Plexciton Modes Guided by an Exciton Slab in a Columnar Thin Film

In this study, we reported plasmon-exciton coupling for excitation the surface plexciton in columnar thin lm with a central exciton slab using the transfer matrix method in Kretschmann conguration. The optical absorption spectra for surface plasmon polariton, surface exciton and surface plexciton was investigated at different structural parameters in proposed structure. The characteristics of surface optical modes were analyzed and there was an anticrossing behavior between polariton branches of plexciton spectra. Localization of surface modes on interfaces and hybridization between plasmons and excitons at both interfaces of exciton slab were proved by the time-averaged Poynting vector. We found that the types of coupling regimes between plasmons and excitons from weak to strong could be achieved. We found a high Rabi splitting energy 840 meV corresponding to the time period 5 fs which includes to the fast energy transfer between surface plasmon polaritons and surface excitons.


I. Introduction
The plexciton modes are new optical hybrid modes that result from a combination of exciton mode and plasmon polariton mode. Polariton modes usually split into two branches when the plasmon polariton frequency is close to the exciton frequency. In a plexcitonic system, the energy transfer between plasmons and excitons is of particular importance. Design a plexcitonic device with ultrafast energy transfer could be a mechanism for switching light by light on the nanoscale [1]. This energy transfer depends on the strength of the coupling between plasmons and excitons. The fast energy transfer can occur in a strong coupling regime with large Rabi splitting energy [2,3]. If the energies of the new optical modes are plotted as a function of the detuning energy (the energy difference between the plasmon and the exciton peaks), an anticrossing behavior is seen in the zero detuning with a minimum energy separation called the Rabi splitting energy, which determines the strength of the coupling between plasmons and excitons [4]. In the strong coupling regime, the excitation energy is shared and oscillates between plasmons and excitons (Rabi oscillations) [5,6]. Some of the applications of the strong coupling between plasmons and excitons have been demonstrated, including Fabry-Perot cavities [7], photonic crystals [8], semiconductor microcavities [9,10], single-atom lasers [11], all-optical switches [12,13] and quantum information processing [14].
In this study, the plasmon-exciton coupling in a columnar thin lm with central exciton slab was theoretically investigated under Kretschmann con guration using the transfer matrix method. In our work, the plexcitonic system consisted of a plasmonic medium (Silver thin lm) and a trilayer structure (Magnesium uoride columnar thin lm-J aggregate slab-Magnesium uoride columnar thin lm). We considered the excitons (Frankel excitons at room temperature) of a cyanine dye [5,5 ,6,6 -tetrachloro-di-(4-sulfobutyl) benzimidazolocarbocyanine(TDBC molecule) were self-assembled as J-aggregate due to strong dipole moment [15]. The theory in brief is presented in Section II, and followed by the results and discussions in Section III.

Ii. Theory In Brief
Let us, consider a Kretschmann con guration according to Fig. 1 for coupling surface plasmon polariton and exciton. The region 0 ≤ z ≤ d 1 is considered by a silver metal thin lm with relative permittivity ϵ Ag .
The region d 1 ≤ z ≤ (d 1 + d 2 ) is occupied by an oblique MgF 2 columnar thin lm, where d 2 = Lsinχ, L and χ is length and tilt angle of nanocolumns of columnar thin lm, respectively. In addition, the region (d 1 + d 2 ) ≤ z ≤ (d 1 + d 2 + d 3 ) is considered by an exciton slab (J-aggregate dyes) with relative permittivity ϵ ja . Again, the region (d 1 + d 2 + d 3 ) ≤ z ≤ (d t = d 1 + 2d 2 + d 3 ) is occupied by oblique MgF 2 columnar thin lm, while the regions z ≤ 0 and z ≥ d t are respectively a prism with relative permittivity ϵ 1 = n 2 1 and an air medium(n 2 =1). The anisotropic dielectric permittivity tensor ϵ MgF 2 for oblique MgF 2 columnar thin lm(CTF) is de ned as: where the superscript T indicates to the transpose of a tensor and χ is tilt angle of columns. The reference relative permittivity and tilt rotation tensors are respectively as: where ϵ a , b , c and u x , y , z are the effective relative permittivity scalars of the oblique MgF 2 columnar thin lm and the Cartesian unit vectors, respectively. Theϵ a , b , c could be obtained from the Bruggeman homogenization formalism [16].
The considered structure ( Fig. 1) exposed by p-linear polarized plane wave from the bottom of the prism at an angle to the z-axis in xz-plane. We can calculate the re ectance (r i = s , p ) and transmittance (t i = s , p ) amplitude using both Maxwell's curl equations, the continuity of the tangential components of electrical and magnetic elds at interfaces and solving following the algebraic matrix equation [17]: (t s t p 00) T = K θ  Fig. 2c that when frequency of the plasmon sates equals to frequency of the exciton states, the polariton mode in a plexciton system is splits to two branches upper polariton (UP) and lower polariton (LP). It is clear that there exist an anticrossing behavior between UP and LP polariton branches at zero detuning frequency. This matter is consistent with the coupled oscillator model (COM) [22] that it express, when two or more oscillators with speci c resonant frequencies are coupled, new hybrid resonant frequencies appear.
The absolute value of Cartesian components of electric and magnetic elds along the z-axis are given in Fig. 3 at λ = 594 nm with the same values used in Fig. 2 for the parameters. It is found in Fig. 3a1& Fig. 3a2 that the intensity of the elds E x , z and H y has increased at interface of metal and columnar thin lm (z=40nm). These elds are damped away from the interface, which it indicates to excitation and localization of surface plasmon polariton wave at metal/CTF interface. Slight roughness in the elds at z= 245nm is related to the removal of exciton slab in structure to propagate the surface plasmon polariton wave, which creates this numerical discontinuity in the calculations. Because we used only one code and this issue could be solved by considering a CTF with length 2L instead of two CTFs of upper and lower each of length L. In order to surface exciton wave, the metal thickness was considered zero in structure, see Fig. 3b1& Fig. 3b2.The z= 205 nm and z= 491 nm refer to CTF/exciton slab (lower interface) and exciton slab/CTF (upper interface), respectively. It is seen that the exciton surface wave is excited at both interfaces of exciton slab. We repeated the same calculations for the whole structure at λ = 594 nm (ω spp = ω x ) as shown in Fig. 3c1& Fig. 3c2. It is observed that the intensity of electromagnetic elds is decreased at metal/CTF interface (z=40 nm) and we expect this energy to be transferred to the excitons and a coupling to take place at both interfaces of exciton slab.
In order to further analyzation the coupling between plasmons and excitons, the Cartesian components of time-averaged Poynting vector and local absorption (LAP) [20] as a function of the depth of the structure are shown in Fig. 4. The increase of the x-component Poynting vector compared to the other components indicates to the localization of the surface plasmon polariton wave at the metal/CTF interface (Fig. 4a1).

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It is well known in local absorption plot (Fig. 4a2) with maximum intensity at z=40nm.The Fig. 4b1&Fig. 4b2 show the excitation of surface exciton wave at both interface of exciton slab. The surface plexciton waves are excited at internal (lower surface) and external interfaces (upper surface) of exciton slab (see Fig. 4c1&Fig. 4c2). A comparison between the plots of the local absorption show the intensity of absorption at z= 40 nm is decreased in Fig. 4c2 [23], where g is the coupling energy, ω up and ω lp are respectively the energy of upper and lower polariton peaks in plexciton spectra. It is seen that the both SPP and X modes were increased slightly by increasing n 1 , while the detuning angle decreased and the Rabi splitting energy increased slightly. Herein, there is an intermediate coupling regime (ICR) ( γ x cript > ) [24] between SPP and X modes. In ICR, the normal mode splitting occurs in the frequency domain, and an anticrossing behavior generally observed in optical absorption spectra [25].
The energy of SPP, X and PLX modes as a function of detuning frequency for different thicknesses of plasmon medium is depicted in Figs. 6a1-6a3.All parameters are the same as Here also, the coupling between SPP and X modes was ICR. The energy of polariton modes as a function of δ at different L, damping of plasmons and excitons, Rabi splitting energy and detuning angle as a function length of nanocolumns of √ dielectric medium(MgF 2 ) are given in Figs. 7a-7c. All parameters are the same as Fig. 6, except that d 1 =40nm.It is clear that the SPP, X and LP modes are almost degenerated at L=200nm while these modes are isolated at L=600nm .Due to the high energy separation between plexciton branches and the close damping of plasmons and excitons, the Rabi splitting energy increased, while the detuning angle was almost constant. It is found that there exist a strong coupling regime (SCR) (2g > (γ spp , γ x )) between plasmons and excitons. In SCR, the energy can be coherently transferred between the two SPP and X modes for at least a few times before their eventual decay [24].Here, Rabi splitting energy was between 430 meV and 840 meV which it corresponds to the the time period T = 2π /Ω for Rabi oscillation as much as 5-10 fs. This time indicates to the fast energy transfer between surface plasmon polaritons and surface excitons. refractive index of CTF and at low porosity, a denser CTF is obtained. We found here too an ICR between SPP and X modes. Finally, we repeated the same quantities as a function of thickness of exciton slab as shown in Figs. 10a-10c, all parameters were same as Fig. 9 except that χ = 25 °, f v = 0.8.In our work, the SPP,X and LP modes were degenerate at d 3 =225 nm while at other thicknesses of exciton slab were isolate. Herein, we found a SCR at d 3 =150, 175 nm and obtained an ICR at else d 3 .Therefore, different coupling regimes can be achieved by adjusting structural parameters in proposed structure. In this eld, researchers are looking for high splitting energy, so we did not pursue the weak coupling regime , where the optical modes of the coupled system remain unchanged from their initial uncoupled states.

Iv. Conclusion
The coupling between surface plasmon polaritons and surface excitons was theoretically studied by a central exciton slab in columnar thin lm in Kretschmann con guration using transfer matrix method. The surface polariton modes were extracted from optical absorption spectra and characteristics of them were investigated at different structural parameters. It is found that that different coupling regimes between plasmons and excitons can be achieved by adjusting structural parameters in proposed structure. More localization of the surface optical modes was proved by the components of the timeaveraged Poynting vector as a function of depth of the structure. The anticrossing behavior was observed between upper and lower polariton branches in plexciton system, while the SPP and X modes had same trend. It is obtained that the damping of plasmons and excitons reduced the Rabi splitting energy. In our work, we achieved to high Rabi splitting energy 840 meV that it showed there is a strong coupling regime between SPP and X modes. Therefore, a fast energy transfer occurs from the order of 5 fs between plasmons and excitons. Figure 1 A schematic of columnar thin lm with a central exciton slab in the Kretschmann con guration for coupling surface plasmon polariton and exciton.

Figure 2
Absorption density plots as functions of wavelength and incident angle for excitation of (a) surface plasmon polariton(SPP), (b) surface exciton(X) and (c) surface plexciton(PLX). The zero detuning frequency is speci ed by a vertical line in each of plots. Inset plots show the depiction of optical absorption versus wavelength at θδ (where the detuning frequency is zero). In plexciton plot exist an anticrossing mode between LP and UP branches of polaritons at δ=0 so that horizontal line shows the location of incident angle at zero detuning frequency. The used parameters were as: d1= 40 nm, L=600 nm, d3 = 286 nm, n1=2.57, χ =20 °, fv = 0.8, θδ =25.5°. See image above for gure legend See image above for gure legend Figure 10 See image above for gure legend