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
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 [5–7].
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.  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 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.
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.
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 and , 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 and degenerate to point A and B, respectively) and all the light rays deflect away from the lower half of resonant circular arc . 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:
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
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 16]
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
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
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
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
To perform variable substitution for the double integral of Eq. (5), we calculate Jacobian determinant
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
We replace the planar integration over kx , ky by a spherical integration over θ, in cylindrical coordinate
We can correspondingly obtain the following relation
In this case, Jacobian determinant will change to be
So that the vertical component of transmitted field on the metal surface is
The angular spectrum representation of the focal field can be expressed in the cylindrical coordinate as
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)  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︱E′t(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.
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   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.
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.
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.
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).
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