Vinck-Posada, A. M. Fox, D. M. Whittaker, M. S. Skolnick, and P. S. S. Guimaraes, Asymmetry tuning of Fano resonances in GaAs photonic crystal cavities, Appl. Phys. Lett. **102**, 111112 (2013). [CrossRef]

4. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, The Fano resonance in plasmonic nanostructures and metamaterials, Nat Mater **9**, 707–715 (2010). [CrossRef]

5. U. Fano, Effects of configuration interaction on intensities and phase shifts, Phys. Rev. **124**, 1866–1878 (1961). [CrossRef]

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

7. 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**, 569–572 (2003). [CrossRef]

8. J. Song, R. P. Zaccaria, M. B. Yu, and X. W. Sun, Tunable Fano resonance in photonic crystal slabs, Opt. Express **14**, 8812–8826 (2006). [CrossRef] [PubMed]

9. L. 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). [CrossRef]

10. W. Suh and S. Fan, Mechanically switchable photonic crystal filter with either all-pass transmission or flat-top reflection characteristics, Opt. Lett. **28**, 1763–1765 (2003). [CrossRef] [PubMed]

11. Y. Nazirizadeh, U. Bog, S. Sekula, T. Mappes, U. Lemmer, and M. Gerken, Low-cost label-free biosensors using photonic crystals embedded between crossed polarizers, Opt. Express **18**, 19120–19128 (2010). [CrossRef] [PubMed]

12. M. E. Beheiry, V. Liu, S. Fan, and O. Levi, Sensitivity enhancement in photonic crystal slab biosensors, Opt. Express **18**, 22702–22714 (2010). [CrossRef] [PubMed]

## 2. Experimental system

*z*-direction. The first polarizer (P1) sets the polarization angle

*θ*of light incident on the photonic crystal (PhC). The photonic crystal (PhC) is assumed to support a guided resonance mode (GRM) that is singly degenerate and couples to normally incident radiation with well-defined polarization. That is, there is a polarization direction for a linearly-polarized, normally-incident plane wave for which the overlap integral with the mode is finite, whereas the overlap integral for a plane wave polarized in the orthogonal direction is zero. We use a Suzuki-phase lattice photonic crystal for demonstration [13

_{i}13. A. R. Alija, L. J. Martínez, P. A. Postigo, J. Sánchez-Dehesa, M. Galli, A. Politi, M. Patrini, L. C. Andreani, C. Seassal, and P. Viktorovitch, Theoretical and experimental study of the Suzuki-phase photonic crystal lattice by angle-resolved photoluminescence spectroscopy, Opt. Express **15**, 704–713 (2007). [CrossRef] [PubMed]

14. L. J. Martínez, A. R. Alija, P. A. Postigo, J. F. Galisteo-López, M. Galli, L. C. Andreani, C. Seassal, and P. Viktorovitch, Effect of implementation of a Bragg reflector in the photonic band structure of the Suzuki-phase photonic crystal lattice, Opt. Express **16**, 8509–8518 (2008). [CrossRef] [PubMed]

*θ*; a similar effect has previously been used to demonstrate polarization conversion in a reflective geometry [15

_{i}15. A. Bristow, V. Astratov, R. Shimada, I. Culshaw, M. S. Skolnick, D. Whittaker, A. Tahraoui, and T. Krauss, Polarization conversion in the reflectivity properties of photonic crystal waveguides, IEEE J. Quantum Electron. **38**, 880–884 (2002). [CrossRef]

*θ′*=

_{p}*θ*+

_{i}*θ*, where

_{p}*θ*is the relative angle between the two polarizers. We assume that the component of transmitted light that is polarized orthogonal to

_{p}*θ′*is lost from the system (e.g. absorbed by a polarizer, or redirected elsewhere by a polarizing beam splitter).

_{p}17. A. R. Alija, L. J. Martínez, A. García-Martín, M. L. Dotor, D. Golmayo, and P. A. Postigo, Tuning of spontaneous emission of two-dimensional photonic crystal microcavities by accurate control of slab thickness, Appl. Phys. Lett. **86**, 141101 (2005). [CrossRef]

*a*= 510 nm, hole diameter

*d*= 346 nm and slab thickness

*t*= 200 nm.

*x*- and

*y*- polarized light; due to the rectangular unit cell of the Suzuki-phase lattice, the two directions are not equivalent. For the

*x*-polarization, there is a resonance mode at 1570 nm with a quality factor of approximately 70. Simulations indicate that this mode corresponds to the fifth TE photonic band. The relatively low quality factor corresponds to a relatively wide peak, which simplifies the characterization of lineshape in experiments. For

*y*-polarization, there is no resonance mode visible in the wavelength range shown.

## 3. Analytical model for transmission lineshape

**t**

*) and guided resonance (*

_{d}**t**

*).*

_{g}6. S. Fan and J. D. Joannopoulos, Analysis of guided resonances in photonic crystal slabs, Phys. Rev. B **65**, 235112 (2002). [CrossRef]

*t*

_{d0}is the transmission coefficient through a homogeneous dielectric slab, and is the transmission coefficient of the guided resonance.

*ε*= (

*ω*−

*ω*

_{0})/

*γ*is the normalized detuning,

*ω*

_{0}is the resonance frequency,

*γ*is the damping coefficient of the resonance, and

*j*is the imaginary unit. The prefactor

*f*is equal to −(

*t*

_{d0}±

*r*

_{d0}) to ensure energy conservation. The plus and minus signs correspond to resonant modes that are even and odd with respect to the

*xy*-mirror plane. The mode we consider in this paper is even, so we keep only the plus sign in the following derivation. Moreover, since the bandwidth of the guided resonance is much narrower than the background Fabry-Pérot features in the spectrum, we can neglect the dispersion of

*t*

_{d0}and assume it to be a complex constant.

**p̂**is the unit vector aligned with the second polarizer. The total transmission coefficient is thus:

*T*is described by the Fano resonance formula: where

*r*

_{d0}/

*t*

_{d0}is always purely imaginary [6

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

*q*is a complex number. This is in contrast to the system without the second polarizer, for which the asymmetry factor is real.

## 4. Comparison of theory and experiment

*t*

_{d0}|, the guided resonance frequency

*ω*

_{0}, the resonance damping coefficient

*γ*, and the incident polarization

*θ*(i.e. the angle between the first polarizer and the mode polarization). The reflection coefficient can be calculated by

_{i}*F*is used to take into account measurement normalization. Figure 3(b) shows the fitted function of Fig. 3(a), with fitting parameters

*F*= 1.90, |

*t*

_{d0}| = 0.633,

*ω*

_{0}

*a*/2

*πc*= 0.319,

*γa*/2

*πc*= 0.0238 and

*θ*= 122°. Visually, the fitted spectrum reproduces the features of the experimental spectrum quite well, while smoothing out experimental noise.

_{i}*θ*= 150°, the transmission spectrum is a straight line. This occurs because the guided resonance path is perfectly cancelled by the second polarizer. The fitting result shows that

_{p}*θ*is approximately 120°, and

_{i}*θ′*=

_{p}*θ*+

_{i}*θ*≈ 270°, indicating that the second polarizer is nearly aligned with the

_{p}*y*-axis, perpendicular to the mode polarization. The non-zero transmission arises from the Fabry-Pérot direct path.

## 5. Discussion

18. I. Avrutsky, R. Gibson, J. Sears, G. Khitrova, H. M. Gibbs, and J. Hendrickson, Linear systems approach to describing and classifying Fano resonances, Phys. Rev. B **87**, 125118 (2013). [CrossRef]

*T*is a complex factor which represents the transmission coefficient far from resonance, and

*ε*is real. However, we are free to consider Eq. (9) as a complex function of a complex variable

*ε*=

*ε*+

^{R}*jε*. The response along the real axis then gives the physical behavior of the system.

^{I}*θ*dependence of the zero point (

_{p}*ε*) of the system shown in Eq. (11) plays an important role in tuning the line shape. For simplicity, we choose

_{z}*θ*to be 45° and

_{i}*θ*. For all cases of

_{p}*θ*, the pole point is fixed at (0, 1) in the complex plane while the zero point varies. In Fig. 6, we choose three special cases when

_{p}*θ*is equal to 0°, 60° and 90°, respectively, to illustrate how zero position affects the system response. Different

_{p}*θ*angles are shown in different rows in Fig. 6. The first and second columns show the system amplitude and phase response in the complex plane.

_{p}*θ*= 0°, when the polarizers are aligned. The zero is at (−0.5, 0.5) in the complex plane and is visible as an amplitude minimum (black region). The pole is at (0, 1) and is visible as an amplitude maximum (white region). The amplitude response along the real axis is shown as a superimposed red line (see red scale bar on right). The transmission first decreases and then increases as a function of frequency. The phase response is shown in Fig. 6(b). The zero and pole are connected by a line of phase discontinuity. The phase response on the real axis is again shown as a superimposed red line. Both the amplitude and phase response change with polarizer angle.

_{p}*θ*= 60°. The pole remains fixed at (0,1), while the zero has moved to (0.36,1.36). The real transmission and phase response reflect this change, showing inverted behavior relative to Fig. 6(a) and Fig. 6(b).

_{p}*θ*is equal to 90°, and the polarizers are perpendicular to one another. In this case, the zero is not visible on the plot. The amplitude lineshape is a symmetric Lorentzian, and the phase response decreases by

_{p}*π*over the resonance.

*τ*= −

*dϕ/dω*. Figure 6(c), Fig. 6(f) and Fig. 6(i) show the dimensionless group delay, (Δ

*τ*)

*c/a*. For a fixed frequency (fixed

*ε*≈ −0.5), the group delay can be tuned from negative to positive by changing

_{R}*θ*.

_{p}*a/c*= 1.7

*fs*. For the relatively low quality factor mode studied here, the delays are in the 1 to 2

*fs*range. The delay increases linearly with quality factor,

*Q*. Recently, a photonic crystal lattice has been designed that supports a coupled, linearly-polarized guided resonance mode with theoretical quality factor as high as 10

^{5}[19

19. J. Ma, L. J. Martínez, and M. L. Povinelli, Optical trapping via guided resonance modes in a Slot-Suzuki-phase photonic crystal lattice, Opt. Express **20**, 6816–6824 (2012). [CrossRef] [PubMed]

*Q*value would increase the delay to the picosecond range.

*θ*with

_{i}*θ′*−

_{p}*θ*, fixing

_{p}*θ′*to be 45°, and tuning

_{p}*θ*from 0° to 180°, the zero point of the system has the same trace as shown in Fig. 5, yielding a similar tuning mechanism for the lineshape.

_{p}## 6. Conclusion

## Acknowledgments

## References and links

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

2. | 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, Appl. Phys. Lett. |

3. | P. T. Valentim, J. P. Vasco, I. J. Luxmoore, D. Szymanski, H. Vinck-Posada, A. M. Fox, D. M. Whittaker, M. S. Skolnick, and P. S. S. Guimaraes, Asymmetry tuning of Fano resonances in GaAs photonic crystal cavities, Appl. Phys. Lett. |

4. | B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, The Fano resonance in plasmonic nanostructures and metamaterials, Nat Mater |

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

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

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

8. | J. Song, R. P. Zaccaria, M. B. Yu, and X. W. Sun, Tunable Fano resonance in photonic crystal slabs, Opt. Express |

9. | L. Babić and M. J. A. de Dood, Interpretation of Fano lineshape reversal in the reflectivity spectra of photonic crystal slabs, Opt. Express |

10. | W. Suh and S. Fan, Mechanically switchable photonic crystal filter with either all-pass transmission or flat-top reflection characteristics, Opt. Lett. |

11. | Y. Nazirizadeh, U. Bog, S. Sekula, T. Mappes, U. Lemmer, and M. Gerken, Low-cost label-free biosensors using photonic crystals embedded between crossed polarizers, Opt. Express |

12. | M. E. Beheiry, V. Liu, S. Fan, and O. Levi, Sensitivity enhancement in photonic crystal slab biosensors, Opt. Express |

13. | A. R. Alija, L. J. Martínez, P. A. Postigo, J. Sánchez-Dehesa, M. Galli, A. Politi, M. Patrini, L. C. Andreani, C. Seassal, and P. Viktorovitch, Theoretical and experimental study of the Suzuki-phase photonic crystal lattice by angle-resolved photoluminescence spectroscopy, Opt. Express |

14. | L. J. Martínez, A. R. Alija, P. A. Postigo, J. F. Galisteo-López, M. Galli, L. C. Andreani, C. Seassal, and P. Viktorovitch, Effect of implementation of a Bragg reflector in the photonic band structure of the Suzuki-phase photonic crystal lattice, Opt. Express |

15. | A. Bristow, V. Astratov, R. Shimada, I. Culshaw, M. S. Skolnick, D. Whittaker, A. Tahraoui, and T. Krauss, Polarization conversion in the reflectivity properties of photonic crystal waveguides, IEEE J. Quantum Electron. |

16. | C. Lin, L. J. Martínez, and M. L. Povinelli, Fabrication of transferrable, fully-suspended silicon photonic crystal membranes exhibiting vivid structural color and high-Q guided resonance, J. Vac. Tech. B , in press (2013). |

17. | A. R. Alija, L. J. Martínez, A. García-Martín, M. L. Dotor, D. Golmayo, and P. A. Postigo, Tuning of spontaneous emission of two-dimensional photonic crystal microcavities by accurate control of slab thickness, Appl. Phys. Lett. |

18. | I. Avrutsky, R. Gibson, J. Sears, G. Khitrova, H. M. Gibbs, and J. Hendrickson, Linear systems approach to describing and classifying Fano resonances, Phys. Rev. B |

19. | J. Ma, L. J. Martínez, and M. L. Povinelli, Optical trapping via guided resonance modes in a Slot-Suzuki-phase photonic crystal lattice, Opt. Express |

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(220.0220) Optical design and fabrication : Optical design and fabrication

(260.5430) Physical optics : Polarization

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 25, 2013

Revised Manuscript: August 14, 2013

Manuscript Accepted: August 16, 2013

Published: August 27, 2013

**Citation**

Ningfeng Huang, Luis Javier Martínez, and Michelle L. Povinelli, "Tuning the transmission lineshape of a photonic crystal slab guided-resonance mode by polarization control," Opt. Express **21**, 20675-20682 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-20675

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### References

- A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Fano resonances in nanoscale structures, Rev. Mod. Phys.82, 2257–2298 (2010). [CrossRef]
- 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, Appl. Phys. Lett.94, 071101 (2009). [CrossRef]
- P. T. Valentim, J. P. Vasco, I. J. Luxmoore, D. Szymanski, H. Vinck-Posada, A. M. Fox, D. M. Whittaker, M. S. Skolnick, and P. S. S. Guimaraes, Asymmetry tuning of Fano resonances in GaAs photonic crystal cavities, Appl. Phys. Lett.102, 111112 (2013). [CrossRef]
- B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, The Fano resonance in plasmonic nanostructures and metamaterials, Nat Mater9, 707–715 (2010). [CrossRef]
- U. Fano, Effects of configuration interaction on intensities and phase shifts, Phys. Rev.124, 1866–1878 (1961). [CrossRef]
- S. Fan and J. D. Joannopoulos, Analysis of guided resonances in photonic crystal slabs, Phys. Rev. B65, 235112 (2002). [CrossRef]
- S. Fan, W. Suh, and J. D. Joannopoulos, Temporal coupled-mode theory for the Fano resonance in optical resonators, J. Opt. Soc. Am. A20, 569–572 (2003). [CrossRef]
- J. Song, R. P. Zaccaria, M. B. Yu, and X. W. Sun, Tunable Fano resonance in photonic crystal slabs, Opt. Express14, 8812–8826 (2006). [CrossRef] [PubMed]
- L. Babić and M. J. A. de Dood, Interpretation of Fano lineshape reversal in the reflectivity spectra of photonic crystal slabs, Opt. Express18, 26569–26582 (2010). [CrossRef]
- W. Suh and S. Fan, Mechanically switchable photonic crystal filter with either all-pass transmission or flat-top reflection characteristics, Opt. Lett.28, 1763–1765 (2003). [CrossRef] [PubMed]
- Y. Nazirizadeh, U. Bog, S. Sekula, T. Mappes, U. Lemmer, and M. Gerken, Low-cost label-free biosensors using photonic crystals embedded between crossed polarizers, Opt. Express18, 19120–19128 (2010). [CrossRef] [PubMed]
- M. E. Beheiry, V. Liu, S. Fan, and O. Levi, Sensitivity enhancement in photonic crystal slab biosensors, Opt. Express18, 22702–22714 (2010). [CrossRef] [PubMed]
- A. R. Alija, L. J. Martínez, P. A. Postigo, J. Sánchez-Dehesa, M. Galli, A. Politi, M. Patrini, L. C. Andreani, C. Seassal, and P. Viktorovitch, Theoretical and experimental study of the Suzuki-phase photonic crystal lattice by angle-resolved photoluminescence spectroscopy, Opt. Express15, 704–713 (2007). [CrossRef] [PubMed]
- L. J. Martínez, A. R. Alija, P. A. Postigo, J. F. Galisteo-López, M. Galli, L. C. Andreani, C. Seassal, and P. Viktorovitch, Effect of implementation of a Bragg reflector in the photonic band structure of the Suzuki-phase photonic crystal lattice, Opt. Express16, 8509–8518 (2008). [CrossRef] [PubMed]
- A. Bristow, V. Astratov, R. Shimada, I. Culshaw, M. S. Skolnick, D. Whittaker, A. Tahraoui, and T. Krauss, Polarization conversion in the reflectivity properties of photonic crystal waveguides, IEEE J. Quantum Electron.38, 880–884 (2002). [CrossRef]
- C. Lin, L. J. Martínez, and M. L. Povinelli, Fabrication of transferrable, fully-suspended silicon photonic crystal membranes exhibiting vivid structural color and high-Q guided resonance, J. Vac. Tech. B, in press (2013).
- A. R. Alija, L. J. Martínez, A. García-Martín, M. L. Dotor, D. Golmayo, and P. A. Postigo, Tuning of spontaneous emission of two-dimensional photonic crystal microcavities by accurate control of slab thickness, Appl. Phys. Lett.86, 141101 (2005). [CrossRef]
- I. Avrutsky, R. Gibson, J. Sears, G. Khitrova, H. M. Gibbs, and J. Hendrickson, Linear systems approach to describing and classifying Fano resonances, Phys. Rev. B87, 125118 (2013). [CrossRef]
- J. Ma, L. J. Martínez, and M. L. Povinelli, Optical trapping via guided resonance modes in a Slot-Suzuki-phase photonic crystal lattice, Opt. Express20, 6816–6824 (2012). [CrossRef] [PubMed]

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