Abstract

Widely used for the design of resonant electronic devices, Mason’s scalar rule is adapted here to the study of resonant subwavelength optical structures. It turns out to be an efficient formalism, especially when dealing with multiple wave interference mechanisms. Indeed it allows to comprehend the underlying physical mechanisms of the structure in a straightforward way and fast analytical formulae can be retrieved. As an illustration, we apply it to the study of dual metallic gratings, which appear to be promissing optical filters as their spectral shape can be tailored according to needs.

© 2012 OSA

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  1. J. Shen and P. Platzman, “Properties of a One-Dimensional Metallophotonic Crystal”, Phys. Rev. B70, 035101 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. A. Hibbins, J. Sambles, C. Lawrence, and J. Brown, “Squeezing millimeter waves into microns, Phys. Rev. Lett.92, 143904 (2004).
    [CrossRef] [PubMed]
  4. Y. Ding and R. Magnusson, “Resonant Leaky-Mode spectral-band engineering and device applications”, Opt. Express12, 5661–5674 (2004).
    [CrossRef] [PubMed]
  5. V. Babicheva and Y. Lozovik, “Extraordinary Transmission and Suppression of Transmission of Dual Metal Gratings with Subwavelength Slits” in “AIP Conference Proceedings”, 291, p. 103 (2010).
    [CrossRef]
  6. P. Lalanne, J. Hugonin, and P. Chavel, “Optical Properties of Deep Lamellar Gratings: A Coupled Bloch-Mode Insight”, J. Lightwave Technol.24, 2442 (2006).
    [CrossRef]
  7. H. B. Chan, Z. Marcet, K. Woo, D. B. Tanner, D. W. Carr, J. E. Bower, R. A. Cirelli, E. Ferry, F. Klemens, J. Miner, C. S. Pai, and J. A. Taylor, “Optical Transmission through Double-Layer Metallic Subwavelength Slit Arrays”, Opt. Lett.31, 516–518 (2006).
    [CrossRef] [PubMed]
  8. T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast Modal Method for Subwavelength Gratings based on B-Spline Formulation”, J. Opt. Soc. Am. A.27, 696–702 (2010).
    [CrossRef]
  11. S. Mason, “Feedback Theory-Some Properties of Signal Flow Graphs”, Proceedings of the IRE41, 1144–1156 (1953).
    [CrossRef]
  12. C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
    [CrossRef]
  13. N. Cotter, T. Preist, and J. Sambles, “Scattering-Matrix Approach to Multilayer Diffraction”, J. Opt. Soc. Am. A.12, 1097–1103 (1995).
    [CrossRef]
  14. E. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic press, 1985).
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    [CrossRef]

2011 (1)

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

2010 (1)

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast Modal Method for Subwavelength Gratings based on B-Spline Formulation”, J. Opt. Soc. Am. A.27, 696–702 (2010).
[CrossRef]

2008 (1)

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

2006 (2)

2004 (4)

R. Bakker, V. Drachev, H. Yuan, and V. Shalaev, “Enhanced Transmission in Near-Field Imaging of Layered Plasmonic Structures”, Opt. Express12, 3701–3706 (2004).
[CrossRef] [PubMed]

Y. Ding and R. Magnusson, “Resonant Leaky-Mode spectral-band engineering and device applications”, Opt. Express12, 5661–5674 (2004).
[CrossRef] [PubMed]

J. Shen and P. Platzman, “Properties of a One-Dimensional Metallophotonic Crystal”, Phys. Rev. B70, 035101 (2004).
[CrossRef]

A. Hibbins, J. Sambles, C. Lawrence, and J. Brown, “Squeezing millimeter waves into microns, Phys. Rev. Lett.92, 143904 (2004).
[CrossRef] [PubMed]

1996 (1)

P. Lalanne and G. Morris, “Highly Improved Convergence of the Coupled-Wave Method for TM Polarization”, J. Opt. Soc. Am. A.13, 779–784 (1996).
[CrossRef]

1995 (1)

N. Cotter, T. Preist, and J. Sambles, “Scattering-Matrix Approach to Multilayer Diffraction”, J. Opt. Soc. Am. A.12, 1097–1103 (1995).
[CrossRef]

1953 (1)

S. Mason, “Feedback Theory-Some Properties of Signal Flow Graphs”, Proceedings of the IRE41, 1144–1156 (1953).
[CrossRef]

1935 (1)

R. W. Wood, “Anomalous Diffraction Gratings”, Phys. Rev.48, 928–936 (1935).
[CrossRef]

Babicheva, V.

V. Babicheva and Y. Lozovik, “Extraordinary Transmission and Suppression of Transmission of Dual Metal Gratings with Subwavelength Slits” in “AIP Conference Proceedings”, 291, p. 103 (2010).
[CrossRef]

Bakker, R.

Bouchon, P.

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast Modal Method for Subwavelength Gratings based on B-Spline Formulation”, J. Opt. Soc. Am. A.27, 696–702 (2010).
[CrossRef]

Bower, J. E.

Brown, J.

A. Hibbins, J. Sambles, C. Lawrence, and J. Brown, “Squeezing millimeter waves into microns, Phys. Rev. Lett.92, 143904 (2004).
[CrossRef] [PubMed]

Carr, D. W.

Chan, H. B.

Chavel, P.

Chen, J.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Cheng, C.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Cirelli, R. A.

Cotter, N.

N. Cotter, T. Preist, and J. Sambles, “Scattering-Matrix Approach to Multilayer Diffraction”, J. Opt. Soc. Am. A.12, 1097–1103 (1995).
[CrossRef]

Derelle, S.

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

Ding, J.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Ding, Y.

Drachev, V.

Estruch, T.

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

Fan, Y.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Ferry, E.

Ghosh, G.

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic press, 1985).

Haidar, R.

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

Haïdar, R.

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast Modal Method for Subwavelength Gratings based on B-Spline Formulation”, J. Opt. Soc. Am. A.27, 696–702 (2010).
[CrossRef]

Hibbins, A.

A. Hibbins, J. Sambles, C. Lawrence, and J. Brown, “Squeezing millimeter waves into microns, Phys. Rev. Lett.92, 143904 (2004).
[CrossRef] [PubMed]

Hugonin, J.

Jaeck, J.

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

Klemens, F.

Lalanne, P.

P. Lalanne, J. Hugonin, and P. Chavel, “Optical Properties of Deep Lamellar Gratings: A Coupled Bloch-Mode Insight”, J. Lightwave Technol.24, 2442 (2006).
[CrossRef]

P. Lalanne and G. Morris, “Highly Improved Convergence of the Coupled-Wave Method for TM Polarization”, J. Opt. Soc. Am. A.13, 779–784 (1996).
[CrossRef]

Lawrence, C.

A. Hibbins, J. Sambles, C. Lawrence, and J. Brown, “Squeezing millimeter waves into microns, Phys. Rev. Lett.92, 143904 (2004).
[CrossRef] [PubMed]

Lozovik, Y.

V. Babicheva and Y. Lozovik, “Extraordinary Transmission and Suppression of Transmission of Dual Metal Gratings with Subwavelength Slits” in “AIP Conference Proceedings”, 291, p. 103 (2010).
[CrossRef]

Magnusson, R.

Marcet, Z.

Mason, S.

S. Mason, “Feedback Theory-Some Properties of Signal Flow Graphs”, Proceedings of the IRE41, 1144–1156 (1953).
[CrossRef]

Miner, J.

Morris, G.

P. Lalanne and G. Morris, “Highly Improved Convergence of the Coupled-Wave Method for TM Polarization”, J. Opt. Soc. Am. A.13, 779–784 (1996).
[CrossRef]

Pai, C. S.

Palik, E.

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic press, 1985).

Pardo, F.

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast Modal Method for Subwavelength Gratings based on B-Spline Formulation”, J. Opt. Soc. Am. A.27, 696–702 (2010).
[CrossRef]

Pelouard, J.

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

Pelouard, J.-L.

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast Modal Method for Subwavelength Gratings based on B-Spline Formulation”, J. Opt. Soc. Am. A.27, 696–702 (2010).
[CrossRef]

Platzman, P.

J. Shen and P. Platzman, “Properties of a One-Dimensional Metallophotonic Crystal”, Phys. Rev. B70, 035101 (2004).
[CrossRef]

Preist, T.

N. Cotter, T. Preist, and J. Sambles, “Scattering-Matrix Approach to Multilayer Diffraction”, J. Opt. Soc. Am. A.12, 1097–1103 (1995).
[CrossRef]

Primot, J.

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

Ren, F.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Sambles, J.

A. Hibbins, J. Sambles, C. Lawrence, and J. Brown, “Squeezing millimeter waves into microns, Phys. Rev. Lett.92, 143904 (2004).
[CrossRef] [PubMed]

N. Cotter, T. Preist, and J. Sambles, “Scattering-Matrix Approach to Multilayer Diffraction”, J. Opt. Soc. Am. A.12, 1097–1103 (1995).
[CrossRef]

Shalaev, V.

Shen, J.

J. Shen and P. Platzman, “Properties of a One-Dimensional Metallophotonic Crystal”, Phys. Rev. B70, 035101 (2004).
[CrossRef]

Shi, D.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Tanner, D. B.

Taylor, J. A.

Wang, H.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Woo, K.

Wood, R. W.

R. W. Wood, “Anomalous Diffraction Gratings”, Phys. Rev.48, 928–936 (1935).
[CrossRef]

Wu, Q.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Xu, J.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Yuan, H.

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A. (3)

N. Cotter, T. Preist, and J. Sambles, “Scattering-Matrix Approach to Multilayer Diffraction”, J. Opt. Soc. Am. A.12, 1097–1103 (1995).
[CrossRef]

P. Lalanne and G. Morris, “Highly Improved Convergence of the Coupled-Wave Method for TM Polarization”, J. Opt. Soc. Am. A.13, 779–784 (1996).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast Modal Method for Subwavelength Gratings based on B-Spline Formulation”, J. Opt. Soc. Am. A.27, 696–702 (2010).
[CrossRef]

Opt. Express (2)

Opt. Lett (1)

T. Estruch, J. Jaeck, F. Pardo, S. Derelle, J. Primot, J. Pelouard, and R. Haidar, “Perfect Extinction in Subwavelength Dual Metallic Transmitting Gratings”, Opt. Lett36, 3160–3162 (2011).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. (1)

R. W. Wood, “Anomalous Diffraction Gratings”, Phys. Rev.48, 928–936 (1935).
[CrossRef]

Phys. Rev. B (2)

J. Shen and P. Platzman, “Properties of a One-Dimensional Metallophotonic Crystal”, Phys. Rev. B70, 035101 (2004).
[CrossRef]

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical Mechanism of Extraordinary Electromagnetic Transmission in Dual-Metallic Grating Structures”, Phys. Rev. B78, 075406 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

A. Hibbins, J. Sambles, C. Lawrence, and J. Brown, “Squeezing millimeter waves into microns, Phys. Rev. Lett.92, 143904 (2004).
[CrossRef] [PubMed]

Proceedings of the IRE (1)

S. Mason, “Feedback Theory-Some Properties of Signal Flow Graphs”, Proceedings of the IRE41, 1144–1156 (1953).
[CrossRef]

Other (2)

V. Babicheva and Y. Lozovik, “Extraordinary Transmission and Suppression of Transmission of Dual Metal Gratings with Subwavelength Slits” in “AIP Conference Proceedings”, 291, p. 103 (2010).
[CrossRef]

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic press, 1985).

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Figures (5)

Fig. 1
Fig. 1

Mason inspired SFGs applied on the FP case. a. Amplitudes for incident light from top (resp. bottom) of the structure. Transmission through the structure with resonant cavity symbolized by a red loop.

Fig. 2
Fig. 2

Mason inspired SFGs applied on the DMG case. a. S matrices containing transmission and reflection amplitudes for the zeroth order and the first couple of evanescent orders that are equal since the system is vertically symmetric. b. Transmission through the structure. Red arrows correspond to the different amplitude transfers from the zeroth order toward itself or the evanescent orders. Green arrows represent amplitude transfers from the + 1 evanescent order toward zeroth or itself.

Fig. 3
Fig. 3

Comparison between the transmission spectra calculated using BMM exact computations (black curves) and the Mason analytical formula (dashed curves) for different gap hg values. Purple curve corresponds to the transmission spectrum when the evanescent orders are considered propagative with hg = 1.0μm (the kz vector becomes real). A FP behaviour is then observed and the high wavelengths extinction vanishes.

Fig. 4
Fig. 4

Mason inspired SFGs applied on the DMG with lateral displacement L case. a. S matrices containing transmission and reflection amplitudes for the zeroth order and the first couple of evanescent orders that are no longer equal since the system is non-symmetric. b. Transmission through the structure. Red arrows correspond to the different amplitude transfers from the zeroth order toward itself or the evanescent orders. Green (resp. blue) arrows represent amplitude transfers from the +1 (resp. −1) evanescent order toward zeroth, −1 (resp. +1) evanescent order or itself.

Fig. 5
Fig. 5

Comparison between the transmission spectra calculated using B-Spline Modal Method (black curves) and with the retrieved Mason analytical formula (dashed curves) for different lag L values. Cyan curve coresponds to the transmission spectrum when the 5 first evanescent orders are taken into account with L = 1.8μm.

Tables (2)

Tables Icon

Table 1 Number of forward paths and loops of each type calculated using Mason’s rule on signal flow graph from Fig. 2.b.

Tables Icon

Table 2 Number of forward paths and loops of each type calculated using Mason’s rule on signal flow graph from Fig. 4.b.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

G = out in = S 32 β ( 1 e i k z 2 h g S 22 α e i k z 2 h g S 22 β ) 1 e i k z 2 h g S 21 α
G = out in = p T p Δ p Δ
Δ = 1 + m = 1 ( 1 ) m Γ m
T F P = G = S 32 e i k z 2 h g S 21 1 ( S 22 β e 2 i k z 2 h g S 22 α )

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