Abstract

For practical applications in quantum electrodynamics, it has been proposed to produce frequency tuning or Q-switching by dynamically changing the dielectric constant around a nano-cavity. Local changes in the dielectric constant of a photonic cavity with finite-lifetime, may affect not only the frequency of electromagnetic cavity modes but also their quality-factor (Q). Thus, it is important to have prediction capability regarding the combined effect of these changes. Here perturbation theory, usually applied to eigenmodes with real eigenvalues, is formulated in the complex domain, in which the eigen-frequency imaginary part is related to the Q-factor. Normalizable leaky modes, and bi-orthogonality in a finite volume are the basis for such a formulation. We introduce such capabilities by presenting semi-analytical expressions of first order perturbation analysis for a 3D cavity with radiation losses. The obtained results are in good agreement with numerical calculations.

©2010 Optical Society of America

1. Introduction

Recent progress in photonic crystal ultra-high-Q cavity design and fabrication [13] makes them very attractive for implementation in the fields of integrated optics (routing and filtering), basic light-matter interaction (cavity quantum electrodynamics) or in quantum-information technologies (quantum-bit-architectures). Dynamic photonic tuning is the phenomenon where electromagnetic cavity is modulated (by a local change in the dielectric constant) while the photonic mode is inside the cavity. Dynamic tuning of an ultra-high-Q cavity would be of a significant advantage, since this allows active tuning of both cavity mode decay and frequency, thus providing a control mechanism over cavity-waveguide, cavity-cavity and light matter interactions in the nano-scale [47]. The realization of dynamic tuning may rely on several physical mechanisms, such as temperature-tuning, non-linear effects, or free carrier induced modulation of the dielectric constant. In the latter a significant progress has been reported recently [8]. In a finite quality-factor (Q) cavity (with radiation losses), the modal fields are resonances, with complex eigenfrequencies. Perturbative tuning affects both the real and the imaginary part of the frequency (ω). Thus, it seems timely to seek for a basic understanding of the parameter space of (complex) frequency perturbative tuning. As a result of the tuning, the system is non-stationary and the frequency is not conserved (a modal field can “change its color” before leaving the cavity, and also its decay rate due to radiation losses is affected). One figure of merit in a quantum electrodynamics context can be the number of detuned line widths, or frequency tunability within the mode bandwidth. A second figure of merit is the degree of Q-switching capability within a (real) frequency range. For that, estimation of both the real frequency and the Q modulation (Δ[Re(ω)]andΔQΔ[Im(ω)], respectively) are needed. The objective of this work is to provide a simple semi-analytical description of the cavity mode complex frequency modulation via dielectric constant tuning. Perturbation can also be used in the design/optimization procedure: once a cavity has been defined with certain geometry (dielectric constant spatial distribution), and the cavity mode calculated by conventional numerical methods, it is possible to use perturbation changes in the dielectric constant to predict the response of Re(ω)and Q.

The effects of perturbative changes in the dielectric constant of a lossless electromagnetic resonator and their influence on an eigenmode frequency are known for some time [9]. Perturbation theory for lossless (closed) systems with only real eigenfrequencies was extensively described in quantum mechanics [1012] and in electromagnetic theory [13,14]. Obviously, the operator describing the wave equation for such systems is Hermitian and the eigenmodes are orthonormal and complete. These are key properties for the application of perturbation theory. In contrast to these systems, optical cavities in a dielectric or semiconductor structure with finite dimensions, exhibit a finite Q due to radiative losses (coupling to radiation modes, or free space propagation). This is characteristic to open systems. The Hamiltonian operator describing such systems is non-Hermitian, the modes are not stationary (leaky), hence, not orthonormal. Mathematically, leaky modes diverge at large values of the position vector. For that reason, traditional perturbation formulation is not applicable in those cases. Nevertheless, perturbative nature treatment of complex eigenvalue problems in microstructures was already introduced in [15], where the eigenvalue is the complex propagation constant of a guided mode, but, here we restrict ourselves to localized modes in all of the problem dimensionalities where the modes are localized (have no propagation constant) and where the governing eigenvalue is the eigenfrequency (or its square). Other important perturbative treatment for example is [16], where the tight-binding method is used to derive a semi-analytical condition for the degeneracy (non-degeneracy) of photonic crystals cavity modes. Nevertheless, we are interested in developing perturbation theory by which we mean an equivalent to perturbation theory of quantum mechanics [1012] in which the solution of the perturbed system is expressed as a linear combination of the unperturbed system eigensolutions. Such perturbation theory in open systems was introduced by using a two-component eigenfunction [17–20]. Bi-orthonormality for non-Hermitian systems was also considered in development of perturbation theory [21,22]. However, in previous studies addressing leaky modes, only scalar fields in one dimension were analyzed. In nanocavity tuning (such as in photonic crystal cavities with a local change in the dielectric constant) it is essential to account for radiation losses (with “free space” propagation) of 2D or 3D structures, and for the vector field nature of the electromagnetic modes. As shown below, the extention to 2D and 3D, and the consideration of the vector fields in electromagnetics, is a non-trivial one.

Here, we introduce normalizable leaky modes (NLM's) in a finite volume to treat 3D electromagnetic problems with vector fields [23]. By applying perturbation theory on the NLM's, the effects on a resonant mode frequency (ω) and Q due to slight changes in the dielectric constant of the system are predicted. As in the case of conservative systems, it is assumed here that adiabatic tuning is well represented by a time independent perturbation. As an example, these results are compared to numerical calculations of a photonic crystal cavity (in 2D).

2. Normalizable leaky modes perturbation theory foundation

Let E and H be the complex electric and magnetic fields related by the Maxwell equations where E and H are their time independent versions. In order to invoke perturbation theory one needs to formulate a wave equation for the desired field using a Hermitian Hamiltonian. Such Hermitian Hamiltonian Θ^ for the magnetic waves and the suitable wave equation cast in a time evolution Hamiltonian form were previously introduced [13] and read:

(a)Θ^×(1εr(r))×(b)Θ^(H)=1c22t2H
where c is the speed of light, εr is the dielectric constant and r is the coordinate in space. Eigen-solutions (modes) of Eq. (1) are labeled with a discrete index. For example, Hn and ωn are the mode n and its eigen-value, respectively. The “Conventional Electromagnetic” (CEM) perturbation theory introduced by [14] relies on the Hermiticity of Θ^ under the inner product:

Hn|HkallspaceHn*Hkd3r.

As conservative modes (existing within cavities with perfectly reflecting boundaries) are zeroed outside the resonator, the integration is actually over a finite volume. However the Hermiticity is lost if the modes exhibit radiation losses, or in other words if there is an electromagnetic flux outwards through the systems boundaries, and integrations like that in Eq. (2) diverge. In order to address problems of optical cavities with radiation losses (finite Q), one should try to overcome this difficulty. We propose here a new parameter space of fields, an Hamiltonian formulation derived from Maxwell's Equations, and a bilinear form, by which a perturbation theory in the complex domain can be formulated.

At first the field M is defined by M=μ0×H where μ0 is the vacuum permeability and M is the time independent version of M. Next the two-component vector field and the Hamiltonian matrix operator are defined by:

(a)|ϕ=(EM)(b)Θ^i(0c2εr(r)××0).

Under such definitions the electromagnetic wave equation may be cast in a time evolution Hamiltonian form:

Θ^|ϕ=it|ϕ.

Note that this time evolution equation resembles the familiar quantum mechanics in which a first time derivative appears. This is unlike the CEM form in Eq. (1) (with second derivative in the RHS) . The cavity is considered to be a part of a finite size dielectric system of volume V embedded in free space (or a different homogeneous dielectric constant environment) where the boundary surface of V (denoted S) exhibits a discontinuity in the dielectric constant. This could be for example a photonic crystal slab of finite lateral size with a cavity, or, a 1D, 2D, or a 3D finite size photonic crystal with embedded cavity. The two component vector space |ϕ is the mathematical entity of the Hamiltonian problem. Physically, the family of leaky cavity modes can be found by solving the wave equation under the outgoing wave boundary condition for the field E:

S[noutctE+(×E)×n^]ds=0
where nout=εr1/2(r)=const:rV [24]. The solutions form an infinite discrete set of eigenvectors (eigenfunctions) |ϕn with corresponding complex eigenvalues (eigenfrequencies) ωn for n=0,±1,±2,.... For the harmonic time evolution (exp(iωt)) we obtain that all ωn have a negative imaginary part, indicating oscillatory and time decaying modes. This set of modes comprises the leaky modes which under the usual inner product are non orthogonal and non normalizable. To overcome this, we define the following bilinear form [25]:
ϕn|ϕkiV(EnMk+EkMn)d3r+inoutcS(EnEk)ds
where En, Mn and Ek, Mk are the two components of vectors |ϕn and |ϕk respectively. Equation (6) is the equivalent to an inner product for our purpose, under which the operator Θ^ is symmetric while the leaky modes are both orthonormal and normalizable (NLM's). Thus, we have defined what can be regarded as a “Hermitian domain” of the operator Θ^ analogous to Hermiticity in conservative systems. The presented formalism is the extension of the 1D scalar problem treatment introduced in [1719] into a 3D vector electromagnetic problem treatment.

3. NLM perturbation theory and complex frequency predictability

3.1 NLM perturbation theory development

The presented framework makes possible the use of perturbation theory on electromagnetic systems with radiation losses. Assuming the dielectric constant of the system is εr(r)=εr0(r)+εr(r) where εr0(r) is the dielectric constant spatial profile of the unperturbed system and εr(r) is the spatial profile of the dielectric constant perturbation with εr(r)εr0(r), the Hamiltonian Θ^ may be split into two parts:

Θ^=(Θ^0+Θ^)=i(0c2εr0(r)××0)+i(0c2εr(r)εr0(r)200)
where Θ^0 is the operator describing the unperturbed system and Θ^ describes only the dielectric perturbation which is small compared to Θ^0. Owing to the linearity of the bilinear from in Eq. (6), one only needs to follow the conventional steps in the first order perturbation theory of the quantum mechanics or CEM to obtain [26]:
|ϕn=|ϕn(0)+knϕk(0)|Θ^|ϕn(0)ϕk(0)|ϕk(0)(ωn(0)ωk(0))|ϕk(0)
Δωn=ϕn(0)|Θ^|ϕn(0)ϕn(0)|ϕn(0)
where |ϕn(0) and ωn(0)are the two-component NLM's and complex eigenfrequencies of the unperturbed system respectively, |ϕn, and ωn are the instantaneous eigenmodes and eigenfrequencies of the new perturbed system (ωn=ωn(0)+Δωn) to the first order approximation and the Dirac notations in Eqs. (8), (9) are finite volume and surface integrations of the known fields, as in Eq. (6). Note that the familiar looking Eqs. (8) and (9) receive entirely new interpretation under the new bilinear form defined in Eq. (6), unlike the result of the conventional electromagnetic (CEM) perturbation theory [14].

3.2 Complex frequency change

Elaborating Eq. (9) leads to:

Δωn=(ωn(0)c)2IVεr(r)EnEnd3r
where I=(1/ϕn(0)|ϕn(0)) which is equal to one with the units [ms1V2] when the mode |ϕn(0) is normalized. From Eq. (10) it follows that induced material perturbation of various spatial profiles may yield different complex Δωn values depending on the overlap between the spatial distribution of εr(r) and the spatial profile of the complex value wnEnEn of the normalized mode. Equation (10) can also be written as:

Δωn=12ωn0Vεr(r)EnEnd3rVεr0(r)EnEnd3r+inout2cωn0SEnEnds.

The most notable differences between Eq. (11) and the expression of Δω in conservative system formalism [9] is the existence of a surface term in the denominator. Another important difference is that in the integrals (such as WV(εr(r)EnEn)d3r in the numerator), both of the electric field terms En appear without conjugation (unlike in former treatments [9,14] where |En|2 (=EnEn*) is used). The last difference is related to the integration volume which, unlike in conservative systems, is over the volume V enclosing the dielectric system and defined by its boundaries.

3.3 Calculation of the change in Q

Concerning the cavity Q, and its perturbative change, using the relation Q=(Re{ω}/2Im{ω}), we obtain:

ΔQnQn0=[Qn0Wi12Wr(c22|ωn(0)|1^)+Qn0WiWr]
where Qn0 and ΔQn are the quality factor of eigenmode n in the unperturbed system and the change in the quality factor induced by the perturbation, respectively, while Wr=Re{W} and Wi=Im{W}. Equations (10) and (12) can be applied, to optimize either the cavity design, or the spatial distribution of the perturbation, in order to obtain the desirable Δω with a certain degree of separate control over its real and imaginary values, (i.e. frequency and Q). This derivation allows a semi-analytical prediction of the dielectric constant modulation effect on the system mode without the need of complex and time consuming numerical calculations. The only requirement is the calculation (e.g. by numerical methods) of the field profile and its eigen-frequency (E n and ωn for eigenmode n, respectively).

4. Comparing present formulation with conventional electromagnetic perturbation and numerical simulations

4.1 Introducing the example

In order to test the accuracy of the perturbative expressions, the values of Δω and ΔQ predicted by Eqs. (10) and (12) where compared to the CEM perturbation theory predicted values and to the results of a finite difference time domain (FDTD) simulation [27] in a two-dimensional (2D) cavity example with chosen resolution of 50 points per unit cell. The simulated structure is depicted in Fig. 1(a) . The nanocavity consists of a finite size rectangular 2D photonic crystal with a 7×7 periodic lattice of air holes (εr=1) in silicon (εr=12.25). The reduced hole size in the middle of the structure defines the nanocavity. The unit cell size is a, the radius of a regular lattice and nanocavity air holes are r=0.48a and r=0.15a, respectively. The structure exhibits a bandgap in the frequency range of 0.2425(2πc/a)ω0.3115(2πc/a) for the TM polarization (E=Ezz^) and supports three cavity modes. The highest cavity NLM |ϕIII(0) is analyzed. This NLM is characterized by ωIII0 = (0.2827320.000168608i)(2πc/a) and QIII0 = 838.429, (unperturbed simulation values). The real and imaginary parts of the NLM's electric field (EIII) and its square (wIII) normalized according to ϕIII(0)|ϕIII(0)=1 are shown in Figs. 1(b)1(e) with normalized intensity [28]. In order to stimulate this cavity mode four sources positioned at the modes absolute peaks with respective symmetry were used. The sources followed a Gaussian frequency dependence centered at ω = 0.2826 (2πc/a)with spectral width of Δω = 0.002(2πc/a). We define two perturbation regions. Region 1 in the middle of the structure near the cavity, marked by dotted green line and region 2 in the cavity surroundings, marked by dashed purple line [see Fig. 1(a)].

 

Fig. 1 (a) Simulated structure. The Perfectly Matched Layer (PML) is marked by light blue dots region at the borders. Green dotted line marks region 1 of the perturbation. Purple dashed line marks region 2 of the perturbation. (b)-(e) Highest frequency normalized cavity mode |ϕIII(0). (b) Re{EIII} .(c) Im{EIII} .(d) Re{wIII} .(e) Im{wIII} [28].

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4.2 Results

At first a dielectric constant perturbation was introduced to the silicon in region 1. For perturbations of εr/εr0 = 0.0001, 0.001 and 0.01, (pert's 1-3) where εr and εr0 are the dielectric region perturbation, and unperturbed values, respectively. The “reference” values for ΔωIII and ΔQIII are obtained by the numerical simulations for each one of the three perturbations. The values of ΔωIII and ΔQIIIwere then estimated using the NLM perturbation theory and the CEM perturbation theory denoted by ΔωIIINLM, ΔQIIINLM and ΔωIIICEM, ΔQIIICEM,respectively. Because we deal with leaky modes which extend to the whole universe and even diverge in amplitude at infinity, the ΔωIIICEM and ΔQIIICEM are actually not defined as is, since the bra and ket operations involve integration over the whole space, in contrary to the NLM formalism which is restricted to volume V by definition. Yet for the purpose of applying the CEM perturbation theory and since the cavity mode is confined, we treat it as a conservative mode and restrict the integration only to volume V at the borders of which the mode's field has undergone substantial attenuation compared to its absolute maximum at the vicinity of the cavity.

Table 1 shows the values of ΔωIII and ΔQIII for each one of the three perturbations (simulation obtained and predicted values) and the corresponding relative prediction errors δQNLM (|ΔQIIIΔQIIINLM|/|ΔQIII|) and δωreNLM (|Re{ΔωIII}Re{ΔωIIINLM}|/|Re{ΔωIII}|) for the NLM perturbation method and similarly δωreCEM for the CEM perturbation method. As it is shown in the table, the NLM method yields a better approximation for Re{Δω} than the CEM perturbation method and, among the two methods only the NLM is capable to predict changes in ΔQ with a reasonable precision (relative error of about 2% or less). The authors are unaware of other techniques for approximating perturbative changes in ΔQ to compare with.

Tables Icon

Table 1. Frequency and Q changes of the highest nanocavity mode obtained by an FDTD simulation and predicted by the Normalizable Leaky Modes and Conventional Electro-Magnetic perturbation treatments accompanied by the relative errors in comparison to the FDTD results.

4.3 Discussion

It is worthwhile to note that, according to Eq. (10), applying the perturbation in a region where Im{wIII} is negative would increase the imaginary part of the eigenfrequency of the mode, i.e. would increase the Q. As one can observe in Fig. 1(e), region 1 is exactly such a region. In the same way because Re{wIII}is positive almost everywhere in volume V, nearly any positive perturbation to the dielectric constant will decrease the real part of the eigenfrequency, as readily observed in Table 1. Hence, one could expect that applying the perturbation in a region in which Im{wIII} is positive will change the Q in the opposite direction. Such example is presented as well. Applying perturbation of εr/εr0=0.01 in ΔωIII = (1.6028×1042.98225i×106) and the Q to decrease ΔQIII = (−15.039). These examples show that there is certain degree of freedom in controlling separately the frequency and the Q-factor, by a proper choice of the geometrical distribution of the dielectric constant perturbation. In a more general sense, the present framework makes the exploration of the parameter space in tuning of leaky cavity resonances possible. A different choice of the dielectric structure would alter the spatial profile of wn which, in turn, is responsible for the system's resonance eigenfrequency sensitivity to a dielectric perturbation, in strength (εr/εr0) and spatial distribution. This may be an important step towards better capabilities of cavity design and resonance tuning by a dielectric perturbation. As an alternative example, and in order to test applicability to high Q systems, we have increased the finite size photonic crystal to 15×15periods instead of 7×7to get an unperturbed cavity with Q = 40867. The inflicted perturbations are of the same intensity and spatial distribution as of (pert's 1-3) and result in virtually the same relative errors when compared to the simulation [29].

5. Parameter space of perturbative changes in Q-factor

The parameter space of complex eigenfrequency response to a dielectric perturbation is further investigated, based on Eq. (12), by defining a “Q-response” fQIII(ΔQIII/QIII0). For a free choice of Wr and Wi values the surface fQIII(Wr,Wi) is depicted in Fig. 2(a) (the areas close to singularity were removed). Two planes, at fQIII=0 (zero Q change) and fQIII=(1) (total nullification of Q) are depicted in light gray and dark gray, respectively.

 

Fig. 2 (a) The ratio fQIII(Wr,Wi)(ΔQIII/QIII0)in the wide range. (b) fQIII(Wr,Wi) in the relevant range with the specific perturbation marked on it. (c) fQIII(Wi) with the four tested perturbations.

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All four tested perturbations in the nanocavity example concentrate very close to the blackcircle in the graph where fQIII deviates only slightly from zero. In the example studied here, the possible values of Wr and Wi are very limited [30] and fQIII in this range is almost linear and negligibly dependent on Wr [see Fig. 2(b)]. The four calculated perturbations are marked by the four points in Figs. 2(b), 2(c). The first three points above the fQIII=0 level and the last one below it. Figure 2(c) presents fQIIIas a function of Wi only for the selected values of Wrcorresponding to the actual tested perturbations.

One important problem with high-Q cavities is the difficulty of out-coupling of excitations from the cavity. One would like a Q-switch, to change the cavity Q from high to low in order to release the electromagnetic cavity mode to the external world, or to do the opposite, to increase the resonator's Q to such an extent that the confined light mode which is about to leave the cavity would be trapped inside it for additional period of time [7]. As seen from Fig. 2, for the first task one need a high positive value ofWi, and for the second task, a negative value of Wi which is close to the surface singularity. In the presented examples only slight changes in Q are possible, but, in other systems the situation can be different. Also note that Qn0 is a multiplicand of Wi in Eq. (12), thus, perturbations in ultra-high-Q systems may produce substantial relative changes in the Q. However, the spatial distribution of Im{wn} may neutralize the effect of a very high Q in such systems. These considerations, related to the parameter space for a perturbative change in the eigenfrequency of cavities with radiation losses may serve as a design framework for the optimization in both: frequency tuning and Q-switching. Higher order perturbation theory, and quantum gate structures can be studied based on the formalism presented here [31].

6. Conclusion

In summary, we have presented the Normalizable Leaky Mode formalism for electromagnetic vector fields in 3D. Then, based on this formalism perturbation theory to electromagnetic waves confined in a dielectric nanocavity with radiation losses was applied, formulas for the changes in the frequency and Q induced by small perturbations of the dielectric constant spatial distribution were derived. A Q-response function was defined for investigation of the dielectric perturbation spatial distribution effect on the Q of the nanocavity. Under some conditions slight changes in the dielectric function may strongly affect the Q while slightly affecting the frequency of an eigenmode and vice-versa [31].

References and links

1. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]  

2. D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express 13(16), 5961–5975 (2005). [CrossRef]   [PubMed]  

3. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006). [CrossRef]  

4. M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006). [CrossRef]  

5. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009). [CrossRef]   [PubMed]  

6. M. Sandberg, C. M. Wilson, F. Persson, G. Johansson, V. Shumeiko, and P. Delsing, “In-situ frequency tuning of photons stored in a high Q microwave cavity”, arXiv:0801.2479v1 (2008)

7. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006). [CrossRef]  

8. A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004). [CrossRef]  

9. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGRAW-HILL, 1961).

10. L. D. Landau, and E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory, 2nd-ed (Addison-Wesley, 1965)

11. L. I. Schiff, Quantum Mechanics, ed-3, (McGraw-Hill, 1968)

12. R. L. Liboff, Introductory Quantum Mechanics, ed-2 (Addison-Wesley, 1992)

13. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995)

14. M. Skorobogatiy, and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University Press, Cambridge, England, 2009)

15. P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005). [CrossRef]  

16. S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009). [CrossRef]  

17. P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997). [CrossRef]  

18. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998). [CrossRef]  

19. P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998). [CrossRef]  

20. A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003). [CrossRef]   [PubMed]  

21. A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001). [CrossRef]  

22. O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009). [CrossRef]   [PubMed]  

23. The Normalizable Leaky Modes may be regarded as a 3D extension of the quasi-normal modes [17–20].

24. Discontinuity of the dielectric constant at surface S is followed by discontinuity of the electric field, therefore by integrating over S we actually mean integrating over a different surface which is outward remote from S but infinitesimally close to it where both the dielectric constant and the electric field are continuous.

25. Bilinear Form is a function B:V×VF mapping from two vectors of the vector space V into the scalar field F. Every inner product is by itself a bilinear map, but for example while every inner product is conjugate symmetric by definition, bilinear map doesn't have to be, and in the presented case it is just symmetric.

26. We define ψ|Θ^|φψ|Θ^φ, where Θ^φ is the matrix by vector multiplication [see Eqs. (3) and (6)].

27. Meep. http://ab-initio.mit.edu/wiki/index.php/Meep (13.10.2009), free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems.

28. The absolute peak value in the computational region of Re{EIII} is roughly 23 times larger than that of Im{EIII}. Similarly the absolute peak value in the computational region of Re{wIII} is roughly 172 times larger than that of Im{wIII}.

29. Tests for applicability to high Q systems suggested by an anonymous reviewer are gratefully acknowledged. Though the simulation resolution was decreased to 20 point per unit cell because of computational time considerations, the relative errors did not increased.

30. In the examples studied here, even under a very large material modulation of εr = 0.15 (εr/εr00.01) at every desired point in the volume V (including air holes), the possible values of Wr and Wi are limited to 0.422103Wr2.341 and 9.615103Wi0.0322 respectively.

31. If, upon cavity design and mode calculation, Wrand Wi receive the adequate magnitude and sign, desired response in Δω or ΔQ can be obtained. This will be shown elsewhere (M. Shlafman and J. Salzman unpublished).

References

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  1. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
    [Crossref]
  2. D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express 13(16), 5961–5975 (2005).
    [Crossref] [PubMed]
  3. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
    [Crossref]
  4. M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
    [Crossref]
  5. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
    [Crossref] [PubMed]
  6. M. Sandberg, C. M. Wilson, F. Persson, G. Johansson, V. Shumeiko, and P. Delsing, “In-situ frequency tuning of photons stored in a high Q microwave cavity”, arXiv:0801.2479v1 (2008)
  7. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
    [Crossref]
  8. A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
    [Crossref]
  9. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGRAW-HILL, 1961).
  10. L. D. Landau, and E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory, 2nd-ed (Addison-Wesley, 1965)
  11. L. I. Schiff, Quantum Mechanics, ed-3, (McGraw-Hill, 1968)
  12. R. L. Liboff, Introductory Quantum Mechanics, ed-2 (Addison-Wesley, 1992)
  13. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995)
  14. M. Skorobogatiy, and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University Press, Cambridge, England, 2009)
  15. P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
    [Crossref]
  16. S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
    [Crossref]
  17. P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
    [Crossref]
  18. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
    [Crossref]
  19. P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
    [Crossref]
  20. A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
    [Crossref] [PubMed]
  21. A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
    [Crossref]
  22. O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009).
    [Crossref] [PubMed]
  23. The Normalizable Leaky Modes may be regarded as a 3D extension of the quasi-normal modes [17–20].
  24. Discontinuity of the dielectric constant at surface S is followed by discontinuity of the electric field, therefore by integrating over S we actually mean integrating over a different surface which is outward remote from S but infinitesimally close to it where both the dielectric constant and the electric field are continuous.
  25. Bilinear Form is a function B:V×V→F mapping from two vectors of the vector space V into the scalar field F. Every inner product is by itself a bilinear map, but for example while every inner product is conjugate symmetric by definition, bilinear map doesn't have to be, and in the presented case it is just symmetric.
  26. We define 〈ψ|Θ^′|φ〉≡〈ψ|Θ^′φ〉, where Θ^′φ is the matrix by vector multiplication [see Eqs. (3) and (6)].
  27. Meep. http://ab-initio.mit.edu/wiki/index.php/Meep (13.10.2009), free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems.
  28. The absolute peak value in the computational region of Re{EIII} is roughly 23 times larger than that of Im{EIII}. Similarly the absolute peak value in the computational region of Re{wIII} is roughly 172 times larger than that of Im{wIII}.
  29. Tests for applicability to high Q systems suggested by an anonymous reviewer are gratefully acknowledged. Though the simulation resolution was decreased to 20 point per unit cell because of computational time considerations, the relative errors did not increased.
  30. In the examples studied here, even under a very large material modulation of ε′r = 0.15 (ε′r/εr0≅0.01) at every desired point in the volume V (including air holes), the possible values of Wr and Wi are limited to −0.422⋅10−3≤Wr≤2.341 and 9.615⋅10−3≤Wi≤0.0322 respectively.
  31. If, upon cavity design and mode calculation, Wrand Wi receive the adequate magnitude and sign, desired response in Δω or ΔQ can be obtained. This will be shown elsewhere (M. Shlafman and J. Salzman unpublished).

2009 (3)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[Crossref] [PubMed]

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[Crossref]

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009).
[Crossref] [PubMed]

2006 (3)

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[Crossref]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
[Crossref]

2005 (3)

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[Crossref]

D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express 13(16), 5961–5975 (2005).
[Crossref] [PubMed]

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[Crossref]

2004 (1)

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

2003 (1)

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

2001 (1)

A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
[Crossref]

1998 (2)

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[Crossref]

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[Crossref]

1997 (1)

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[Crossref]

Akahane, Y.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[Crossref]

Asano, T.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[Crossref]

Bartal, G.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009).
[Crossref] [PubMed]

Bertolotti, M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Bloemer, M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Botten, L.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[Crossref]

Bowden, C. M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Brink, A. M.

A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
[Crossref]

Centini, M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Ching, E. S. C.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[Crossref]

Christodoulides, D. N.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009).
[Crossref] [PubMed]

Cohen, O.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

D’Aguanno, G.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

de Sterke, C.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[Crossref]

Dossou, K.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[Crossref]

Englund, D.

Fushman, I.

Greentree, A. D.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[Crossref]

Hollenberg, L. C. L.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[Crossref]

Jones, R.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

Kristensen, P.

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[Crossref]

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[Crossref] [PubMed]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Lagsgaard, J.

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[Crossref]

Leung, P. T.

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[Crossref]

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[Crossref]

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[Crossref]

Liao, L.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

Liu, A.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

Maassen van den Brink, A.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[Crossref]

Mahmoodian, S.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[Crossref]

Mattiucci, N.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

McPhedran, R.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[Crossref]

Mitsugi, S.

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
[Crossref]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Moiseyev, N.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009).
[Crossref] [PubMed]

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

Nielsen, T.

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[Crossref]

Noda, S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[Crossref]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[Crossref] [PubMed]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
[Crossref]

Paniccia, M.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

Peleg, O.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009).
[Crossref] [PubMed]

Poulton, C.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[Crossref]

Prawer, S.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[Crossref]

Riishede, J.

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[Crossref]

Rubin, D.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

Rubio, D. S.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

Salzman, J.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[Crossref]

Scalora, M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Segev, M.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009).
[Crossref] [PubMed]

Settimi, A.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Severini, S.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Shinya, A.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Sibilia, C.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Song, B. S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[Crossref]

Suen, W. M.

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[Crossref]

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[Crossref]

Sun, C. P.

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[Crossref]

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[Crossref] [PubMed]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[Crossref] [PubMed]

Tong, S. S.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[Crossref]

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[Crossref]

Vuckovic, J.

Watanabe, T.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Young, K.

A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
[Crossref]

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[Crossref]

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[Crossref]

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[Crossref]

Appl. Phys. Lett. (1)

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

J. Phys. A (1)

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[Crossref]

J. Phys. Math. Gen. (1)

A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
[Crossref]

Nat. Mater. (1)

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[Crossref]

Nature (1)

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[Crossref]

Opt. Express (1)

Photonics Nanostruct. Fundam. Appl. (1)

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[Crossref]

Phys. Rev. (1)

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
[Crossref]

Phys. Rev. A (2)

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[Crossref]

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[Crossref]

Phys. Rev. Lett. (2)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[Crossref] [PubMed]

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. 102(16), 163902 (2009).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[Crossref]

Other (16)

M. Sandberg, C. M. Wilson, F. Persson, G. Johansson, V. Shumeiko, and P. Delsing, “In-situ frequency tuning of photons stored in a high Q microwave cavity”, arXiv:0801.2479v1 (2008)

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGRAW-HILL, 1961).

L. D. Landau, and E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory, 2nd-ed (Addison-Wesley, 1965)

L. I. Schiff, Quantum Mechanics, ed-3, (McGraw-Hill, 1968)

R. L. Liboff, Introductory Quantum Mechanics, ed-2 (Addison-Wesley, 1992)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995)

M. Skorobogatiy, and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University Press, Cambridge, England, 2009)

The Normalizable Leaky Modes may be regarded as a 3D extension of the quasi-normal modes [17–20].

Discontinuity of the dielectric constant at surface S is followed by discontinuity of the electric field, therefore by integrating over S we actually mean integrating over a different surface which is outward remote from S but infinitesimally close to it where both the dielectric constant and the electric field are continuous.

Bilinear Form is a function B:V×V→F mapping from two vectors of the vector space V into the scalar field F. Every inner product is by itself a bilinear map, but for example while every inner product is conjugate symmetric by definition, bilinear map doesn't have to be, and in the presented case it is just symmetric.

We define 〈ψ|Θ^′|φ〉≡〈ψ|Θ^′φ〉, where Θ^′φ is the matrix by vector multiplication [see Eqs. (3) and (6)].

Meep. http://ab-initio.mit.edu/wiki/index.php/Meep (13.10.2009), free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems.

The absolute peak value in the computational region of Re{EIII} is roughly 23 times larger than that of Im{EIII}. Similarly the absolute peak value in the computational region of Re{wIII} is roughly 172 times larger than that of Im{wIII}.

Tests for applicability to high Q systems suggested by an anonymous reviewer are gratefully acknowledged. Though the simulation resolution was decreased to 20 point per unit cell because of computational time considerations, the relative errors did not increased.

In the examples studied here, even under a very large material modulation of ε′r = 0.15 (ε′r/εr0≅0.01) at every desired point in the volume V (including air holes), the possible values of Wr and Wi are limited to −0.422⋅10−3≤Wr≤2.341 and 9.615⋅10−3≤Wi≤0.0322 respectively.

If, upon cavity design and mode calculation, Wrand Wi receive the adequate magnitude and sign, desired response in Δω or ΔQ can be obtained. This will be shown elsewhere (M. Shlafman and J. Salzman unpublished).

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Figures (2)

Fig. 1
Fig. 1 (a) Simulated structure. The Perfectly Matched Layer (PML) is marked by light blue dots region at the borders. Green dotted line marks region 1 of the perturbation. Purple dashed line marks region 2 of the perturbation. (b)-(e) Highest frequency normalized cavity mode | ϕ I I I ( 0 ) . (b) Re { E I I I } .(c) Im { E I I I } .(d) Re { w I I I } .(e) Im { w I I I } [28].
Fig. 2
Fig. 2 (a) The ratio f Q I I I ( W r , W i ) ( Δ Q I I I / Q I I I 0 ) in the wide range. (b) f Q I I I ( W r , W i ) in the relevant range with the specific perturbation marked on it. (c) f Q I I I ( W i ) with the four tested perturbations.

Tables (1)

Tables Icon

Table 1 Frequency and Q changes of the highest nanocavity mode obtained by an FDTD simulation and predicted by the Normalizable Leaky Modes and Conventional Electro-Magnetic perturbation treatments accompanied by the relative errors in comparison to the FDTD results.

Equations (12)

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( a ) Θ ^ × ( 1 ε r ( r ) ) × ( b ) Θ ^ ( H ) = 1 c 2 2 t 2 H
H n | H k a l l s p a c e H n * H k d 3 r
( a ) | ϕ = ( E M ) ( b ) Θ ^ i ( 0 c 2 ε r ( r ) × × 0 )
Θ ^ | ϕ = i t | ϕ
S [ n o u t c t E + ( × E ) × n ^ ] d s = 0
ϕ n | ϕ k i V ( E n M k + E k M n ) d 3 r + i n o u t c S ( E n E k ) d s
Θ ^ = ( Θ ^ 0 + Θ ^ ) = i ( 0 c 2 ε r 0 ( r ) × × 0 ) + i ( 0 c 2 ε r ( r ) ε r 0 ( r ) 2 0 0 )
| ϕ n = | ϕ n ( 0 ) + k n ϕ k ( 0 ) | Θ ^ | ϕ n ( 0 ) ϕ k ( 0 ) | ϕ k ( 0 ) ( ω n ( 0 ) ω k ( 0 ) ) | ϕ k ( 0 )
Δ ω n = ϕ n ( 0 ) | Θ ^ | ϕ n ( 0 ) ϕ n ( 0 ) | ϕ n ( 0 )
Δ ω n = ( ω n ( 0 ) c ) 2 I V ε r ( r ) E n E n d 3 r
Δ ω n = 1 2 ω n 0 V ε r ( r ) E n E n d 3 r V ε r 0 ( r ) E n E n d 3 r + i n o u t 2 c ω n 0 S E n E n d s .
Δ Q n Q n 0 = [ Q n 0 W i 1 2 W r ( c 2 2 | ω n ( 0 ) | 1 ^ ) + Q n 0 W i W r ]

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