Evaluation of surface roughness of metal films using plasmonic Fano resonance in attenuated total reflection

Attenuated total reflection (ATR) by surface plasmon polariton (SPP) is a method for evaluating the dispersion relation of SPP from the position of a dip in the reflection spectrum. However, recent studies have shown that the dips are displaced from SPP resonance because they are produced by a type of Fano resonance, i.e., the interference between the resonant reflection process accompanied by resonant excitation of SPP and the direct reflection process without resonant excitation. This result suggests that the system properties difficult to be achieved in the dispersion relation of SPP can be characterized using the ATR method. In this study, we investigate the effect of surface roughness due to nanosized dimples created in the initial stage of pitting corrosion on the ATR spectrum, from the viewpoint of Fano resonance. Using the temporal coupled-mode method, it is shown that the Fano resonance in ATR is caused by the phase change of direct reflection because of the absorption on the metal surface, and the spectral shape is determined by this phase, along with the ratio of the external (radiative) decay rate to the total decay rate of the resonant mode. Moreover, it is clarified that the internal and external decay rates extracted from the ATR spectrum provide information on corrosion, such as the effective thickness of the metal film and the randomness in dimple distribution.

Figure 1

System configuration. (a) Schematic of a cylindrical nanodimple array created on an aluminum film. (b) A cross section of the system. (c) Configuration of a periodic dimple array. (d), (e) Random arrangements of dimples that produce the smallest (d) and largest (e) internal losses in 100 samples.

Figure 2

Characteristics of spectral profile of Fano resonance. (a) Characteristics of the reflection spectrum determined by SSF $f^{\pm }$. (b), (c) Dependence of SSF on the direct reflection phase $\varphi$ and the ratio of decay rates $\chi$. The dashed line in (c) denotes the critical coupling condition  (10).

Figure 3

Extracted parameters for a flat metal film. Solid lines denote the results obtained by parameter extraction by ZPS. Dashed lines denote the results obtained by the parameter extraction by using the TCM method. (a), (b) Wave number $k_{\text{r}}$ and decay rate $\gamma$ of the resonant mode as functions of (a) the metal film thickness $h$ and (b) metal loss factor $\eta$. (c), (d) Direct reflection phase $\varphi$ and ratio of decay rates $\chi$ as functions of (c) $h$ and (d) $\eta$.

Figure 4

Resonant and dip angles extracted for a flat metal film. (a), (b) Dependence of angular spectra of reflection on (a) the metal film thickness $h$ and (b) metal loss factor $\eta$. Dotted lines depict the resonant angles extracted by the TCM method. Arrows denote the dip positions. (c), (d) Resonant angle extracted by ZPS (blue squares) and by the TCM method (orange circles) and dip angle (green triangles) as functions of (c) $h$ and (d) $\eta$. Dotted line depicts the resonant angle obtained using the dispersion relation of SPP.

Figure 5

Parameters extracted by the TCM method from the reflection spectrum of a flat metal film. (a), (b) Total decay rate $\gamma$, external decay rate ${\gamma }_{\text{e}}$, internal decay rate ${\gamma }_{\text{i}}$, and direct reflection phase $\varphi$ as functions of (a) metal film thickness $h$ and (b) metal loss factor $\eta$. (c) Relation between direct reflection phase $\varphi$ and the ratio of decay rates $\chi$ depending on $\eta$ and $h$ plotted on a color map of spectral shape factor $f^{-}$, where $h$ is varied from 22.2 nm (lower right) to 7.2 nm (upper left) by 1 nm.

Figure 6

Resonant and dip angles extracted for the nanodimple periodic array. (a)–(c) Dependence of the reflection spectrum on (a) dimple radius $r$, (b) period of dimple array $p$, and (c) depth of dimple $d$. Dotted lines depict the resonant angles extracted by the TCM method. Arrows denote the dip positions. (d)–(f) Resonant and dip angles as functions of (d) $r$, (e) $p$, and (f) $d$. Dotted line depicts the resonant angle obtained using the dispersion relation of SPP. Here, we assume that the thickness of the aluminum film outside the dimple is $h_0=22.2$ nm, and the fixed values of $r,p$, and $d$ are 80, 400, and 10.2 nm, respectively.

Figure 7

Parameters extracted by the TCM method from the reflection spectrum of the nanodimple periodic array. Total decay rate $\gamma$, external decay rate ${\gamma }_{\text{e}}$, internal decay rate ${\gamma }_{\text{i}}$, and direct reflection phase $\varphi$ as functions of (a) dimple radius $r$, (b) period of dimple array $p$, and (c) depth of dimple $d$. The system configuration condition is the same as that in Fig. 6.

Figure 8

Parameters extracted by the TCM method from the reflection spectrum of the nanodimple random array. (a) Dependence of the reflection spectrum on the depth of dimple $d$. Dotted lines depict the resonant angles. Arrows denote the dip positions. (b) Resonant angle and dip angle as functions of $d$. Dotted line depicts the resonant angle obtained using the dispersion relation of SPP. (c) Total decay rate $\gamma$, external decay rate ${\gamma }_{\text{e}}$, internal decay rate ${\gamma }_{\text{i}}$, and direct reflection phase $\varphi$ as functions of $d$. Here, we assume that the thickness of the aluminum film outside the dimple is $h_0=22.2$ nm, and the radius of the dimple is set to 80 nm.

Figure 9

Relation between the external decay rate ${\gamma }_{\text{e}}$ and the effective aluminum thickness $h_{\text{eff}}$ for nanodimple periodic array (red dots), nanodimple random array (green squares), and flat aluminum film (black solid line). The light-blue dashed line depicts the exponential dependence on $h_{\text{eff}}$ based on the value at $h_{\text{eff}}=22.2$ nm.

Figure 10

Parameters extracted by the TCM method from the reflection spectra for various configurations of nanodimples. (a) Direct reflection phase $\varphi$ and ratio of decay rates $\chi$ for periodic array (yellow circles) and random array (green squares) plotted on the color map of the spectral shape factor $f^{-}$. (b), (c) Thickness $h_{\text{SSF}}$ and the metal loss factor ${\eta }_{\text{SSF}}$ for the flat metal film that gives the same shape of the normalized reflection spectrum as the dimple array system with the effective thickness $h_{\text{eff}}$. (d) Internal decay rate ${\gamma }_{\text{i}}$ as a function of $h_{\text{eff}}$. Red dots, green squares, and the black solid line denote the results for nanodimple periodic array, nanodimple random array, and flat aluminum film, respectively.