Abstract

Microstructured photonic crystal membranes are of great interest as compact filters in a variety of fields, including telecommunication and sensing. Here we present an analysis of the transmission and resonance behavior of a dielec tric membrane featuring periodically arranged holes using the frequency domain finite element analysis. We solve both the source as well as eigenvalue problem for a given geometry. Applying symmetry considerations, we discard modes that cannot be excited by incident transverse electro-magnetic (TEM) light simultaneously. We relate the results of the source problem to the modal analysis and obtain strong correspondence in both frequency and resonance quality factor. We find that the sharp resonances can be related to the eigenmodes of a rectangular dielectric resonator formed by the membrane and the holes of the photonic crystal. Analyzing noncircular holes for polarization-dependent transmittivity, the modal analysis presented herein constitutes a powerful tool to understand and identify occurrence of pronounced wideband polarization selectivity.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2009 (2)

2008 (2)

2007 (1)

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

2004 (2)

2003 (3)

S. Fan, W. Suh, and J. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

P. Alivisatos, “The use of nanocrystals in biological detection,” Nat. Biotechnol. 22, 47–52 (2003).
[CrossRef]

J. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[CrossRef] [PubMed]

2002 (1)

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

1999 (1)

T. Krauss and R. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[CrossRef]

1997 (1)

S. Fan, P. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. Lett. 124, 1866–1878 (1961).

Alivisatos, P.

P. Alivisatos, “The use of nanocrystals in biological detection,” Nat. Biotechnol. 22, 47–52 (2003).
[CrossRef]

Arbenz, P.

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

Bjarklev, A.

Campopiano, S.

Carlsen, A.

Castaldi, G.

Chinellato, O.

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

Craven, G.

G. Craven and R. Skedd, Evanescent Mode Microwave Components (Artech House, 1987).

Cusano, A.

De La Rue, R.

T. Krauss and R. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[CrossRef]

Digonnet, M.

Fan, S.

O. Kilic, S. Fan, and O. Solgaard, “Analysis of guided-resonance-based polarization beam splitting in photonic crystal slabs,” J. Opt. Soc. Am. A 25, 2680–2692 (2008).

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518(2004).
[CrossRef]

S. Fan, W. Suh, and J. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

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

S. Fan, P. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. Lett. 124, 1866–1878 (1961).

Folkenberg, J.

Galdi, V.

Gallina, I.

Hansen, T.

Hillmer, H.

T. Kusserow, M. Wulf, R. Zamora, T. Vengatesan, B. Witzigmann, and H. Hillmer, Photonic Crystal Polarizer Element on InP/Air Membranes for Optical MEMS Applications, Compound Semiconductor Photonics (Pan Stanford, 2010).

Hoiby, P.

Jensen, J.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics(Wiley, 1993).

Joannopoulos, J.

S. Fan, W. Suh, and J. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

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

S. Fan, P. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Kilic, O.

Kino, G.

Knight, J.

J. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[CrossRef] [PubMed]

Krauss, T.

T. Krauss and R. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[CrossRef]

Kusserow, T.

T. Kusserow, M. Wulf, R. Zamora, T. Vengatesan, B. Witzigmann, and H. Hillmer, Photonic Crystal Polarizer Element on InP/Air Membranes for Optical MEMS Applications, Compound Semiconductor Photonics (Pan Stanford, 2010).

Nielsen, K.

Nielsen, L.

Noordegraaf, D.

Pedersen, L.

Pisco, M.

Ricciardi, A.

Riishede, J.

Römer, F.

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

Schubert, E.

S. Fan, P. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

Skedd, R.

G. Craven and R. Skedd, Evanescent Mode Microwave Components (Artech House, 1987).

Solgaard, O.

Suh, W.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518(2004).
[CrossRef]

S. Fan, W. Suh, and J. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

Vengatesan, T.

T. Kusserow, M. Wulf, R. Zamora, T. Vengatesan, B. Witzigmann, and H. Hillmer, Photonic Crystal Polarizer Element on InP/Air Membranes for Optical MEMS Applications, Compound Semiconductor Photonics (Pan Stanford, 2010).

Villeneuve, P.

S. Fan, P. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

Wang, Z.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518(2004).
[CrossRef]

Witzigmann, B.

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

T. Kusserow, M. Wulf, R. Zamora, T. Vengatesan, B. Witzigmann, and H. Hillmer, Photonic Crystal Polarizer Element on InP/Air Membranes for Optical MEMS Applications, Compound Semiconductor Photonics (Pan Stanford, 2010).

Wulf, M.

T. Kusserow, M. Wulf, R. Zamora, T. Vengatesan, B. Witzigmann, and H. Hillmer, Photonic Crystal Polarizer Element on InP/Air Membranes for Optical MEMS Applications, Compound Semiconductor Photonics (Pan Stanford, 2010).

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Zamora, R.

T. Kusserow, M. Wulf, R. Zamora, T. Vengatesan, B. Witzigmann, and H. Hillmer, Photonic Crystal Polarizer Element on InP/Air Membranes for Optical MEMS Applications, Compound Semiconductor Photonics (Pan Stanford, 2010).

IEEE J. Quantum Electron. (1)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518(2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nat. Biotechnol. (1)

P. Alivisatos, “The use of nanocrystals in biological detection,” Nat. Biotechnol. 22, 47–52 (2003).
[CrossRef]

Nature (1)

J. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

Phys. Rev. B (1)

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

Phys. Rev. Lett. (4)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. Lett. 124, 1866–1878 (1961).

S. Fan, P. Villeneuve, J. Joannopoulos, and E. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997).
[CrossRef]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

T. Krauss and R. De La Rue, “Photonic crystals in the optical regime—past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[CrossRef]

Other (3)

G. Craven and R. Skedd, Evanescent Mode Microwave Components (Artech House, 1987).

T. Kusserow, M. Wulf, R. Zamora, T. Vengatesan, B. Witzigmann, and H. Hillmer, Photonic Crystal Polarizer Element on InP/Air Membranes for Optical MEMS Applications, Compound Semiconductor Photonics (Pan Stanford, 2010).

J. Jin, The Finite Element Method in Electromagnetics(Wiley, 1993).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the PC slab and the computational domain. Top left, schematic illustration of a PC slab featuring elliptical holes arranged in a square array; bottom left, definition of the dimensions of the hole and of the unit cell; right, computational domain of the 3-D quarter unit cell, terminated by PML.

Fig. 2
Fig. 2

Top, illustration of the nontrivially degenerate BC pairs for circular holes and their nomenclature. Because of the symmetry of a circular hole, six out of the 16 possible BC pairs are redundant. Bottom, transmission curve (SP) of a plain slab and PC filter illuminated by normally incident plane waves related to the resonances of the structure (results of the EVP) for all nontrivially degenerate BC pairs. Hole radius 0.1 a 0 , slab thickness 0.5 a 0 . Green curve indicates the semianalytical solution obtained from the solution of the XLXU EVP and Eq. (6). XLXU and YLYU resonances are equivalent; both are provided for the purpose of verification as the mesh does not exhibit the symmetry of the geometry; however, their properties are identical and can be excited by x-polarized and y-polarized normally incident plane waves, respectively. Note that the position of the markers with respect to the ordinate has no relevance and only serves the spreading of the points for better legibility.

Fig. 3
Fig. 3

Normalized frequencies of resonances for varying circular hole radius; color axis represents the base-10 logarithm of the resonance quality factor. The existence of all modes at vanishing hole size suggests that the modes are features of the slab that are perturbed by the presence of the hole. For vanishing hole diameter, the quality factors for modes other than the trivial TEM slab modes increase. The dotted–dashed curves refer to the effective refractive index approximation described in Eq. (5). At higher frequencies, the density of modes increases significantly. In case their spectral proximity with respect to their quality factor is too tight, their influence on the transmission spectra cannot be viewed separately.

Fig. 4
Fig. 4

Electric field distribution of the five lowest frequency resonances for the dielectric slab with no hole and various hole diameters. The normalized angular frequency ω and quality factor Q are given. Asterisks bring to attention that the numeric value of the quality factor of the noncoupling slab modes is limited by the presence of the lossy PML at finite distance from the slab. The modal fields are the solutions of the EVP, and their scaling with respect to each other is arbitrary. The illustration shall depict the modal fields compared to the slab, thus a quantification of the color bar was omitted. Even with increasing hole diameter, modes remain odd or even, while their order on the frequency scale may swap (see Fig. 3) indicated by the crossing lines.

Fig. 5
Fig. 5

Strongly polarization-dependent transmittivity synthesized employing Eq. (6) for various modal arrangements for common reference frequency ω c : (a) x polarization ( ω = 0.95 ω c , Q = 20 , odd, XLXU); y polarization ( ω = 1.05 ω c , Q = 20 , odd, YLYU); the modal resonance frequencies are separated by 2 ω c Q so that the minimum and maximum overlap, giving rise to polarization dependence at ω c . (b) x polarization ( ω = ω c , Q = 20 , odd, XLXU); y polarization ( ω = ω c , Q = 20 , even, YLYU); leads to high polarization selectivity at approximately ω c ± ω c Q . (c) Scenario (B) involving four modes, leading to wideband polarization selectivity. Note: The 20 dB suppression bandwidth is significantly larger, even though the modal quality factors are higher, which is more realistically encountered in those structures. Illustration has a qualitative purpose, as the transmittivity curves were synthesized, and no geometry is given featuring the assumed modes. Common background assumed. Markers indicate modes on frequency axis; position on ordinate bears no information.

Fig. 6
Fig. 6

Polarization-dependent modal resonances of elliptical hole PC membrane. On the ordinate, the normalized semimajor axis u and semiminor axis v evolve from u = v = 0.1 to u = v = 0.45 by increasing u first, while leaving v unchanged. The marker color specifies the base-10 logarithm of the resonance quality factor. The marker shape to the BC permutation. For noncircular holes, the modal characteristic separates for both polarizations as the trivial degeneracy vanishes. A polarization-selective behavior can be encountered near all crossings of round- and square-markered curves. A highly favorable constellation occurs when a window of even–odd and odd–even modes for the two polarizations spans across a common frequency range, giving rise to wide bandwidth frequency-selective behavior (black rectangle). Modes commencing for small circular holes u = v = 0.1 beyond ω > 0.41 were omitted in this illustration for the purpose of clarity.

Fig. 7
Fig. 7

Transmittivity of a PC membrane filter featuring elliptical holes of normalized semiaxes u = 0.45 and v = 0.28 , computed with 3-D FEM (solid curves) and by the semianalytical approach of Eq. (6) (see [16]). The normalized slab thickness is 0.5 . The 20 dB selectivity bandwidth of the four-mode scenario at ω = 0.45 is approximately three times larger than the two-mode scenario at ω = 0.505 . The attenuation of x-polarized light at ω = 0.45 is caused by an unfavorably high quality factor of the XLXU resonance at ω = 0.441 . Calculations based on the semianalytical method are approximative at large hole diameters due to uncertain fitting of the background, i.e., determining an effective refractive index of the slab and the validity of Eq. (6).

Equations (9)

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× ( μ ¯ ¯ 1 × E ) k 0 2 ϵ ¯ ¯ E = j k 0 Z 0 J ,
[ E x E y E z ] [ cos ( l π x x 0 ) cos ( m π y y 0 ) cos ( n π z z 0 ) f x ( z ) sin ( l π x x 0 ) sin ( m π y y 0 ) cos ( n π z z 0 ) f y ( z ) q sin ( l π x x 0 ) cos ( m π y y 0 ) sin ( n π z z 0 ) f z ( z ) ] ,
ω = l 2 + m 2 + n 2 n slab .
z = 0 S ( x , y , z ) · e z d a = z = z 0 S ( x , y , z ) · e z d a = 0 ,
ω ( r a 0 ) = ω slab n slab n eff with n eff = π r 2 a 0 2 n air + ( 1 π r 2 a 0 2 ) n slab ,
t ( ω ) = t d ( ω ) n = 1 N t d ( ω ) ± r d ( ω ) 2 i Q n ( ω ω n ) + 1 ,
r d ( ω ) = i 1 n ( ω ) 2 2 n ( ω ) sin ( 2 π n ( ω ) ω d 0 ) cos ( 2 π n ( ω ) ω d 0 ) i n ( ω ) 2 + 1 2 n ( ω ) sin ( 2 π n ( ω ) ω d 0 ) ,
t d ( ω ) = 1 cos ( 2 π n ( ω ) ω d 0 ) i n ( ω ) 2 + 1 2 n ( ω ) sin ( 2 π n ( ω ) ω d 0 ) ,
n eff ( ω 0 ) = 1 ω 0 and n eff ( ω 1 ) = 2 ω 1 .

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