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Optics Express

  • Editor: Martijn de Sterke
  • Vol. 16, Iss. 20 — Sep. 29, 2008
  • pp: 15506–15513
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Strong influence of nonlinearity and surface plasmon excitations on the lateral shift

Kihong Kim, D. K. Phung, F. Rotermund, and H. Lim  »View Author Affiliations


Optics Express, Vol. 16, Issue 20, pp. 15506-15513 (2008)
http://dx.doi.org/10.1364/OE.16.015506


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Abstract

When surface plasmons are excited at a metal-dielectric interface, the electromagnetic field takes a very large value near the interface. If the dielectric is a nonlinear Kerr medium, then the effect of nonlinearity can be greatly amplified due to the field enhancement. In this paper, we calculate the lateral shift of p wave beams incident on metal-dielectric multilayer systems in the Otto configuration in a numerically exact manner, using the invariant imbedding method of wave propagation in nonlinear stratified media. In the linear case, we find that the lateral shift becomes very large at the incident angles where the surface plasmons are excited. As the nonlinearity is turned on, the value of the lateral shift changes rapidly. We find that even a small change of the intensity of the incident wave can cause a huge change of the lateral shift. We propose that this phenomenon can be applied to designing precise optical switches operating at small powers.

© 2008 Optical Society of America

1. Introduction

During the last decade, there has been a tremendous increase of interest in surface plasmons, which are waves that propagate along the surface of an electrical conductor. Due to their applicability to such diverse areas as nanotechnology, photonics, spectroscopy and sensing, researches in surface plasmons have been conducted in many scientific disciplines including physics, chemistry, biology and electrical engineering [1

1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

-7

7. X. D. Hoa, A. G. Kirk, and M. Tabrizian, “Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress,” Biosens. Bioelectron. 23, 151–160 (2007). [CrossRef] [PubMed]

]. As is well-known, when surface plasmons are excited at a metal-dielectric interface, the electromagnetic field becomes very large near the interface. Therefore, if the dielectric in contact with the metal has optical nonlinearity, its effect can be greatly amplified due to the field enhancement. The second-harmonic generation and other nonlinear optical processes associated with the surface plasmon excitation have been studied for a long time [1

1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

, 4

4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). [CrossRef]

]. However, there have been only a limited number of studies on the nonlinear effects caused by the intensity-dependent refractive index such as the optical bistability [8

8. J. R. Sambles and R. A. Innes, “A comment on nonlinear optics using surface plasmon-polaritons,” J. Mod. Opt. 35, 791–797 (1988). [CrossRef]

-12

12. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007). [PubMed]

].

Recently, it has been pointed out that when surface plasmons are excited, the Goos-Hänchen-type lateral shift of an incident p wave beam, which is the spatial shift of the reflected beam along the interface with respect to the incident beam, can take extremely large values [13

13. X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006). [CrossRef]

, 14

14. L. Chen, Z. Cao, F. Ou, H. Li, Q. Shen, and H. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. 32, 1432–1434 (2007). [CrossRef] [PubMed]

]. In this paper, we combine this effect with the enhancement of optical nonlinearity near the interface and study the strong influence of both nonlinearity and surface plasmon excitations on the lateral shift. We consider a bilayer system made of a nonlinear dielectric layer and a metal layer in the Otto configuration and calculate the reflectance, the phase of the reflection coefficient, the magnetic field profile and the lateral shift of p wave beams in a numerically exact manner, using the invariant imbedding method of wave propagation in stratified nonlinear media [15

15. G. I. Babkin and V. I. Klyatskin, “Theory of wave propagation in nonlinear inhomogeneous media,” Sov. Phys. JETP 52, 416–420 (1980).

-17

17. K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16, 1150–1164 (2008). [CrossRef] [PubMed]

]. We find that even for a small change of the intensity of the incident wave, the value of the lateral shift can change by a very large amount. We propose that this phenomenon can be applied to designing precise optical switches operating at small powers.

Our motivation for this study is in the application of nonlinear media in nano- or micro-structured photonic devices, where the length scale of inhomogeneity is comparable to or smaller than the wavelength. We point out that in these devices, the interference effects between the forward and backward propagating waves are important and the usual approximate theoretical methods such as the slowly varying envelope approximation, which ignores them, are often inadequate [15

15. G. I. Babkin and V. I. Klyatskin, “Theory of wave propagation in nonlinear inhomogeneous media,” Sov. Phys. JETP 52, 416–420 (1980).

]. Therefore it is highly desirable to use an accurate theoretical method such as the method used here for the analysis of nonlinear photonic devices.

2. Invariant imbedding method

We consider the propagation of p waves of frequencyω and vacuum wave number k 0=ω/c= 2π/λ incident on layered structures where the dielectric permittivity ε varies only in the z direction. We assume that the medium lies in 0≤zL and the wave propagates in the xz plane. The complex amplitude of the magnetic filed, H=H(z), satisfies:

d2Hdz21ε(z)dεdzdHdz+[k02ε(z)q2]H=0,
(1)

where q is the x component of the wave vector. We assume that the wave is incident from the region where z>L and is transmitted to the region where z<0. The dielectric permittivity is given by

ε(z)={ε1ifz>LεL(z)+α(z)E(z)2if0zL,ε2ifz<0
(2)

where εL(z) and α(z) are arbitrary complex functions of z and E(z) is the electric field. α(z) describes the strength of the Kerr-type nonlinearity.

The reflection coefficient r is defined by the wave function in the incident region (z>L):

H(z)=νε1[eip(Lz)+r(L)eip(zL)],
(3)

where |ν|2 (≡w) is the electric field intensity of the incident wave and p is the z component of the wave vector. When θ is defined as the incident angle, p and q are given by p=ε1k0cosθ and q=ε1k0sinθ. Using the invariant imbedding method [15

15. G. I. Babkin and V. I. Klyatskin, “Theory of wave propagation in nonlinear inhomogeneous media,” Sov. Phys. JETP 52, 416–420 (1980).

-25

25. K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008). [CrossRef]

], we derive exact ordinary differential equations satisfied by the reflection coefficient r and the imbedding parameter w:

1pdr(l)dl=2iε(l)ε1r(l)i2[ε(l)ε11][1ε1ε(l)tan2θ][1+r(l)]2,
1pdw(l)dl=Im{2ε(l)ε1[ε(l)ε11][1ε1ε(l)tan2θ][1+r(l)]}w(l),
(4)

where ε(l) is obtained by solving the cubic equation

ε(l)=εL(l)+α(l)w(l)[ε12ε(l)21+r(l)2sin2θ+1r(l)2cos2θ].
(5)

A detailed derivation of these equations can be found in [17

17. K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16, 1150–1164 (2008). [CrossRef] [PubMed]

]. They are supplemented with the initial conditions for r and w:

r(0)=ε2ε1cosθε1ε2ε1sin2θε2ε1cosθ+ε1ε2ε1sin2θ,w(0)=w0.
(6)

The constant w0 is chosen such that the final solution for w(L) is the same as the physical input intensity. The fact that in general, there are several w0 values corresponding to a given w(L) value gives a natural explanation of optical multistability. The reflectivity R is given by R=|r(L)|2 and the lateral shift Δ is obtained from

Δ=dΦdq=λ2πε1cosθdΦdθ,
(7)

where Φ is the phase of the reflection coefficient r=ReiΦ.

The invariant imbedding method can also be used in calculating the normalized magnetic field amplitude u(z)=H(z)/v inside the inhomogeneous medium. We consider the u field as a function of both z and l: u=u(z, l). Then we obtain

1pu(z,l)l=iε(l)ε1u(z,l)i2[ε(l)ε11][1ε1ε(l)tan2θ][1+r(l)]u(z,l).
(8)

For a given z(0<z<L), u(z,L) is obtained by integrating this equation, together with Eq. (4), from l=z to l=L using the initial condition u(z, z)=1+r(z).

3. Numerical results

We have calculated the lateral shifts of p wave beams in the Otto configuration. The wave is incident from a prism onto a dielectric-metal bilayer which lies on a dielectric substrate. The linear refractive indices of the prism and the nonlinear dielectric layer are 1.77 and 1.46 respectively. The dielectric constant of the metal layer at 633 nm wavelength is assumed to be -16+i, which is the value for silver. The refractive index of the substrate, ns(=ε2), is assumed to be equal to 1.77, but the main phenomenon discussed in this paper occurs when ns<1.77, too. The thicknesses of the dielectric and metal layers are 320 nm and 50 nm respectively. The critical incident angle in the present case is 55.57°.

αw=24π2c107χ(3)I0.079χ(3)I,
(9)

One important point we want to emphasize is that in the presence of strong external radiation, the nonlinear medium, which was originally uniform, can no longer be considered to be so. The effective dielectric permittivity inside the medium depends on the strength of the electric field, which is highly nonuniform as can be seen in Fig. 2. This is the main mechanism for obtaining greatly enhanced nonlinear effects due to the interplay between surface plasmon excitations and the Kerr effect. One should not assume that the refractive index of the nonlinear medium is replaced uniformly by nL+n2I, where nL is the linear refractive index.

In Fig. 1, we plot the reflectance versus incident angle in the linear (αw=0), focusing (αw=0.003) and defocusing (αw=-0.003) cases when λ=633 nm. The reflectance minimum occurring due to the surface plasmon excitation at θ=62.285° in the linear case shifts to larger (smaller) angles as the focusing (defocusing) nonlinearity is turned on. For αw=±0.003, the minima occur at θ=63.564° and 60.79°.

Fig. 1. Reflectance versus incident angle in the linear (αw=0), focusing (αw=0.003) and defocusing (αw=-0.003) cases when λ=633 nm.

In Fig. 2, we show the spatial distributions of the normalized magnetic field intensity corresponding to the three reflectance minima shown in Fig. 1. The fact that the field is strongly enhanced near the metal-dielectric boundary in all three cases demonstrates that the surface plasmons are indeed excited at those angles. In the defocusing case with αw=-0.003, the field enhancement is somewhat bigger than that in the linear case, whereas in the focusing case with αw=0.003, it becomes smaller.

In Fig. 3, we plot the phase of the reflection coefficient Φ and the lateral shift Δ versus incident angle when αw=0, ±0.0015 and ±0.003. We find that Φ changes very rapidly near the angles where the surface plasmons are excited. This, according to Eq. (7), causes the magnitude of the lateral shift to become very large. In the present configuration, Δ is negative in the linear case and the maximum shift is about -77.23λ. The maximum lateral shift occurs at an angle which is roughly, but not exactly, the same as the angle where R is smallest. We also note that the curve for the lateral shift is much sharper than the reflectance dip.

As αw increases from zero in the focusing case, the position of the lateral shift peak moves to larger angles and the maximum value of the negative shift associated with the surface plasmon excitation decreases monotonically. For αw=0.0015 and 0.003, the maximum shifts are about -44.5λ and -31.8λ respectively. In the defocusing case, as |α|w increases from zero, the position of the lateral shift peak moves to smaller angles and the maximum value of the negative shift initially increases rapidly. As |α|w is increased further, the sign of the lateral shift changes abruptly at a critical value of αw, across which the maximum lateral shift changes from a large negative value to a large positive value. When αw=-0.0015, Δ is negative and the maximum of |Δ| is about 311λ. When αw=-0.003, Δ is positive and the maximum of Δ is about 153λ.

In order to examine more closely how the sign of the lateral shift changes at the critical value of αw, we show the behavior of the phase of the reflection coefficient at the boundary between positive and negative lateral shifts in Fig. 4(a). When αw=-0.00199, the phase increases very rapidly near θ=61.317° as θ increases and the lateral shift is negative, whereas when αw=-0.002, it decreases very rapidly near θ=61.312° as θ increases and the lateral shift is positive. This behavior can be understood easily by examining the real and imaginary parts of the reflection coefficient shown in Fig. 4(b).

Fig. 2. Spatial distributions of the normalized magnetic field intensity corresponding to the three reflectance minima shown in Fig. 1. The vertical dotted line represents the position of the metal-dielectric boundary. A p wave is assumed to be incident from the right side where z>L.
Fig. 3. Dependencies of (a) the phase of the reflection coefficient and (b) the lateral shift on the incident angle when αw=0, ±0.0015, ±0.003.
Fig. 4. (a) Phase of the reflection coefficient and (b) the real and imaginary parts of the reflection coefficient versus incident angle, when αw=-0.00199 and -0.002.

The behavior of the lateral shift peaks shown in Fig. 3(b) suggests that when the incident angle is fixed, the lateral shift will show a highly sensitive dependence on the intensity of the incident wave. In Fig. 5, we plot the lateral shift versus the nonlinearity parameter |α|w when θ=62.285°, 61°, 62° and 63° for both the focusing and defocusing cases. When θ is fixed to 62.285°, which corresponds to the angle where the surface plasmon is excited in the linear case, |Δ| decreases rapidly as |α|w increases from zero, both in the focusing and defocusing cases. We find that the change of Δ when |α|w increases from 0 to 10-4 is about 12.4λ (6.0λ) in the focusing (defocusing) case, which is sufficiently large to be observable. For other incident angles, the dependence of the lateral shift on the nonlinearity parameter can be more complicated. In Figs. 5(d), 5(f) and 5(g), we show some examples of a strong nonmonotonic dependence of Δ on αw.

In order to cause an observable change of the lateral shift, we have found that we need to have the parameter |α|w larger than about 10-4. If we use a highly nonlinear optical glass with χ (3)≈10-12 esu as the nonlinear dielectric [27

27. A. S. L. Gomes, E. L. F. Filho, C. B. de Araújo, D. Rativa, R. E. de Araujo, K. Sakaguchi, F. P. Mezzapesa, I. C. S. Carvalho, and P. G. Kazansky, “Third-order nonlinear optical properties of bismuth-borate glasses measured by conventional and thermally managed eclipse Z scan,” J. Appl. Phys. 101, 033115 (2007). [CrossRef]

], we will be able to see the desired effect for I≈1 GW/cm2. If we use a nonlinear polymer material with χ (3)≈10-8 esu [28

28. T. E. O. Screen, K. B. Lawton, G. S. Wilson, N. Dolney, R. Ispasoiu, T. Goodson III, S. J. Martin, D. D. C. Bradley, and H. L. Anderson, “Synthesis and third order nonlinear optics of a new soluble conjugated porphyrin polymer,” J. Mater. Chem. 11, 312–320 (2001). [CrossRef]

], we will see the effect for I as small as 0.1 MW/cm2. These estimates suggest that it is feasible to design sensitive optical switches operating at small powers based on the effect discussed in this paper.

Finally, we have verified that the effect discussed above still persists when the thickness of the metal layer is much bigger than the value used here. Therefore Joule heating of the metal layer encountered in an experiment using the Kretschmann-Raether configuration will not be a serious problem in our case [8

8. J. R. Sambles and R. A. Innes, “A comment on nonlinear optics using surface plasmon-polaritons,” J. Mod. Opt. 35, 791–797 (1988). [CrossRef]

].

4. Conclusion

For the parameter values used in the present paper, the optical nonlinearity is sufficiently weak that no bistability or multistability effect is observed. We have verified numerically that bistability occurs when αw<-0.0057 in the defocusing case and when αw>0.007 in the focusing case. In the parameter regime where bistability occurs, there will be two physical solutions for the lateral shift as well as for the reflectance and the absorptance. As is well-known, which of these two solutions is observed experimentally depends on the history of the experiment. A detailed analysis of bistability associated with the lateral shift will be discussed in a separate paper.

Fig. 5. Lateral shift versus the nonlinearity parameter |α|w when θ=62.285°, 61°, 62°, 63° for both the focusing (a–d) and defocusing (e–h) cases.

In summary, we have considered the strong influence of optical nonlinearity and surface plasmon excitations on the lateral shift of p wave beams. Using the invariant imbedding method, we have calculated the reflectance and the lateral shift in a numerically exact manner for a bilayer system made of a nonlinear dielectric layer and a metal layer in the Otto configuration. We have found that even for a very small change of the intensity of incident light, the value of the lateral shift can change by a very large amount. We propose that this effect can be applied to designing highly sensitive optical switches operating at small powers.

Acknowledgments

This work has been supported by the Korea Science and Engineering Foundation grant (No. R0A-2007-000-20113-0) funded by the Korean Government.

References and links

1.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

2.

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

3.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef] [PubMed]

4.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). [CrossRef]

5.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]

6.

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58, 267–297 (2007). [CrossRef]

7.

X. D. Hoa, A. G. Kirk, and M. Tabrizian, “Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress,” Biosens. Bioelectron. 23, 151–160 (2007). [CrossRef] [PubMed]

8.

J. R. Sambles and R. A. Innes, “A comment on nonlinear optics using surface plasmon-polaritons,” J. Mod. Opt. 35, 791–797 (1988). [CrossRef]

9.

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002). [CrossRef] [PubMed]

10.

N.-C. Panoiu and R. M. Osgood, Jr., “Subwavelength nonlinear plasmonic nanowire,” Nano Lett. 4, 2427–2430 (2004). [CrossRef]

11.

G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. 97, 057402 (2006). [CrossRef] [PubMed]

12.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007). [PubMed]

13.

X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006). [CrossRef]

14.

L. Chen, Z. Cao, F. Ou, H. Li, Q. Shen, and H. Qiao, “Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides,” Opt. Lett. 32, 1432–1434 (2007). [CrossRef] [PubMed]

15.

G. I. Babkin and V. I. Klyatskin, “Theory of wave propagation in nonlinear inhomogeneous media,” Sov. Phys. JETP 52, 416–420 (1980).

16.

B. Doucot and R. Rammal, “Invariant-imbedding approach to localization. II. Non-linear random media,” J. Phys. (Paris) 48, 527–546 (1987). [CrossRef]

17.

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16, 1150–1164 (2008). [CrossRef] [PubMed]

18.

R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).

19.

V. I. Klyatskin, “The imbedding method in statistical boundary-value wave problems,” Prog. Opt. 33, 1–127 (1994). [CrossRef]

20.

R. Rammal and B. Doucot, “Invariant-imbedding approach to localization. I. General framework and basic equations,” J. Phys. (Paris) 48, 509–526 (1987). [CrossRef]

21.

K. Kim, “Reflection coefficient and localization length of waves in one-dimensional random media,” Phys. Rev. B 58, 6153–6160 (1998). [CrossRef]

22.

K. Kim, D.-H. Lee, and H. Lim, “Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media,” Europhys. Lett. 69, 207–213 (2005). [CrossRef]

23.

K. Kim and D.-H. Lee, “Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas,” Phys. Plasmas 12, 062101 (2005). [CrossRef]

24.

K. Kim and D.-H. Lee, “Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity,” Phys. Plasmas 13, 042103 (2006). [CrossRef]

25.

K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method,” Phys. Rev. B 77, 024203 (2008). [CrossRef]

26.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003).

27.

A. S. L. Gomes, E. L. F. Filho, C. B. de Araújo, D. Rativa, R. E. de Araujo, K. Sakaguchi, F. P. Mezzapesa, I. C. S. Carvalho, and P. G. Kazansky, “Third-order nonlinear optical properties of bismuth-borate glasses measured by conventional and thermally managed eclipse Z scan,” J. Appl. Phys. 101, 033115 (2007). [CrossRef]

28.

T. E. O. Screen, K. B. Lawton, G. S. Wilson, N. Dolney, R. Ispasoiu, T. Goodson III, S. J. Martin, D. D. C. Bradley, and H. L. Anderson, “Synthesis and third order nonlinear optics of a new soluble conjugated porphyrin polymer,” J. Mater. Chem. 11, 312–320 (2001). [CrossRef]

OCIS Codes
(190.1450) Nonlinear optics : Bistability
(190.5940) Nonlinear optics : Self-action effects

ToC Category:
Nonlinear Optics

History
Original Manuscript: July 9, 2008
Revised Manuscript: September 12, 2008
Manuscript Accepted: September 12, 2008
Published: September 17, 2008

Citation
Kihong Kim, D. K. Phung, F. Rotermund, and H. Lim, "Strong influence of nonlinearity and surface plasmon excitations on the lateral shift," Opt. Express 16, 15506-15513 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15506


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References

  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
  3. E. Ozbay, "Plasmonics: Merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006). [CrossRef] [PubMed]
  4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005). [CrossRef]
  5. S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005). [CrossRef]
  6. K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007). [CrossRef]
  7. X. D. Hoa, A. G. Kirk, and M. Tabrizian, "Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress," Biosens. Bioelectron. 23, 151-160 (2007). [CrossRef] [PubMed]
  8. J. R. Sambles and R. A. Innes, "A comment on nonlinear optics using surface plasmon-polaritons," J. Mod. Opt. 35, 791-797 (1988). [CrossRef]
  9. I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, "Single-photon tunneling via localized surface plasmons," Phys. Rev. Lett. 88, 187402 (2002). [CrossRef] [PubMed]
  10. N.-C. Panoiu and R. M. Osgood, Jr., "Subwavelength nonlinear plasmonic nanowire," Nano Lett. 4, 2427-2430 (2004). [CrossRef]
  11. G. A. Wurtz, R. Pollard, and A. V. Zayats, "Optical bistability in nonlinear surface-plasmon polaritonic crystals," Phys. Rev. Lett. 97, 057402 (2006). [CrossRef] [PubMed]
  12. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength discrete solitons in nonlinear metamaterials," Phys. Rev. Lett. 99, 153901 (2007). [PubMed]
  13. X. Yin and L. Hesselink, "Goos-Hänchen shift surface plasmon resonance sensor," Appl. Phys. Lett. 89, 261108 (2006). [CrossRef]
  14. L. Chen, Z. Cao, F. Ou, H. Li, Q. Shen, and H. Qiao, "Observation of large positive and negative lateral shifts of a reflected beam from symmetrical metal-cladding waveguides," Opt. Lett. 32, 1432-1434 (2007). [CrossRef] [PubMed]
  15. G. I. Babkin and V. I. Klyatskin, "Theory of wave propagation in nonlinear inhomogeneous media," Sov. Phys. JETP 52, 416-420 (1980).
  16. B. Doucot and R. Rammal, "Invariant-imbedding approach to localization. II. Non-linear random media," J. Phys. (Paris) 48, 527-546 (1987). [CrossRef]
  17. K. Kim, D. K. Phung, F. Rotermund, and H. Lim, "Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses," Opt. Express 16, 1150-1164 (2008). [CrossRef] [PubMed]
  18. R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).
  19. V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994). [CrossRef]
  20. R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987). [CrossRef]
  21. K. Kim, "Reflection coefficient and localization length of waves in one-dimensional random media," Phys. Rev. B 58, 6153-6160 (1998). [CrossRef]
  22. K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005). [CrossRef]
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