Coherence and aberration effects in surface plasmon polariton imaging

We study theoretically and experimentally coherent imaging of surface plasmon polaritons using either leakage radiation microscopy through a thin metal film or interference microscopy through a thick metal film. Using a rigorous modal formalism based on scalar Whittaker potentials, we develop a systematic analytical and vectorial method adapted to the analysis of coherent imaging involving surface plasmon polaritons. The study includes geometrical aberrations due index mismatch which played an important role in the interpretation of recent experiments using leakage radiation microscopy. We compare our theory with experiments using classical or quantum near-field scanning optical microscopy probes and show that the approach leads to a full interpretation of the recorded optical images.

Figure 1

Sketch of the optical microscope including a high numerical aperture objective. Rays originating from the focus $F_1$ are collimated along the optical axis after crossing the reference sphere ${\mathrm{\Sigma }}_1$. The plane $\mathrm{\Pi }$ is mapped onto ${\mathrm{\Pi }}^{\prime }$. ${\mathrm{\Pi }}_1$ is the back focal plane of the objective and ${\mathrm{\Pi }}_2$, ${\mathrm{\Sigma }}_2$ play for the ocular or tube lens the same role played by ${\mathrm{\Pi }}_1$, ${\mathrm{\Sigma }}_1$ for the objective. The metal sample of thickness $d$ is located between the plane $z=0$ and $d$ (object plane, $\mathrm{\Pi }$).

Figure 2

Sketch illustrating how rays originating from $F_1$ are transformed after crossing the reference sphere ${\mathrm{\Sigma }}_1$. The unit vectors used in the text are indicated.

Figure 3

Simulation of the SPP field imaged in the ${\mathrm{\Pi }}^{\prime }$ plane for, respectively, a radiating vertical dipole (a) or horizontal dipole aligned with the $x$ direction (b) and located on the air side on top of a 50 nm thick metal film at a distance $h=20$ nm of the air-gold interface. The wavelength is $\lambda =633$ nm and the optical constants for gold are taken from [58]. The insets show zooms of the central regions.

Figure 4

Fourier space images associated with the real space images of Fig. 3. (a) For the vertical dipole and (b) for the horizontal one. The dashed white circle represents unit circle $\text{NA}=1$.

Figure 5

Experimental LRM images obtained using an aperture NSOM. (a), (b) Correspond to a perfect matching of the glass optical indexes of substrate oil and objective while (c), (d) are obtained when the glass substrate is replaced by a fused silica substrate with the oil adapted to the new index. (a), (c) For the direct space while (b), (d) stand for the Fourier space SPP images. (a) A low $k$-filtered image showing only the SPP contribution [we included an inset in (a) and (c) to show the unfiltered signal].

Figure 6

Experimental LRM images obtained with a NV-based NSOM tip over a 50 nm gold film. (a) Shows the low $k$-filtered direct space SPP image while (b) shows the Fourier space.

Figure 7

Sketch of the geometrical rays and interface involved in the spherical aberration modeling (details in the text).

Figure 8

Recorded LRM intensity of a dipole in the image plane (at the tip image position) as a function of the defocusing $FS$. The maximum is obtained for a distance $FS=d-z_F\simeq 14.5\mu \mathrm{m}$. Here, the dipole is along the $x$ direction and $h=20$ nm while the gold film is 50 nm thick.

Figure 9

Simulation of the LRM aberrated images in the ${\mathrm{\Pi }}^{\prime }$ plane due to objective and glass-oil index mismatch (see text). (a) For the vertical dipole case and (b) for the horizontal dipole aligned with the $x$ direction (b) and located on the air side on top of a 50 nm thick metal film at a distance $h=20$ nm of the air-gold interface (compare with Fig. 3). The wavelength is $\lambda =633$ nm and the optical constants for gold are taken from [58]. The insets show unsaturated zooms of the central regions.

Figure 10

Sketch of the geometrical aberrations involved in LRM when an index mismatch modifies the refraction of light at the oil-objective interface (see also Fig. 7).

Figure 11

Experimental images of SPP scattering by a circular slit (6 $\mu \mathrm{m}$ diameter) milled on a 200 nm thick gold film. The SPPs are excited by a NSOM tip located at the center of the structure (a) and (b) or at 12 $\mu \mathrm{m}$ outside the cavity (c) and (d). (a), (c) The direct space images and (b), (d) the associated Fourier space images. The inset in (a) shows a scanning electron image of the structure (scale bar 1 $\mu \mathrm{m}$). The white arrow indicates the tip polarization effective dipole. The tip position $T$ is visible in (c) the distance $AT=9\mu \mathrm{m}$ (scale bar 5 $\mu \mathrm{m}$).

Figure 12

Simulations of the images obtained in Fig. 11 using the aberration theory developed in this work. (a), (c) The direct space images and (b), (d) the associated Fourier space images. The parameters are the same as for Fig. 11.