## Abstract

In this paper, a high-*Q/V _{eff}* gap-mode plasmonic Fabry-Perot nanocavity, which is composed of a silver nanowire on a flat silver substrate spaced by patterned dielectric distributed Bragg gratings, is investigated both analytically and numerically. The design parameters and properties of the nanocavity are exploited with the use of generalized Fabry-Perot model. The

*V*~0.0026 (λ/

_{eff}*n*)

^{3}and

*Q/V*~1.4 × 10

_{eff}^{5}/μm

^{3}of the nanocavity can be achieved. Such a gap-mode plasmonic Fabry-Perot nanocavity design provides a promising realization for wide novel band filters and spaser.

©2013 Optical Society of America

## 1. Introduction

Optical micro-resonant-cavities have received great attention due to their control light-matter interactions and the ability to confine and store optical energy in small volumes. These high cavity quality factor (*Q*) devices are useful in many applications such as wavelength filter, switch, especially, low-threshold lasers, brighter single-photon sources, etc., in quantum optics [1–6]. Traditional optical micro-cavities made of dielectric or semiconducting materials have large footprints due to the rapidly increasing radiated loss with the decreasing cavities dimensions. Plasmonic cavities that integrate surface plasmon polariton (SPP) with optical cavities can hugely reduce the effective mode volume (*V _{eff}*) to achieve high

*Q/V*. Owing to their many excellent characteristics and potential applications, various plasmonic cavities, especially utilizing the gap mode of SPP, have been investigated recently [7–10]. Among their many characteristics, the high

_{eff}*Q/V*is a very important figure of merit for their practical applications. However, the confinement of these cavities utilizing the gap mode of SPP is extremely strong in two dimensions due to the plasmonic gap-mode between nanowire and surface, but it weakens along the nanowire direction due to lower reflections between the end-facets of nanowire. Recently, a nanoscale plasmon resonator composed of a silver nanowire surrounded by patterned dielectric distributed high-reflection Bragg reflectors [11] can be used to improve the reflections and convert a broadband quantum emitter to narrow-band single-photon source.

_{eff}In this work, we propose and investigate a high-*Q/V _{eff}* gap-mode plasmonic FP nanocavity. Both generalized Fabry-Perot model theory and a finite-difference time-domain (FDTD) numerical model are developed and utilized to study and optimize the nanocavity to achieve a smaller effective mode volume (

*V*) and high

_{eff}*Q/V*

_{eff}.## 2.The design, numerical simulation results and theoretical model

The proposed high-*Q*/*V _{eff}* gap-mode plasmonic FP nanocavity is illustrated in Fig. 1
. The design consists of a silver nanowire and a patterned polymethylmethacrylate (PMMA) layer and a flat silver substrate. The patterned polymethylmethacrylate (PMMA) serve as spacer and Bragg reflectors. This silver nanowires-PMMA-silver film structure supports a SPP gap-mode whose electric field intensity is strongly confined within the gap formed between Ag nanowire and planar Ag substrate. And, the gap-mode along the nanowire direction is also confined and selected by two high-reflection distributed Bragg reflectors (DBRs), as constructing a Fabry-Perot cavity.

There are several distinct advantages of our design, which combine the benefits of SPP gap-mode [7,8] and pronounced Fabry-Perot mode [11]. The confinement of this architecture is excellent in all three directions, which greatly surpassed other photonic and plasmonic structures [7,8,11]. Especially, extraordinary enhancement of light-matter interactions can be achieved by engineering dielectric high-reflection Bragg reflectors and constructing SPP nanogap to confine light well below the diffraction limit. Moreover, the structure can be easily fabricated in the experiment. The gap spacing (*h*) can be precisely controlled by spin-coating PMMA film on planar Ag substrate [12, 13]. The patterned PMMA slabs, which can be fabricated by electron beam lithography or photoetching technology, do not suffer high scattering losses. Chemically synthesized single-crystalline silver nanowires with lower loss [11,14] can be drop-assembled on the PMMA. The architecture can also be used to explore active plasmonic physics and devices by mixing quantum dots or fluorescent molecules doped PMMA before fabrication.

The design parameters of the structures are as the follows. The radius of the silver nanowire is *R* = 80nm, the vertical thickness of gap spacing (PMMA) is *h* = 20nm, with refractive index *n* of 1.5, and the length of cavity between two DBRs gratings is *L* = 0.7μm, 0.28μm, 0.14μm, respectively. The periodicity of grating is *P* = 160nm, the width of air section is *d* = 80nm, the number of each grid is 6. And, the Drude model is used to describe the dispersion of Ag [15].

So that, the transmission spectrums, which are simulated with various cavity lengths by FDTD, are shown in Fig. 2(a) .

With no cavity (*L* = *d*) situation, the stop band with a bandwidth exceeding 100 nm is presented (black line in Fig. 2(a)). When the length of the cavity *L* is 0.7μm, 0.28μm, 0.14μm, respectively, a sharp resonant feature appears within the stop band at λ = 665nm, λ = 657nm, λ = 650nm. Figure 2(b)–2(d) are the simulated electric field intensity of the resonance at different *L*. As *L* is 0.14μm, the mode volume at 650nm wavelength can get

by using the Drude model result from the energy density of dispersive medium [16], and quality factor *Q*≈30 can be achieved.

For the structure, we can develop a generalized Fabry-Perot model to describe transmission spectrum of the nanocavity. The schematic diagram of a generalized Fabry-Perot model is as illustrated in Fig. 3 .

In this model, *L* also is the cavity length between the two DBRs. *A* is the amplitude of the light incident into the left DBRs. *U _{1}* and

*U*are the amplitudes of the light transmission from the left and right DBRs. We use $\tilde{r}$ and $\tilde{t}$(in general frequency dependent) to denote the complex reflection and transmission coefficient, respectively.

_{2}With *U*_{+} and *U*_{-} denoting the amplitudes of the left and right propagating SPP modes, one obtains

This leads to

Intensity of transmitted light from left and right at input and output DBRs are ${I}_{1}={U}_{1}{U}_{1}^{*}$ and ${I}_{2}={U}_{2}{U}_{2}^{*}$, respectively. When $\phi $is used to denote the reflection phase shift of DBRs, the ${I}_{2}$ can be expressed as

As well known, the relationship of light path shift and phase shift is $\phi =2\pi /\lambda \cdot \Delta L$, We can use the light path shift $\Delta L$(similar to half wave loss, it can be called as generalized wave loss) instead of $\phi $ to describe the reflection phase shift of DBRs. Then Eq. (8) becomes

_{sp}, $\tilde{t}$ and $\Delta L$ got.

So, we plot the complex SPP–gap mode dispersion relation (*k _{sp}*) by 2D finite element method (FEM) [14] and the normalized transmission spectrum of the DBRs by FDTD simulation, as shown in Fig. 4(a)
–4(c), respectively. The electric permittivity of silver is obtained from Refs [17] in FEM. The polynomial fitting curve of

*k*is shown as red line in Fig. 4(a), 4(b) and the inset of Fig. 4(a) is the electric field intensity on the cross section at 650nm wavelength. The value of the transmission coefficient $\tilde{t}$ can be obtained from Fig. 4(c), and $\left|\tilde{r}\right|=\sqrt{1-{\left|\tilde{t}\right|}^{2}}$(energy conservation). The transmission coefficient $\tilde{t}$is chosen as

_{sp}*L*= 0.14μm, the light path shift (generalized wave loss) is$\Delta L=-6(nm)$, shown in Fig. 4(d). We also have obtained the light path shift, which are all around −6nm, at different

*L*.

The above analyses are also verified by the generalized Fabry-Perot model. Applying these values to Eq. (9), we plot the normalized transmission power spectrum of the design nanocavity at various wavelengths by the generalized Fabry-Perot model, which is shown in Fig. 5
. It is clear that the resonant feature appears within the stop band at λ = 664nm, 658nm, and 648nm, respectively, which is in good qualitative correspondence with the result of simulation (*λ* = 665nm, 657nm, and 650nm). And, when the cavity length is *L* = 0.14μm, a quality factor *Q≈*26 can be achieved from the theoretical spectrum, which is also in good qualitative correspondence with the result of simulation (*Q*≈30).

## 3. Conclusions

In conclusion, we study the properties of SPP modes and ratio of *Q* factor to mode volume of silver nanowire-PMMA-silver film Fabry-Perot cavities. The characteristics of the device make the SPP mode localized in the dielectric gap layer, and SPP mode along the nanowire direction is confined and selected by dielectric Bragg reflectors. By developing a generalized Fabry-Perot model and using transmission $\tilde{t}$(frequency dependent), a sharp resonant feature of the transmission spectrum can be anticipated, and good qualitative correspondence is observed between theory and simulation. When the cavity length is *L* = 0.14μm, the *V _{eff}* of 0.0026 (λ/

*n*)

^{3},

*Q*≈30 and

*Q/V*of 1.4 × 10

_{eff}^{5}/μm

^{3}can be achieved. The cavity can be used to convert a broadband quantum emitter to a narrow-band single-photon source with color-selective emission enhancement by the cavity design. Furthermore, this silver nanowires-PMMA-silver structure is a practical transmission device, which has great advantages for highly integrated optical interconnects.

## Acknowledgments

This work is supported by the National Key Basic Research Program of China under grant no. 2012CB921900; National Natural Science Foundation of China under grant no. 61036005,60977019,11004182; Specialized Research Fund for the Doctoral Program of Higher Education (20113402110039).

## References and links

**1. **S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics **1**(8), 449–458 (2007). [CrossRef]

**2. **P. Bianucci, X. Wang, J. G. Veinot, and A. Meldrum, “Silicon nanocrystals on bottle resonators: Mode structure, loss mechanisms and emission dynamics,” Opt. Express **18**(8), 8466–8481 (2010). [CrossRef] [PubMed]

**3. **K. J. Vahala, “Optical microcavities,” Nature **424**(6950), 839–846 (2003). [CrossRef] [PubMed]

**4. **A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature **450**(7168), 402–406 (2007). [CrossRef] [PubMed]

**5. **R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stöhr, A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Wave–particle duality of single surface plasmon polaritons,” Nat. Phys. **5**(7), 470–474 (2009). [CrossRef]

**6. **D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**(5), 053002 (2006). [CrossRef] [PubMed]

**7. **K. J. Russell and E. L. Hu, “Gap-mode plasmonic nanocavity,” Appl. Phys. Lett. **97**(16), 163115 (2010). [CrossRef]

**8. **K. J. Russell, T. L. Liu, S. Cui, and E. L. Hu, “Large spontaneous emission enhancement in plasmonic nanocavities,” Nat. Photonics **6**(7), 459–462 (2012). [CrossRef]

**9. **A. Hohenau, P. Kusar, C. Gruber, and J. R. Krenn, “Analysis of damping-induced phase flips of plasmonic nanowire modes,” Opt. Lett. **37**(4), 746–748 (2012). [CrossRef] [PubMed]

**10. **R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**(7264), 629–632 (2009). [CrossRef] [PubMed]

**11. **N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. **108**(22), 226803 (2012). [CrossRef] [PubMed]

**12. **M. Yi, D. Zhang, X. Wen, Q. Fu, P. Wang, Y. Lu, and H. Ming, “Fluorescence enhancement caused by plasmonics coupling between silver nano-cubes and silver film,” Plasmonics **6**(2), 213–217 (2011). [CrossRef]

**13. **X. Wen, M. Yi, D. Zhang, P. Wang, Y. Lu, and H. Ming, “Tunable plasmonic coupling between silver nano-cubes and silver nano-hole arrays,” Nanotechnology **22**(8), 085203 (2011). [CrossRef] [PubMed]

**14. **C. L. Zou, F. W. Sun, Y. F. Xiao, C. H. Dong, X. D. Chen, J. M. Cui, Q. Gong, Z. F. Han, and G. C. Guo, “Plasmon modes of silver nanowire on a silica substrate,” Appl. Phys. Lett. **97**(18), 183102 (2010). [CrossRef]

**15. **P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

**16. **R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A **299**(2-3), 309–312 (2002). [CrossRef]

**17. ***Handbook of Optical Constants of Solids*, E. D. Palik, ed. (Academic Press, 1985).