OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 16992–17000
« Show journal navigation

Analysis of hybrid plasmonic-photonic crystal structures using perturbation theory

Ishita Mukherjee and Reuven Gordon  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 16992-17000 (2012)
http://dx.doi.org/10.1364/OE.20.016992


View Full Text Article

Acrobat PDF (913 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A perturbation theory approach for the analysis of hybrid plasmonic- photonic crystal structures is presented. This theory allows for accurate calculation of the resonance frequency shift and quality factor change when introducing a resonant plasmonic structure into a photonic crystal microcavity. An example calculation is shown, agreeing to within 5% with comprehensive finite difference time domain simulations but taking an order of magnitude less time. This theoretical approach overcomes the challenge of poor scaling in computations with hybrid plasmonic-photonic crystal structures, allowing for rapid design optimization in such hybrid geometries.

© 2012 OSA

1. Introduction

Dielectric photonic cavities offer light confinement by introducing bandgaps in the radiation zone [1

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

3

3. S. Hughes, “Coupled cavity QED using planar photonic crystals,” Phys. Rev. Lett. 98(8), 083603 (2007). [CrossRef]

], resulting in high quality factors for localized states. Limits on the mode volume pose a challenge to coupling to the subwavelength regime when using only dielectrics [4

4. H. Ryu, M. Notomi, and Y. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003). [CrossRef]

, 5

5. S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quantum Electron. 38(1-3), 257–267 (2006). [CrossRef]

]. Metal nanoparticles, on the other hand, due to their extremely small “mode” volume, are able to provide intense near-field (plasmonic) confinement but suffer from losses which give a poor quality factor [6

6. S. A. Maier, “Plasmonics: metal nanostructures for subwavelength photonic devices,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1214–1220 (2006). [CrossRef]

10

10. Y. X. Ni, D. L. Gao, Z. F. Sang, L. Gao, and C. W. Qiu, “Influence of spherical anisotropy on the optical properties of plasmon resonant metallic nanoparticles,” Appl. Phys., A Mater. Sci. Process. 102(3), 673–679 (2011). [CrossRef]

]. Recent works have attempted to reach a compromise between the two scenarios with the use of the hybrid approach, where a resonantly tuned nanoparticle is introduced into a photonic crystal cavity [11

11. M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010). [CrossRef]

14

14. C. Grillet, C. Monat, C. L. Smith, B. L. Eggleton, D. J. Moss, S. Frédérick, D. Dalacu, P. J. Poole, J. Lapointe, G. Aers, and R. L. Williams, “Nanowire coupling to photonic crystal nanocavities for single photon sources,” Opt. Express 15(3), 1267–1276 (2007). [CrossRef]

]. With such an approach, even though a loss in the quality is expected, the local field enhancement can increase [11

11. M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010). [CrossRef]

,15

15. I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express 19(23), 22462–22469 (2011). [CrossRef]

]. For many applications that do not require an ultra-high quality factor, such as high-bandwidth high-efficiency extraction of single photons [16

16. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78(17), 3294–3297 (1997). [CrossRef]

,17

17. M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient sources of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002). [CrossRef]

], this compromise preserves the best of both worlds.

Recently, photonic crystal nanobeams have been shown to possess high quality factors and have been found to retain a reasonably good quality even after being coupled to a metal nanoparticle [15

15. I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express 19(23), 22462–22469 (2011). [CrossRef]

,18

18. M. Kim, S. H. Lee, M. Choi, B. Ahn, N. Park, Y. H. Lee, and B. Min, “Low-loss surface-plasmonic nanobeam cavities,” Opt. Express 18(11), 11089–11096 (2010). [CrossRef]

]. Attempts at theoretical analyses of these nanobeams, however, have proven to be quite tedious. So far, photonic crystals have been comprehensively analyzed using techniques like finite difference time domain (FDTD), relying on brute force calculations for the entire cavity [19

19. M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal cavity with a quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16(23), 19136–19144 (2008). [CrossRef]

21

21. G. D. Kondylis, F. D. Flaviis, G. J. Pottie, and T. Itoh, “A memory efficient formulation of the finite difference time domain method for the solution of Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 49(7), 1310–1320 (2001). [CrossRef]

]. Changing the shape of the cavity or changing the metal particle requires a repeat computation.

To solve this problem for hybrid cavities, we propose a hybrid approach. This approach is based on perturbation theory. While the word “perturbation” is usually associated with an approximate formulation [22

22. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(6), 066611 (2002). [CrossRef]

28

28. L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B 76(4), 041102 (2007). [CrossRef]

], we stress that the perturbation theory presented here (following past microwave approaches [29

29. R. A. Waldron, Theory of Waveguides and Cavities (Maclaren & Sons, 1967).

]) is an exact formulation. With this hybrid approach, we calculate the fields of the bare cavity followed by a separate calculation of the local fields of the plasmonic particle on the waveguide (without the cavity) and combine the contributions from both the global and local fields to obtain the changed frequency as well as the quality factor, due to the introduction of the nanoparticle.

In this work, we present the analytical results obtained from applying perturbation theory to a silicon nitride photonic crystal nanobeam cavity. The source of perturbation is considered to be an ellipsoidal silver nanoparticle with the maximum extinction frequency tuned to the resonant frequency of the cavity. The frequency shifts and the new quality factors of the cavities with the nanoparticle obtained from perturbation theory are found to agree well with the FDTD results.

2. Perturbation theory

Considering E1 and H1 as the electric and magnetic fields in an unperturbed cavity with a resonant frequency ωsuch that:
E=E1eiωt
(1)
H=H1eiωt
(2)
Then, we can writeD1and B1as:
D1=ε0ε1E1
(3)
B1=μ0μ1H1
(4)
whereε0andμ0denote absolute permittivity and absolute permeability while ε1andμ1denote relative permittivity and permeability values of the cavity. On introducing a particle (perturbation) into the cavity, the electric and magnetic fields in the cavity undergo a change, thus changing the resonant frequency (ω+δω). The field expressions for the perturbed cavity can be given as [29

29. R. A. Waldron, Theory of Waveguides and Cavities (Maclaren & Sons, 1967).

]:
E'=(E1+E2)ei(ω+δω)t
(5)
H'=(H1+H2)ei(ω+δω)t
(6)
where E2 and H2are the corrections to the field from the perturbation. Assuming ε2 and μ2to be the relative permittivity and permeability of the perturbing particle respectively, D2 and B2 can be written as:
D2=ε0[ε2(E1+E2)ε1E1]
(7)
B2=μ0[μ2(H1+H2)μ1H1]
(8)
From Maxwell’s differential equations, we can derive from Eqs. (1) and (5):
×E1=jωB1
(9)
×E2=j{ωB2+δω(B1+B2)}
(10)
Similarly, the expressions for magnetic fields can be obtained as:
×H1=jωD1
(11)
×H2=j{ωD2+δω(D1+D2)}
(12)
Now we use the vector identitydiv{(H1×E2)+(E1×H2)}=E2.×H1H1.×E2+H2.×E1E1.×H2, which can be rewritten putting δδt=jωas:
H1.×E2+E1.×H2=jωE2.D1jωH2.B1div{(H1×E2)+(E1×H2)}
(13)
Using Eqs. (9) through (12), the right side of the above Eq. (13) can also be expressed as:
H1.×E2+E1.×H2=jω{E1.D2H1.B2}+jδω{(E1.D1H1.B1)+(E1.D2H1.B2)}
(14)
Substituting Eqs. (13) to (14) and integrating over the volume V1 of the cavity after some rearrangement, we obtain:
jδωV1{(E1.D1H1.B1)+(E1.D2H1.B2)}dV=jωV1{(E2.D1E1.D2)(H2.B1H1.B2)}dVV1div{(H1×E2)+(E1×H2)}dV
(15)
If S1is the surface enclosing V1, the divergence integral of the above equation can be replaced by the divergence theorem so that:
jδωV1{(E1.D1H1.B1)+(E1.D2H1.B2)}dV=jωV1{(E2.D1E1.D2)(H2.B1H1.B2)}dVS1{(H1×E2)+(E1×H2)}.dS
(16)
For the introduction of a small particle of volumeV2, the relative change in frequency,δω, in V1with respect to the cavity resonant frequency ω is now given by:

δωω=V2{(E2.D1E1.D2)(H2.B1H1.B2)}dV1jωS1{(H1×E2)+(E1×H2)}.dSV1(E1.D1H1.B1)+(E1.D2H1.B2)dV
(17)

The most common forms of this perturbation formula have been derived for cavities with perfectly conducting walls in case of which, the surface integral term vanishes [29

29. R. A. Waldron, Theory of Waveguides and Cavities (Maclaren & Sons, 1967).

]. However, in a dielectric waveguide with non-conducting walls, such as ours, there would be some energy flow through the waveguide which cannot be accounted for if we assume the surface integral to be zero. It is also commonly assumed that since D2 and B2 are much smaller than D1and B1respectively (as particle size V2 << cavity volume V1), the contribution of the second integral in the denominator may be neglected except in the neighborhood of the particle [28

28. L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B 76(4), 041102 (2007). [CrossRef]

, 29

29. R. A. Waldron, Theory of Waveguides and Cavities (Maclaren & Sons, 1967).

]. For higher accuracy, we have not made this assumption. Instead, in order to capture the local field corrections accurately while considering that D2and B2decay to zero at a finite distance, their effects have been taken into account to within a distance of 150 nm from the nanoparticle. This distance was chosen by inspection of the simulation domain over which the field from the nanoparticle has decayed to below106times its maximum value.

Clearlyδωωobtained from Eq. (17) is a complex term in the presence of loss. Now, assuming Ω to be the complex resonance frequency and Q1 to be the quality factor of the unperturbed cavity Ω can be expressed as [29

29. R. A. Waldron, Theory of Waveguides and Cavities (Maclaren & Sons, 1967).

]:
Ω=ω(1+i2Q1)
(18)
The complex resonance frequency of the perturbed cavity with a new resonance of Qa'can therefore be written as:
Ω+δΩ=(ω+δω)(1+i2Qa')
(19)
Using Eqs. (18) and (19), we can separateδωωobtained from Eq. (17) in its real and imaginary components with the following expression [26

26. A. Morand, K. Phan-Huy, Y. Desieres, and P. Benech, “Analytical study of the microdisk’s resonant modes coupling with a waveguide based on the perturbation theory,” J. Lightwave Technol. 22(3), 827–832 (2004). [CrossRef]

]:
δΩω=δω'ω+i{12Qa'12Q1}
(20)
From Eq. (20), it is clear that the real part of the expression obtained from Eq. (17) (written as δω'ω for convenience) gives the frequency shift, whereas the imaginary part (that we can write asδω''ω) gives the new quality factor. Using Eq. (20), we can therefore write the new quality factor of the perturbed cavity as:

Qa,=ω2(δω''+ω2Q1)
(21)

It must be considered that the above expression for the new quality factor only accounts for the absorption losses due to the perturbing particle. In order to calculate the additional contribution from scattering losses of the added particle, we separate the absorption and scattering times using absorption and scattering cross sections of the particle, represented by Cabs and Cscat respectively.τscatcan be calculated from the following expression [30

30. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall Inc., 1984).

]:
τscat=τabs(CabsCscat)
(22)
where τabsis the absorption time given by Qa'ω.

The modified quality factor accounting for both scattering and absorption,Q', is now given by the expressionQ'=ωτ' where τ' can be obtained as [30

30. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall Inc., 1984).

]:

1τ'=1τscat+1τabs
(23)

Hence, provided that the fields of the resonant cavity before the perturbation and the fields inside the perturbing sample after the perturbation are known, we can calculate the shift in the frequency of the cavity and its quality factor readily. For solving the problems of resonant hybrid photonic crystal cavities theoretically, we show below that this property of perturbation theory can save simulation time and memory, while predicting accurate changes in resonant frequencies and quality factors.

3. Perturbation theory applied to a photonic cavity with an Ag nanoparticle

Figure 1
Fig. 1 A FIB fabricated photonic crystal nanobeam cavity on silicon nitride [15].
shows an example of a previously fabricated 300 nm wide photonic crystal nanobeam cavity on a 200 nm thick silicon nitride membrane using focused ion beam milling (FIB). Typically, these cavities support a resonant mode at ~600 nm and have a quality factor of ~55,000 (theoretically) forn = 2.0. Inserting a small silver nanoparticle at the center of the nanobeam has been found to lower the resonant frequency as well as the quality factor of the cavity, the latter by a factor of 20 [15

15. I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express 19(23), 22462–22469 (2011). [CrossRef]

]. To achieve the same using perturbation theory, we approached the problem in the manner shown in Fig. 2
Fig. 2 (a) 3D schematic showing the photonic crystal nanobeam with a nanoparticle inside the simulation region. The pink arrow shows the direction of the propagation of the mode. The inset is a close-up on the nanoparticle at the center (b) Top view of the nanobeam in the simulation region, forming an unperturbed cavity (c) Zoom-in to the center of the nanobeam after the introduction of the nanoparticle. The shaded area shows the perturbed region considered.
.

The 3D schematic of Fig. 2(a) has been broken down into 2D cavities shown before (Fig. 2(b)) and after (Fig. 2(c)) perturbation. After exciting the 6800 × 300 × 200 nm3 nanobeam with a mode source centered at the resonant frequency of the cavity (shown in Fig. 2(a)), we obtain the unperturbed electric and magnetic field values (E1 andH1) of the cavity over the entire FDTD simulation region (6800 × 4300 × 1800 nm3) that represents the cavity volume (Fig. 2(b)). Next, a silver nanoparticle is placed at the center of the nanobeam and the perturbed cavity fields (E' andH') are obtained from the vicinity of the nanoparticle (white shaded region in Fig. 2(c)) while it still rests on the beam. The subsequent frequency shifts and reduced quality factors can now be calculated from the equations described in section 2.

To have a resonant enhancement, we tuned the plasmonic resonance of the silver nanoparticle on silicon nitride to that of the cavity. We used Johnson and Christy Ag database which offers lower losses, to conduct the simulations [31

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

]. It was found that a particle of size 60 × 52 × 10 nm3 produces an extinction peak at ~600 nm as shown in Fig. 3(a)
Fig. 3 (a) The scattering cross section of a 60 × 52 × 10 nm3 silver nanoparticle on a 200 nm thick blank Si3N4 substrate. (b)A transverse cross section through the nanoparticle showing the electric field distribution inside.
, when simulated on a silicon nitride substrate. Figure 3(b) shows a cross section of the particle on silicon nitride indicating non-uniformity of fields inside the particle due to the effect of the substrate.

4. Theory results and FDTD verification

SinceE1,H1andQ1 have to be known in order to implement our theory, we ran FDTD simulations (using Lumerical FDTD Solutions 7.5) on the bare nanobeam cavity to collect the fields, while noting the corresponding resonant frequency and quality factor. The other two known quantities,E'andH'were similarly collected from FDTD simulations on the nanoparticle.

We initially attempted a perturbation approach on our test case that simply assumed the standard polarizability of the ellipsoid [32

32. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, Inc., 1983).

] in the cavity field. In this case, the frequency shift and quality factor were found to be 0.457 THz and 808.6 which was not in good agreement with the FDTD results (2.32 THz and 679.08).

To verify the results, the nanoparticle was meshed with a mesh step size 5 times smaller than the cavity in x and y directions and 10 times smaller in z direction and FDTD simulations were conducted. The simulations showed a frequency shift of 0.36 THz and a quality factor of 1501.9 which agrees within 5% of the results predicted by our perturbation theory. The sources of error may involve finite domain mesh size when simulating the field change around the particle, numerical errors and the ad hoc method of calculating the quality factor.

In terms of calculation time, the bare cavity simulations took about 3.3 hours to run while the nanoparticle simulation took about 0.7 hours. A full cavity simulation with the nanoparticle took 48 hours to run. This indicates that our theory was able to cut down the total computation time by a factor of 12.

To show that a full recalculation was not necessary if the nanoparticle was changed, we replaced the ellipsoidal particle with a rectangular silver nanorod of dimensions 60 × 100 × 10 nm3, applied perturbation theory and verified the results using FDTD. Our theory predicted a frequency shift of 1.39 THz and a quality factor of 1270.17 as compared to FDTD which calculated a frequency shift of 1.42 THz and a quality factor of 1327.67. Similarly, to test the effect of our theory on other hybrid structures, we replaced the nanobeam cavity with a reduced hybrid cavity using perfect electric conductor walls as mirrors instead of the photonic crystal holes. Calculations were repeated using the ellipsoidal particle as the source of perturbation and a mesh step size of 1 nm in all directions to achieve the best possible convergence in a reasonable simulation time.

To set up the reduced cavity, the intended resonant wavelength (λ0≈600 nm) is at first used to calculate the wavelength of the mode (λwg) propagating through a given block of silicon nitride (neff = 1.569).λwgcan thus be expressed as:
λwg=λ0neff
(24)
Setting x=12λwg, y = 300 nm and z = 200 nm of the silicon nitride block, we allowed perfectly conducting boundaries in x direction (direction of propagation of the mode) and PML boundaries inyand z directions. A small amount of loss was introduced to the dielectric in the above lossless system to bring down the quality factor to a value comparable to that of the photonic nanobeam cavity. Simulations were conducted with and without the nanoparticle to obtain the fields as before.

Perturbation theory when applied to this cavity produced a frequency shift of 2.29 THz and a quality factor of 1157.9. These results were supported to within 5% by the FDTD simulations showing a frequency shift of 2.23 THz and a quality factor of 1208.86.

Figure 4
Fig. 4 The shift in the photonic nanobeam cavity frequency and the change in the quality factor with and without (blue) the addition of the nanoparticle from FDTD simulations (black) and perturbation theory (red).
shows the cavity resonant frequencies and quality factors for the photonic crystal nanobeam cavities with and without the nanoparticle in terms of wavelengths.

5. Conclusions

In this work, we calculated the resonance frequency shifts and quality factors of hybrid plasmonic nanoparticle/photonic crystal cavities after the introduction of the metal nanoparticle using perturbation theory. For an example calculation, the results were tested against comprehensive FDTD, showing agreement to within 5%. The technique based on a hybrid approach is found to be particularly insightful for analyzing the global and local field contributions in hybrid photonic crystal cavities, allowing for an order of magnitude speed up in the calculations. The results are promising for future applications especially involving large geometries, resulting in faster design process.

Acknowledgment

This work is supported by the British Columbia Innovation Council NRAS grant.

References and links

1.

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

2.

S. Hughes and P. Yao, “Theory of quantum light emission from strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express 17(5), 3322–3330 (2009). [CrossRef]

3.

S. Hughes, “Coupled cavity QED using planar photonic crystals,” Phys. Rev. Lett. 98(8), 083603 (2007). [CrossRef]

4.

H. Ryu, M. Notomi, and Y. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003). [CrossRef]

5.

S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quantum Electron. 38(1-3), 257–267 (2006). [CrossRef]

6.

S. A. Maier, “Plasmonics: metal nanostructures for subwavelength photonic devices,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1214–1220 (2006). [CrossRef]

7.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(1), 011101 (2005). [CrossRef]

8.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef]

9.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010). [CrossRef]

10.

Y. X. Ni, D. L. Gao, Z. F. Sang, L. Gao, and C. W. Qiu, “Influence of spherical anisotropy on the optical properties of plasmon resonant metallic nanoparticles,” Appl. Phys., A Mater. Sci. Process. 102(3), 673–679 (2011). [CrossRef]

11.

M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. 10(3), 891–895 (2010). [CrossRef]

12.

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic photonic nanodevice for label free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008). [CrossRef]

13.

P. E. Barclay, K. M. Fu, C. Santori, and R. G. Beausoleil, “Hybrid photonic crystal cavity and waveguide for coupling to diamond NV centers,” Opt. Express 17(12), 9588–9601 (2009). [CrossRef]

14.

C. Grillet, C. Monat, C. L. Smith, B. L. Eggleton, D. J. Moss, S. Frédérick, D. Dalacu, P. J. Poole, J. Lapointe, G. Aers, and R. L. Williams, “Nanowire coupling to photonic crystal nanocavities for single photon sources,” Opt. Express 15(3), 1267–1276 (2007). [CrossRef]

15.

I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express 19(23), 22462–22469 (2011). [CrossRef]

16.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78(17), 3294–3297 (1997). [CrossRef]

17.

M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient sources of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002). [CrossRef]

18.

M. Kim, S. H. Lee, M. Choi, B. Ahn, N. Park, Y. H. Lee, and B. Min, “Low-loss surface-plasmonic nanobeam cavities,” Opt. Express 18(11), 11089–11096 (2010). [CrossRef]

19.

M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal cavity with a quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16(23), 19136–19144 (2008). [CrossRef]

20.

Y. Liu, Z. Yang, Z. Liang, and L. Qi, “A memory efficient strategy for FDTD implementation applied to photonic crystal problems,” Prog. Electromagn. Res. 3, 374–378 (2007).

21.

G. D. Kondylis, F. D. Flaviis, G. J. Pottie, and T. Itoh, “A memory efficient formulation of the finite difference time domain method for the solution of Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 49(7), 1310–1320 (2001). [CrossRef]

22.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(6), 066611 (2002). [CrossRef]

23.

A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett. 8(1), 63–65 (1983). [CrossRef]

24.

A. Parkash, J. K. Vaid, and A. Mansingh, “Measurement of dielectric parameters at microwave frequencies by cavity-perturbation technique,” IEEE Trans. Microw. Theory Tech. 27(9), 791–795 (1979). [CrossRef]

25.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163(1-3), 86–94 (1999). [CrossRef]

26.

A. Morand, K. Phan-Huy, Y. Desieres, and P. Benech, “Analytical study of the microdisk’s resonant modes coupling with a waveguide based on the perturbation theory,” J. Lightwave Technol. 22(3), 827–832 (2004). [CrossRef]

27.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601 (2002). [CrossRef]

28.

L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B 76(4), 041102 (2007). [CrossRef]

29.

R. A. Waldron, Theory of Waveguides and Cavities (Maclaren & Sons, 1967).

30.

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

31.

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

32.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, Inc., 1983).

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(230.5298) Optical devices : Photonic crystals
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Photonic Crystals

History
Original Manuscript: May 15, 2012
Revised Manuscript: June 29, 2012
Manuscript Accepted: July 3, 2012
Published: July 11, 2012

Citation
Ishita Mukherjee and Reuven Gordon, "Analysis of hybrid plasmonic-photonic crystal structures using perturbation theory," Opt. Express 20, 16992-17000 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16992


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. K. J. Vahala, “Optical microcavities,” Nature424(6950), 839–846 (2003). [CrossRef]
  2. S. Hughes and P. Yao, “Theory of quantum light emission from strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express17(5), 3322–3330 (2009). [CrossRef]
  3. S. Hughes, “Coupled cavity QED using planar photonic crystals,” Phys. Rev. Lett.98(8), 083603 (2007). [CrossRef]
  4. H. Ryu, M. Notomi, and Y. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett.83(21), 4294–4296 (2003). [CrossRef]
  5. S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quantum Electron.38(1-3), 257–267 (2006). [CrossRef]
  6. S. A. Maier, “Plasmonics: metal nanostructures for subwavelength photonic devices,” IEEE J. Sel. Top. Quantum Electron.12(6), 1214–1220 (2006). [CrossRef]
  7. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98(1), 011101 (2005). [CrossRef]
  8. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science311(5758), 189–193 (2006). [CrossRef]
  9. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010). [CrossRef]
  10. Y. X. Ni, D. L. Gao, Z. F. Sang, L. Gao, and C. W. Qiu, “Influence of spherical anisotropy on the optical properties of plasmon resonant metallic nanoparticles,” Appl. Phys., A Mater. Sci. Process.102(3), 673–679 (2011). [CrossRef]
  11. M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett.10(3), 891–895 (2010). [CrossRef]
  12. F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic photonic nanodevice for label free detection of a few molecules,” Nano Lett.8(8), 2321–2327 (2008). [CrossRef]
  13. P. E. Barclay, K. M. Fu, C. Santori, and R. G. Beausoleil, “Hybrid photonic crystal cavity and waveguide for coupling to diamond NV centers,” Opt. Express17(12), 9588–9601 (2009). [CrossRef]
  14. C. Grillet, C. Monat, C. L. Smith, B. L. Eggleton, D. J. Moss, S. Frédérick, D. Dalacu, P. J. Poole, J. Lapointe, G. Aers, and R. L. Williams, “Nanowire coupling to photonic crystal nanocavities for single photon sources,” Opt. Express15(3), 1267–1276 (2007). [CrossRef]
  15. I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express19(23), 22462–22469 (2011). [CrossRef]
  16. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett.78(17), 3294–3297 (1997). [CrossRef]
  17. M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient sources of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett.89(23), 233602 (2002). [CrossRef]
  18. M. Kim, S. H. Lee, M. Choi, B. Ahn, N. Park, Y. H. Lee, and B. Min, “Low-loss surface-plasmonic nanobeam cavities,” Opt. Express18(11), 11089–11096 (2010). [CrossRef]
  19. M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal cavity with a quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express16(23), 19136–19144 (2008). [CrossRef]
  20. Y. Liu, Z. Yang, Z. Liang, and L. Qi, “A memory efficient strategy for FDTD implementation applied to photonic crystal problems,” Prog. Electromagn. Res.3, 374–378 (2007).
  21. G. D. Kondylis, F. D. Flaviis, G. J. Pottie, and T. Itoh, “A memory efficient formulation of the finite difference time domain method for the solution of Maxwell’s equations,” IEEE Trans. Microw. Theory Tech.49(7), 1310–1320 (2001). [CrossRef]
  22. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002). [CrossRef]
  23. A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett.8(1), 63–65 (1983). [CrossRef]
  24. A. Parkash, J. K. Vaid, and A. Mansingh, “Measurement of dielectric parameters at microwave frequencies by cavity-perturbation technique,” IEEE Trans. Microw. Theory Tech.27(9), 791–795 (1979). [CrossRef]
  25. M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun.163(1-3), 86–94 (1999). [CrossRef]
  26. A. Morand, K. Phan-Huy, Y. Desieres, and P. Benech, “Analytical study of the microdisk’s resonant modes coupling with a waveguide based on the perturbation theory,” J. Lightwave Technol.22(3), 827–832 (2004). [CrossRef]
  27. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.66(5), 055601 (2002). [CrossRef]
  28. L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B76(4), 041102 (2007). [CrossRef]
  29. R. A. Waldron, Theory of Waveguides and Cavities (Maclaren & Sons, 1967).
  30. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall Inc., 1984).
  31. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
  32. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, Inc., 1983).

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.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited