Analysis of SPP conversion efficiency
To extract the conversion efficiency of SPPs in far-field imaging measurements, a slit-groove structure for SPP generation and scattering measurements was used [14, 15]. A schematic of the slit-groove structure is shown in Fig. 1. A transmission measurement setup was used. The metal for the SPPs was deposited on the substrate, and it is assumed that the metal film is thick enough to not transmit the incident light. By exposing the back of the sample to the incident light, SPPs can be launched by propagating toward the normal direction of the length of the slit with the aid of slit geometry. When light is incident from the substrate, some of the incident light (intensity I0) is reflected and converted as SPPS at the bottom side of the substrate because of the edge of the slit structure. Additionally, light transmitting through the slit structure is absorbed by the metal film. If the coefficient of light reaching the top side of the substrate is written as γ, some of γI0 is also converted to SPPs at the top side of the substrate because it meets the edge structure of the slit. When the conversion coefficient of SPPs by the edge structure is denoted as Cs, the SPP-converted intensity of incident light at the top surface of the substrate can be written as 2CsγI0 owing to two edge structures. Therefore, the intensity transmitted through the slit was measured by the detector located above the slit.
Is = I0γ(1 - 2Cs) (1)
Similarly, the intensity measured by the detector located above the groove structure can be analyzed as follows: SPPs generated by the top edge of the slit propagate in the direction normal to the slit length at the front metal interface. The initial intensity of the SPPs propagating toward the groove corresponded to CsγI0. Then, SPPs are decayed by the exp(-αd) factor, while they propagate the distance of d on the metal interface. The parameter of α is the double of the imaginary part of the wavevector (kspp= kn,spp + i ki,spp) of the SPP mode formed at the interface, and the propagation length of the SPPs at the interface of the metal and air can be denoted as 1/α. When the propagated SPPs meets a groove located at a distance d from the slit, they scatter and convert into light because of the edge of the groove structure, as in the case of the slit. When the coefficient of which light is converted from SPPs to light and scattered by the edge of the groove is written as Cg, the intensity obtained at the groove position is as follows:
(2)
At that time, the coefficient of light-to-plasmon conversion, Cs, and plasmon-to-light conversion, Cg, should be the same because the edge geometries of the slit and groove are the same. Therefore, Eq. (2) can be expressed as follows:
(3)
Finally, the SPP conversion coefficient can be obtained as Eq. (5) by solving the quadratic Eq. (4) related to the ratio of Is and Ig.
(4)
(5)
In Eq. (5), the parameter M can be extracted directly from the scattered intensities measured in the far-field imaging experiment. In particular, the decay parameter α of the SPP propagation can be obtained from the propagation measurement experiment and simultaneously calculated by solving the dispersion of the SPP formed at the interface of the metal and air [16]. Additionally, it is worth noting that this analysis of the conversion coefficient does not depend on the shape of the scattering structure. Therefore, if the shapes of the scattering structure in both light-to-plasmon and plasmon-to-light conversions are the same, the suggested approach can be applied in any plasmon experiment irrespective of the geometry of scattering.
Comparison of two FDTD simulations
The FDTD simulation can solve Maxwell equations sequentially and directly in an interesting structure. Therefore, two types of FDTD simulations (Lumerical FDTD simulator) were executed. One mimicked the real experiment of measuring the far-field intensities and propagation decay length. From this simulation, the parameter M involving slit scattering intensity, groove scattering intensity, and propagation decay length were extracted directly by analyzing the far-field images. The other simulation determined the SPP conversion coefficient of the slit structure directly by simulating only one slit structure.
Figure 2(a) shows a schematic of the slit-groove structure used in the first FDTD simulation. In this simulation, a gold layer with a thickness of 300 nm was used as the metal layer for the SPP generation, and the width of the slit was 150 nm, the same as that of the groove. The depth of the groove was 150 nm, and the wavelength of the incident light was 600 nm. Detection monitors that measure scattering intensities in front of the sample and the propagation decay length in the simulation are also shown in the schematic. Monitor 1, indicated by the black line, is located 1.2 µm away from the top surface of the metal layer and measures scattering intensities by using the slit and groove. Monitor 2, indicated by the gray dashed line, is located 250 nm away from the top surface of the metal layer and measures the intensity of the propagating SPP at the interface of the metal and air. From the results obtained by monitors 1 and 2, parameter M can be extracted. When a real experiment is performed, M can also be obtained by measuring the intensities scattered at the slit and groove in far-field imaging, and by fitting the propagation length in the intensity plot of SPP propagation as an exponential function of distance. Figure 2(b) shows the distribution of the electric field (E-field) intensity obtained in monitor 2 of the FDTD simulation. Figure 2(c) is an E-field distribution detected at monitor 3 normal to the x-direction located 1.5 µm away from the slit structure. From Fig. 2(b), it is clear that the light converted into SPPs was scattered by the groove structure. Figure 3 shows the intensity plot obtained using monitor 2. The scattering intensity of Is at the slit structure can be obtained by summing the whole E-field intensities in the left area enclosed by yellow color in Fig. 3. Similarly, the scattering intensity of Ig at the groove structure could be extracted by summing the intensities in the right area enclosed by cyan color. In addition to Is and Ig, the parameter of α could also be obtained from this plot by fitting the decay intensities of the plot as an exponential function of x, as shown by the red line in Fig. 3. As shown in Fig. 3, the decay parameter of α was fitted as ~ 2.105. Finally, the parameter M in Eq. (4) was calculated as ~ 1.768, and the coupling coefficient of the SPP conversion was obtained as ~ 0.232 from Eq. (5). By comparing the obtained number with those previously reported, it is clear that the coupling efficiency obtained from this simulation agreed with values previously reported in some literatures, thus the proposed analysis method is useful [10, 11, 17–22].
The second FDTD simulation was executed to verify the usefulness of the suggested analysis. In this case, the coupling coefficient was extracted directly by simulating a single-slit structure. Figure 4(a) shows the schematic of the single-slit structure used in the second FDTD simulation. For the single-slit simulation, a gold layer with a thickness of 300 nm and a slit with a width of 150 nm were used. In addition, the wavelength of the incident light was 600 nm. The schematic shows detection monitors enclosing the slit structure in the analysis. In a single-slit structure, the incident light from the substrate is divided into four parts. The incident light is reflected (reflection R), transmitted (transmission T), absorbed (absorption A), and converted into SPPs at four edges of the slit structure (conversion Cs). R and T can be calculated from Monitors 1 and 2, which are located apart by 1.2 µm away from the metal layer. The conversion coefficient of Cs can be extracted by multiplying the intensity of monitor 3 or 4 normal to the x direction, as shown in Fig. 4, by the factor of exp(βs). Here, s is the distance of the monitoring location in the x-direction and β is the decay parameter of the SPPs. Since SPPs generated at the slit decay because of absorption by the metal, the intensity ISPP measured by the monitor apart by s should be compensated by multiplying the factor of exp(βs). Therefore, Cs can be calculated from ISPP∙exp(βs). The absorption A can also be extracted from the simulation by calculating the boxed region that is shown by the dashed gray box in Fig. 4(a). The absorption at the slit structure can be obtained by directly calculating the absorption power of Pabs = -0.5real(∇ ∙ P) = -0.5ω|E|2imag(ε) from simulated E-fields in monitor 3 [23]. Figure 4(b) also shows the bound SPP mode supported by the metal layer and air, which was obtained by monitor 3 located at the position apart by 1.5 µm from the slit structure. Figure 4(c) shows the spectra of the T and Cs obtained from the simulation. The calculated Cs was ~ 0.220, which is close to the result obtained from the first slit-groove simulation and the previously reported data. This also verified the validity of the suggested analysis. Two conversion coefficient Cs values calculated from two FDTD simulations and the data reported in some references are listed in Table 1 for the comparison.
Table 1
Coefficients of SPP conversion extracted from slit-groove simulation, single slit simulation, and reference data.
|
FDTD simulation
|
Reported experimental data
|
slit-groove
|
single slit
|
Ref. 18, 19
|
Ref. 20
|
Ref. 22
|
Cs
|
~ 0.232
|
~ 0.220
|
~ 0.38
|
~ 0.20
|
~ 0.28
|