Abstract

A four-layer waveguide structure comprising a dielectric substrate, a metal layer, a left-handed material (LHM) as a guiding layer, and a cladding is investigated as a metal-clad waveguide sensor. Fresnel reflection coefficients are used to study the resonance dips at which the reflectance minimizes. Our calculations show that the proposed structure has a preference over the surface-plasmon resonance structure since it gives a much sharper reflectance dip and can achieve considerable sensitivity improvement. The effects of the LHM permittivity, permeability, and thickness on the reflectance curves is studied.

© 2012 Optical Society of America

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References

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  1. N. Skivesen, R. Horvath, and H. Pedersen, “Optimization of metal-clad waveguide sensors,” Sens. Actuators B 106, 668–676 (2005).
    [CrossRef]
  2. N. Skivesen, R. Horvath, and H. Pedersen, “Peak-type and dip-type metal-clad waveguide sensing,” Opt. Lett. 30, 1659–1661 (2005).
    [CrossRef]
  3. G. Tollin and Z. Salamon, “Optical anisotropy in lipid bilayer membranes: coupled plasmon waveguide resonance measurements of molecular orientation, polarizability and shape,” Biophys. J. 80, 1557–1567 (2001).
    [CrossRef]
  4. Z. Salamon, G. Lindblom, and G. Tollin, “Plasmon-waveguide resonance and impedance spectroscopy studies of the interaction between penetratin and supported lipid bilayer membranes,” Biophys. J. 84, 1796–1807 (2003).
    [CrossRef]
  5. M. Zourob and N. Goddard, “Metal clad leaky waveguides for chemical and biosensing applications,” Biosens. Bioelectron. 20, 1718–1727 (2005).
    [CrossRef]
  6. S. Taya, M. Shabat, H. Khalil, and D. Jäger, “Theoretical analysis of TM nonlinear asymmetrical waveguide optical sensors,” Sens. Actuators A 147, 137–141 (2008).
    [CrossRef]
  7. S. Taya, M. Shabat, and H. Khalil, “Enhancement of Sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
    [CrossRef]
  8. S. Taya, M. Shabat, and H. Khalil, “Nonlinear planar asymmetrical optical waveguides for sensing applications,” Optik 121, 860–865 (2010).
    [CrossRef]
  9. S. Taya and T. El-Agez, “Comparing optical sensing using slab waveguides and total internal reflection ellipsometry,” Turk. J. Phys. 35, 31–36 (2011).
  10. T. El-Agez and S. Taya, “Theoretical spectroscopic scan of the sensitivity of asymmetric slab waveguide sensors,” Opt. Appl. 41, 89–95 (2011).
  11. S. Taya and T. El-Agez, “Reverse symmetry optical waveguide sensor using plasma substrate,” J. Opt. 13, 075701(2011).
    [CrossRef]
  12. V. Veselago, “The electrodynamics of subctance with simultaneously negative index values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  13. K. Park, B. Lee, C. Fu, and Z. Zhang, “Study of the surface and bulk polaritons with a negative index metamaterial,” J. Opt. Soc. Am. B 22, 1016–1023 (2005).
    [CrossRef]
  14. A. Grbic and G. Eleftheriadesm, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
    [CrossRef]
  15. W. Cai, D. Genov, and V. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).
  16. D. Schurig, J. Mock, and B. Justice, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
    [CrossRef]
  17. P. Markoš and C. Soukoulis, “Wave propagation,” in From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).
  18. V. Podolskiyand and E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005).
    [CrossRef]
  19. P. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
    [CrossRef]
  20. A. Otto and W. Sohler, “Modification of the total reflection modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
    [CrossRef]
  21. A. S. Vioktalamo, R. Watanabe, and T. Ishihara, “Permeability enhancement of stratified metal dielectric metamaterial in optical regime,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.005 (to be published).

2011 (3)

S. Taya and T. El-Agez, “Comparing optical sensing using slab waveguides and total internal reflection ellipsometry,” Turk. J. Phys. 35, 31–36 (2011).

T. El-Agez and S. Taya, “Theoretical spectroscopic scan of the sensitivity of asymmetric slab waveguide sensors,” Opt. Appl. 41, 89–95 (2011).

S. Taya and T. El-Agez, “Reverse symmetry optical waveguide sensor using plasma substrate,” J. Opt. 13, 075701(2011).
[CrossRef]

2010 (1)

S. Taya, M. Shabat, and H. Khalil, “Nonlinear planar asymmetrical optical waveguides for sensing applications,” Optik 121, 860–865 (2010).
[CrossRef]

2009 (1)

S. Taya, M. Shabat, and H. Khalil, “Enhancement of Sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

2008 (1)

S. Taya, M. Shabat, H. Khalil, and D. Jäger, “Theoretical analysis of TM nonlinear asymmetrical waveguide optical sensors,” Sens. Actuators A 147, 137–141 (2008).
[CrossRef]

2006 (1)

D. Schurig, J. Mock, and B. Justice, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

2005 (6)

V. Podolskiyand and E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005).
[CrossRef]

K. Park, B. Lee, C. Fu, and Z. Zhang, “Study of the surface and bulk polaritons with a negative index metamaterial,” J. Opt. Soc. Am. B 22, 1016–1023 (2005).
[CrossRef]

N. Skivesen, R. Horvath, and H. Pedersen, “Optimization of metal-clad waveguide sensors,” Sens. Actuators B 106, 668–676 (2005).
[CrossRef]

N. Skivesen, R. Horvath, and H. Pedersen, “Peak-type and dip-type metal-clad waveguide sensing,” Opt. Lett. 30, 1659–1661 (2005).
[CrossRef]

M. Zourob and N. Goddard, “Metal clad leaky waveguides for chemical and biosensing applications,” Biosens. Bioelectron. 20, 1718–1727 (2005).
[CrossRef]

W. Cai, D. Genov, and V. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).

2004 (1)

A. Grbic and G. Eleftheriadesm, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef]

2003 (1)

Z. Salamon, G. Lindblom, and G. Tollin, “Plasmon-waveguide resonance and impedance spectroscopy studies of the interaction between penetratin and supported lipid bilayer membranes,” Biophys. J. 84, 1796–1807 (2003).
[CrossRef]

2001 (1)

G. Tollin and Z. Salamon, “Optical anisotropy in lipid bilayer membranes: coupled plasmon waveguide resonance measurements of molecular orientation, polarizability and shape,” Biophys. J. 80, 1557–1567 (2001).
[CrossRef]

1977 (1)

P. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[CrossRef]

1971 (1)

A. Otto and W. Sohler, “Modification of the total reflection modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
[CrossRef]

1968 (1)

V. Veselago, “The electrodynamics of subctance with simultaneously negative index values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Cai, W.

W. Cai, D. Genov, and V. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).

El-Agez, T.

S. Taya and T. El-Agez, “Comparing optical sensing using slab waveguides and total internal reflection ellipsometry,” Turk. J. Phys. 35, 31–36 (2011).

T. El-Agez and S. Taya, “Theoretical spectroscopic scan of the sensitivity of asymmetric slab waveguide sensors,” Opt. Appl. 41, 89–95 (2011).

S. Taya and T. El-Agez, “Reverse symmetry optical waveguide sensor using plasma substrate,” J. Opt. 13, 075701(2011).
[CrossRef]

Eleftheriadesm, G.

A. Grbic and G. Eleftheriadesm, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef]

Fu, C.

Genov, D.

W. Cai, D. Genov, and V. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).

Goddard, N.

M. Zourob and N. Goddard, “Metal clad leaky waveguides for chemical and biosensing applications,” Biosens. Bioelectron. 20, 1718–1727 (2005).
[CrossRef]

Grbic, A.

A. Grbic and G. Eleftheriadesm, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef]

Horvath, R.

N. Skivesen, R. Horvath, and H. Pedersen, “Peak-type and dip-type metal-clad waveguide sensing,” Opt. Lett. 30, 1659–1661 (2005).
[CrossRef]

N. Skivesen, R. Horvath, and H. Pedersen, “Optimization of metal-clad waveguide sensors,” Sens. Actuators B 106, 668–676 (2005).
[CrossRef]

Ishihara, T.

A. S. Vioktalamo, R. Watanabe, and T. Ishihara, “Permeability enhancement of stratified metal dielectric metamaterial in optical regime,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.005 (to be published).

Jäger, D.

S. Taya, M. Shabat, H. Khalil, and D. Jäger, “Theoretical analysis of TM nonlinear asymmetrical waveguide optical sensors,” Sens. Actuators A 147, 137–141 (2008).
[CrossRef]

Justice, B.

D. Schurig, J. Mock, and B. Justice, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Khalil, H.

S. Taya, M. Shabat, and H. Khalil, “Nonlinear planar asymmetrical optical waveguides for sensing applications,” Optik 121, 860–865 (2010).
[CrossRef]

S. Taya, M. Shabat, and H. Khalil, “Enhancement of Sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

S. Taya, M. Shabat, H. Khalil, and D. Jäger, “Theoretical analysis of TM nonlinear asymmetrical waveguide optical sensors,” Sens. Actuators A 147, 137–141 (2008).
[CrossRef]

Lee, B.

Lindblom, G.

Z. Salamon, G. Lindblom, and G. Tollin, “Plasmon-waveguide resonance and impedance spectroscopy studies of the interaction between penetratin and supported lipid bilayer membranes,” Biophys. J. 84, 1796–1807 (2003).
[CrossRef]

Markoš, P.

P. Markoš and C. Soukoulis, “Wave propagation,” in From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

Mock, J.

D. Schurig, J. Mock, and B. Justice, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Narimanov, E.

Otto, A.

A. Otto and W. Sohler, “Modification of the total reflection modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
[CrossRef]

Park, K.

Pedersen, H.

N. Skivesen, R. Horvath, and H. Pedersen, “Peak-type and dip-type metal-clad waveguide sensing,” Opt. Lett. 30, 1659–1661 (2005).
[CrossRef]

N. Skivesen, R. Horvath, and H. Pedersen, “Optimization of metal-clad waveguide sensors,” Sens. Actuators B 106, 668–676 (2005).
[CrossRef]

Podolskiyand, V.

Salamon, Z.

Z. Salamon, G. Lindblom, and G. Tollin, “Plasmon-waveguide resonance and impedance spectroscopy studies of the interaction between penetratin and supported lipid bilayer membranes,” Biophys. J. 84, 1796–1807 (2003).
[CrossRef]

G. Tollin and Z. Salamon, “Optical anisotropy in lipid bilayer membranes: coupled plasmon waveguide resonance measurements of molecular orientation, polarizability and shape,” Biophys. J. 80, 1557–1567 (2001).
[CrossRef]

Schurig, D.

D. Schurig, J. Mock, and B. Justice, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Shabat, M.

S. Taya, M. Shabat, and H. Khalil, “Nonlinear planar asymmetrical optical waveguides for sensing applications,” Optik 121, 860–865 (2010).
[CrossRef]

S. Taya, M. Shabat, and H. Khalil, “Enhancement of Sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

S. Taya, M. Shabat, H. Khalil, and D. Jäger, “Theoretical analysis of TM nonlinear asymmetrical waveguide optical sensors,” Sens. Actuators A 147, 137–141 (2008).
[CrossRef]

Shalaev, V.

W. Cai, D. Genov, and V. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).

Skivesen, N.

N. Skivesen, R. Horvath, and H. Pedersen, “Optimization of metal-clad waveguide sensors,” Sens. Actuators B 106, 668–676 (2005).
[CrossRef]

N. Skivesen, R. Horvath, and H. Pedersen, “Peak-type and dip-type metal-clad waveguide sensing,” Opt. Lett. 30, 1659–1661 (2005).
[CrossRef]

Sohler, W.

A. Otto and W. Sohler, “Modification of the total reflection modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
[CrossRef]

Soukoulis, C.

P. Markoš and C. Soukoulis, “Wave propagation,” in From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

Taya, S.

T. El-Agez and S. Taya, “Theoretical spectroscopic scan of the sensitivity of asymmetric slab waveguide sensors,” Opt. Appl. 41, 89–95 (2011).

S. Taya and T. El-Agez, “Reverse symmetry optical waveguide sensor using plasma substrate,” J. Opt. 13, 075701(2011).
[CrossRef]

S. Taya and T. El-Agez, “Comparing optical sensing using slab waveguides and total internal reflection ellipsometry,” Turk. J. Phys. 35, 31–36 (2011).

S. Taya, M. Shabat, and H. Khalil, “Nonlinear planar asymmetrical optical waveguides for sensing applications,” Optik 121, 860–865 (2010).
[CrossRef]

S. Taya, M. Shabat, and H. Khalil, “Enhancement of Sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

S. Taya, M. Shabat, H. Khalil, and D. Jäger, “Theoretical analysis of TM nonlinear asymmetrical waveguide optical sensors,” Sens. Actuators A 147, 137–141 (2008).
[CrossRef]

Tien, P.

P. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[CrossRef]

Tollin, G.

Z. Salamon, G. Lindblom, and G. Tollin, “Plasmon-waveguide resonance and impedance spectroscopy studies of the interaction between penetratin and supported lipid bilayer membranes,” Biophys. J. 84, 1796–1807 (2003).
[CrossRef]

G. Tollin and Z. Salamon, “Optical anisotropy in lipid bilayer membranes: coupled plasmon waveguide resonance measurements of molecular orientation, polarizability and shape,” Biophys. J. 80, 1557–1567 (2001).
[CrossRef]

Veselago, V.

V. Veselago, “The electrodynamics of subctance with simultaneously negative index values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vioktalamo, A. S.

A. S. Vioktalamo, R. Watanabe, and T. Ishihara, “Permeability enhancement of stratified metal dielectric metamaterial in optical regime,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.005 (to be published).

Watanabe, R.

A. S. Vioktalamo, R. Watanabe, and T. Ishihara, “Permeability enhancement of stratified metal dielectric metamaterial in optical regime,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.005 (to be published).

Zhang, Z.

Zourob, M.

M. Zourob and N. Goddard, “Metal clad leaky waveguides for chemical and biosensing applications,” Biosens. Bioelectron. 20, 1718–1727 (2005).
[CrossRef]

Biophys. J. (2)

G. Tollin and Z. Salamon, “Optical anisotropy in lipid bilayer membranes: coupled plasmon waveguide resonance measurements of molecular orientation, polarizability and shape,” Biophys. J. 80, 1557–1567 (2001).
[CrossRef]

Z. Salamon, G. Lindblom, and G. Tollin, “Plasmon-waveguide resonance and impedance spectroscopy studies of the interaction between penetratin and supported lipid bilayer membranes,” Biophys. J. 84, 1796–1807 (2003).
[CrossRef]

Biosens. Bioelectron. (1)

M. Zourob and N. Goddard, “Metal clad leaky waveguides for chemical and biosensing applications,” Biosens. Bioelectron. 20, 1718–1727 (2005).
[CrossRef]

J. Opt. (1)

S. Taya and T. El-Agez, “Reverse symmetry optical waveguide sensor using plasma substrate,” J. Opt. 13, 075701(2011).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Appl. (1)

T. El-Agez and S. Taya, “Theoretical spectroscopic scan of the sensitivity of asymmetric slab waveguide sensors,” Opt. Appl. 41, 89–95 (2011).

Opt. Commun. (1)

A. Otto and W. Sohler, “Modification of the total reflection modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
[CrossRef]

Opt. Lett. (2)

Optik (2)

S. Taya, M. Shabat, and H. Khalil, “Enhancement of Sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

S. Taya, M. Shabat, and H. Khalil, “Nonlinear planar asymmetrical optical waveguides for sensing applications,” Optik 121, 860–865 (2010).
[CrossRef]

Photon. Nanostr. Fundam. Appl. (1)

A. S. Vioktalamo, R. Watanabe, and T. Ishihara, “Permeability enhancement of stratified metal dielectric metamaterial in optical regime,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.005 (to be published).

Phys. Rev. B (1)

W. Cai, D. Genov, and V. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).

Phys. Rev. Lett. (1)

A. Grbic and G. Eleftheriadesm, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef]

Rev. Mod. Phys. (1)

P. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[CrossRef]

Science (1)

D. Schurig, J. Mock, and B. Justice, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Sens. Actuators A (1)

S. Taya, M. Shabat, H. Khalil, and D. Jäger, “Theoretical analysis of TM nonlinear asymmetrical waveguide optical sensors,” Sens. Actuators A 147, 137–141 (2008).
[CrossRef]

Sens. Actuators B (1)

N. Skivesen, R. Horvath, and H. Pedersen, “Optimization of metal-clad waveguide sensors,” Sens. Actuators B 106, 668–676 (2005).
[CrossRef]

Sov. Phys. Usp. (1)

V. Veselago, “The electrodynamics of subctance with simultaneously negative index values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Turk. J. Phys. (1)

S. Taya and T. El-Agez, “Comparing optical sensing using slab waveguides and total internal reflection ellipsometry,” Turk. J. Phys. 35, 31–36 (2011).

Other (1)

P. Markoš and C. Soukoulis, “Wave propagation,” in From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

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Figures (9)

Fig. 1.
Fig. 1.

Metal-clad waveguide structure configuration.

Fig. 2.
Fig. 2.

Calculated reflectance for a TM wave reflected from a silver-clad waveguide with and without an LHM film layer for d2=60nm, λ=632.8nm, ε1=2.3, ε2=16+i0.52, ε3=4+i0.001, ε4=1.77, μ1=1, μ2=1, μ3=2.4+i0.001, and μ4=1.

Fig. 3.
Fig. 3.

Calculated reflectance for TM wave reflected from a gold-clad waveguide with and without an LHM film layer for d2=60nm, λ=632.8nm, ε1=2.3, ε2=10.22+i0.96, ε3=4+i0.001, ε4=1.77, μ1=1, μ2=1, μ3=2.4+i0.001, and μ4=1.

Fig. 4.
Fig. 4.

Calculated reflectance for a TE wave reflected from (a) a silver-clad waveguide and (b) a gold-clad waveguide for different values of the metal thickness for d3=250nm, λ=632.8nm, ε1=2.3, ε2=16+i0.52 (silver), ε2=10.22+i0.96 (gold), ε3=4+i0.001, ε4=1.77, μ1=1, μ2=1, μ3=3+i0.001, and μ4=1.

Fig. 5.
Fig. 5.

Calculated reflectance for TE wave reflected from (a) a silver-clad waveguide and (b) a gold-clad waveguide for different values of LHM film thickness for d2=40nm, λ=632.8nm, ε1=2.3, ε2=16+i0.52 (silver), ε2=10.22+i0.96 (gold), ε3=4+i0.001, ε4=1.77, μ1=1, μ2=1, μ3=3+i0.001, and μ4=1.

Fig. 6.
Fig. 6.

Calculated reflectance for TE waves reflected from (a) a silver-clad waveguide and (b) a gold-clad waveguide for different values of permittivity of the LHM film for d2=40nm, d3=250nm, λ=632.8nm ε1=2.3, ε2=16+i0.52 (silver), ε2=10.22+i0.96 (gold), ε4=1.77, μ1=1, μ2=1, μ3=3+i0.001, and μ4=1.

Fig. 7.
Fig. 7.

Calculated reflectance for TE waves reflected from (a) a silver-clad waveguide and (b) a gold-clad waveguide for different values of permeability of the LHM film for d2=40nm, d3=250nm, λ=632.8nm ε1=2.3, ε2=16+i0.52 (silver), ε2=10.22+i0.96 (gold), ε3=4+i0.001, ε4=1.77, μ1=1, μ2=1, and μ4=1.

Fig. 8.
Fig. 8.

Calculated reflectance for MCWG (TE and TM) and SPR modes and for a silver-clad waveguide for two values of n4 for d2=40nm, d3=350nm, λ=632.8nm, ε1=2.3, ε2=16+i0.52, ε3=4+i0.001, μ1=1, μ2=1, μ3=3+i0.001, and μ4=1.

Fig. 9.
Fig. 9.

Calculated reflectance for TE waves reflected from a structure with an LHM guiding layer and a structure with an RHM guiding layer for two values of n4 with λ=632.8nm, d2=40nm, d3=250nm, ε1=2.3, ε2=16+i0.52, μ1=1, μ2=1. LHM layer parameters are ε3=4+i0.001, μ3=3+i0.001 and RHM layer parameters are ε3=2.53, and μ3=1.

Equations (17)

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

ε3(ω)=1ωp2ω2+iγω,
μ3(ω)=1Fω2ω2ωo2+iγω,
R123=|r123|2=|r12+r23exp(2iβ2d2)1+r12r23exp(2iβ2d2)|2,
rij=μjβiμiβjμjβi+μiβj,
βj=±koεjμjN2,
N=kxko=njsin(Θj),
R1234=|r1234|2=|r12+r234exp(2iβ2d2)1+r12r234exp(2iβ2d2)|2,
Ey={a4+ei(kxx+β4z)Cladding Layera3+ei(kxx+β3z)+a3ei(kxxβ3z)Film Layera2+ei(kxx+β2z)+a2ei(kxxβ2z)Metal Layera1ei(kxxβ1z)Substrate Layer,
Hx={a4+β4ωμ4ei(kxx+β4z)Cladding Layerβ3ωμ3(a3+ei(kxx+β3z)+a3ei(kxxβ3z))Film Layerβ2ωμ2(a2+ei(kxx+β2z)+a2ei(kxxβ2z))Metal Layera1β1ωμ1ei(kxxβ1z)Substrate Layer.
Mψ=0,
ψ={a4+,a3+,a3,a2+,a2,a1},
M=[eiβ4(d2+d3)eiβ3(d2+d3)eiβ3(d2+d3)000β4μ4eiβ4(d2+d3)β3μ3eiβ3(d2+d3)β3μ3eiβ3(d2+d3)0000eiβ3d2eiβ3d2eiβ2d2eiβ2d200β3μ3eiβ3d2β3μ3eiβ3d2β2μ2eiβ2d2β2μ2eiβ2d20000111000β2μ2β2μ2β1μ1].
2β3d3+Φ34+Φ321=±2πm,
Φ34=2arctan(iμ3β4μ4β3),
Φ321=2arctan(i(1r32)(1+r32)(1r21e2iβ2d2)(1+r21e2iβ2d2)),
r32=μ2β3μ3β2μ2β3+μ3β2,
r21=μ1β2μ2β1μ1β2+μ2β1,

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