Abstract

We present an iterative method for calculating complex-frequency eigenmodes of photonic crystal slabs with 1D periodicity based on the aperiodic Fourier modal method. By comparison with the known methods, we show that the proposed method is efficient for studying resonant properties of long-period photonic crystal slabs and diffraction gratings. We demonstrate that the method can be used to calculate the eigenmodes of the structures with periods up to at least 500λ. We discuss different aspects of the mode calculation, including convergence of the method, mode field and dispersion analysis. Potential applications of the presented method include investigation of periodic structures with defects and of quasiperiodic and random structures within the super-cell approach.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]

2017 (1)

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

2016 (1)

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18, 115006 (2016).
[Crossref]

2015 (2)

D. A. Bykov and L. L. Doskolovich, “On the use of the Fourier modal method for calculation of localized eigenmodes of integrated optical resonators,” Comput. Opt. 39, 663–673 (2015).
[Crossref]

D. A. Bykov and L. L. Doskolovich, “Spatiotemporal coupled-mode theory of guided-mode resonant gratings,” Opt. Express 23, 19234–19241 (2015).
[Crossref] [PubMed]

2014 (3)

M. Pisarenco and I. Setija, “On the complexity of aperiodic Fourier modal methods for finite periodic structures,” J. Comput. Phys. 261, 130–144 (2014).
[Crossref]

S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Rep. Prog. Phys. 77, 126402 (2014).
[Crossref] [PubMed]

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref] [PubMed]

2013 (2)

Q. Bai, M. Perrin, C. Sauvan, J.-P. Hugonin, and P. Lalanne, “Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure,” Opt. Express 21, 27371–27382 (2013).
[Crossref] [PubMed]

D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightw. Technol. 31, 793–801 (2013).
[Crossref]

2012 (3)

E. A. Bezus and L. L. Doskolovich, “Stable algorithm for the computation of the electromagnetic field distribution of eigenmodes of periodic diffraction structures,” J. Opt. Soc. Am. A 29, 2307–2313 (2012).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[Crossref]

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photon. 4, 379–440 (2012).
[Crossref]

2011 (2)

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
[Crossref]

B. Vial, M. Commandré, F. Zolla, A. Nicolet, and S. Tisserand, “Resonances determination in microstructured films embedded in multilayered stacks,” Proc. SPIE 8168, 816822 (2011).
[Crossref]

2010 (3)

2009 (1)

N. A. Gippius and S. G. Tikhodeev, “The scattering matrix and optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[Crossref]

2006 (1)

T. Zentgraf, A. Christ, J. Kuhl, N. A. Gippius, S. G. Tikhodeev, D. Nau, and H. Giessen, “Metallodielectric photonic crystal superlattices: Influence of periodic defects on transmission properties,” Phys. Rev. B 73, 115103 (2006).
[Crossref]

2005 (1)

2003 (1)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure Appl. Opt. 5, 345–355 (2003).
[Crossref]

2002 (2)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[Crossref]

2001 (2)

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[Crossref]

D. Felbacq, “Numerical computation of resonance poles in scattering theory,” Phys. Rev. E 64, 047702 (2001).
[Crossref]

2000 (1)

1999 (2)

1997 (1)

1996 (2)

1995 (3)

1993 (2)

S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[Crossref] [PubMed]

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[Crossref]

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Alpeggiani, F.

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

Alù, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref] [PubMed]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Bai, Q.

Bezus, E. A.

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18, 115006 (2016).
[Crossref]

E. A. Bezus and L. L. Doskolovich, “Stable algorithm for the computation of the electromagnetic field distribution of eigenmodes of periodic diffraction structures,” J. Opt. Soc. Am. A 29, 2307–2313 (2012).
[Crossref]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Bykov, D. A.

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18, 115006 (2016).
[Crossref]

D. A. Bykov and L. L. Doskolovich, “On the use of the Fourier modal method for calculation of localized eigenmodes of integrated optical resonators,” Comput. Opt. 39, 663–673 (2015).
[Crossref]

D. A. Bykov and L. L. Doskolovich, “Spatiotemporal coupled-mode theory of guided-mode resonant gratings,” Opt. Express 23, 19234–19241 (2015).
[Crossref] [PubMed]

D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightw. Technol. 31, 793–801 (2013).
[Crossref]

Cao, Q.

Castaldi, G.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref] [PubMed]

Chandezon, J.

Chang-Hasnain, C. J.

Christ, A.

T. Zentgraf, A. Christ, J. Kuhl, N. A. Gippius, S. G. Tikhodeev, D. Nau, and H. Giessen, “Metallodielectric photonic crystal superlattices: Influence of periodic defects on transmission properties,” Phys. Rev. B 73, 115103 (2006).
[Crossref]

Collin, S.

S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Rep. Prog. Phys. 77, 126402 (2014).
[Crossref] [PubMed]

Commandré, M.

B. Vial, M. Commandré, F. Zolla, A. Nicolet, and S. Tisserand, “Resonances determination in microstructured films embedded in multilayered stacks,” Proc. SPIE 8168, 816822 (2011).
[Crossref]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Doskolovich, L. L.

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18, 115006 (2016).
[Crossref]

D. A. Bykov and L. L. Doskolovich, “On the use of the Fourier modal method for calculation of localized eigenmodes of integrated optical resonators,” Comput. Opt. 39, 663–673 (2015).
[Crossref]

D. A. Bykov and L. L. Doskolovich, “Spatiotemporal coupled-mode theory of guided-mode resonant gratings,” Opt. Express 23, 19234–19241 (2015).
[Crossref] [PubMed]

D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightw. Technol. 31, 793–801 (2013).
[Crossref]

E. A. Bezus and L. L. Doskolovich, “Stable algorithm for the computation of the electromagnetic field distribution of eigenmodes of periodic diffraction structures,” J. Opt. Soc. Am. A 29, 2307–2313 (2012).
[Crossref]

Engheta, N.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref] [PubMed]

Felbacq, D.

D. Felbacq, “Numerical computation of resonance poles in scattering theory,” Phys. Rev. E 64, 047702 (2001).
[Crossref]

Flach, S.

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

Galdi, V.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref] [PubMed]

Gaylord, T. K.

Giessen, H.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18, 7569–7574 (2010).
[Crossref] [PubMed]

T. Zentgraf, A. Christ, J. Kuhl, N. A. Gippius, S. G. Tikhodeev, D. Nau, and H. Giessen, “Metallodielectric photonic crystal superlattices: Influence of periodic defects on transmission properties,” Phys. Rev. B 73, 115103 (2006).
[Crossref]

Gippius, N. A.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18, 7569–7574 (2010).
[Crossref] [PubMed]

N. A. Gippius and S. G. Tikhodeev, “The scattering matrix and optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[Crossref]

T. Zentgraf, A. Christ, J. Kuhl, N. A. Gippius, S. G. Tikhodeev, D. Nau, and H. Giessen, “Metallodielectric photonic crystal superlattices: Influence of periodic defects on transmission properties,” Phys. Rev. B 73, 115103 (2006).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Granet, G.

Grann, E. B.

Hugonin, J. P.

Hugonin, J.-P.

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Kivshar, Y. S.

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

Kuhl, J.

T. Zentgraf, A. Christ, J. Kuhl, N. A. Gippius, S. G. Tikhodeev, D. Nau, and H. Giessen, “Metallodielectric photonic crystal superlattices: Influence of periodic defects on transmission properties,” Phys. Rev. B 73, 115103 (2006).
[Crossref]

Kuipers, L.

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

Lalanne, P.

Li, L.

Magnusson, R.

Mattheij, R.

Maubach, J.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[Crossref]

Miroshnichenko, A. E.

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

Moharam, M. G.

Monticone, F.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref] [PubMed]

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Nau, D.

T. Zentgraf, A. Christ, J. Kuhl, N. A. Gippius, S. G. Tikhodeev, D. Nau, and H. Giessen, “Metallodielectric photonic crystal superlattices: Influence of periodic defects on transmission properties,” Phys. Rev. B 73, 115103 (2006).
[Crossref]

Nevière, M.

Nicolet, A.

B. Vial, M. Commandré, F. Zolla, A. Nicolet, and S. Tisserand, “Resonances determination in microstructured films embedded in multilayered stacks,” Proc. SPIE 8168, 816822 (2011).
[Crossref]

Parappurath, N.

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

Perrin, M.

Pisarenco, M.

M. Pisarenco and I. Setija, “On the complexity of aperiodic Fourier modal methods for finite periodic structures,” J. Comput. Phys. 261, 130–144 (2014).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
[Crossref]

Plumey, J.-P.

Pommet, D. A.

Popov, E.

Reinisch, R.

Sauvan, C.

Setija, I.

M. Pisarenco and I. Setija, “On the complexity of aperiodic Fourier modal methods for finite periodic structures,” J. Comput. Phys. 261, 130–144 (2014).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
[Crossref]

Silberstein, E.

Silva, A.

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[Crossref]

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[Crossref]

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N. A. Gippius and S. G. Tikhodeev, “The scattering matrix and optical properties of metamaterials,” Phys. Usp. 52, 967–971 (2009).
[Crossref]

Proc. SPIE (1)

B. Vial, M. Commandré, F. Zolla, A. Nicolet, and S. Tisserand, “Resonances determination in microstructured films embedded in multilayered stacks,” Proc. SPIE 8168, 816822 (2011).
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Science (1)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 (a) Photonic crystal slab geometry. (b) “Rotated” PCS geometry. The adjacent periods are optically isolated by means of PML. Both structures are uniform in the y direction.
Fig. 2
Fig. 2 Computation time vs. the super-period size m for the proposed method (dashed lines) and the method of paper [20] (solid lines). Different lines correspond to different relative tolerances δ.
Fig. 3
Fig. 3 Mode dispersion of the structures S 1 (a) (solid lines), S 5 (a) (solid and dashed lines), and S ¯ 5 (b).
Fig. 4
Fig. 4 Transmission spectra of the structures S 1 (dashed line) and S ¯ 20 (solid line).

Tables (1)

Tables Icon

Table 1 Performance of the proposed method in comparison to the method of paper [20]

Equations (11)

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A ( x + d , y , z ) = A ( x , y , z ) e i k x d ,
S ( k x , ω ) [ P Q ] = [ T R ] .
1 / det S ( k x , ω p ) = 0 .
P ˜ e i k x d = T ˜ , R ˜ e i k x d = Q ˜ .
S ˜ ψ ˜ = [ I e i k x d 0 0 I e i k x d ] ψ ˜ ,
S ˜ ( ω ) = S ˜ ( ω n ) + S ˜ ( ω n ) ( ω ω n ) .
[ I e i k x d 0 0 I e i k x d ] ψ ˜ = S ˜ ( ω n ) ψ ˜ + S ˜ ( ω n ) ( ω p ω n ) ψ ˜ .
( S ˜ ( ω n ) [ I e i k x d 0 0 I e i k x d ] ) ψ ˜ = ( ω n ω p ) S ˜ ( ω n ) ψ ˜ .
ω n + 1 = ω n mineig ( S ˜ ( ω n ) [ I e i k x d 0 0 I e i k x d ] , S ˜ ( ω n ) ) .
R m = 1 d 0 d A y ( x ˜ 0 , z ˜ ) exp { i ( k x + 2 π d m ) z ˜ } d z ˜ .
| ω n ω p | | ω p | δ .

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