Large Plasmonic Resonance Shifts from Metal Loss in Slits

The impact of loss on the plasmonic resonances in metal–insulator–metal slits is analyzed, particularly the significant effect of loss on the reflection phase. The reflection is calculated analytically using single mode matching theory with the unconjugated form of the orthogonality relation. This theoretical calculation agrees well with comprehensive simulations, but differs substantially from the conjugated orthogonality result, as was used in past analytical works. This reflection phase has a large impact on the plasmonic resonance wavelengths, which are calculated using a Fabry–Pérot theory and compared with past experiment and finite-difference time-domain simulations.

The theory of the transmission of light through single slit in a perfect electric conductor has been studied extensively in the past [22,23]. It was shown that the maxima in the transmission spectrum are different from the normal Fabry-Pérot resonances, with zero phase shift upon reflection, as confirmed by experiment [24]. In MIM structures, the plasmon wavevector and the phase of reflection from the end faces determine the resonant length of the cavity. The resonance length is not equal to the simple multiple of the half-wavelength in general because of large reflection phase [25,26]. Therefore, to predict MIM cavity resonances, calculation of the reflection phase is necessary. SPP reflection has been investigated for single slit with different dielectric and metallic materials [27][28][29]. For MIM cavities, analytical calculations of the reflection phase have been done for lossless and dispersion-free metals by our group [23]. Another group advanced this theory for a real metal while still using the conjugated form of the orthogonality relation [30].
In this paper, we calculate analytically the reflection coefficient and phase acquired by SPPs upon reflection from the slit end-face using a simple mode matching model for real metals exhibiting both dispersion and loss. The approach is similar to past works [23,30] except that we employ the unconjugated form of the orthogonality relation. Our method shows good agreement with comprehensive finite-difference time-domain (FDTD) simulations, even for thin slits at the nanometer scale. By comparison, we show that the conjugated form of the orthogonality, as used in past works, differs considerably from the simulation results at shorter wavelengths, especially for narrower slits. We use the reflection phase to calculate plasmon resonances and plot the dispersion. The resulting dispersion shows good agreement with past experimental results [1]. The unconjugated approach is more accurate than the conjugated form because the conjugated form is not a proper orthogonality relation (i.e., it gives nonzero values for different modes) when there is loss or gain [31] The unconjugated form is a proper orthogonality relation for the loss/gain cases and it isolates each mode given a value of zero for different modes-for the case of single mode matching approximation, the unconjugated form is therefore more accurate. It is possible to calculate the phase for non-resonant cases. We attempted this approach to calculate the phase [32]; however, we found that it is numerically challenging and does not give a clear physical interpretation. Our theory shows a clear physical peak that makes the analysis more transparent. Figure 1 shows a schematic of the MIM waveguide resonator. Two infinitely thick metal layers are separated by a thin dielectric. The thickness of the dielectric is denoted by w and the SPPs propagate in the z direction. The metal permittivity and the insulator (dielectric) permittivity are m and d .

Unconjugated Orthogonality Reflection Theory
Loss affects the cavity resonance by changing the phase of reflection, in addition to the quality factor. In general, MIM waveguides support SPP modes with both symmetric and antisymmetric electric field profiles. If the dielectric is sufficiently thin, only the mode with a symmetric field profile is propagating; such cavities are considered here. We calculate the propagation constant and the reflection coefficient at the interface to free space for the fundamental TM mode within a slit.
The spatial dependence of the x component of the electric field for the lowest order TM mode within the slit can be formulated as: , where the superscript s denotes the MIM surface plasmon electric field, k z is the propagation constant in the z direction, k 0 is the free-space wave vector. The corresponding y component of the magnetic field can be found using the Maxwell-Ampère relation.
The propagation constant is given by: We solve this equation for complex k z by an iterative approach.
The reflection coefficient is calculated using single mode matching theory with the unconjugated orthogonality. Continuity of the tangential electric and magnetic fields at the boundary, z=0, gives: where the electric field in the free space region is written as an infinite sum of plane wave modes and t and r are transmission and reflection coefficients.
Using the unconjugated form of the orthogonality relation, we multiply both side of Eqs. 3 Fig. 1 Schematic representation of the slit with complex permittivity d in a real metal with complex permittivity m . The electric field profile of the symmetric SPP mode is shown solving Eqs. 6 and 7 for r, the reflection coefficient can be found: where H s y is the surface plasmon magnetic field in the cavity, Z 0 is the characteristic impedance of free space. We calculate k z from Eq. 2 for lossy and lossless cases (with and without considering the imaginary part of the relative permitivity of the metal) and use the values to calculate the electric and magnetic field using Eq. 1 and then calculate the reflection coefficient in Eq. 8. Comparing Eq. 8 with past work [30], the unconjugated orthogonality is different here and appears in the denominator of G. This has a significant impact on the reflection coefficient when loss is considered. Figure 2 shows the impact of loss for an Au-air-Au structure. As it is clear for shorter wavelengths and shorter width, the metal loss affects phase of reflection. In a lossy MIM structure, the mode index does not get as large as in the lossless case at short wavelengths. Consequently, the smaller index and the decreased confinement lead to the difference for lossy and lossless case. At longer wavelengths, because the SPP mode is pushed out of the metal, losses are negligible and the reflection characteristics are similar to the lossless case. A huge difference can also be seen when considering the unconjugated form of the orthogonality relation. To make a comparison with past work [30], the phase of reflection has been calculated for three different metals (Au, Al and Ag). Figure 3 compares phase of reflection for slit width of 50 nm considering loss of the metal using both conjugated and unconjugated forms. Comparing the results for Ag with the results past works [30], the difference of reflection phase in the two methods is obvious. This shows the importance of considering both the metal loss and conjugated form of the orthogonality relation in the calculation of the reflection phase. Figure 4 compares reflection coefficient phase for an Au-air-AU structure for different slit width using both the conjugated [30] and unconjugated forms of the orthogonality. The unconjugated form results are plotted with solid lines, and the conjugated results are plotted with dashed lines. A clear and significant difference is seen between the two approaches, which is most profound for narrower slits and shorter wavelengths. The reflection phase decreases with increasing wavelength and is larger for a thicker slit. The phase is clearly affected by the introduction of losses in the metal, which is different from what was suggested in past works [30].

Comparison with FDTD Simulations
To determine the validity of this theory, we used commercial FDTD simulations (Lumerical FDTD ver. 2020 R2.3). Perfectly matched layer boundary conditions were used to prevent reflection of the outgoing waves. The simulation cross section area was chosen to surround the whole structure. The mesh sizes along the x and z directions were reduced in size to ensure convergence (which was achieved at 0.01 nm for the smallest slit). A normally incident plane wave source was used to excite the structure. The simulation domain was 1.6 × 1.5 m. A frequency-domain power monitor was placed at the exit side of the slit to record the transmission. In addition to convergence tests, the simulations were compared with past simulations for the same conditions. Figure 5 shows the power distribution inside the slit for the slit width of 20 nm and L=250 nm. A TMpolarized plane wave of =1.08 m is incident from the left. The white dashed lines are the borders of the slit. Figure 6a and b shows the electric and magnetic fields inside the structure along the z and x directions. The figure compares well with past simulations [20]. Energy confinement is clearly visualized in the figures. Furthermore, the field distribution shows a standing wave inside the MIM region. Note that the electric field is maximized near the entrance and the exit surfaces. The strong energy confinement of MIM cavities can give strong light-matter interaction, even with losses considered. Figure 7 shows the numerically simulated transmission through slit width of 20 nm for different values of L. Resonant transmission peaks are observed with a wavelength that increases as the slit length increases. The propagation constant, k z , at the resonance wavelength was calculated in FDTD using the mode-source feature. The phase of reflection, Φ r , was calculated using (9) Φ r = m − k z L where L is the length of the slit, m is integer order of resonance. Figure 8 compares the phase of reflection calculated using the formulation above and the FDTD simulations with the theory presented in the last section. Equation 9 is the usual formula for the FP case. Here the difference is that Φ r is modified for the lossy case following the theory outlined above. Good agreement is seen between the analytic theory and the comprehensive FDTD results, thus supporting the use of the unconjugated orthogonality formulation. Although a slight difference can be seen for shorter wavelengths and especially for the shortest slit width, comparing this figure with Fig. 4 shows that the unconjugated method results are close to the simulation results. The difference between our results and the simulation results comes from simulation accuracy and the approximation in the single mode matching theory.

Comparison with Experimental Data
We compare the theory with a past experiment of narrow slits using a silica spacer layer [1]. A dielectric constant of d = 2.1 was used in Eq. 2 to match the experimental conditions. Figure 9 shows the experimental data of the MIM waveguides for different slit widths and lengths, digitized from [1]. The theoretical dispersion curves are shown for three different cases: the dashed line ignores the phase of reflection, the thick solid line is from the theory presented above, and the dotted line is our FDTD simulation. Clearly, the phase of reflection plays an important role in determining the plasmonic resonances seen in experiment, and the simple theory presented above accurately accounts for this phase.

Conclusion
In conclusion, we have provided a theory for the phase of reflection from a plasmonic slit that incorporates the real loss of the metal using the unconjugated form of the orthogonality relation. The impact of the real loss on the reflection phase is significant. Thereby, the imaginary part of the permittivity impacts not only the quality factor, but also the plasmonic resonance wavelength. The results for the unconjugated form differ significantly from the conjugated form that was presented in past works. The theory presented here agrees well with full-field FDTD simulations and past experiments, showing promise for simple theoretical investigation of future plasmonic MIM structures.
Author Contributions ZS performed all calculations. ZS and RG conceived of approach, developed theory and wrote manuscript. Data Availability All relevant data is presented in the manuscript.
Code Availability Lumerical and MATLAB code can be made available at reasonable request.

Declarations
Ethics Approval Not applicable.
Consent to Participate Not applicable.  Dispersion relation for an Au-silica-Au structure. Solid lines: calculation of dispersion considering the effect of the reflection phase calculated using unconjugated form of the orthogonality relation. Dashed lines: calculation of dispersion ignoring the effect of the reflection phase as was done in the past analysis. Dotted lines: FDTD simulation. Discrete points: experimental results using the experimentally measured plasmonic resonances [1]. The theory presented above shows good agreement with both FDTD and experiment