Analysis of hybrid plasmonic-photonic crystal structures using perturbation theory

Ishita Mukherjee; Reuven Gordon

doi:10.1364/OE.20.016992

1. Introduction

Dielectric photonic cavities offer light confinement by introducing bandgaps in the radiation zone [1–3], 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, 5]. 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 [6–10]. 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 [11–14]. With such an approach, even though a loss in the quality is expected, the local field enhancement can increase [11,15]. For many applications that do not require an ultra-high quality factor, such as high-bandwidth high-efficiency extraction of single photons [16,17], 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 [15,18]. 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 [19–21]. 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 [22–28], we stress that the perturbation theory presented here (following past microwave approaches [29]) 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.

Unlike comprehensive simulation techniques, perturbation theory offers physical insight related to the hybrid approach by splitting up the problem naturally and considering the cavity and local field effects separately. Multiple designs with different nanoparticles can be pursued with recalculations needed for only the local part. Similarly, if different cavities are chosen with the same local contribution from a particular nanoparticle, then only the cavity calculation need be repeated. Overall, this results in a faster design process. FDTD simulations for hybrid cavities require the smallest mesh step to be 1 nm for an accurate calculation of fields in the vicinity of the nanoparticle but can use a 10 × larger mesh step for bare cavities. This accounts for a speed up factor of 10 from time step (Courant condition) and a factor of 10³ from spatial (dx, dy, dz) step size. Even with a non-uniform mesh, the Courant stability requires that the time step be matched by the smallest grid size in the simulation.

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 $E_{1}$ and $H_{1}$ as the electric and magnetic fields in an unperturbed cavity with a resonant frequency $ω$ such that:

E = E_{1} e^{i ω t}

H = H_{1} e^{i ω t}

Then, we can write

D_{1}

and

B_{1}

as:

D_{1} = ε_{0} ε_{1} E_{1}

B_{1} = μ_{0} μ_{1} H_{1}

where

ε_{0}

and

μ_{0}

denote absolute permittivity and absolute permeability while

ε_{1}

and

μ_{1}

denote 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 + δ ω

). The field expressions for the perturbed cavity can be given as [29]:

E' = (E_{1} + E_{2}) e^{i (ω + δ ω) t}

H' = (H_{1} + H_{2}) e^{i (ω + δ ω) t}

where

E_{2}

and

H_{2}

are the corrections to the field from the perturbation. Assuming

ε_{2}

and

μ_{2}

to be the relative permittivity and permeability of the perturbing particle respectively, D₂ and B₂ can be written as:

D_{2} = ε_{0} [ε_{2} (E_{1} + E_{2}) - ε_{1} E_{1}]

B_{2} = μ_{0} [μ_{2} (H_{1} + H_{2}) - μ_{1} H_{1}]

From Maxwell’s differential equations, we can derive from Eqs. (1) and (5):

\nabla \times E_{1} = - j ω B_{1}

\nabla \times E_{2} = - j {ω B_{2} + δ ω (B_{1} + B_{2})}

Similarly, the expressions for magnetic fields can be obtained as:

\nabla \times H_{1} = j ω D_{1}

\nabla \times H_{2} = j {ω D_{2} + δ ω (D_{1} + D_{2})}

Now we use the vector identity

d i v {(H_{1} \times E_{2}) + (E_{1} \times H_{2})} = E_{2} . \nabla \times H_{1} - H_{1} . \nabla \times E_{2} + H_{2} . \nabla \times E_{1} - E_{1} . \nabla \times H_{2}

, which can be rewritten putting

\frac{δ}{δ t} = j ω

as:

H_{1} . \nabla \times E_{2} + E_{1} . \nabla \times H_{2} = j ω E_{2} . D_{1} - j ω H_{2} . B_{1} - d i v {(H_{1} \times E_{2}) + (E_{1} \times H_{2})}

Using Eqs. (9) through (12), the right side of the above Eq. (13) can also be expressed as:

H_{1} . \nabla \times E_{2} + E_{1} . \nabla \times H_{2} = j ω {E_{1} . D_{2} - H_{1} . B_{2}} + j δ ω {(E_{1} . D_{1} - H_{1} . B_{1}) + (E_{1} . D_{2} - H_{1} . B_{2})}

Substituting Eqs. (13) to (14) and integrating over the volume

V_{1}

of the cavity after some rearrangement, we obtain:

\begin{array}{l} j δ ω \underset{V_{1}}{∭} {(E_{1} . D_{1} - H_{1} . B_{1}) + (E_{1} . D_{2} - H_{1} . B_{2})} d V = \\ j ω \underset{V_{1}}{∭} {(E_{2} . D_{1} - E_{1} . D_{2}) - (H_{2} . B_{1} - H_{1} . B_{2})} d V - \underset{V_{1}}{∭} d i v {(H_{1} \times E_{2}) + (E_{1} \times H_{2})} d V \end{array}

If

S_{1}

is the surface enclosing

V_{1}

, the divergence integral of the above equation can be replaced by the divergence theorem so that:

\begin{array}{l} j δ ω \underset{V_{1}}{∭} {(E_{1} . D_{1} - H_{1} . B_{1}) + (E_{1} . D_{2} - H_{1} . B_{2})} d V = \\ j ω \underset{V_{1}}{∭} {(E_{2} . D_{1} - E_{1} . D_{2}) - (H_{2} . B_{1} - H_{1} . B_{2})} d V - \iint_{S_{1}} {(H_{1} \times E_{2}) + (E_{1} \times H_{2})} . d S \end{array}

For the introduction of a small particle of volume

V_{2}

, the relative change in frequency,

δ ω

, in

V_{1}

with respect to the cavity resonant frequency

ω

is now given by:

\frac{δ ω}{ω_{}} = \frac{\underset{V_{2}}{∭} {(E_{2} . D_{1} - E_{1} . D_{2}) - (H_{2} . B_{1} - H_{1} . B_{2})} d V - \frac{1}{j ω} \iint_{S_{1}} {(H_{1} \times E_{2}) + (E_{1} \times H_{2})} . d S}{\underset{V_{1}}{∭} (E_{1} . D_{1} - H_{1} . B_{1}) + (E_{1} . D_{2} - H_{1} . B_{2}) d V}

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]. 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 $D_{2}$ and $B_{2}$ are much smaller than $D_{1}$ and $B_{1}$ respectively (as particle size $V_{2}$ << cavity volume $V_{1}$ ), the contribution of the second integral in the denominator may be neglected except in the neighborhood of the particle [28, 29]. For higher accuracy, we have not made this assumption. Instead, in order to capture the local field corrections accurately while considering that $D_{2}$ and $B_{2}$ decay 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 below $10^{- 6}$ times its maximum value.

Clearly $\frac{δ ω}{ω}$ obtained from Eq. (17) is a complex term in the presence of loss. Now, assuming $Ω$ to be the complex resonance frequency and $Q_{1}$ to be the quality factor of the unperturbed cavity $Ω$ can be expressed as [29]:

Ω = ω (1 + \frac{i}{2 Q_{1}})

The complex resonance frequency of the perturbed cavity with a new resonance of

Q_{a}^{'}

can therefore be written as:

Ω_{} + δ Ω = (ω + δ ω) (1 + \frac{i}{2 Q_{a}^{'}})

Using Eqs. (18) and (19), we can separate

\frac{δ ω}{ω}

obtained from Eq. (17) in its real and imaginary components with the following expression [26]:

\frac{δ Ω}{ω_{}} = \frac{δ ω'}{ω_{}} + i {\frac{1}{2 Q_{a}^{'}} - \frac{1}{2 Q_{1}}}

From Eq. (20), it is clear that the real part of the expression obtained from Eq. (17) (written as

\frac{δ ω'}{ω}

for convenience) gives the frequency shift, whereas the imaginary part (that we can write as

\frac{δ ω''}{ω}

) gives the new quality factor. Using Eq. (20), we can therefore write the new quality factor of the perturbed cavity as:

Q_{a}^{,} = \frac{ω}{2 (δ ω'' + \frac{ω}{2 Q_{1}})}

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 $C_{a b s}$ and $C_{s c a t}$ respectively. $τ_{s c a t}$ can be calculated from the following expression [30]:

τ_{s c a t} = τ_{a b s} (\frac{C_{a b s}}{C_{s c a t}})

where

τ_{a b s}

is the absorption time given by

\frac{Q_{a}^{'}}{ω}

.

The modified quality factor accounting for both scattering and absorption, $Q^{'}$ , is now given by the expression $Q^{'} = ω τ^{'}$ where $τ^{'}$ can be obtained as [30]:

\frac{1}{τ^{'}} = \frac{1}{τ_{s c a t}} + \frac{1}{τ_{a b s}}

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 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) for $n$ = 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]. To achieve the same using perturbation theory, we approached the problem in the manner shown in Fig. 2 .

Fig. 1 A FIB fabricated photonic crystal nanobeam cavity on silicon nitride [15].

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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.

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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 ( $E_{1}$ and $H_{1}$ ) 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 ( $E'$ and $H'$ ) 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]. 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) , 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.

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.

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4. Theory results and FDTD verification

Since $E_{1}$ , $H_{1}$ and $Q_{1}$ 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, $E'$ and $H'$ 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] 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).

Applying the modified perturbation theory next, $E_{2}$ and $H_{2}$ were obtained using Eqs. (5) and (6). $D_{1}$ and $B_{1}$ were calculated from Eqs. (3) and (4) while $D_{2}$ and $B_{2}$ from Eqs. (7) and (8). Similar to our simulations, we used Johnson and Christy database to determine the $ε_{2}$ of the particle, which at the resonance wavelength of 600 nm, is −16.0859 + 0.4429i. $μ_{2}$ was assumed to be 1.The calculated values were then substituted in Eq. (17) to obtain the complex resonance shift. As shown in Eq. (17), only the real component of this expression can give the frequency shift, which was calculated to be 0.32 THz. The quality factor due to absorption, $Q_{a}^{'}$ , from Eq. (21) was calculated to be 2469.4. $C_{a b s}$ and $C_{s c a t}$ were next obtained from nanoparticle simulations by subtracting the incident source field from the scattered field outside the source region. Using Eq. (22), $Q_{s c a t}$ therefore calculated from the expression $Q_{s c a t} = ω τ_{s c a t}$ was found to be 3431.3. Finally, following Eq. (23), $Q^{'}$ was calculated to be 1436.

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 ( $λ_{0}$ ≈600 nm) is at first used to calculate the wavelength of the mode ( $λ_{w g}$ ) propagating through a given block of silicon nitride ( $n_{e f f}$ = 1.569). $λ_{w g}$ can thus be expressed as:

λ_{w g} = \frac{λ_{0}}{n_{e f f}}

Setting

x = \frac{1}{2} λ_{w g}

,

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 in

y

and

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.

From the discussion above, we can conclude that both the cases of changing nanoparticle and changing cavity showed good agreement between the numerical simulation and the perturbation method. It was also observed that with a larger mesh size of 2 nm in the z-direction, the difference between the FDTD and perturbation method was 11%. This can be improved to 4.4% at the expense of increased simulation time using a finer grid of 1 nm, which is found to be a good compromise between reasonable computation times and accuracy. It is possible to consider extending this approach to whispering gallery mode cavities and multiple metal nanoparticles, but so far we have not attempted such a diversity of calculations.

Figure 4 shows the cavity resonant frequencies and quality factors for the photonic crystal nanobeam cavities with and without the nanoparticle in terms of wavelengths.

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).

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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).

Analysis of hybrid plasmonic-photonic crystal structures using perturbation theory

Abstract

1. Introduction

2. Perturbation theory

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

4. Theory results and FDTD verification

5. Conclusions

Acknowledgment

References and links

Cited By

Figures (4)

Equations (24)

Optics Express