## Strong influence of nonlinearity and surface plasmon excitations on the lateral shift

Optics Express, Vol. 16, Issue 20, pp. 15506-15513 (2008)

http://dx.doi.org/10.1364/OE.16.015506

Acrobat PDF (303 KB)

### 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

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]

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

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

*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. 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]

*p*wave beams in a numerically exact manner, using the invariant imbedding method of wave propagation in stratified nonlinear media [15-17

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]

## 2. Invariant imbedding method

*p*waves of frequencyω and vacuum wave number

*k*

_{0}=

*ω*/

*c*= 2

*π*/

*λ*incident on layered structures where the dielectric permittivity

*ε*varies only in the

*z*direction. We assume that the medium lies in 0≤

*z*≤

*L*and the wave propagates in the

*xz*plane. The complex amplitude of the magnetic filed,

*H*=

*H*(

*z*), satisfies:

*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

*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.

*r*is defined by the wave function in the incident region (

*z*>

*L*):

*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

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]

*exact*ordinary differential equations satisfied by the reflection coefficient r and the imbedding parameter

*w*:

*l*) is obtained by solving the cubic equation

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]

*r*and

*w*:

_{0}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

*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

*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

*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,

*n*<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°.

_{s}*n*+

_{L}*n*, where

_{2}I*n*is the linear refractive index.

_{L}*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°.

*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.

*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 -77.23

*λ*. 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.

*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 -44.5

*λ*and -31.8

*λ*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 311

*λ*. When

*αw*=-0.003, Δ is positive and the maximum of Δ is about 153

*λ*.

*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).

*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 12.4

*λ*(6.0

*λ*) 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*.

*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]

*I*≈1 GW/cm

^{2}. 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]

*I*as small as 0.1 MW/cm

^{2}. These estimates suggest that it is feasible to design sensitive optical switches operating at small powers based on the effect discussed in this paper.

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

*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.

*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

## References and links

1. | H. Raether, |

2. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

3. | E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science |

4. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. |

5. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

6. | K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. |

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. |

8. | J. R. Sambles and R. A. Innes, “A comment on nonlinear optics using surface plasmon-polaritons,” J. Mod. Opt. |

9. | I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. |

10. | N.-C. Panoiu and R. M. Osgood, Jr., “Subwavelength nonlinear plasmonic nanowire,” Nano Lett. |

11. | G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. |

12. | Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. |

13. | X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. |

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. |

15. | G. I. Babkin and V. I. Klyatskin, “Theory of wave propagation in nonlinear inhomogeneous media,” Sov. Phys. JETP |

16. | B. Doucot and R. Rammal, “Invariant-imbedding approach to localization. II. Non-linear random media,” J. Phys. (Paris) |

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 |

18. | R. Bellman and G. M. Wing, |

19. | V. I. Klyatskin, “The imbedding method in statistical boundary-value wave problems,” Prog. Opt. |

20. | R. Rammal and B. Doucot, “Invariant-imbedding approach to localization. I. General framework and basic equations,” J. Phys. (Paris) |

21. | K. Kim, “Reflection coefficient and localization length of waves in one-dimensional random media,” Phys. Rev. B |

22. | K. Kim, D.-H. Lee, and H. Lim, “Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media,” Europhys. Lett. |

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 |

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 |

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 |

26. | R. W. Boyd, |

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. |

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. |

**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

- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- E. Ozbay, "Plasmonics: Merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006). [CrossRef] [PubMed]
- A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005). [CrossRef]
- 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]
- K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007). [CrossRef]
- 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]
- J. R. Sambles and R. A. Innes, "A comment on nonlinear optics using surface plasmon-polaritons," J. Mod. Opt. 35, 791-797 (1988). [CrossRef]
- 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]
- N.-C. Panoiu and R. M. Osgood, Jr., "Subwavelength nonlinear plasmonic nanowire," Nano Lett. 4, 2427-2430 (2004). [CrossRef]
- 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]
- Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, "Subwavelength discrete solitons in nonlinear metamaterials," Phys. Rev. Lett. 99, 153901 (2007). [PubMed]
- X. Yin and L. Hesselink, "Goos-Hänchen shift surface plasmon resonance sensor," Appl. Phys. Lett. 89, 261108 (2006). [CrossRef]
- 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]
- G. I. Babkin and V. I. Klyatskin, "Theory of wave propagation in nonlinear inhomogeneous media," Sov. Phys. JETP 52, 416-420 (1980).
- B. Doucot and R. Rammal, "Invariant-imbedding approach to localization. II. Non-linear random media," J. Phys. (Paris) 48, 527-546 (1987). [CrossRef]
- 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]
- R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).
- V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994). [CrossRef]
- R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987). [CrossRef]
- K. Kim, "Reflection coefficient and localization length of waves in one-dimensional random media," Phys. Rev. B 58, 6153-6160 (1998). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, 2003).
- 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]
- T. E. O. Screen, K. B. Lawton, G. S. Wilson, N. Dolney, R. Ispasoiu, T. GoodsonIII, 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]

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