## Theoretical analysis of obliquely excited surface plasmon self-interference |

Optics Express, Vol. 21, Issue 15, pp. 18572-18581 (2013)

http://dx.doi.org/10.1364/OE.21.018572

Acrobat PDF (1193 KB)

### Abstract

We present the theoretical analysis of surface plasmon polaritons induced by a tightly focused light beam at oblique incidence. Firstly, we propose a geometrical model to explain the evolution of SPPs effect as light deviating from normal incidence, and introduce a concept of *critical oblique angle* (*θ _{co}*) which is one of the key factors affecting the stability, efficiency and lateral resolution of SPPs. Secondly, the integral expressions for the transmitted SPP field excited by a linearly polarized vortex beam are derived, using angular spectrum representation and rotation matrix trans-formation, for the oblique directions as parallel and perpendicular to polarization plane. An interesting finding is that the system completely goes out of SPP self-interference resonance at an incident angle smaller than

*θ*at parallel obliquity, while larger than

_{co}*θ*at perpendicular obliquity.

_{co}© 2013 Optical Society of America

## 1. Introduction

2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

3. G. E. Cragg and P. T. C. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett. **25**(1), 46–48 (2000). [CrossRef] [PubMed]

4. E. Chung, D. K. Kim, and P. T. C. So, “Extended resolution wide-field optical imaging: objective-launched standing-wave total internal reflection fluorescence microscopy,” Opt. Lett. **31**(7), 945–947 (2006). [CrossRef] [PubMed]

5. B. Bailey, D. L. Farkas, D. L. Taylor, and F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature **366**(6450), 44–48 (1993). [CrossRef] [PubMed]

7. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. **85**(24), 5833 (2004). [CrossRef]

8. H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B **15**(4), 1381–1386 (1998). [CrossRef]

9. A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J.-C. Weeber, A. Dereux, G. P. Wiederrecht, and L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. **32**(17), 2535–2537 (2007). [CrossRef] [PubMed]

*et al*. [10

10. P. S. Tan, X. C. Yuan, J. Lin, Q. Wang, and R. E. Burge, “Analysis of surface Plasmon interference pattern formed by optical vortex beams,” Opt. Express **16**(22), 18451–18456 (2008). [CrossRef] [PubMed]

11. D. Ganic, X. S. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express **11**(21), 2747–2752 (2003). [CrossRef] [PubMed]

12. Q. W. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. **31**(11), 1726–1728 (2006). [CrossRef] [PubMed]

## 2. Theoretical analysis

*ε*) deposited on a glass substrate (

_{2}*ε*), with air (

_{1}*ε*)above, is illuminated by a tightly focused linearly-polarized vortex beam through the glass substrate. Vortex beam can be generated by passing a plane wave through an azimuthally mo- dulated phase mask and getting a phase of

_{3}*l*is topological charge. The tight focalization of light is achieved by the use of an aplanatic objective lens with large numerical aperture (N.A.>1). At normal incidence, we choose z axis of the Cartesian coordinate system to coincide with the optical axis of the light beam and the metal/glass interface to be located at z = 0, i.e. the

*xoy*plane, as depicted in Fig. 1.

*θ*excite two counter-propagating SPP waves of

_{sp}*k*, as shown in Fig. 2(a). However, if the focused light beam deviates from the normal, supposing to tilt towards x axis in xoz plane [Fig. 2(b)], the oblique incidence gives rise to intensity inequality of counter-propagating waves due to the intensity law to ensure energy conservation across each interface, as well as the radial intensity distribution of a Gaussian beam. Subsequently, the interference fringe contrast will be reduced. Besides, the oblique incidence of the focused light beam also causes asymmetrical distribution of the incident angle. Fig. 3(a) depicts a case in incident plane, and by simple calculation we have

_{sp}*θ*50° and

_{A}=*θ*60°. In this example, two different incidentangle of light beams excite SPP waves with different initial phases, as we can observe in Fig. 3(b). So, the SPP interference field undergoes a phase shift, leading to the spatial shift of interference fringes. Summing up the above two aspects, we deduce that oblique incidence causes a deterioration in the SPP interference pattern.

_{B}=*critical oblique angle θ*. Via the analysis of solid geometrical involvement of optical rays in a focused beam deviating from normal incidence as shown in Fig. 2(c), we can prove that

_{co}*θ*is decided by:Here,

_{co}*θ*

_{max}= sin

^{−1}(NA/n

_{1}) is the maximum convergence angle of the focused light beam in glass.

*θ*. If the focused beam of light is oblique in the plane perpendicular to the polarization, the interference will be significantly affected when the value of oblique angle is in the vicinity of

_{co}*θ*. So far, we’ve assumed that SPR occurs only at the very angle of

_{co}*θ*. However, the SPPs are actually excited by a collection of waves with incident angles centered around

_{sp}*θ*, having angular full width at half maximum (FWHM) of Δ

_{sp}*θ*. Consequently, the interference actually still exists at an incident angle, to some extent, larger than

_{sp}*θ*. We can infer that the oblique angle would have exceeded

_{co}*θ*before the system completely went out of SPP self-interference resonance.

_{co}13. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

14. E. Wolf, “Electromagnetic diffraction in optical systems. I. an integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 349–357 (1959). [CrossRef]

**E**

*, entirely polarized along x-axis, is formed by a collimated fundamental Gaussian laser beam with filling factor of*

_{inc}*f*

_{0}= 1, then the incoming focused far-field can be denoted in terms of the spatial frequencies aswhere

*E*is the incident field amplitude, and

_{0}*k*,

_{x}*k*are the transverse components of the wavevector

_{y}**k**.

*f*is the focal length of the lens,

**n**

*is the unit vector in*

_{ρ}*ρ*(radial distance from the optical axis) direction of a cylindrical coordinate system, whereas

**n**

*the unit vector in the direction of increasing zenith angle of a spherical coordinate system.*

_{θ}*t*(

^{p}*k*) is the Fresnel transmission coefficients of the p-polarized light for the glass/metal/air configuration which is given by [16]

_{zj}*d*is the thickness of the gold layer,

*i*,

*j*= 1, 2, 3, represents the individual medium, i.e. 1 = glass, 2 = metal and 3 = air.

*k*is the longitudinal component of the wavevector

_{zj}**k**in medium

*j*.

*x*,

*y*,

*z*), then the incidence obliquity is equivalent to the reverse rotation of the metal film to a new Cartesian coordinate (

*x′*,

*y′*,

*z′*) with film surface being in the

*x′oy′*plane and its normal along the z′ axis. Correspondingly, the expression for the transmitted field vector near the focal plane will change to be

*x*,

*y*,

*z*) to Cartesian coordinate (

*x′*,

*y′*,

*z′*). For rotation about an arbitrary axis passing through origin o, the transformation of position vector and incident light wavevector in medium 1 (glass) can be expressed by equations as followswhere

*R*is the rotation matrix. For simplicity, we consider only two particular cases

*Case1*: the focused light beam tilts at a counter-clockwise angle of

*α*in the polarization plane, i.e. the metal film rotate -

*α*about y axis, the three-dimensional rotation matrix can be denoted as

**n**

*can be expressed in terms of the Cartesian unit vectors*

_{ρ}**x**,

**y**,

**z**and the spatial frequencies, in combination with Eqs. (6) and (7), the unit vector

**n**

*can be derived as*

_{ρ’}*x*,

*y*,

*z*). As the surface plasmon interference is generated by the diametrically opposed plane waves, only the vertical component of the transmitted field on the metal surface need to be considered and calculated by choosing

*z*′

*= d*as

*k*,

_{x}*k*by a spherical integration over

_{y}*θ*,

*CaseII*: the focused light beam tilts at a positive angle of

*β*perpendicular to the polarization plane, i.e. the metal film rotate by -

*β*about x axis, the rotation matrix will be

## 3. Numerical calculation

*ε*, gold film

_{1}*ε*and air

_{2}*ε*are 2.31, −5.28 + 2.04

_{3}*i*(at the wavelength of 532nm) [10

10. P. S. Tan, X. C. Yuan, J. Lin, Q. Wang, and R. E. Burge, “Analysis of surface Plasmon interference pattern formed by optical vortex beams,” Opt. Express **16**(22), 18451–18456 (2008). [CrossRef] [PubMed]

*l*= 1. The oil-immersion objective lens has a numerical aperture of 1.4, corresponding to a

*θ*of 68°, well beyond the SPP resonant angles

_{max}*θ*~47°. Under normal illumination, vertical transmitted field strength︱

_{sp}*E*′

*(*

_{t}*x′, y′*)

*︱*

_{z′}*on the film surface has a symmetric, two-lobe pattern. When focused vortex beam is obliquely incident in the plane parallel to its polarization (i.e. in xoz plane), the interference pattern still maintains symmetry about the x-axis, but no longer it does about the y-axis, as shown in Fig. 4(a). Figure 5(a) is the distributions of normalized surface plasmon intensity along midline in the x-direction under the illumination of focused beam at different incident angles. If we tilt incident vortex beam in the direction perpendicular to its polarization (i.e. in the yoz plane), the interference pattern is still symmetric about the y-axis, but not about the x-axis anymore, as in Fig. 4(b), and a coma-shaped interference pattern is formed. Figure 5(b) shows the distributions of normalized surface plasmon field intensity along the central line in y-direction under the illumination of focused beam at different incident angles.*

^{2}*θ*at parallel obliquity, while larger than

_{co}*θ*at perpendicular obliquity. Here, we understand that going out of SPP self-interference resonance will bring about a significant decrease in peak intensity and sudden increase in full width at half maximum (FWHM) of the transmitted focal field. To carry out numerical calculation, let us consider a glass/silver/air structure illuminated by a 532nm vortex beam with

_{co}*l*= 1. With the same parameters as above except for the dielectric constant of silver

*ε*−11.24 + 0.30

_{2}=*i*[17

17. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

*θ*to be 43.58°. Using Eq. (1), we get

_{sp}*θ*≈31°. On the other hand, we use Eqs. (12) and (17) to calculate the normalized central maximum intensity and FWHM of SPP main peak at different oblique angles, and the results are shown in Figs. 6 and 7.

_{co}*θ*) at parallel obliquity and 50°(>

_{co}*θ*) at perpendicular obliquity, respectively. Therefore, the numerical calculation results agree well with the predictions based on geometrical model, which shows the validity of our theory.

_{co}*θ*) for oblique incidence corresponds to the condition that the system completely goes out of SPP self-interference resonance.

_{l}*θ*is decided by various factors as

_{l}*θ*, dispersion properties of SPP, polarization of incident light beam and it’s oblique direction. When considering the differential oblique angle

_{co}*Δθ*, we can’t neglect the contributions from the SPR components whose detuning angles are larger than the FWHM. In addition to

_{co}*Δθ*, the polarization of incident light beam coupled with dispersion properties of SPP may have significant impact on the deviation between

_{co}*θ*and

_{l}*θ*. Therefore, a considerable discrepancy in oblique angle presents in the above calculation.

_{co}## 4. Conclusion

## Acknowledgments

## References and links

1. | H. Raether, |

2. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

3. | G. E. Cragg and P. T. C. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett. |

4. | E. Chung, D. K. Kim, and P. T. C. So, “Extended resolution wide-field optical imaging: objective-launched standing-wave total internal reflection fluorescence microscopy,” Opt. Lett. |

5. | B. Bailey, D. L. Farkas, D. L. Taylor, and F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature |

6. | H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Two-dimensional optics with surface plasmon polaritons,” Appl. Phys. Lett. |

7. | T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. |

8. | H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B |

9. | A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J.-C. Weeber, A. Dereux, G. P. Wiederrecht, and L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. |

10. | P. S. Tan, X. C. Yuan, J. Lin, Q. Wang, and R. E. Burge, “Analysis of surface Plasmon interference pattern formed by optical vortex beams,” Opt. Express |

11. | D. Ganic, X. S. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express |

12. | Q. W. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. |

13. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

14. | E. Wolf, “Electromagnetic diffraction in optical systems. I. an integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

15. | L. Novotny and B. Hetch, |

16. | J. A. Kong, |

17. | P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B |

18. | T. M. Hsu, C. C. Chang, Y. F. Hwang, and K. C. Lee, “The Dielectric Function of Silver by ATR Technique,” Chin. J. Phys. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(310.6860) Thin films : Thin films, optical properties

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: June 10, 2013

Revised Manuscript: July 14, 2013

Manuscript Accepted: July 19, 2013

Published: July 26, 2013

**Citation**

Wendong Zou, Pinbo Huang, Wenjuan Ma, and Fei Guo, "Theoretical analysis of obliquely excited surface plasmon self-interference," Opt. Express **21**, 18572-18581 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-15-18572

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### References

- H. Raether, Surface-Plasmons on Smooth and Rough Surfaces and on Grating, Springer Tracts in Modern Physics (Springer Berlin, 1988).
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]
- G. E. Cragg, P. T. C. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett. 25(1), 46–48 (2000). [CrossRef] [PubMed]
- E. Chung, D. K. Kim, P. T. C. So, “Extended resolution wide-field optical imaging: objective-launched standing-wave total internal reflection fluorescence microscopy,” Opt. Lett. 31(7), 945–947 (2006). [CrossRef] [PubMed]
- B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366(6450), 44–48 (1993). [CrossRef] [PubMed]
- H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, F. R. Aussenegg, “Two-dimensional optics with surface plasmon polaritons,” Appl. Phys. Lett. 81(10), 1762 (2002). [CrossRef]
- T. Nikolajsen, K. Leosson, S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833 (2004). [CrossRef]
- H. Kano, S. Mizuguchi, S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B 15(4), 1381–1386 (1998). [CrossRef]
- A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J.-C. Weeber, A. Dereux, G. P. Wiederrecht, L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32(17), 2535–2537 (2007). [CrossRef] [PubMed]
- P. S. Tan, X. C. Yuan, J. Lin, Q. Wang, R. E. Burge, “Analysis of surface Plasmon interference pattern formed by optical vortex beams,” Opt. Express 16(22), 18451–18456 (2008). [CrossRef] [PubMed]
- D. Ganic, X. S. Gan, M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11(21), 2747–2752 (2003). [CrossRef] [PubMed]
- Q. W. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31(11), 1726–1728 (2006). [CrossRef] [PubMed]
- B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]
- E. Wolf, “Electromagnetic diffraction in optical systems. I. an integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959). [CrossRef]
- L. Novotny and B. Hetch, Principle of Nano-optics (Cambridge U. Press, 2006).
- J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge MA, 2005).
- P. B. Johnson, R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
- T. M. Hsu, C. C. Chang, Y. F. Hwang, K. C. Lee, “The Dielectric Function of Silver by ATR Technique,” Chin. J. Phys. 21(1), 26–32 (1983).

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