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This paper describes the image formation process in optical leakage radiation microscopy of surface plasmon-polaritons with diffraction limited spatial resolution. The comparison of experimentally recorded images with simulations of point-like surface plasmon-polariton emitters allows for an assignment of the observed fringe patterns. A simple formula for the prediction of the fringe periodicity is presented and practically relevant effects of abberations in the imaging system are discussed.
© 2011 Optical Society of America
1. Introduction
Surface plasmon-polaritons (SPP) are electromagnetic modes guided by a metal-dielectric interface [1]. Perpendicular to the interface they show a very tight mode confinement and their propagation can be controlled by the interaction with nano-structures on the metal surface [2,3]. Over the last years growing research efforts led to the demonstration of the feasibility of surface bound, two-dimensional SPP based photonics – plasmonics. A major motivation therefore is the high potential of plasmonic devices for miniaturization to match the dimensions of micro(opto)electronic devices [4, 5].
For the development and understanding of plasmonic devices, it is often beneficial to directly observe SPP propagation. This can be done by mapping the optical near fields close to the metal surface, either with an optical near field microscope [6,7] or by fluorescence microscopic imaging [8] of the emission of dyes deposited on the metal surface and excited by the SPP near-field. The first method is relatively demanding and slow, whereas the latter suffers from dye bleaching and therefore does not allow quantitative analysis. However, a third possibility opens up if the SPPs are bound to a metal film which is sufficiently thin (i.e in the range of ∼ 50 nm) and placed on a transparent substrate. The SPP modes of e.g. an air-metal interface are then leaky and emit leakage radiation (LR) to a very narrow angular range into the substrate [9]. The emission angle of this LR is above the critical angle of total internal reflection but can be collected by an usual oil immersion objective and imaged in a standard microscope setup [10–12]. The obtained LR microscopic images are related quantitatively to the local SPP intensity [3, 13].
Due to its simplicity, LR microscopy is widely used to analyze plasmonic devices [2,14–16], however a clear and detailed description of the LR images obtained with diffraction limited resolution is still missing and shall be given below. After a review of the image formation process, we analyze and explain experimentally recorded LR images of a point-like SPP source by comparing them to simulations. Then, practically relevant effects of abberations in the imaging system on the LR image formation are discussed.
2. Leakage radiation emission and imaging
We start by considering a SPP on a metal-air interface of a thin metal film on glass substrate, which is excited somehow at the position x0 (Fig. 1(a)). The thickness of the metal (silver or gold) film has to be sufficiently thin, so that the SPP field can extend through the metal into the glass substrate. As the real part of the SPP wavenumber (Re[kSPP]) is smaller than the wavenumber of light in the glass substrate (kg = ngk0; ng is the refractive index of glass and k0 the vacuum wave number), the SPP is leaky and emits LR to a narrow angular range around the phase-match angle α defined by Re[kSPP] = ngk0 sinα. In a LR microscope this radiation is collected by an oil immersion objective (Fig. 1(b)) and, in the most simple case, imaged directly onto a camera (IP2). In a more elaborate setup for example, two lenses (Fig. 1(c), L1 and L2) are added between the objective and the tube lens (L3) in a way that the back focal plane (BFP) which usually lies within the objective lens assembly is imaged to give an accessible second diffraction image (DI) outside the objective (see below).
Fig. 1 (a) A SPP is excited, e.g. by a point-like emitter (grey ellipse at x0) above the metal-air interface and propagates from there to the left and the right. If the metal film is sufficiently thin, leakage radiation (LR) is emitted under the phase-match angle α. At every point x1 the phase and strength of the LR electric field ELR have the same, constant relation to the electromagnetic near fields of the SPP above the metal-air interface (Ez, Ep and Hp; the index p indicates field components parallel to the metal-air interface). The amplitudes and phases of Ez, Ep and ELR in the diagrams are calculated for a SPP on a 35 nm thick gold film on BK7 substrate. (b) The leakage radiation is then collected by an oil immersion objective. After the objective the transversal polarization of the LR (blue arrows) is turned to be basically perpendicular to the optical axis (dash-dotted line) as the further beam propagation is paraxial. (c) After the objective the LR is imaged (image plane IP2) onto a camera by the tube lens L3. Two additional lenses (L1 and L2 between the objective and L3) lead to an intermediate image (image plane IP1) and an accessible diffraction image (DI) of the back focal plane (BFP) of the objective to allow for e.g. spatial filtering. L3 can be replaced by a lens of shorter focal length to image the DI plane on the camera (not sketched).
We assume for now, that at a sufficiently large distance from the source, the SPP propagation within a small surface area can be described as a plane SPP wave to a good approximation and we follow the mathematical description of plane SPP wave propagation on laterally infinite interfaces [1, 12, 17] can be applied. Then, the LR fields at the metal-glass interface have a constant, position-independent ratio and phase to the optical near fields at the metal-air interface [12] at every point x1 (Fig. 1(a)). Thus also the intensity of one point in the LR image of the metal-glass interface is proportional to the intensity of the SPP and its optical near fields at the corresponding point on the metal-air interface. Both, the proportionality factor and phase are wavelength and material dependent.
Let us now consider the LR microscopy image of the interference of two plane SPP waves incident under azimuthal (in-plane) angle of ±ϕ to the x-coordinate (Fig. 2(a)). To understand the resulting image for this case one has to take into account the redirection of the transversal magnetic polarized (with respect to the SPP propagation direction) LR “beams” by the immersion objective and the related rotation of the electric field polarization direction (blue arrows above and below the objective lens in Fig. 1(b)). Typical values (e.g. for a 60x oil immersion objective) for the field of view (∼ 0.2mm), the objective focal length (∼ 3mm) and the tube-lens focal length (200mm) lead to angles γ (Fig. 1(b)) and β (Fig. 1(c)) smaller than 2°. The image formation after the objective is thus well described within the paraxial limit: the axial (i.e. z) components of the electromagnetic fields are negligible, i.e “lost” for the image formation and the LR electric fields in the image plane are polarized parallel to kSPP and to the electric near field component parallel to the metal surface (Ep, Fig. 2(a)). The LR image intensity of two (and more) interfering SPP beams thus resembles that of Ip ∝ |Ep|2 [18].
Fig. 2 (a) Sketch of the top view of two interfering SPP beams incident under angles ϕ to the x-axis (parallel to the metal air interface). Ez, Ep and Hp are the perpendicular (parallel to z) and parallel (index p) SPP electromagnetic field components close to the interface. (b) Calculated interference pattern along the y direction of the electric near field intensities Iz ∝ |Ez,1 + Ez,2|2 (black, dotted curves), Ip ∝ |Ep,1 + Ep,2|2 (blue dashed curves; for better visibility 10 × Ip is plotted) and I = Iz + Ip (green, solid curves). The LR image intensity follows qualitatively that of Ip.
For low angles ϕ the qualitative difference between the electric near field intensities of the perpendicular (Iz ∝ |Ez|2), and parallel (Ip) field components and the total electric near field intensity (I = Ip + Iz) is barely visible (ϕ = 15°, Fig. 2(b), top graph). The difference between these intensities is most striking if the two SPP waves propagate perpendicular to each other (ϕ = 45°, Fig. 2(b) middle graph). Then, in the SPP’s electromagnetic near field above the air-metal interface, the field components parallel to the interface (Ep, Hp) of the two SPP waves are perpendicular to each other and cannot interfere. Ip and the image intensity thus show no interference pattern but a constant intensity (Fig. 2(b), middle graph) [18]. However, the total electric near field intensity I shows a strong modulation due to the interference of the SPP’s dominant Ez components.
For angles ϕ > 45° the interference pattern in the image appears again, but the antiparallel orientated Ep y-components now dominate over the Ep x-components (Fig. 2(a)). Thus their interference maxima are shifted compared to the maxima of I and Iz by half a periodicity, i.e. I and Iz have a minimum where Ip and the image intensity are maximum (Fig. 2(b), lower graph for ϕ = 90°, i.e., counter propagating SPP beams).
It has to be pointed out, that in case of two SPP beams only, the interference pattern of Iz could be qualitatively imaged by placing a polarizer oriented parallel to x between objective and camera [19]. Then the image intensity resembles the near field intensity of the x-components of Ep which are in phase with Ez and thus show a pattern identical to that of Iz. Due to the polarizer, this method reduces the image intensity by a factor of cos2(ϕ) and thus cannot be applied at ϕ = 90° and close by.
3. Diffraction limited imaging of locally excited SPPs
Up to now we discussed the LR imaging of SPP beams propagating along the metal-air interface, but did not include effects due to their excitation or the finite objective’s numerical aperture (NA). These shall be analyzed in the following.
As the propagation length of SPP for e.g. gold or silver films is in the range of only 10 to 100μm in the visible spectral range, usually the SPP source will also be in the objective’s field of view. SPP sources suitable for LR microscopy are usually local “defects” in or close to the metal surface, e.g. particles on the surfaces [10, 12], holes in the metal film [13, 20], scanning tunneling microscope (STM) tips [21] and fluorescent quantum emitters coupled to scanning near field optical microscope (SNOM) [22, 23]. In many cases the defects are much smaller than the SPP wavelength, and, as shown below, they can be considered to a first approximation as point-like dipole emitters. If placed close enough to a smooth metal-air interface, the dipole emission couples to the SPP and acts as a local SPP source. However, there will also be direct dipole emission and neither in the diffraction image (DI) nor in the image (due to diffraction limited imaging) contributions from SPP-LR and direct dipole emission can be sharply separated. In this context, recent work using either SNOM (coupled to color-centers in single diamond nanocrystal) [22] and STM tips [24] showed that SPPs launched from a point source and imaged through a leakage microscopy contains two families of circular fringes around the SPP launching point, with respectively small (Λs) and large (Λl) spatial periods. Λs is roughly half the optical wavelength while Λl can reach several micrometers in the visible spectral range. As a first example, Fig. 3 depicts a typical LR image of a SPP excited locally by a homemade SNOM aperture tip [22,25–27] on a 60nm thick gold film, showing pronounced circular fringes.
Fig. 3 LR image of a SPP excited by a homemade SNOM aperture tip (100 nm diameter) placed at h ≃ 20 nm above a 60 nm thick gold metal film evaporated on a fused silica substrate. The optical set up is described in refs. [22, 27]. The arrow indicates the polarization of the light-mode which excites the tip apex and the central spot shows the tip position. The optical wavelength of the laser injected in the tip used here is λ = 647nm. The inset shows the SNOM image obtained when the tip is retracted few microns from the surface showing that no SPP can be launched. This is confirmed by images of the BFP (not shown).
In the following we study more systematically these fringes and show their relation with the existence of the two contributions mentioned above (i.e. SPP-LR and direct dipole emission) to the image formation in LR microscopy.
To emphasize the generality of the phenomena considered here, we give a second example in Fig. 4: LR image and diffraction image of a SPP launched from a subwavelength cylindrical hole (diameter d = 300 nm). The LR microscopy setup, described in further detail in [28], relies on SPP excitation by a weakly focused laser beam (wavelength λ = 785 nm) using an illumination objective of low numerical aperture (NA= 0.13). The leakage radiation is collected by an oil immersion objective of high numerical aperture NA= 1.49. The hole was milled using focused ion beam lithography through a 80 nm thick gold film evaporated on a glass substrate (thickness 0.17 mm). Periodic fringes are clearly seen in the real-space image and are associated with the long periodicity fringes Λl. The short periodicity fringes can be observed when the image is further optically magnified.
Fig. 4 Experimentally recorded LR image (a) and diffraction image (b) of a SPP launched on a subwavelength d = 300 nm single hole with a laser wavelength of λ = 785 nm. The laser polarization is indicated by the white arrow. Fringes of long periodicity Λl are clearly seen in (a). The inner disk in (b) corresponds to the Fourier transform of the illumination Airy spot with a diameter therefore given by the numerical aperture NA= 0.13, the bright circle associated to the SPP has a diameter equal to 2Re[kSPP] and the outer disk diameter is set by the collection NA= 1.49.
Finally, we consider a third configuration with the example of a SPP excited by a point like defect in an otherwise smooth 55 nm thin silver film on a smooth glass substrate. From now we will focus exclusively on this experiment. The defect itself consists in a protrusion in the silver film due to a electron beam lithographically defined 45 nm high SiO2 disk of ∼ 100 nm diameter below the silver film of constant thickness. In our experiment, the defect is excited to a horizontal dipole oscillation by a weakly focused laser beam (wavelength 725 nm) incident from the air side perpendicular to the silver-air interface. The LR was collected by a Zeiss Plan-Neofluoar, 63x, 1.25NA. In our setup the lenses L1 and L2 had the same focal length f = 200 mm and the distances and were both 2f. L3 had a focal length of f = 300 mm and the images were recorded by a Vosskühler COOL-1300Q camera.
Figure 5 depicts the experimentally recorded image and diffraction image. The diffraction image (Fig. 5(b)) clearly shows the two main signal contributions, i.e. the central spot from the directly transmitted excitation laser and the outer ring-like pattern, which stems from the radially excited SPP’s LR. The modulation of the ring intensity with azimuthal angle is caused by the TM character of the SPP which is excited more strongly parallel to the defect dipole orientation as defined by the incident laser polarization.
Fig. 5 Experimentally recorded LR image (a) and diffraction image (b) of a SPP excited by a 725nm laser weakly focused on a electron beam lithographically defined defect in a 55nm thin silver film on glass substrate. The laser polarization is indicated by the white arrow. The pattern in (a) is the image of the laser focus on the sample surface with the circular fringes coming from the defect which is located at their center. In the diffraction image, the central spot is from the directly transmitted excitation light whereas the thin outer ring is a signature of the LR of the excited SPP. The inset in (b) shows a 3.5× magnified view of a part of the ring
In the image circular fringes around the defect can be seen, but no clear signature of an SPP excitation is discernible. To more clearly image the propagation of the excited SPP, one has to place in this case a central beam stop in the plane of the diffraction image (DI in Fig. 1(c)) to block the directly transmitted light of the excitation laser. The corresponding LR images are shown in Fig. 7(a) and (b), which will be discussed later in this paper.
Fig. 7 Experimental (a,b) and simulated (c,d) LR images. The experimental images are recorded from the LR emitted from a SPP excited by a laser focused on a lithographically defined defect in an otherwise smooth 55 nm thick Ag film. The simulations are for a point dipole emitter placed 1nm above a smooth silver film of 55nm thickness. In the experiment and the simulations, a central beam block for kp/k0 < 0.083 was used and the wavelength of the excitation was 725 nm. (b) and (d) are closeups of (a) and (c), respectively, but less saturated to reveal more details around the SPP launching point.
To gain an understanding of the observed features, we compare our observations to the results of simple simulations. The scheme of the calculations is as follows. The fields of an oscillating point dipole are analytically expanded to a plane wave representation [29]. Then the emission diagram, i.e. the field strength as a function of the in-plane wavevector is calculated from this expansion in combination with the Fresnel transmission coefficients for the air-silver-glass system [30, 31] and filtered to account for the finite NA of the objective (and for a possible central beam stop or other spatial filter). The intensity of this filtered emission diagram plotted vs. the in-plane wavevector components ky and kx represents the diffraction image of the LR microscope. The image intensity can then be calculated by a simple numerical Fourier transformation of the electric field distribution E in the emission diagram. For this purpose we consider the full electromagnetic field transmitted by the metal film by using a plane wave expansion and by taking into account the Fresnel coefficients for transmission and reflection through the layered system [29]. Unlike in Ref. [21] which deals with a vertical dipole the field distribution of an horizontal dipole involves both p and s polarizations (the general method due to Sommerfeld is discussed in Ref. [29], see also [32]). We point out that this approach is actually a simplification since for a high-NA aplanatic microscope objective we should also consider the so-called ‘sine’ correction [29] associated with the refraction of light rays emitted from the focus region and going through the microscope objective. This induces a correction in the BFP (and its image DI) of the microscope and therefore the intensity I(k) in this plane is actually not just |FT[E(r)](k)|2 (FT denotes the Fourier-transform, r is the in-plane position vector and k the in-plane wave vector) but should be written as:
Remarkably, this correction does not play a significant role here because the Fourier transformed field associated with the launched SPP is a narrow ring-shaped function sharply peaked on the circle of radius K = Re[kSPP] and with typical width 2Im[kSPP] << K. We however point out that this is not necessarily always the case in LR microscopy. In particular this is not true if the SPP source is an extended one-dimensional ridge or slit since then the SPP Fourier field FT[E](k) will indeed contain considerable contributions crossing the circle of radius k0NA ∼ k0ng (see for example [33]).Neglecting these small corrections in the following, we performed calculations for a point like horizontal dipole emitter placed 1 nm above a smooth silver film of 55 nm on glass (n = 1.52) substrate. Literature values for the dielectric function of Ag were used [34] and the vacuum wavelength was set to 725 nm. To achieve a reasonable spatial resolution for the calculated image intensities, the diffraction diagram was calculated with 4250 × 4250 points within a range of −5ngk0 < (kx, ky) < 5ngk0. Apart from an arbitrary intensity scale, there are no free parameters in the model.
Let us first analyze the diffraction image in Fig. 5(b). Figure 6 depicts a cut through it along a horizontal line through the center, where the distance from the center is a function of the in-plane wave vector kx. In this case, the directly transmitted intensity of the excitation laser (red curves, central maximum in Fig. 6(a)) is about 1000 times stronger than the intensity in the SPP-peaks (outer maxima at kx/k0 = ±1.02). Both features are well separated and the intensity is close to zero everywhere else. Apart from the central peak due to the transmitted excitation light and a slight intensity asymmetry of the left and right SPP peaks we find good agreement with the calculated intensity distribution (blue dashed curve in Fig. 6(a)). However, with a closeup of the region around the SPP peak plotted on a logarithmic scale, some discrepancies get obvious (Fig. 6(b)). Beside a smaller width of the calculated SPP peak (which we attribute to slightly larger damping of our silver film compared to the literature data used for the simulations [34]) the simulated data show a zero at kx/k0 = ±1 which is here clearly not observed experimentally (it is nicely observed, e.g., in Fig. 4 or Ref. [20] for SPP’s excited at nanoholes). Importantly, the simulations do not include the excitation laser and if we assume a coherent background from a possible “tail” of the central peak with only 10−6 of its maximum intensity overlayed to the diffraction image, we see the qualitatively same behavior as recorded experimentally (green, dotted curve, Fig. 6(b)). As these discrepancies are on a scale of some percent of the SPP peak intensity only, we expect no major effects on the image formation.
Fig. 6 (a) Line profile of the diffraction image in Fig. 5(b) along a horizontal line through the center (red, solid line) and the same but recorded with an additional neutral density filter of optical density 3 in front of the camera to reveal the shape and intensity of the central peak. The blue-dashed line is the result from the simulation (vertically offset by −1). The central peak corresponds to the direct transmission of the excitation laser whereas the peaks at kx/k0 = ±1.02 are from the SPP LR. (b) A closeup of (a) around the SPP peak. The green, dotted line depicts the result of the simulations if an additional coherent background of 10−6 of the maximum central peak intensity is considered.
We now turn to the LR images (Fig. 7(a),(b)) and compare them to the calculated images (Fig. 7(c),(d)). Before performing the Fourier transform of the calculated diffraction image, it was filtered to account for the experimental numerical aperture (1.25) and the used central beam stop (blocking all emission with k/k0 < 0.83, where k denotes the modulus of th in-plane wave-vector k). Apart from some minor irregularities in the experimentally recorded image (which can be attributed partly to an etaloning effect of the camera), one finds again very good agreement between simulations and experiment.
In both experiment and simulation circular fringes are observed, where the finite NA of the objective leads to two effects. First, the point dipole emitter not only excites a SPP, but also has its own dipolar emission diagram (including the allowed light [29]) and its diffraction limited image (i.e. the point dipole diffraction pattern) that overlays coherently on the leakage radiation image of the SPP. This leads to rather strong circular fringes at the image center (for a vertical dipole this contribution is much lower [21]). Second, the spatial frequency spectrum collected by the objective is insufficient to fully image the field (intensity) distribution of the locally excited SPP
(r is the in-plane position vector and θ the azimuthal angle with respect to the polarization direction of the horizontal point dipole). Thus the image intensity of one point on the sample is a result of the coherent superposition of the diffraction limited LR images of the points of an extended area surrounding that specific point on the sample. As a consequence of the large SPP spatial coherence, fringes appear in the image which depend on the objective NA (Fig. 8).Fig. 8 Calculated LRM image intensities along the x axis for a horizontal point dipole emitter. The objective NA is varied and no central beam block is considered.
For an estimation of the fringe periodicity, one can consider a simple model which can mimic the main result of the full electromagnetic calculation we showed earlier. We here suppose that the Fourier field obtained in the back focal plane of the objective can be written as
where the background field B(k) is a function which we approximate for this estimation as a constant over the numerical aperture (that is B(k) = a if k ≤ k0NA and B(k) = 0 otherwise). It represents the residual part of the field which is not necessarily associated with SPPs. However, it can also represent the tail of the SPP field in the diffraction image, which extends far beyond the ring at k ≃ kSPP. For qualitatively describing the signal field S(k), we choose the highly singular function Such a S(k) field has the interesting properties that its inverse Fourier transform FT−1[S](r) is a good representation of a lossless SPP asymptotic field (Im[kSPP] = 0). Indeed, from the well known properties of Hankel transforms we get Therefore, using the asymptotic expansion of the Bessel function for large |r| we get after a straightforward calculation: that is the usual 2D field representation of a point source for [12]. Adding S(k) and B(k) lead to a simple image field FT−1[E](−r/M) where M is the microscope magnification. We obtain that is the sum of the SPP field and of a Airy like function. For k0NA|r| ≫ 1 we get also . In this model the imaged field is therefore the sum of two oscillating fields with different spatial wavelengths (i.e. 2π/(NAk0), 2π/kSPP) which are both multiplied by envelope amplitudes whose variations are much slower than the oscillations mentioned before. Calculating the intensity |FT−1[E](r)|2 in this approach one thus expects to see mode beating, i.e. fringes with periodicity Λ corresponding to Λl = 2π/(NAk0 − kSPP) and Λs = 2π/(NAk0 + kSPP). Comparing to the data we deduced from Fig. 8 we actually find good agreement of this simple estimation with the values obtained from the simulated LR images (Tab. 1).
Table 1. Short (Λs) and long (Λl) fringe periodicities derived from the calculated LR microscope images compared to the values estimated by the formula derived in the text.
For a vertical dipole emitter, the simulations reveal qualitatively similar fringes. However, as the vertical dipole does not emit in the direction of the optical axis, the diffraction image has a zero at k = 0 (k is the modulus of the in-plane wave-vector). There is no zero at k/k0 = 1 and the intensity distribution is more localized around the SPP-peaks. Accordingly, a lower fraction of the emitted intensity is cut by the finite NA and thus the fringes in the image are slightly less apparent. The image is rotational symmetric and, due to the imaging of the horizontal field component only, the image intensity is zero at the position of the dipole emitter.
The results discussed above for a point-like emitter can be generalized to extended SPP sources, which can be viewed as a coherent superposition of SPP point sources.
4. Effects of abberations
LR microscopy in principle does not differ from conventional immersion microscopy either in bright field or in dark field mode (if a central beam block is inserted in the diffraction image plane). However, in contrast to light scattered and diffracted from a “usual” object, SPP LR used for imaging in the LR microscope is emitted around a very small range of the SPP phase match angle only. Thus, if the central beam stop is used, good LR images can be obtained, even if spherical abberation is present (for example if the objective is not perfectly corrected for the used substrate thickness and refractive index), albeit at a different focus position. As due to this abberation the phase of the LR tails towards larger and smaller angles will be shifted in the image plane, the positions of the fringes caused by diffraction will change compared to the abberation-free case.
In particular if the LR microscope is operated in the red to near infra-red spectral range, chromatic abberation has to be taken into account, as standard microscope objectives are not corrected for that spectral range. If the spectral range used for LR-imaging is sufficiently narrow (as in the experimental case discussed above, where a monochromatic laser was used as excitation source) this also only leads to a shift of the focal position, but does not deteriorate the LR images.
Together both, spherical and chromatic abberations, mainly lead to a shift of the focal position, which raises a practical problem. In particular if the SPPs are excited by a laser, the focal position is not easily found experimentally as pronounced fringe patterns are observed within a large focal range. Additionally, due to the abberations, the focal positions found with a white-light bright-field image cannot be used. This situation is well illustrated by the quasi two-dimensional example of a SPP launched by a laser weakly focused on a line-defect (all other experimental conditions are as for the point defect above). In Fig. 9(a), the middle image shows the LR image recorded for the case, where the camera is placed at the correct focal plane (IP, Fig. 9(b)). It shows the two SPP beams propagating from the line source to the left and to the right. As the LR beams are incident under a (small) angle on the image plane, below the focal plane the LR beams are separated and a dark region appears in the image center if the camera is placed at IP+ (bottom image in (a)). In contrast, if the camera is placed above the image plane (IP−, Fig. 9(a), upper image), the images of the left- and right- propagating SPP start to overlap. For both cases the fringe positions (most notably of the low frequency fringes) are laterally shifted compared to the IP image.
Fig. 9 (a) LR images of a SPP excited at a line-shaped defect of a 55 nm thick silver film excited by a weakly focused laser (wavelength 725 nm) and recorded at the image plane (IP), above (IP−), and below (IP+) the image plane. (b) Sketch of the LR beams and their overlap in the image region. The vertical dashed-dotted line is the optical axis and the angles to the optical axis are exaggerated for clarity. LR1, LR2: leakage radiation from the SPP propagating to the right and the left, respectively. The solid curves sketch roughly the expected image intensity (fringes neglected).
One can overcome this problem by focusing the sample surface when illuminated with a more incoherent source but in the same spectral range and with a sufficiently large central beam stop in place. In practice, for horizontal dipole emitters this focus position coincides usually with those found by adjusting the focus to the position where the image intensity of the SPP launching point is at a maximum.
We finally discuss the relation of the diffraction image coordinate with k/k0, i.e. the in-plane wave vectors of the LR emission collected by the immersion objective. The light collected by the objective lens can be described as a superposition of plane waves with the wavenumber kw = nk0 (where n is the refractive index of the immersion oil) emitted under the angles to the optical axis. Within the ray optical description, these plane waves are focused in the back focal plane to points at distance d = f tan(α̃) (f is the objective focal length) from the optical axis. However, in practice it turns out that this approximation is not valid and usual objectives are well corrected to a linear relationship d ∝ k/k0. Figure 10 depicts the measurements for the distance N (the number of camera pixels) from the center of the DI in dependence of k/k0. For the measurements a parallel laser beam was incident on the sample plane (a clean cover slide) under a defined angle δ to the optical axis, either from air (red crosses) or through a 60° glass prism (green crosses, δ was corrected for the refraction when entering the prism). In the DI, the laser is focused to a “point” and its pixel distance N to the center of the DI (δ = 0) was measured. We find a linear relationship within the experimental uncertainty. This is very practical, as once the linearity is established, one can use the very defined kSPP from the SPP-LR to calibrate the diffraction image.
Fig. 10 Relation between the pixel-distance N from the center of the diffraction image to the ratio k/k0. The crosses represent the experimental values measured for a parallel laser incident on the sample from air (red) and BK7 (green), the small vertical lines within the crosses are the experimental uncertainty in k/k0. The black, dotted line represents a linear fit to the experimental data and the black circle indicates the position of the SPP-peak with kSPP/k0 = 1.023 found in the diffraction image (Fig. 6).The blue dashed line is the calculated result from the ray-optical approximation. The inset shows the linear fit and the ray-optical approximation in an extended range.
5. Conclusion
LR microscopy is a simple and quantitative method for imaging SPP propagation. By a detailed analysis of the image formation process, we explain that the image intensity has to be proportional to the SPP near field intensity of the electric (or magnetic) field component parallel to the metal surface. A comparison of experimentally recorded LR microscopy images of SPPs excited by point-like defects in an otherwise smooth metal film with simple analytical calculations of SPP coupled emission from point dipoles shows excellent agreement. The observed fringe patterns can be attributed to the diffraction limited imaging of the LR in combination with the SPP’s coherent propagation. Practical aspects of LR microscopy as chromatic and spherical abberations are discussed and found to be of minor importance, as the LR is emitted only to a very narrow range around the SPP phase match angle. Also the diffraction images are in good agreement with simulations and the linearity of the spatial coordinates of the diffraction image with respect to the in-plane wave-vector components is demonstrated.
Acknowledgments
AH and JRK acknowledge support from the Austrian Science Fund (FWF-project P21235-N20). AD, OM and SH acknowledge support from the Agence Nationale de la Recherche, France through the Napho and Plastips projects. CG, BS and TWE acknowledge support from the ERC (grant 227557).
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