OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 20 — Sep. 24, 2012
  • pp: 22172–22180
« Show journal navigation

Orders of magnitude enhancement of mode splitting by plasmonic intracavity resonance

Chao-Yi Tai and Wen-Hsiang Yu  »View Author Affiliations


Optics Express, Vol. 20, Issue 20, pp. 22172-22180 (2012)
http://dx.doi.org/10.1364/OE.20.022172


View Full Text Article

Acrobat PDF (1388 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

we report on significant mode splitting in plasmonic resonators induced by intracavity resonance. In contrast to traditional dielectric resonators where only picometer range of splitting was achieved, splitting over several hundred nanometers can be obtained without using ultrahigh quality resonators. We show that by appropriately choosing the coupling length, minute reflection is sufficient to establish intracavity resonance, which effectively lifts the degeneracy of the counterpropagating modes in the resonator. The mode splitting provides two self-referenced channels enabling simultaneous monitoring of the position and the polarizability of nano-scatterers in the resonator.

© 2012 OSA

1. Introduction

In the present study, we analyze the spectral responses based on plasmonic waveguide side coupled to resonators with various geometries. Of particular interest, we found that giant mode splitting can be achieved by minute reflections due to bent corners of resonators with moderate Q factors. Under critical coupling condition, it is found that the transmission spectra exhibit singlet or doublet dips depending on whether the Fabry-Pérot (FP) resonance was established in the coupling zone. The result calculated by finite difference time domain (FDTD) method is compared with coupled mode theory (CMT) and the discrepancy is well justified by considering the structural dispersion in bent corners. The time averaged Poynting vector was mapped out showing that with the absorption takes into account, nonzero power flow of the standing wave type resonance was formed. It should be noted that the power flow exhibits universal distributions of the power source, sink or saddle in regardless of the mode order rendering the system a promising platform for nano-object trapping, sorting, and sizing applications. A practical example where simultaneous detection of the position and the polarizability of a nano-scatterer exploiting the mode splitting as two monitor channels is illustrated.

2. Modeling

Figure 2
Fig. 2 The external quality factor as functions of the coupling length and gap width.
shows the critical coupling achieved with a variety of combinations of structural parameters. As expected, the Qex decreases monotonically with the increase of the perturbation, for instance, the coupling length. Normally, the larger the gap, the weaker the perturbation, and a longer coupling length is required to transfer the energy completely into the resonator.

3. Correspondence with coupled mode theory

To explain the splitting induced by intermodal coupling, coupled mode theory Eq. (2) was applied for the calculation of the net energy transferring between the clockwise (CW) and counter-clockwise (CCW) modes in time domain [15

15. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27(19), 1669–1671 (2002). [CrossRef] [PubMed]

, 16

16. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17(6), 1051–1057 (2000). [CrossRef]

, 20

20. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

]. The acw and accw are the complex amplitude of the CW and CCW modes, respectively. 1/τin and 1/τex are the decay rate due to the intrinsic and external loss associated with the quality factor Q = ω0τ/2 at the resonant frequency ω0. β denotes the backward coupling coefficient, and κ represents the coupling strength between the incident wave and the cavity modes. In steady state, the transmittance and reflectance subject to critical coupling condition can be represented by Eq. (3). If β<<1/τex, Eq. (3) can be reduced to Eq. (1), corresponding to the off-resonance condition in the coupling zone and traveling wave resonance occurs in the resonator. On the contrary, if β>>1/τex, intracavity resonance occurs in the coupling zone, and standing wave resonance results in the resonator. Due to the strong coupling between the CW and CCW waves, the unperturbed singlet resonant frequency ω0 splits into doublets and shifts to ω0 + β and ω0-β. The results are compared with our full vectorial FDTD simulation in Fig. 3.

dacwdt=i(ω-ω0)acw+(1τin+1τex)acw+βaccw+κs+daccwdt=i(ω-ω0)accw+(1τin+1τex)accw+βacw
(2)
T(ω)=|β2(ω-ω0)2+i(2/τin)(ω-ω0)(ω-ω0)2-i2(ω-ω0)(2/τin)+(2/τin)2+β2|2R(ω)=|i(2/τin)β(ω-ω0)2-i2(ω-ω0)(2/τin)+(2/τin)2+β2|2
(3)

To verify the type of resonance, distributions of time-average Poynting vector S are calculated. As shown in Fig. 4(a)
Fig. 4 Time-average Poynting vector in plasmonic rectangular resonator. (a) Standing wave resonance, which is composed of two counterpropagating traveling waves on resonance. (b) Traveling wave resonance.
, when the coupling zone is on-resonance, S is composed of the CW and CCW modes. Due to the intercoupling between the two modes, a power source forms at the center of the lower side, and a power saddle forms at the center of the upper side. The power flow exhibits a standing wave resonance. On the other hand, when the coupling zone is off-resonance, a purely traveling wave resonance is obtained, as shown in Fig. 4(b).

4. Phase front acceleration

5. Angle dependent reflectivity and mode splitting

Based on our analysis, the maximum mode splitting depends on the backward coupling coefficient (more precisely, the backward coupling ideality [20

20. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

]), which is associated with the bent angle of the plasmonic resonator. In order to clarify the interplay between the bent angle, backward reflectivity and the associated mode splitting, full vectorial FDTD calculation was applied to three bent angles: 120°, 90°, and 60°. The mode splitting for resonators constructed by the abovementioned bents subject to a common resonator length are calculated, as shown in Fig. 6(a)
Fig. 6 (a) Mode splitting of plasmonic resonators with rectangular, triangular, and parallelogram shapes. The dashed lines refer to the transmittance and the dotted lines refer to the reflectance. (b)-(e) The instantaneous distribution of the magnetic field intensity for straight, 120°, 90° and 60° bent waveguides, respectively.
. And the reflectivity for the bents at the wavelength of 1550nm are all referenced to the straight waveguide and calculated to be R120≅0.001, R90≅0.012 and R60≅0.051 respectively, as shown in Fig. 6(b)-6(e). As in our illustration, significant mode splitting ≅141nm was achieved in triangular-shaped resonator. Compared to the rectangular-shaped resonator, the mode splitting is twice larger which is attributed to the doubled ratio of the reflected amplitude (R60/ R90)0.5≅2. While for parallelogram-shaped resonator with uneven reflectivity at both ends of the coupling zone, the mode splitting valued in between the two extremes.

The on-resonance time averaged magnetic field intensity |Hy|2 of the plasmonic resonator is shown in Fig. 7
Fig. 7 The spatial distribution of the time-averaged magnetic field intensity |Hy|2 for various geometries at resonant condition.
. It is found that the energy distributions between the two standing waves are orthogonal, i.e., the nodes of energy density for the symmetric mode lie at the antinodes of antisymmetric mode.

6. Illustration of practical applications

The significant mode splitting provides two self-referenced channels which may facilitate the detection of the polarizability and the position of a nano-object simultaneously and independently. To illustrate how it works, a nano-object with size 100 nm × 100 nm and the refractive index of n is placed at the position z in the coupling zone, as shown in Fig. 8(a)
Fig. 8 (a) Schematic diagram of a nano-object in the rectangular ring cavity. (b) Spectrum shift of the nano-object with fixed normalized polarizability αs = 0.18 at various positions in the resonator. (c) Spectrum shift of the nano-object located at z = 0 with various polarizabilities. Note that the position and the polarizability can be determined simultaneously and independently, when the polarizability and the position are both variables, as shown in the contour map of (b) and (c).
. To generalize the sensing capability, the index variation can be correlated to the polarizability by [24

24. J. Avelin, R. Sharma, I. Hänninen, and A. H. Sihvola, “Polarizability analysis of cubical and square-shaped dielectric scatterers,” IEEE Trans. Antenn. Propag. 49(3), 451–457 (2001). [CrossRef]

]. Essentially the mode splitting is proportional to the spatial overlapping between the resonant mode and the nano-object, the position can therefore be determined. Since the field distribution of the symmetric and the antisymmetric modes form complementary set in spatial domain, presumably, the position of the nano-object can be monitored linearly and continuously. Here, the following cases are considered: when the nano-object with fixed normalized polarizability (αs = 0.18) shifts from the node (antinode) at z = 0 to the anti-node (node) at z = 324 of the symmetric (anti-symmetric) mode, the resonant wavelength of the symmetric (anti-symmetric) mode exhibits blue (red) shift δλ1 (δλ2), as shown in Fig. 8(b). The position of the nano-object can be determined by the normalized shift of the resonant wavelength defined by δλ2/(δλ1 + δλ2). On the other hand, when the nano-object is placed at z = 0 and the polarizability is varied, the electric field of the symmetric mode is disturbed and the corresponding resonant wavelength shifted, as shown in Fig. 8(c). It should be noted that the position can be independently determined in regardless of the polarizability of the nano-object which is merely a function of the total wavelength shift (δλ1 + δλ2), as shown in the contour map in Fig. 8(b) and 8(c). Thus by measuring the wavelength shift at the two individual resonances, the sensitivity of the polarizability and the position were estimated to be δαs = (δλ1 + δλ2)/50 and δz = 324 × δλ2/(δλ1 + δλ2), respectively.

7. Conclusion

We show that significant mode splitting can be achieved by plasmonic resonators without the necessity of ultrahigh quality factors. By tailoring the coupling length so as to establish the intracavity resonance in the coupling zone, mode splitting can be maximized. The mode splitting as large as 140 nm is achieved by the triangular resonator with a moderate quality factor Q = 120 in combination of corner reflections as low as 5%. Simultaneous detection of the position and the polarizability of a nano-object exploiting the splitted self-referenced resonances are illustrated. It is expected that with the orders of magnitude improvement of the mode splitting, the stringent demand for highly spectral resolution can be much relaxed, and low-cost handheld spectrometers will be sufficient to cope with sensing applications.

Acknowledgments

References and links

1.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54(3), R1777–R1780 (1996). [CrossRef] [PubMed]

2.

T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett. 34(4), 482–484 (2009). [CrossRef] [PubMed]

3.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004). [CrossRef] [PubMed]

4.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005). [CrossRef]

5.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007). [CrossRef] [PubMed]

6.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008). [CrossRef] [PubMed]

7.

A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low Q limit,” Appl. Phys. B 90(3-4), 561–567 (2008). [CrossRef]

8.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]

9.

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]

10.

D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Opt. Lett. 30(10), 1186–1188 (2005). [CrossRef] [PubMed]

11.

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010). [CrossRef] [PubMed]

12.

J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009). [CrossRef] [PubMed]

13.

A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90(18), 181102 (2007). [CrossRef]

14.

J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Plasmon flow control at gap waveguide junctions using square ring resonators,” J. Phys. D Appl. Phys. 43(5), 055103 (2010). [CrossRef]

15.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27(19), 1669–1671 (2002). [CrossRef] [PubMed]

16.

M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17(6), 1051–1057 (2000). [CrossRef]

17.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U.S.A. 108(15), 5976–5979 (2011). [CrossRef] [PubMed]

18.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010). [CrossRef]

19.

M. A. Ordal, R. J. Bell, R. W. Alexander Jr, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985). [CrossRef] [PubMed]

20.

A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

21.

M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering gallery modes,” J. Opt. Soc. Am. B 16(1), 147–154 (1999). [CrossRef]

22.

G.-X. Fan and Q. H. Liu, “An FDTD algorithm with perfectly matched layers for general dispersive media,” IEEE Trans. Antenn. Propag. 48(5), 637–646 (2000). [CrossRef]

23.

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000). [CrossRef]

24.

J. Avelin, R. Sharma, I. Hänninen, and A. H. Sihvola, “Polarizability analysis of cubical and square-shaped dielectric scatterers,” IEEE Trans. Antenn. Propag. 49(3), 451–457 (2001). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 17, 2012
Revised Manuscript: September 8, 2012
Manuscript Accepted: September 9, 2012
Published: September 12, 2012

Citation
Chao-Yi Tai and Wen-Hsiang Yu, "Orders of magnitude enhancement of mode splitting by plasmonic intracavity resonance," Opt. Express 20, 22172-22180 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22172


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54(3), R1777–R1780 (1996). [CrossRef] [PubMed]
  2. T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett. 34(4), 482–484 (2009). [CrossRef] [PubMed]
  3. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004). [CrossRef] [PubMed]
  4. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005). [CrossRef]
  5. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007). [CrossRef] [PubMed]
  6. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008). [CrossRef] [PubMed]
  7. A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low Q limit,” Appl. Phys. B 90(3-4), 561–567 (2008). [CrossRef]
  8. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]
  9. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]
  10. D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Opt. Lett. 30(10), 1186–1188 (2005). [CrossRef] [PubMed]
  11. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010). [CrossRef] [PubMed]
  12. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009). [CrossRef] [PubMed]
  13. A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90(18), 181102 (2007). [CrossRef]
  14. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Plasmon flow control at gap waveguide junctions using square ring resonators,” J. Phys. D Appl. Phys. 43(5), 055103 (2010). [CrossRef]
  15. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27(19), 1669–1671 (2002). [CrossRef] [PubMed]
  16. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17(6), 1051–1057 (2000). [CrossRef]
  17. T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U.S.A. 108(15), 5976–5979 (2011). [CrossRef] [PubMed]
  18. J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010). [CrossRef]
  19. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985). [CrossRef] [PubMed]
  20. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  21. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering gallery modes,” J. Opt. Soc. Am. B 16(1), 147–154 (1999). [CrossRef]
  22. G.-X. Fan and Q. H. Liu, “An FDTD algorithm with perfectly matched layers for general dispersive media,” IEEE Trans. Antenn. Propag. 48(5), 637–646 (2000). [CrossRef]
  23. J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000). [CrossRef]
  24. J. Avelin, R. Sharma, I. Hänninen, and A. H. Sihvola, “Polarizability analysis of cubical and square-shaped dielectric scatterers,” IEEE Trans. Antenn. Propag. 49(3), 451–457 (2001). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited