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
Recent progress in photonic crystal ultra-high-Q cavity design and fabrication [1–3] 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 [4–7]. 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 . 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 (and, 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 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 . Perturbation theory for lossless (closed) systems with only real eigenfrequencies was extensively described in quantum mechanics [10–12] 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 , 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 , 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 [10–12] 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 . 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 for the magnetic waves and the suitable wave equation cast in a time evolution Hamiltonian form were previously introduced  and read:Eq. (1) are labeled with a discrete index. For example, and are the mode n and its eigen-value, respectively. The “Conventional Electromagnetic” (CEM) perturbation theory introduced by  relies on the Hermiticity of under the inner product:
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 where 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:
Under such definitions the electromagnetic wave equation may be cast in a time evolution Hamiltonian form:
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 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:24]. The solutions form an infinite discrete set of eigenvectors (eigenfunctions) with corresponding complex eigenvalues (eigenfrequencies) for . For the harmonic time evolution () we obtain that all 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 :Equation (6) is the equivalent to an inner product for our purpose, under which the operator 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 analogous to Hermiticity in conservative systems. The presented formalism is the extension of the 1D scalar problem treatment introduced in [17–19] 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 where is the dielectric constant spatial profile of the unperturbed system and is the spatial profile of the dielectric constant perturbation with , the Hamiltonian may be split into two parts:Eq. (6), one only needs to follow the conventional steps in the first order perturbation theory of the quantum mechanics or CEM to obtain :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 .
3.2 Complex frequency change
Elaborating Eq. (9) leads to:Eq. (10) it follows that induced material perturbation of various spatial profiles may yield different complex values depending on the overlap between the spatial distribution of and the spatial profile of the complex value of the normalized mode. Equation (10) can also be written as:
The most notable differences between Eq. (11) and the expression of in conservative system formalism  is the existence of a surface term in the denominator. Another important difference is that in the integrals (such as in the numerator), both of the electric field terms appear without conjugation (unlike in former treatments [9,14] where () 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 , we obtain: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 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 and 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  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 periodic lattice of air holes () in silicon (). 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 and , respectively. The structure exhibits a bandgap in the frequency range of for the TM polarization () and supports three cavity modes. The highest cavity NLM is analyzed. This NLM is characterized by = and = 838.429, (unperturbed simulation values). The real and imaginary parts of the NLM's electric field () and its square () normalized according to are shown in Figs. 1(b)–1(e) with normalized intensity . 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 with spectral width of = 0.002. 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)].
At first a dielectric constant perturbation was introduced to the silicon in region 1. For perturbations of = 0.0001, 0.001 and 0.01, (pert's 1-3) where and are the dielectric region perturbation, and unperturbed values, respectively. The “reference” values for and are obtained by the numerical simulations for each one of the three perturbations. The values of and were then estimated using the NLM perturbation theory and the CEM perturbation theory denoted by , and , ,respectively. Because we deal with leaky modes which extend to the whole universe and even diverge in amplitude at infinity, the and 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 and for each one of the three perturbations (simulation obtained and predicted values) and the corresponding relative prediction errors and for the NLM perturbation method and similarly for the CEM perturbation method. As it is shown in the table, the NLM method yields a better approximation for than the CEM perturbation method and, among the two methods only the NLM is capable to predict changes in with a reasonable precision (relative error of about 2% or less). The authors are unaware of other techniques for approximating perturbative changes in to compare with.
It is worthwhile to note that, according to Eq. (10), applying the perturbation in a region where 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 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 is positive will change the Q in the opposite direction. Such example is presented as well. Applying perturbation of 0.01 in = () and the Q to decrease = (−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 which, in turn, is responsible for the system's resonance eigenfrequency sensitivity to a dielectric perturbation, in strength () 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 periods instead of to 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 .
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” . For a free choice of and values the surface is depicted in Fig. 2(a) (the areas close to singularity were removed). Two planes, at (zero Q change) and (total nullification of Q) are depicted in light gray and dark gray, respectively.
All four tested perturbations in the nanocavity example concentrate very close to the blackcircle in the graph where deviates only slightly from zero. In the example studied here, the possible values of and are very limited  and in this range is almost linear and negligibly dependent on [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 level and the last one below it. Figure 2(c) presents as a function of only for the selected values of corresponding 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 . As seen from Fig. 2, for the first task one need a high positive value of, and for the second task, a negative value of 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 is a multiplicand of in Eq. (12), thus, perturbations in ultra-high-Q systems may produce substantial relative changes in the Q. However, the spatial distribution of 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 .
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 .
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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 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 , where 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 is roughly 23 times larger than that of . Similarly the absolute peak value in the computational region of is roughly 172 times larger than that of .
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 = 0.15 () at every desired point in the volume V (including air holes), the possible values of and are limited to and respectively.
31. If, upon cavity design and mode calculation, and receive the adequate magnitude and sign, desired response in or can be obtained. This will be shown elsewhere (M. Shlafman and J. Salzman unpublished).