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 (θco) 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 θco at parallel obliquity, while larger than θco at perpendicular obliquity.

© 2013 Optical Society of America

1. Introduction

Surface plasmon polaritons (SPPs) are excited by evanescent wave in incident light of photons which interact with free electrons in a metal and generate collective oscillation propagating along the metal/dielectric interface [1,2]. Consequently, they possess the character of not only ultra-high resolution beyond the diffraction limit, but substantially enhancement of the evanescent field intensity. The surface plasmon interference can further increase the lateral resolution as it was proposed to use in the standing wave surface plasmon resonance fluorescence(SW-SPRF) microscopy [3,4]. Due to their extraordinary nature, SPPs open up new exciting applications in sub-diffraction imaging system as well as plasmonic devices such as sub-wavelength waveguide components, modulators, switches and so on [57].

The surface plasmon interference can be excited by various approaches. One convenient and straightforward type is the so-called surface plasmon self-interference which is excited by a focused laser beam at normal incidence on a metal film without any protrusions and holes, while the intensity distribution on the metal surface is partly dominated by interference between counter-propagating plasmons [8,9]. Recently, P. S. Tan et al. [10] reported the analysis of surface plasmon interference pattern formed by a focused optical vortex beam. They've demonstrated that optical vortex beams is a simple methodology to excited SPP standing wave, and can be served as resolution enhancement technique for sub-diffraction imaging system.

From the viewpoint of image quality evaluation as well as instrument engineering, it is scientifically valuable for us to investigate the problems as piston, tilt, aberration, jitter, and so on, which we may face in a practical SPP imaging system. To do this, theoretically under-standing the evolution of SPP interference in the event of beam deviating from normal incidence, which corresponds to an off-axis condition, is fundamentally important. To our knowledge, until recently no report about study on these problems has been found elsewhere. Therefore, this paper focuses on the case of tilt incidence of focused beam on a glass/metal/air structure, as a basis and starting point for further study. In this work, we consider a linearly polarized vortex beam. Vortex beam has an extraordinary intensity profile that is a ring of primary intensity accompanied by concentric outer rings of diminishing intensity [11,12], so it can be used to overcome the problem of the SPP interference ambiguous. To reach our goal, we manage to investigate via two different approaches as geometrical model and angular spectrum representation which have shown a good consistency.

2. Theoretical analysis

Let us suppose that a metallic thin film (ε2) deposited on a glass substrate (ε1), with air (ε3)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 exp(ilφ) where 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.

 

Fig. 1 Schematic diagram of SPPs excitation by a tightly focused vortex beam obliquely incident on a metal film.

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Qualitatively, two sets of diametrically opposed plane waves with incident angles of ± θsp excite two counter-propagating SPP waves of ksp, 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 θA = 50° and θB = 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.

 

Fig. 2 Schematic of focused light beam converging towards the geometric focus o on the metal/glass interface located at the z = 0 plane. The cross-sectional circle in dash line represents the light rays with cone semi-angle being equal to SPPs resonant angle θsp in the vicinity of the focal plane, while the circle in solid line represents light rays with cone semi-angle being equal to the maximum convergence angle θmax. Cn and Co are the positions of the optical axis on the cross-sectional plane at normal and oblique incidence, respectively. (a) normal incidence; (b)oblique incidence in the xoz plane;(c) 3-D plot of a focused beam which is geometrically at critical obliquity for SPPs excitation.

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Fig. 3 (a) Schematic of oblique incidence of light focused on a gold film deposited on glass, θmax = 55°, α = 5° ; (b)The SPP waves excited by light with incident angle of θA = 50° and θB = 60°, respectively.

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It is obvious that the SPP wave propagating in the same direction as obliquity will detune from SPR first and as a result the self-interference in this orientation will disappear. With the increase of incident angle, the cone of light exciting SPP interference will gradually shrink down to two conical side stripes passing through sector arc A1AA2 and B1BB2, as illustrated in Fig. 2(b), until eventually self-interference in the orientation perpendicular to the obliquity also disappears when the beam is tilted up to a critical angle so that the radical axis of two circles moves upward to coincide with the diameter AB of dash circle (i.e. the A1AA2 and B1BB2 degenerate to point A and B, respectively) and all the light rays deflect away from the lower half of resonant circular arc ALB. We define this angle as critical oblique angle θco. 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 θco is decided by:

θco=cos1(cosθmax/cosθsp)
Here, θmax = sin−1(NA/n1) is the maximum convergence angle of the focused light beam in glass.

In the above analysis, we ignore the influences from the polarization of light. It is well-known that SPPs are only excited by the p-polarized light, linear polarization of focused beam of light produces anisotropy in the interference pattern which is azimuthally modulated. Under normal illumination, the interference effect is strongest in the radial orientation parallel to the polarization, whereas disappears in the perpendicular radial orientation. Under oblique illumination, if the incident plane is parallel to the polarization, the interference will be significantly affected by small angle obliquity, so the system would have completely gone out of SPP self-interference resonance before the oblique angle reached the value of θco. 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 θco. So far, we’ve assumed that SPR occurs only at the very angle of θsp. However, the SPPs are actually excited by a collection of waves with incident angles centered around θsp, having angular full width at half maximum (FWHM) of Δθsp. Consequently, the interference actually still exists at an incident angle, to some extent, larger than θco. We can infer that the oblique angle would have exceeded θco before the system completely went out of SPP self-interference resonance.

To quantitatively describe the oblique excitation of SPP self-interference near metal surface, we follow the vectorial Debye integral theory established by Richards and Wolf [13,14], and solve for the transmitted focal field distribution by the use of the integral angular spectrum representation.

We assume that the optical vortex beam field Einc, entirely polarized along x-axis, is formed by a collimated fundamental Gaussian laser beam with filling factor of f0 = 1, then the incoming focused far-field can be denoted in terms of the spatial frequencies as

Einc(kx,ky)=Eincx=E0exp(-kx2+ky2k12NA2n12+ilϕ)x
where E0 is the incident field amplitude, and kx, ky are the transverse components of the wavevector k.

By taking into account of multiple reflections of three layers configuration, integral formulation for the transmitted field vector near the metal surface under normal illumination of a focused light beam can be expressed as [15]

Et(x,y,z)=ifeik1f2πkxkytp(kzj)(Eincnρ)nθkz1/k1kz1exp[i(kxx+kyy+kz3z)]dkxdky
where f is the focal length of the lens, nρ is the unit vector in ρ(radial distance from the optical axis) direction of a cylindrical coordinate system, whereas nθ the unit vector in the direction of increasing zenith angle of a spherical coordinate system. tp(kzj) is the Fresnel transmission coefficients of the p-polarized light for the glass/metal/air configuration which is given by [16]
tp(kzj)=4exp[i(kz2kz3)d](1+p12)(1+p23)[1+r12r23exp(i2kz2d)]pij=εikzjεjkzi,rij=1pij1+pij,kzj=kj2(kx2+ky2)
here, 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. kzj is the longitudinal component of the wavevector k in medium j.

Now we consider the situation of oblique incidence. For the sake of calculation convenience, we keep the incident light beam axis to be fixed along the z axis of Cartesian coordinate (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

Et(x,y,z)=ifeik1f2πkxkytp(kzj)(Eincnρ)nθkz1/k1kz1exp[i(kxx+kyy+kz3z)]dkxdky

To calculate Eq. (5), we need to employ the rotation matrix to describe the rotational transformation from Cartesian coordinate (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 follows

[xyz]=R[xyz],[kxkykz1]=R[kxkykz1]
where 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

R=[cosα0sinα010sinα0cosα]

It is worth of noting that the sign and its physical meaning of oblique angle still comply with the symbol convention in geometrical optics. As the unit vector nρ can be expressed in terms of the Cartesian unit vectors x, y, z and the spatial frequencies, in combination with Eqs. (6) and (7), the unit vector nρ’ can be derived as

nρ=kxkx2+ky2x+kykx2+ky2y=kxcosαkx2+ky2x+kykx2+ky2ykxsinαkx2+ky2z

Hence,

Eincnρ=Einckxcosαkx2+ky2

To perform variable substitution for the double integral of Eq. (5), we calculate Jacobian determinant

|(kx,ky)(kx,ky)|=cosα+kxkz1sinα

Substitute Eqs. (9) and (10) into Eq. (5), we transform the integration back to be performed in Cartesian coordinate (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

Et(x,y)z=ifeik1f2πkxkyEinc(kx,ky)tp[kzj(kx,ky)]exp(ikz3d)(kxcosαkz1sinα)×exp{i[(kxcosαkz1sinα)x+kyy]}cosαkz1/k1k1kz1dkxdky

We replace the planar integration over kx , ky by a spherical integration over θ,ϕ in cylindrical coordinate

Et(x,y)z=ifk1eik1f2π0θmax02πEinc(θ,ϕ)tp[kzj(θ,ϕ)]exp(ikz3d)(sinθcosϕcosαcosθsinα)×exp{ik1[(sinθcosϕcosαcosθsinα)x+sinθsinϕy]}cosαcosθsinθdθdϕ
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

R=[1000cosβsinβ0sinβcosβ]

We can correspondingly obtain the following relation

Eincnρ=Einckxkx2+ky2

In this case, Jacobian determinant will change to be

|(kx,ky)(kx,ky)|=cosβkykz1sinβ

So that the vertical component of transmitted field on the metal surface is

Et(x,y)z=ifeik1f2πkxkyEinc(kx,ky)tp[kzj(kx,ky)]exp(ikz3d)×kxexp{i[(kxx+(kycosβ+kz1sinβ)y]}kz1/k1k1kz1dkxdky

The angular spectrum representation of the focal field can be expressed in the cylindrical coordinate as

Et(x,y)z=ifk1eik1f2π0θmax02πEinc(θ,ϕ)tp[kzj(θ,ϕ)]exp(ikz3d)sin2θcosθ×exp{ik1[sinθcosϕx+(sinθsinϕcosβ+cosθsinβ)y]}cosϕdθdϕ

It’s noteworthy that we only consider tilt incidence of a focused paraxial optical field on a glass/film interface. For more complicated cases, we need to further investigate on the situations of off-axial focusing, but these are beyond the scope of this paper.

3. Numerical calculation

For simulation, we consider a glass/gold/air three-layer structure and assume that the dielectric constant of glass ε1, gold film ε2 and air ε3 are 2.31, −5.28 + 2.04 i (at the wavelength of 532nm) [10] and 1, respectively. The gold film is 45nm thick and illuminated by a 532nm linearly polarized vortex beam with topological charge l = 1. The oil-immersion objective lens has a numerical aperture of 1.4, corresponding to a θmax of 68°, well beyond the SPP resonant angles θsp~47°. Under normal illumination, vertical transmitted field strength︱Et(x′, y′)z′2 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.

 

Fig. 4 SPP interference pattern maps on the Au film surface excited by a x-polarized 532nm vortex beam at (a) 10° of parallel obliquity and (b) 30° of perpendicular obliquity.

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Fig. 5 SPP intensity profiles (a) along x-axis at different angle of parallel obliquity and (b) along y-axis at different angle of perpendicular obliquity.

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From Figs. 4 and 5, we can clearly see the reduction in visibility of the interference fringes under oblique illumination. Besides, we can also see that the main peak of SPPs deviates from the center, which is due to the spatial shift of SPP interference fringes.

According to the theoretical analysis in previous subsection, the system completely goes out of SPP self-interference resonance at an incident angle smaller than θco at parallel obliquity, while larger than θco 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 l = 1. With the same parameters as above except for the dielectric constant of silver ε2 = −11.24 + 0.30i [17] [18] and supposing the objective lens numerical aperture N.A. = 1.19, we calculate θsp to be 43.58°. Using Eq. (1), we get θco≈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.

 

Fig. 6 The normalized peak intensity of SPPs self-interference on Ag film excited by a 532nm vortex beam at different angle of (a) parallel and (b) perpendicular obliquity.

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Fig. 7 The FWHM of the SPPs self-interference intensity profile on Ag film excited by a 532nm vortex beam at different angle of (a) parallel and (b) perpendicular obliquity.

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It is worth of noting that the impacts of going out of SPP self-interference resonance on FWHM are different from that on the peak intensity of the transmitted focal field. In the former case a step jump process must happen, while it’s more like an analog attenuation inthe latter situation, and this difference can also be observed through comparison of Figs. 6 and 7. Therefore, the FWHM curve is more suitable for characterizing the disappearance of interference effect.

From Figs. 7(a) and (b), one can see that the turning point for FWHM curve of SPP intensity profile is about 16°(<θco) at parallel obliquity and 50°(>θco) 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.

The limiting angle (θl) for oblique incidence corresponds to the condition that the system completely goes out of SPP self-interference resonance. θl is decided by various factors as θco, dispersion properties of SPP, polarization of incident light beam and it’s oblique direction. When considering the differential oblique angle Δθco, we can’t neglect the contributions from the SPR components whose detuning angles are larger than the FWHM. In addition to Δθco, the polarization of incident light beam coupled with dispersion properties of SPP may have significant impact on the deviation between θl and θco . Therefore, a considerable discrepancy in oblique angle presents in the above calculation.

Through further comparison of the results from the calculations, we can see that oblique incidence impacts on the SPP pattern anisotropically, depending on its orientation with respect to the polarization of light beam. To maintain effective excitation of SPPs, the incident directions of the focused light beam should be limited within an elliptical cone of divergence centered around the normal of dielectric/metal interface.

4. Conclusion

In conclusion, we both qualitatively and quantitatively analyze the oblique excitation of the SPP self-interference. We propose a geometrical model and introduce a concept of critical oblique angle for the first time. This angle, in together with other factors as the light polarization, angular FWHM of SPR, determine the stability, efficiency and lateral resolution of SPP self-interference. By the use of angular spectrum representation, we derive rigorous formulas for the vertical component of transmitted SPP interference field obliquely excited by a linearly polarized vortex beam. Numerical calculation results of analytic equations are in a good agreement with the theoretical model inferences. The results based on our theory reveal that SPPs excited by linearly polarized beam of light is more sensitive to the obliquity in the plane parallel to light polarization than that perpendicular to. We can preliminarily predict that the tilt incidence will cause error, influence the lateral resolution and intensity distribution which affect the perfect SPP image formation.

The properties of SPP self-interference under oblique-incidence excitation can be used to investigate optical distortion, aberration in a practical SPP imaging instrumentation. The theory we propose here can provide guidance for experimental manipulation as well as the design of SPP interference microscope in future.

Acknowledgments

This work is supported by National Natural Science Foundation of China under grant No. 51241005, Natural Science Foundation of Jiangxi Province, China (Grant No. 2008GZW0011, No. 20122BAB202010), The Industry Project of Science & Technology Pillar Program of Jiangxi Province, China (Grant No. 20122BBE500040).

References and links

1. H. Raether, Surface-Plasmons on Smooth and Rough Surfaces and on Grating, Springer Tracts in Modern Physics (Springer Berlin, 1988).

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]  

6. H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Two-dimensional optics with surface plasmon polaritons,” Appl. Phys. Lett. 81(10), 1762 (2002). [CrossRef]  

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]  

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]  

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]  

15. L. Novotny and B. Hetch, Principle of Nano-optics (Cambridge U. Press, 2006).

16. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge MA, 2005).

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

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. 21(1), 26–32 (1983).

References

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  1. H. Raether, Surface-Plasmons on Smooth and Rough Surfaces and on Grating, Springer Tracts in Modern Physics (Springer Berlin, 1988).
  2. 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]
  3. G. E.  Cragg, 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, 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, F.  Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366(6450), 44–48 (1993).
    [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  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]
  13. 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]
  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]
  15. L. Novotny and B. Hetch, Principle of Nano-optics (Cambridge U. Press, 2006).
  16. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge MA, 2005).
  17. P. B.  Johnson, R. W.  Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [CrossRef]
  18. 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).

2008 (1)

2007 (1)

2006 (2)

2004 (1)

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]

2003 (1)

2002 (1)

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]

2000 (1)

1998 (2)

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]

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]

1993 (1)

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]

1983 (1)

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).

1972 (1)

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

1959 (2)

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]

Aussenegg, F. R.

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]

Bailey, B.

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]

Bouhelier, A.

Bozhevolnyi, S. I.

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]

Bruyant, A.

Burge, R. E.

Chang, C. C.

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).

Christy, R. W.

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

Chung, E.

Colas des Francs, G.

Cragg, G. E.

Dereux, A.

Ditlbacher, H.

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]

Ebbesen, T. W.

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]

Farkas, D. L.

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]

Gan, X. S.

Ganic, D.

Ghaemi, H. F.

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]

Gu, M.

Hsu, T. M.

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).

Huang, C.

Hwang, Y. F.

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).

Ignatovich, F.

Johnson, P. B.

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

Kano, H.

Kawata, S.

Kim, D. K.

Krenn, J. R.

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]

Lanni, F.

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]

Lee, K. C.

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).

Leitner, A.

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]

Leosson, K.

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]

Lezec, H. J.

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]

Lin, J.

Mizuguchi, S.

Nikolajsen, T.

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]

Novotny, L.

Richards, B.

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]

Schider, G.

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]

So, P. T. C.

Tan, P. S.

Taylor, D. L.

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]

Thio, T.

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]

Wang, Q.

Weeber, J.-C.

Wiederrecht, G. P.

Wolf, E.

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]

Wolff, P. A.

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]

Yuan, X. C.

Zhan, Q. W.

Appl. Phys. Lett. (2)

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]

Chin. J. Phys. (1)

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).

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

Nature (2)

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]

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]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. B (1)

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

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

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]

Other (3)

L. Novotny and B. Hetch, Principle of Nano-optics (Cambridge U. Press, 2006).

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge MA, 2005).

H. Raether, Surface-Plasmons on Smooth and Rough Surfaces and on Grating, Springer Tracts in Modern Physics (Springer Berlin, 1988).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of SPPs excitation by a tightly focused vortex beam obliquely incident on a metal film.

Fig. 2
Fig. 2

Schematic of focused light beam converging towards the geometric focus o on the metal/glass interface located at the z = 0 plane. The cross-sectional circle in dash line represents the light rays with cone semi-angle being equal to SPPs resonant angle θsp in the vicinity of the focal plane, while the circle in solid line represents light rays with cone semi-angle being equal to the maximum convergence angle θmax. Cn and Co are the positions of the optical axis on the cross-sectional plane at normal and oblique incidence, respectively. (a) normal incidence; (b)oblique incidence in the xoz plane;(c) 3-D plot of a focused beam which is geometrically at critical obliquity for SPPs excitation.

Fig. 3
Fig. 3

(a) Schematic of oblique incidence of light focused on a gold film deposited on glass, θmax = 55°, α = 5° ; (b)The SPP waves excited by light with incident angle of θA = 50° and θB = 60°, respectively.

Fig. 4
Fig. 4

SPP interference pattern maps on the Au film surface excited by a x-polarized 532nm vortex beam at (a) 10° of parallel obliquity and (b) 30° of perpendicular obliquity.

Fig. 5
Fig. 5

SPP intensity profiles (a) along x-axis at different angle of parallel obliquity and (b) along y-axis at different angle of perpendicular obliquity.

Fig. 6
Fig. 6

The normalized peak intensity of SPPs self-interference on Ag film excited by a 532nm vortex beam at different angle of (a) parallel and (b) perpendicular obliquity.

Fig. 7
Fig. 7

The FWHM of the SPPs self-interference intensity profile on Ag film excited by a 532nm vortex beam at different angle of (a) parallel and (b) perpendicular obliquity.

Equations (17)

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

θ co = cos 1 ( cos θ max / cos θ sp )
E inc ( k x , k y )= E inc x= E 0 exp(- k x 2 + k y 2 k 1 2 NA 2 n 1 2 +ilϕ)x
E t (x,y,z)= if e i k 1 f 2π k x k y t p ( k zj )( E inc n ρ ) n θ k z1 / k 1 k z1 exp[i( k x x+ k y y+ k z3 z)]d k x d k y
t p ( k zj )= 4exp[i( k z2 k z3 )d] (1+ p 12 )(1+ p 23 )[1+ r 12 r 23 exp(i2 k z2 d)] p ij = ε i k zj ε j k zi , r ij = 1 p ij 1+ p ij , k zj = k j 2 ( k x 2 + k y 2 )
E t ( x , y , z )= if e i k 1 f 2π k x k y t p ( k z j )( E inc n ρ ) n θ k z1 / k 1 k z 1 exp[i( k x x + k y y + k z 3 z )]d k x d k y
[ x y z ]=R[ x y z ],[ k x k y k z 1 ]=R[ k x k y k z1 ]
R=[ cosα 0 sinα 0 1 0 sinα 0 cosα ]
n ρ = k x k x 2 + k y 2 x+ k y k x 2 + k y 2 y= k x cosα k x 2 + k y 2 x+ k y k x 2 + k y 2 y k x sinα k x 2 + k y 2 z
E inc n ρ = E inc k x cosα k x 2 + k y 2
| ( k x , k y ) ( k x , k y ) |=cosα+ k x k z1 sinα
E t ( x , y ) z = if e i k 1 f 2π k x k y E inc ( k x , k y ) t p [ k z j ( k x , k y )]exp(i k z 3 d)( k x cosα k z1 sinα ) ×exp{ i[( k x cosα k z1 sinα) x + k y y ] }cosα k z1 / k 1 k 1 k z1 d k x d k y
E t ( x , y ) z = if k 1 e i k 1 f 2π 0 θ max 0 2π E inc (θ,ϕ) t p [ k z j (θ,ϕ)]exp(i k z 3 d)(sinθcosϕcosαcosθsinα)× exp{ i k 1 [(sinθcosϕcosαcosθsinα) x +sinθsinϕ y ] }cosα cosθ sinθdθdϕ
R=[ 1 0 0 0 cosβ sinβ 0 sinβ cosβ ]
E inc n ρ = E inc k x k x 2 + k y 2
| ( k x , k y ) ( k x , k y ) |=cosβ k y k z1 sinβ
E t ( x , y ) z = if e i k 1 f 2π k x k y E inc ( k x , k y ) t p [ k z j ( k x , k y )]exp(i k z 3 d) × k x exp{ i[( k x x +( k y cosβ+ k z1 sinβ) y ] } k z1 / k 1 k 1 k z1 d k x d k y
E t ( x , y ) z = if k 1 e i k 1 f 2π 0 θ max 0 2π E inc (θ,ϕ) t p [ k z j (θ,ϕ)] exp(i k z 3 d) sin 2 θ cosθ ×exp{ i k 1 [sinθcosϕ x +(sinθsinϕcosβ+cosθsinβ) y ] }cosϕdθdϕ

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