Abstract

We present a novel analytical approach for efficient sensitivity analysis of surface plasmon polaritons (SPPs) waveguide-based structures using the beam propagation method (BPM). Our approach exploits the adjoint variable technique to extract the response sensitivities with respect to all the design parameters regardless of their number. No extra BPM simulations are required. The accuracy of the results are comparable to those obtained using the expensive central finite difference approximations applied at the response level. Our approach is successfully applied to different SPPs structures for different applications.

© 2008 Optical Society of America

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References

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  1. S. A. Maier, Plasmonics: Fundamentals and Applications, (Springe, 2007).
  2. P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B, Condens. Matter. 61, 10484-10503 (2000).
    [CrossRef]
  3. G. Veronis and S. Fan, "Modes of Subwavelength Plasmonic Slot Waveguides,"J. Lightwave Technol. 25, 2511-2521 (2007).
    [CrossRef]
  4. W. L. Barnes, A. Dereux, and T. W. Ebbesen "Surface plasmon subwavelength optics," Nature. 424, 824-830 (2003).
    [CrossRef] [PubMed]
  5. I. Breukelaar, R. Charbonneau, and P. Berini, " Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides," J. Appl. Phys. 100, 043104-1-043104-9 (2006).
    [CrossRef]
  6. N. N. Feng, M. L. Brongersma, and L. D Negro, "Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 μm," IEEE J.Quantum Electron. 43, 479 - 485 (2007).
    [CrossRef]
  7. J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
    [CrossRef]
  8. R. D. Harris and J. S. Wilkinson, "Waveguide surface plasmon resonance sensors," Sens. Actuators B 29, 261-267 (1995).
    [CrossRef]
  9. J. Homola, "Present and future of surface plasmon resonance biosensors," Anal. Bioanal. Chem. 377, 528-539 (2003).
    [CrossRef] [PubMed]
  10. J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron.  E90-C, 95-100 (2007).
    [CrossRef]
  11. J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533- 1539 (2005).
    [CrossRef]
  12. G. Veronis, R. W. Dutton, and S. Fan, "Method for sensitivity analysis of photonic crystal devices," Opt. Lett. 29, 2288-2290 (2004).
    [CrossRef] [PubMed]
  13. M. A. Swillam, M. H. Bakr, and X. Li, "Accurate sensitivity analysis of photonic devices exploiting the finite-difference time-domain central adjoint variable method," Appl. Opt. 46, 1492-1499.(2007).
    [CrossRef] [PubMed]
  14. M. A. Swillam, M. H. Bakr, and X. Li, "Efficient adjoint sensitivity analysis exploiting the FD-BPM," J. Lightwave Technol. 25, 1861 - 1869 (2007).
    [CrossRef]
  15. M. A. Swillam, M. H. Bakr, and X. Li, "Full vectorial 3D sensitivity analysis and design optimization using BPM," J. Lightwave Technol. 26, 528-536 (2008).
    [CrossRef]
  16. Y. Hsueh, M. Yang, and H. Chang," Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method," J. Lightwave Technol. 19, 2389-2397 (1999).
    [CrossRef]
  17. P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, "Efficient estimation of sensitivities in TLM with dielectric discontinuities," IEEE Microwave Wirel. Compon. Lett. 15, 89-91 (2005).
    [CrossRef]
  18. J. W. Brown and R. V. Churchil, Complex Variables and Applications (McGraw-Hill, 2003).
  19. FEMLAB, 2.3 ed. COMSOL AB, Sweden, 2002. http://www.comsol.com
  20. L. B. Soldano and E. C. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
    [CrossRef]

2008

2007

M. A. Swillam, M. H. Bakr, and X. Li, "Accurate sensitivity analysis of photonic devices exploiting the finite-difference time-domain central adjoint variable method," Appl. Opt. 46, 1492-1499.(2007).
[CrossRef] [PubMed]

M. A. Swillam, M. H. Bakr, and X. Li, "Efficient adjoint sensitivity analysis exploiting the FD-BPM," J. Lightwave Technol. 25, 1861 - 1869 (2007).
[CrossRef]

G. Veronis and S. Fan, "Modes of Subwavelength Plasmonic Slot Waveguides,"J. Lightwave Technol. 25, 2511-2521 (2007).
[CrossRef]

N. N. Feng, M. L. Brongersma, and L. D Negro, "Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 μm," IEEE J.Quantum Electron. 43, 479 - 485 (2007).
[CrossRef]

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron.  E90-C, 95-100 (2007).
[CrossRef]

2006

I. Breukelaar, R. Charbonneau, and P. Berini, " Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides," J. Appl. Phys. 100, 043104-1-043104-9 (2006).
[CrossRef]

2005

P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, "Efficient estimation of sensitivities in TLM with dielectric discontinuities," IEEE Microwave Wirel. Compon. Lett. 15, 89-91 (2005).
[CrossRef]

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533- 1539 (2005).
[CrossRef]

2004

2003

J. Homola, "Present and future of surface plasmon resonance biosensors," Anal. Bioanal. Chem. 377, 528-539 (2003).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen "Surface plasmon subwavelength optics," Nature. 424, 824-830 (2003).
[CrossRef] [PubMed]

2000

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B, Condens. Matter. 61, 10484-10503 (2000).
[CrossRef]

1999

Y. Hsueh, M. Yang, and H. Chang," Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method," J. Lightwave Technol. 19, 2389-2397 (1999).
[CrossRef]

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

1995

R. D. Harris and J. S. Wilkinson, "Waveguide surface plasmon resonance sensors," Sens. Actuators B 29, 261-267 (1995).
[CrossRef]

L. B. Soldano and E. C. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Bakr, M. H.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen "Surface plasmon subwavelength optics," Nature. 424, 824-830 (2003).
[CrossRef] [PubMed]

Basl, P. A. W.

P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, "Efficient estimation of sensitivities in TLM with dielectric discontinuities," IEEE Microwave Wirel. Compon. Lett. 15, 89-91 (2005).
[CrossRef]

Berini, P.

I. Breukelaar, R. Charbonneau, and P. Berini, " Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides," J. Appl. Phys. 100, 043104-1-043104-9 (2006).
[CrossRef]

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B, Condens. Matter. 61, 10484-10503 (2000).
[CrossRef]

Breukelaar, I.

I. Breukelaar, R. Charbonneau, and P. Berini, " Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides," J. Appl. Phys. 100, 043104-1-043104-9 (2006).
[CrossRef]

Brongersma, M. L.

N. N. Feng, M. L. Brongersma, and L. D Negro, "Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 μm," IEEE J.Quantum Electron. 43, 479 - 485 (2007).
[CrossRef]

Chang, H.

Y. Hsueh, M. Yang, and H. Chang," Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method," J. Lightwave Technol. 19, 2389-2397 (1999).
[CrossRef]

Charbonneau, R.

I. Breukelaar, R. Charbonneau, and P. Berini, " Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides," J. Appl. Phys. 100, 043104-1-043104-9 (2006).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen "Surface plasmon subwavelength optics," Nature. 424, 824-830 (2003).
[CrossRef] [PubMed]

Dutton, R. W.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen "Surface plasmon subwavelength optics," Nature. 424, 824-830 (2003).
[CrossRef] [PubMed]

Fan, S.

Feng, N. N.

N. N. Feng, M. L. Brongersma, and L. D Negro, "Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 μm," IEEE J.Quantum Electron. 43, 479 - 485 (2007).
[CrossRef]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Harris, R. D.

R. D. Harris and J. S. Wilkinson, "Waveguide surface plasmon resonance sensors," Sens. Actuators B 29, 261-267 (1995).
[CrossRef]

Homola, J.

J. Homola, "Present and future of surface plasmon resonance biosensors," Anal. Bioanal. Chem. 377, 528-539 (2003).
[CrossRef] [PubMed]

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Hsueh, Y.

Y. Hsueh, M. Yang, and H. Chang," Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method," J. Lightwave Technol. 19, 2389-2397 (1999).
[CrossRef]

Li, X.

Nakano, H.

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron.  E90-C, 95-100 (2007).
[CrossRef]

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533- 1539 (2005).
[CrossRef]

Negro, L. D

N. N. Feng, M. L. Brongersma, and L. D Negro, "Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 μm," IEEE J.Quantum Electron. 43, 479 - 485 (2007).
[CrossRef]

Nikolova, N. K.

P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, "Efficient estimation of sensitivities in TLM with dielectric discontinuities," IEEE Microwave Wirel. Compon. Lett. 15, 89-91 (2005).
[CrossRef]

Pennings, E. C.

L. B. Soldano and E. C. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Shibayama, J.

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron.  E90-C, 95-100 (2007).
[CrossRef]

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533- 1539 (2005).
[CrossRef]

Soldano, L. B.

L. B. Soldano and E. C. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Swillam, M. A.

Takagi, S.

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron.  E90-C, 95-100 (2007).
[CrossRef]

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533- 1539 (2005).
[CrossRef]

Veronis, G.

Wilkinson, J. S.

R. D. Harris and J. S. Wilkinson, "Waveguide surface plasmon resonance sensors," Sens. Actuators B 29, 261-267 (1995).
[CrossRef]

Yamauchi, J.

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron.  E90-C, 95-100 (2007).
[CrossRef]

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533- 1539 (2005).
[CrossRef]

Yamazaki, T.

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron.  E90-C, 95-100 (2007).
[CrossRef]

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533- 1539 (2005).
[CrossRef]

Yang, M.

Y. Hsueh, M. Yang, and H. Chang," Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method," J. Lightwave Technol. 19, 2389-2397 (1999).
[CrossRef]

Yee, S. S.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Anal. Bioanal. Chem.

J. Homola, "Present and future of surface plasmon resonance biosensors," Anal. Bioanal. Chem. 377, 528-539 (2003).
[CrossRef] [PubMed]

Appl. Opt.

IEEE J.Quantum Electron.

N. N. Feng, M. L. Brongersma, and L. D Negro, "Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 μm," IEEE J.Quantum Electron. 43, 479 - 485 (2007).
[CrossRef]

IEEE Microwave Wirel. Compon. Lett.

P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, "Efficient estimation of sensitivities in TLM with dielectric discontinuities," IEEE Microwave Wirel. Compon. Lett. 15, 89-91 (2005).
[CrossRef]

IEICE Trans. Electron

J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron.  E90-C, 95-100 (2007).
[CrossRef]

J. Appl. Phys.

I. Breukelaar, R. Charbonneau, and P. Berini, " Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides," J. Appl. Phys. 100, 043104-1-043104-9 (2006).
[CrossRef]

J. Lightwave Technol.

Nature.

W. L. Barnes, A. Dereux, and T. W. Ebbesen "Surface plasmon subwavelength optics," Nature. 424, 824-830 (2003).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. B, Condens. Matter.

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B, Condens. Matter. 61, 10484-10503 (2000).
[CrossRef]

Sens. Actuators B

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

R. D. Harris and J. S. Wilkinson, "Waveguide surface plasmon resonance sensors," Sens. Actuators B 29, 261-267 (1995).
[CrossRef]

Other

J. W. Brown and R. V. Churchil, Complex Variables and Applications (McGraw-Hill, 2003).

FEMLAB, 2.3 ed. COMSOL AB, Sweden, 2002. http://www.comsol.com

S. A. Maier, Plasmonics: Fundamentals and Applications, (Springe, 2007).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic diagram of the metal loaded waveguide

Fig. 2.
Fig. 2.

Effective index of the fundamental mode of the metal loaded waveguide at wavelength λ=1.55µm and t=0.2 µm.

Fig. 3.
Fig. 3.

Schematic diagram of the metal loaded channel waveguide.

Fig. 4.
Fig. 4.

Normalized Sensitivity of the propagation length of the fundamental mode in a metal loaded on channel waveguide structure at wavelength λ=1.55µm, Wm =Ws =0.5 µm, and tm =0.1 µm.

Fig. 5.
Fig. 5.

Normalized Sensitivity of the propagation length of the fundamental mode in a metal loaded on channel waveguide structure at wavelength λ=1.55µm, Ws =0.5 µm s tm =0.1 µm and ts =0.5 µm.

Fig. 6.
Fig. 6.

Normalized Sensitivity of the propagation length of the fundamental mode in a metal loaded on channel waveguide structure at wavelength λ=1.55µm, Wm =0.5 µm s tm =0.1 µm and ts =0.5 µm.

Fig. 7.
Fig. 7.

Schematic diagram of 1×3 power splitter SPPs structure.

Fig. 8.
Fig. 8.

Normalized Sensitivity of the power coupling coefficient of 1×3 power splitter at wavelength λ=1.55µm, Lm =78.2 µm, and tm =ts =0.2 µm

Fig. 9.
Fig. 9.

Normalized Sensitivity of the power coupling coefficient of 1×3 power splitter at wavelength λ=1.55µm, Lm =78.2 µm, and tm =ts =0.2 µm

Equations (19)

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

ψ y z = P yy ψ y = j 2 n o k ( y ( 1 n 2 y ( n 2 ψ y ) ) + 2 ψ y x 2 + ( n 2 n o 2 ) k 2 ψ y )
ψ x , y l + 1 = R l ψ x , y l = [ ( Γ 4 l ) 1 Γ 3 l ( Γ 2 l ) 1 Γ 1 l ] Ψ x , y l .
Γ 1 = I + Δ z 2 B x , Γ 2 = I Δ z 2 B y , Γ 3 = I + Δ z 2 B y , and Γ 4 = I Δ z 2 B x
B y ψ y = j 2 n o k ( y ( 1 n 2 y ( n 2 ψ y ) ) + 1 2 ( n 2 n o 2 ) k 2 ψ y )
B x ψ y = j 2 n o k ( 2 ψ y x 2 + 1 2 ( n 2 n o 2 ) k 2 ψ y ) .
f p i = k I ( f ψ y k ) T ψ y k p i
ψ y l + 1 p i = [ R l p i ψ y l + R l ψ y l p i ]
R l p i = ( Γ 4 l ) 1 Γ 4 l p i ( Γ 4 l ) 1 Γ 3 l ( Γ 2 l ) 1 Γ 1 l + ( Γ 4 l ) 1 Γ 3 l p i ( Γ 2 l ) 1 Γ 1 l
( Γ 4 l ) 1 Γ 3 l ( Γ 2 l ) 1 Γ 2 l p i ( Γ 2 l ) 1 Γ 1 l + ( Γ 4 l ) 1 Γ 3 l ( Γ 2 l ) 1 Γ 1 l p i
Γ 1 p i = Δ z 2 B x p i = Γ 4 p i and Γ 3 p i = Δ z 2 B y p i = Γ 2 p i
f p i = k I ( f ψ t k ) T · ( R k 1 p i ψ t k 1 + R k 1 ψ t k 1 p i )
R k p i = q = 1 Q R k n q n q p i
B x = j 2 n o k [ b 1 a 0 a b q a 0 a b M ] ,
a = 1 Δ x 2 and b q = 2 Δ x 2 + k 2 2 ( n q 2 n o 2 ) .
B x n q = j k n q n o I
n = n r j n i m
R k n = R r k n r + j R m k n r
R k n im = j R k n r
C = Φ N · Ψ d x 2

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