The simple method for modeling of circuits of weakly coupled lossy resonant cavities, previously developed in quantum mechanics, is generalized to enable calculation of the transmission and reflection amplitudes and group delay of light. Our result is the generalized Breit-Wigner formula, which has a clear physical meaning and is convenient for fast modeling and optimization of complex resonant cavity circuits and, in particular, superstructure gratings in a way similar to modeling and optimization of electric circuits. As examples, we find the conditions when a finite linear chain of cavities and a linear chain with adjacent cavities act as bandpass and double bandpass filters, and the condition for a Y-shaped structure to act as a bandpass 50/50 light splitter. The group delay dependencies of the considered structures are also investigated.
©2003 Optical Society of America
Recently, investigation of light propagation through a series of weakly coupled photonic resonant cavities became of special interest [1–12]. Being mostly considered as the devices, which are based on the non-one-dimensional photonic crystal and some other lithographically fabricated structures [1–10], the resonant cavity structures can be also implemented in the more conventional fiber Bragg gratings [11,12]. The relative simplicity of the model used for description of these structures, which was previously developed in solid state physics and known as tight-binding approximation , makes them all the more attractive for investigation and applications. The linear interaction of light with photonic cavities is analogous to interaction of electrons with quantum dots in solid state physics  once one ignore the multi-particle nature in the latter and the polarization effects in the former. If the cavities are relatively small and contain a few eigenstates, the light propagation through series of them is essentially one-dimensional. Therefore, a device built on these cavities is analogous to an electric circuit. The present paper develops a simple method for modeling of resonant cavity circuit devices. As examples, we find a condition when a finite linear chain and a linear chain coupled to adjacent cavities act as a bandpass and double bandpass filters, and a similar condition for a Y-shaped structure to act as a bandpass 50/50 light splitter. Generally, the suggested approach allows for simple and fast modeling of resonant cavity circuits containing hundreds of coupled lossy cavities with complex topology.
2. Generalized Breit-Wigner formula
Figure 1 illustrates a type of devices studied in this paper, which consist of several resonant cavities (i, j, k, …) and ports (l, m, …). It is essential to consider a photonic device not as an isolated structure but as a structure coupling to the rest of the optical system through ports [15–18]. Below the ports are assumed to be the waveguides transmitting light with wavelength λ and having no internal losses and reflection except for the reflection from the interface between the port and the structure under consideration. According to the basic concept of the Breit-Wigner approximation  a single cavity near the resonant wavelength can be described by introducing its complex wavelength eigenvalue, λn + ½iγn, where the lossy component γn defines the decay time of the corresponding eigenstate at resonant wavelength λn, τ = /(πcγn), and c is the speed of light. Generalizing this concept, if N cavities and M ports are weakly coupled to each other then the whole structure can be described by introducing λn, γn, and also the coupling coefficient between cavities i and k, δ ik , and the coupling coefficient between cavity j and the port m, γ jm , (see Fig.1). The resonant condition assumes that the eigenwavelengths λn are close to each other and the wavelength λ is close to λn. Below we also assume that the coupling between cavities and ports is relatively weak, so that the characteristic bandwidths are relatively small: γi ~ δkl ~ |λ i – λ j | ≪λi. The single particle theory of resonant propagation of electrons based on this approach was developed in [20–22]. Papers [20–22] deal with calculation of the absolute value of propagation amplitude, which is proportional to electron current. In photonics, calculation of the phase shift, which determines the delay time, is of equal importance. In this paper the results  are generalized to enable calculation of the complex-valued transmission and reflection coefficients and, in particular, the group delay of light.
Physically, the complex eigenvalues are introduced as follows. Assume that the cavity j is close to the port m. In the absence of port m, the cavity eigenvalue is λ j . If the port m is weakly coupling to this cavity then the corresponding eigenstate is no longer stationary and decays in time. Formal continuation of the eigenstate calculated in the absence of the port into the port region will generate there an outgoing wave as well as an incoming wave. Because there are no external sources, the incoming wave should be absent. Often it can be eliminated by introducing the complex quasi-eigenwavelength λ j + Δ jm + ½iγ jm and choosing it so that the coefficient in front of the incoming wave becomes zero. In our calculations we include the real shift Δλ jm into λ j writing down the quasi-eigenwavelength in the form λ j + ½iγ j where γ j summarizes the contribution of all ports coupled to the cavity and also other possible losses.
Within the tight-binding approximation  we look for the field of the resonant cavity system as a linear combination of the normalized quasi-eigenstates un (r) corresponding to quasi-eigenwavelengths λ j + ½iγ j and the incident wave (r) in port m:
According to the definition of the quasi-eigenstates, the outgoing waves, which are included into un (r), can be found by direct continuation of un (r) into the region of the corresponding ports. The principal difference of our approach compared to the conventional tight-binding approximation [13,11] is that we are solving here the scattering problem, which determines the transmission and reflection amplitudes, but not only the eigenvalues of the system. The total field u(r) satisfy the equation:
where H is the Hamiltonian of the considered system. For example, for the scalar wave equation H=n(r)-2Δr where n(r) is the refractive index. Similar as it is done in derivation of the equations of the tight-binding approximation, we chose un (r) and (r) to satisfy the equations:
where and are the Hamiltonians describing the field in port m and in cavity n with adjacent ports, respectively, in the absence of the rest of the system. We substitute Eq. (1) into Eq. (2), use Eqs. (3) for each term of Eq. (1), and then multiply both sides of Eq.(2) by uk (r) and integrate over space. As the result, we arrive at the following equation for Cn :
and the expressions for λ j , χ jm and δ ij are given in the Appendix. In this paper we consider systems of coupling cavities with very close resonant eigenvalues, which obey the scalar wave equation. Then, in the tight-binding approximation, matrix Λ is symmetric, δij=δji, and has real non-diagonal coefficients δ ij . Physically, the coefficient χ jm determines coupling between cavity j and port m, and coefficient δ ij determines coupling between two cavities, i and j. For convenience, we introduce the coupling coefficient between a cavity and a port having the same units as λ j , γ j , and δ jk :
The transmission and reflection amplitudes for the considered system can be expressed through the solution of Eq. (4). Assume that the port l is coupled only to cavity i and the port m is coupled only to cavity j as shown in Fig. 1. Assume also, that the ports are the single mode waveguides, for which the incoming wave (like the one introduced in Eq. (1)) and the outgoing wave into the same port are conjugate to each other within a constant factor. Then, the transmission amplitude from port l to port m, Alm (λ), and the reflection amplitude to port l, Rll (λ), are defined by equations:
where matrix T(λ) = ‖tij (λ)‖ is inverse to Λ(λ), T(λ) = Λ(λ)-1. Generally, the imaginary component of the cavity eigenvalue is γ n = Σ m γ nm + γ int where γ int is the part determined by internal losses and γ nm defined by Eq. (6) is the partial width of the eigenwavelength n due to decay into port m. For the lossless device material and the absent of hidden drains, γ int = 0. Eqs. (7) generalize the results obtained in  allowing for calculation of both phase and amplitude of reflection and transmission. The group delay associated with transmission between ports and reflection back to the port is simply expressed through the derivatives of phases of amplitudes from Eq. (6) as follows (see e.g.):
3. A single-cavity structure
and Eqs. (7) give:
In this case the delay time in transmission and in reflection is the same. This structure is fully transparent, i.e. A 12(λ) = 1, if there is no internal loss, γ1 =γ11 + γ12, if it is symmetric, γ11 = γ12, and if the wavelength of incident light is resonant, λ = λ1. This condition corresponds to the maximum delay time τ = /(πcγ1). In a glass with refractive index n = 1.5 the speed of light c = 2·108 m/s and for λ1 = 1500 nm and γ1 = 2 nm, the delay time τ = 1.2 ps. Figure 2 shows the corresponding transmission spectrum (b) and group delay (c) for γ11 = γ12 = 1 nm. By decreasing γ1 we increase the cavity Q-factor, Q = λλ1/γ1, and, proportionally, increase the delay time which, physically, is the light dwell time in the cavity. If the second port is absent, this structure works as an all-reflecting filter with the group delay characteristic defined by Eq. (10).
4. A few cavity bandpass filters and a Y-splitter
In this section the generalized Breit-Wigner formula is used to optimize the structures of resonant cavities to act as bandpass filters and to determine their group delay characteristics. We first optimize linear chain structures shown in Fig. 3 and 4 in order to make them the bandpass filters and examine the effect of internal losses on the spectrum characteristics of the structure shown in Fig. 4. Then we consider a structure shown in Fig. 5 consisting of a linear chain of cavities coupled to additional adjacent cavities. We demonstrate that this structure may become a double-channel bandpass filter with the channel spacing proportional to the coupling between the linear chain cavities and the adjacent cavities. Next, the Y-splitter structure shown in Fig. 6 is optimized to become a bandpass 50/50 light splitting device. In our design of filters we used a kind of numerical trial and error approach. We consider the ripple magnitude as a function of the varying coupling parameters, δ ij and γ im , and minimize it in several iterations. The examples of this section do not pretend to determine the smallest possible transmission ripple with minimum reflection in a certain bandwidth but rather demonstrate some designs having very flat passbands.
4.1. Three cavity all-pass filter
Let us optimize a structure consisting of 3 cavities and make it a bandpass filter. Fig. 3 compares the results of simple calculations using Eqs. (7) and (8) for the structure having all equal coupling coefficients (a) and for the structure with the optimized ratio of external and internal coupling coefficients equal to 2.3 (b). It is seen that, in case a, the transmission coefficient oscillates and has three resonances corresponding to the eigenvalues of the coupling three-cavity system. In case b, the ratio 2.3 was chosen so that in the interval | λ – λ0 |<δ0 the transmission coefficient |A12|2 > 0.99.
Though the transmission in case b is practically constant for | λ – λ0 |<δ0, the group delay is varying. The characteristic value of the group delay has the same order of magnitude as the group delay defined by Eqs. (10). For example, for the characteristic bandwidth δ0 = 1nm, the group delay is of order 1 ps, while for δ0 = 0.1nm it is of order 10 ps. Note that similar procedure of optimization of the linear chain of resonant cavities was performed in Ref.  using the transfer matrix method. The transmission spectrums found in this paper have minor asymmetry. It can be roughly explained by the fact that the Q-factor of the structure considered in  is approximately 30 and is not large enough for the Breit-Wigner approximation we are using to have a good accuracy.
4.2. Long linear chain of resonant cavities
If the linear chain is not lossy and consists of many resonant cavities, it can be made a bandpass by the appropriate variation of a first few coupling coefficients adjacent to the ports. In order to make it, we optimize the first three coupling coefficients and arrive at their values shown in Fig. 4. Numerical simulation shows that for these parameters, independently of the number of cavities > 4, the transmission coefficient | A 12 |2> 0.97 in the interval | λ – λ0 |<1.5δ0. The transmission (curve a) and group delay spectrums shown in Fig. 4 correspond to the linear chain consisting of 50 cavities. The port-chain matching condition determined here is analogous to the apodization condition commonly used in fiber Bragg grating fabrication in order to suppress the Fabry-Perot-like oscillations in transmission and reflection spectrum . However, while in the case of weak Bragg gratings the apodization region includes very large number of grating periods, we were able to accurately solve the problem by variation of 3 coupling coefficients only. Note, that the resonant cavity chain experimentally investigated in Ref.  was not optimized and for this reason the Fabry-Perotlike oscillations in transmission spectrum were observed.
The group delay spectrum shown in Fig. 4 is similar to the one in Fig. 3b. It has the characteristic group delay enhancement near the edges of the transmission band. Generally, the group delay in the apodized linear chain is proportional to the number of cavities, which can be also observed by comparison of group delays for 3 and 50 cavity structures shown in Fig. 3b and Fig. 4, respectfully.
Generally, the realistic cavities are lossy and it is important to investigate the effect of internal dissipation of light on the transmission and group delay spectrums. As an example, in Fig. 4 we compare the transmission spectrum of the lossless chain of 50 cavities (a) to the one having cavities with internal loss γ int = 0.01δ0 (b). While the decrease in transmission is significant and grows with the number of cavities, the effect of this loss on the group delay spectrum is negligible.
4.3. Double-channel bandpass filter
Consider now the structure shown in Fig. 5. It consists of a linear chain of cavities 1, 2, and 3 similar to the ones shown in Fig. 3 coupled to the adjacent cavities 4, 5 and 6. If coupling to cavities 4, 5, and 6 is negligible then the behavior of transmission coefficient for this structure is similar to the one shown in Fig. 3b. If coupling δ1 is turning on then the all-pass band in Fig. 3b is splitting into two all-pass bands as shown in Fig. 5. Interestingly, the distance between the centers of the created bands is approximately equal to the coupling coefficient δ1. For the transmission and group delay spectrums shown in Fig. 5, we put δ1 = 2δ0.
4.4. Bandpass 50/50 Y-splitter
The Y-splitting device consisting of resonant cavities can be first optimized by variation of the first three coupling coefficients adjacent to the ports. The values of these coefficients, which are the same for each of three ports, are shown in Fig. 6(a). In order to make this structure bandpassing, it is also necessary to determine the coupling coefficients adjacent to the center of the Y-splitter. Interestingly, this device can be made bandpassing with total transmission on 2 |A 12 |2> 0.9994 in the interval | λ – λ0 |< 0.75δ0 by variation of only a single coupling coefficient to the center cavity as shown in Fig. 6(a). The behavior of the transmission and the group delay spectrum is shown in Fig. 6(b). The fine behavior of transmission spectrum in the all-pass band is shown in the insert of the first plot of Fig. 6(b).
Many other interesting photonic devices can be investigated using the suggested technique. A single port structure like shown in Fig. 7(a) is among them. If the cavities are not lossy then the single port device is always all-reflecting. The group delay spectrum of this device can be optimized by variation of its parameters, similar to optimization of the all-pass etalon devices [23,25].
The apodization condition obtained for a chain of cavities in section 4 is universally applicable to a one-dimensional superstructure grating [11,12] and to a series of photonic cavities of a more general nature. It is essential that the optimization of a structure, like a fiber grating filter considered in , can be significantly simplified by introducing the coupling parameters and benefiting from the local character of coupling between cavities.
In non-one-dimensional case, the resonant eigenstate of photonic cavity is usually asymmetric and have dimensions comparable to the distances between cavities. In particular, the resonant cavity may inherit the symmetry of the photonic crystal it is built in and its eigenvalues may be degenerated. Then it may be not accurate to model a cavity using a single complex-valued wavelength λ k +iγ k and to model coupling between cavities using the single coupling parameter δ jk independent of λ. However, the formalism developed in this paper can be applied to this general case too. Actually, Fig. 7b illustrates how to make a simple model of a device consisting of cavities having internal hexagonal symmetry. The complexstructured cavities (large circles) can be assembled of elementary cavities (small circles). The photonic device is then built of these complex cavities as shown in Fig. 7b. The basic concept of such modeling originates from similar concept in quantum mechanics .
Equations (7) and (8) are valid if there is only one cavity coupled to a port. They can be generalized to the case when several cavities are coupled to a port by introducing an additional cavity, which is coupled to these cavities and also strongly coupled to this port. In this case the port is modeled as a port coupled to one or more cavities as illustrated in Fig. 7c. The theory generalizing the approach of Ref.  to the case of systems having more general structure of ports and cavities was developed in [27,28]. The latter allows for general simplification of the solution of the wave equation by expressing the total Green function of the problem through the Green function of the internal system taken at the complex-valued self-eigenvalues (similar to the ones introduced in Eqs. (3)) and the matrix elements describing coupling between the internal system and the ports.
Having a large Q-factor the resonant cavity circuits are slowing the group velocity of light and enhancing the non-linear effects . It is interesting to generalize the Breit-Wigner formulae obtained in this paper to account for the non-linear effects, like the bi-stability, in a way similar to the one suggested in Ref. .
We have demonstrated a simple formalism allowing for fast quantitative and qualitative evaluation of complex high-Q resonant cavity circuits. In the case when the device is coupled to ports through linear chains of resonant cavities, we have determined a simple apodization condition under which the port-chain link becomes non-reflecting independently of the chain length. We have shown that a Y-splitter, properly apodized near its ports, can be made a bandpass device by modification of the coupling coefficient to its central cavity only. Generally, the developed method may become a powerful tool for design and investigation of resonant cavity circuits.
We are grateful to J. Fini, N.M. Litchnitser, and P.S. Westbrook for valuable discussions of the results presented in this paper.
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