## Abstract

We analyze transmission characteristics of two coupled identical cavities, of either standing-wave (SW) or traveling-wave (TW) type, based on temporal coupled mode theory. Mode splitting is observed for both directly (cavity-cavity) and indirectly (cavity-waveguide-cavity) coupled cavity systems. The effects of direct and indirect couplings, if coexisting in one system, can offset each other such that no mode splitting occurs and the original single-cavity resonant frequency is retained. By tuning the configuration of the coupled cavity system, one can obtain different characteristics in transmission spectra, including splitting in transmission, zero transmission, Fano-type transmission, electromagnetically-induced-transparency (EIT)-like transmission, and electromagnetically-induced-absorption (EIA)-like transmission. It is also interesting to notice that a side-coupled SW cavity system performs similarly to an under-coupled TW cavity. The results are useful for the design of cavity-based devices for integration in nanophotonics.

©2010 Optical Society of America

## 1. Introduction

The optical cavity has been a subject of great interest as it is a vital functional building block for filtering, modulating, buffering, and switching for integrated optical processing in nanophotonics [1-4]. Optical cavities can also be applied in the field of quantum information processing [5], optomechanics [6], sensing [7], etc. Generally speaking, optical cavities can be categorized into two groups: standing-wave (SW) cavities and traveling-wave (TW) cavities. Both types of cavities have been researched extensively. The photonic crystal cavity, distributed feed-back cavity are considered as SW cavities while the ring cavity is a typical example of a TW cavity. Interaction between a single optical cavity and a waveguide has been theoretically studied based on the temporal coupled mode theory (CMT) in Refs.[8-12]. When two or more coupled identical cavity modes exist in a system, mode splitting will naturally occur. Previous study on mode splitting in such systems was however restricted to the directly (cavity-cavity) coupled TW cavity mode for add-drop filter application [13]. SW cavity systems or systems with other types of inter-cavity coupling mechanisms are not studied. In this paper, we systematically analyze mode splittings and transmission characteristics for both directly and indirectly (cavity-waveguide-cavity) coupled cavity systems. The cavities are of either SW or TW type. Altogether we treat coupled SW cavity systems with up to ten different configurations as well as coupled TW cavities with six different configurations using CMT. The transmission amplitude, phase and corresponding group delay are considered in all the configurations. It will be shown that both direct and indirect coupling can lead to mode-splitting, and a coupled SW cavity system and a coupled TW cavity system can have the same characteristics in some cases. A coupling quality factor Qc is introduced to characterize the direct coupling between two cavity modes, which is more convenient to analyze the behavior of cavity system compared with the conventional matrix approach. The coupling between two cavity modes through waveguide is treated as indirect coupling. The mode-splitting resulted from the coupling between cavity modes can lead to different transmission spectra, including splitting in the transmission, zero transmission, Fano-transmission, electromagnetically-induced-transmission (EIT)-like transmission [14-16] and electromagnetically-induced-absorption(EIA)-like transmission. The mode splitting can also affect the phase characteristics of the transmission and lead to different dispersion features. Among our findings, it is interesting to notice that a side-coupled SW cavity system can perform the same functions as an under-coupled TW cavity system.

It is meaningful to point out that mode splitting can also occur in a single TW-cavity system by introducing structural perturbations on the cavity [17, 18]. This is however not within the scope of our study.

This paper is organized as follows. Section 2 gives a detailed analysis of a single SW and TW cavities, which forms the basis for our later discussions. Section 3 analyzes the mode-splitting characteristics of two directly coupled cavity modes. Both symmetrical and asymmetrical waveguide coupling configurations are considered in this section. Section 4 analyzes the mode-splitting characteristics of two indirectly coupled cavity modes. Section 5 presents our findings on two cavity mode with both direct and indirect coupling. Conclusion is given in Section 6.

## 2. Single SW and TW cavities

In CMT, an overall system is described in terms of a set of weakly coupled components, each of which can be analyzed using general principles [19]. A cavity, for example, is treated as an oscillator in time. For a SW cavity coupled to a waveguide, its cavity mode decays equally into the forward and backward propagating waveguide mode; while for a TW cavity, the cavity mode decays in only one direction due to momentum matching. For characterizing a cavity, we define the following parameters : *ω*_{0} (*λ*_{0}) is the resonant frequency (wavelength); *a* is the cavity mode amplitude; *s _{i}*/

*s*/

_{t}*s*/

_{r}*s*are the incident/transmitted/reflected/dropped waveguide mode amplitudes, which are normalized such that their squared values correspond to incident/transmitted/reflected/dropped power; 1/

_{d}*τ*and 1/

_{i}*τ*are decay rates due to intrinsic loss and waveguide coupling loss, respectively;

_{w}*Q*and

_{i}*Q*are cavity quality factors related to intrinsic loss and waveguide coupling loss, respectively;

_{w}*Q*is total quality factor (1/

_{t}*Q*=1/

_{t}*Q*+ 1 /

_{i}*Q*). Note that in this paper,

_{w}*Q*denotes the cavity-waveguide coupling and is referred to as “waveguide coupling quality factor”, whereas

_{w}*Q*represents the cavity-cavity and is referred to as “cavity coupling quality factor”. The decay rates are related to the cavity quality factors by

_{c}*Q*=

_{i}*ω*

_{0}

*τ*/2 and

_{i}*Q*=

_{w}*ω*

_{0}

*τ*/2. We use

_{w}*δ*to normalize the frequency

*ω*, which is defined by

*δ*= (

*ω*-

*ω*

_{0})/

*ω*

_{0}, and

*t*(=

*s*/

_{t}*s*),

_{i}*r*(=

*s*/

_{r}*s*),

_{i}*d*(=

*s*/

_{d}*s*) to denote transfer functions for transmitted port, reflected port and dropped port, respectively.

_{i}Figure 1 gives the schematics of a single cavity. In this paper, we use a disk and a ring to denote a SW and a TW cavity, respectively. For SW cavities, two different waveguide coupling configurations are considered, including the side-coupling configuration (Fig. 1(a)) and the shoulder-coupling configuration (Fig. 1(b)). Since *Q _{w}* denotes the total waveguide coupling quality factor for the mode considered in this paper, it is related to

*τ*by

_{w}*Q*=

_{w}*ω*

_{0}

*τ*for the

_{w}*S*2 configuration in Fig. 1(b)(and also for

*S*4,

*S*7,

*S*9,

*T*4 and

*T*6 in the following discussions). Without loss of generality, we assume the cavity mode is symmetric about the vertical mirror plane perpendicular to the waveguide for

*S*1. For the other systems mentioned in this paper, similar assumptions are made to ensure that all waveguide coupling coefficients are the same. According to CMT [11, 12, 20], the transfer functions at the transmitted ports of

*S*1,

*S*2 and

*T*1, denoted by

*t*

_{S1},

*t*

_{S2}and

*t*

_{T1}, are given in Table 1. For other ports, we can use the relations

*r*

_{S1}= -

*t*

_{S2}and

*r*

_{S2}= -

*t*

_{S1}. Therefore the transfer functions for the transmitted ports of

*S*1 and

*S*2 are identical to those for the reflected ports of

*S*2 and

*S*1, respectively, except with a

*π*phase shift. The transmission

*T*, effective phase shift

*θ*and group delay

*τ*can be calculated as

*T*=

*abs*(

*t*)

^{2},

*θ*= arg(

*t*) and

*τ*=

*dθ*(

*ω*)/

*dω*, respectively; their values at resonance are also given in Table 1.

From the Table 1 and Fig. 2, we can draw the following conclusions:

- When
*Q*_{i-over-T1}=*Q*_{w-under-T1}= 2*Q*_{w-S1}and 1/*Q*_{w-over-T1}= 1/_{Qi-under-T2}= 1/*Q*_{i-S1}+ 1/(2*Q*_{w-S1}), the over-coupled*T*1 (the TW cavity with*Q*>_{i}*Q*), under-coupled_{w}*T*1 (the TW cavity with*Q*<_{i}*Q*) and_{w}*S*1 have identical transmission spectra, as is shown in Fig. 2(a). Especially, the phase shift and group delay are completely the same for under-coupled*T*1 and*S*1. Therefore, we can treat the under-coupled TW cavity as a side-coupled SW cavity. In the following analyses, a system based on under-coupled TW cavities can be replaced with its counterpart based on side-coupled SW cavities. - For the SW cavity, the transmission approaches zero for
*S*1 and unity for*S*2 when*Q*≫_{i}*Q*. For the TW cavity, the transmissivity is unity when_{w}*Q*≫_{i}*Q*and zero when_{w}*Q*=_{i}*Q*, i.e, the critical coupling condition satisfied._{w} - The over-coupled
*T*1 exhibits the largest phase shift range (~ 2*π*). For*S*2, the phase shift range is only*π*. For*S*1 and under-coupled*T*1, the phase shift ranges are 2*θ*_{1}and 2*θ*_{2}, respectively, where ${\theta}_{1}={\mathrm{tan}}^{-1}[\sqrt{{Q}_{i}{Q}_{t}}/\left({2Q}_{w}\right)]\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}{\theta}_{2}={\mathrm{tan}}^{-1}\left(1/\sqrt{{Q}_{w}^{2}/{Q}_{i}^{2}-1}\right).$The phase shift range can achieve*π*only when*Q*≫_{i}*Q*for_{w}*S*1 and*Q*approaches_{w}*Q*for under-coupled_{i}*T*1. - The dispersions at resonance for
*S*2 and over-coupled*T*1 are normal and thus slow-light can be obtained. While for S1 and under-coupled*T*1, abnormal dispersion occurs at resonance and thus fast-light takes place.

## 3. Two directly coupled cavity modes

In this section, we analyze the mode-splitting characteristics of two identical cavity modes with direct coupling. Two situations are considered, where the coupling waveguide(s) is(are) placed either symmetrically or asymmetrically. The coupling coefficient between the two cavity modes is denoted by *μ* and is related to coupling quality factor by *Q _{c}* = (

*ω*

_{0}/(2

*μ*).

#### 3.1. Two directly coupled cavity modes with symmetric coupling waveguides

Figure 3 shows two identical cavity modes with direct coupling. The coupling waveguides are in symmetric placement. In these cases, the degeneracy of the two cavity modes are lifted due to the coupling, hence a split in the resonant frequencies. For two degenerate SW modes shown in Fig. 3(a), the evolution of fields *a*_{1} and *a*_{2} can be described from the CMT

Therefore, the transfer functions can be given as

Following the same procedure, the transfer functions for the transmitted ports of *S*4, *S*5 and *T*2 can be given as

The transfer functions for the other ports can be obtained through the relations *r*_{S3} = *t*_{S4}, *d*1_{S3} = *d*2_{S3} = *d*1_{S4} = *d*2*S*4 = -*t*_{S5}/2, *r*_{S4} = - *t*_{S3}, *r*_{S5} = -*t*_{T2} and *d*_{T2} = -*t*_{S5}. From Eqs.(3)–(6), we can see that the two degenerate modes with frequency *ω*_{0} are split into two resonant frequencies, namely *ω*_{0} − *ω*_{0}/(2*Q _{c}*) and

*ω*

_{0}+

*ω*

_{0}/(2

*Q*). The waveguide coupling quality factor for the two split modes still keeps at

_{c}*Q*. The separation of the two split resonances is solely determined by the coupling factor

_{w}*Q*for a fixed

_{c}*ω*

_{0}. Figures 4 and 5 illustrate the transmission, phase shift and group delay for the transmitted ports of

*S*3-

*S*5 and

*T*2, respectively.

For the SW cavity system *S*3, a decrease in *Q _{c}* reduces the depth of resonance notch and further lifts the degeneracy of the two resonances, as shown in Fig. 4(a). The splitting in the transmission turns more and more obvious with increasing coupling. However, the dispersions at the two splittings in the transmission still keep anomalous and thus fast light always takes place, as shown in Fig. 4(b) and (c), respectively. We must point out that although the mode-splitting occurs as long as the coupling exists, it doesn’t necessarily mean that the splitting occurs in the transmission spectrum. This is because that the overall transmission is determined by the superposition of the two split modes with different amplitudes and phases. Here the two splittings can be seen in the transmission only when the coupling is large enough.

For the SW cavity system *S*4, the peak transmission at resonance decreases initially as *Q _{c}* decreases. When

*Q*decreases further, splitting takes place in the transmission despite that the dispersion remains normal in the two splittings in the transmission, where slow light occurs therefore. The same trends in the transmission, phase shift and group delay occur for the SW cavity system

_{c}*S*5, which are shown in Fig. 4(g)-(i). The coupling gradually raises the transmission at original resonant wavelength. With decreasing

*Q*, splitting manifests itself in the transmission but normal dispersion and slow-light always take place at the two splittings in the transmission.

_{c}For the TW cavity system *T*2, we consider two cases: (1) over-coupling case (*Q _{i}* >

*Q*) shown in Fig. 5. As

_{w}*Q*decreases gradually, the coupling increases resonance notch depth. When 1/

_{c}*Q*

_{c}^{2}= 1/

*Q*

_{i}^{2}− 1/

*Q*

_{w}^{2}, the transmission is zero at resonance and there is an abrupt

*π*jump in the phase shift. Once

*Q*decreases further, splitting takes place in the transmission and the dispersion at resonances shifts from normal to anomalous. For the dispersion response, slow-light occurs at the enhanced resonance and fast-light at the two splittings in the transmission. (2) under-coupling case (

_{c}*Q*<

_{i}*Q*). The under-coupled TW cavity can be regarded as a side-coupled SW cavity, as has been pointed out in Section 2. Therefore, the transmission, phase shift and group delay characteristics for

_{w}*T*2 in this case are completely the same as those for

*S*3.

#### 3.2. Two directly-coupled cavity modes with asymmetric coupling waveguide(s)

In this part, we analyze the cases that the waveguide-coupling for the two coupled cavity modes are asymmetric, which are shown in Fig. 6. The resonant frequencies for *a*1 and *a*2 are *ω*_{0} while the intrinsic quality factors are *Q*_{i1} and *Q*_{i2}, respectively. The waveguide-coupling occurs only for mode *a*_{1} in all three configurations. For the S6 configuration, the CMT equations are given as follows:

Therefore, the transfer function can be obtained as follows:

Following the same procedure, we can obtain the transfer functions for *S*7 and *T*3:

The transfer functions for other ports can be related to those for the transmitted port: *r*_{S6} -*t*_{S7}, *r*_{S7} = −*t*_{S6}.

Table 2 provides the mode-splitting characteristic for *S*6 for different coupling strengths.
Here (1/*Q*_{0})^{2} = ∣(1/*Q _{c}*)

^{2}- [1/(2

*Q*

_{i1}) - 1 /(2

*Q*

_{i2}) + 1/(2

*Q*)]∣

_{W}^{2}. We can see from Table 2 that the mode-splitting depends greatly on the coupling strength: (1) when 2/

*Q*> 1/

_{c}*Q*

_{i1}-1/

*Q*

_{i2}+ 1/

*Q*, the resonant frequency for the two split modes are

_{w}*ω*

_{0}−

*ω*

_{0}/(2

*Q*

_{0}) and

*ω*

_{0}+

*ω*

_{0}/(2

*Q*

_{0}), respectively, while the intrinsic quality factor and the waveguide coupling quality factor are the same; (2) when 2/

*Q*< 1/

_{c}*Q*

_{i1}- 1/

*Q*

_{i2}+ 1/

*Q*, the resonant frequencies for the two split modes are still

_{w}*ω*

_{0}but the waveguide coupling quality factors are different, which are 2

*Q*

_{w}Q_{0}/(

*Q*

_{0}+ 2

*Q*) and 2

_{w}*Q*

_{w}*Q*

_{0}/(

*Q*

_{0}- 2

*Q*), respectively; (3) when 2/

_{w}*Q*= 1/

_{c}*Q*

_{i1}− 1/

*Q*

_{i2}+ 1/

*Q*, the two split modes are degenerate, with the same resonant frequency and intrinsic quality factor and waveguide coupling quality factor. The transmissions for

_{w}*S*6,

*S*7 and

*T*4 when

*Q*

_{i2}→ ∞ are given by the following expressions:

We first recall EIT equation from Ref. [21], *Tab* = Ω_{1}^{2}ΓΔ^{2} /[Δ^{2}Γ^{2} + 4(Δ^{2} - Ω^{2}_{2}/4)^{2}] , where Ω_{1} and Ω_{2} are respective Rabi frequencies of probe field and pump field, Γ is decay rate and Δ is detuning of probe field from atomic resonance. It can be seen that Eq. 12 is identical to the EIT equation if we regard *ω*_{0}
*δ* → Δ, *ω*_{0}/*Q*_{i1}+*ω*_{0}/*Q _{w}* → Γ,

*ω*

_{0}/

*Q*→ Ω

_{c}_{2}Therefore, the EIT-like transmission is a special case of mode-splitting due to the coupling between cavity modes. From Eq. 12, two conclusions can be drawn: (1)

*T*(

*ω*

_{0}) = 1, meaning that complete transparency can be obtained in the transmission spectrum; (2)

*T*

_{S6}achieves minimum at

*δ*= 1/(2

*Q*) and this minimum can be zero if

_{c}*Q*

_{i1}≫

*Q*.

_{w}Figure 7 plot the transmission, phase shift and group delay for *S*6 and *S*7. For *S*6, without direct coupling, the dispersion is abnormal and fast-light occurs at resonance. As the direct coupling increases, the EIT-resonance becomes obvious and the dispersion changes to normal in the EIT-like resonance. It can be seen that the delay is quite large at the EIT-like resonance, which is obviously demonstrated in Fig. 7(c) when *Q _{c}* = 2 × 10

^{5}. No complete transparency is achieved because we consider the loss of mode

*a*

_{2}here.

For *S*7, the transmission at resonance forms a dip with increased direct coupling, which is called EIA. The EIA is an opposite effect of EIT and enhancement of absorption in EIA results from from atomic coherence induced by optical radiation [22]. Here, it originates from the direct coupling between two modes. From Eq.13, we can see that *T*(*ω*_{0}) = 0, meaning that complete absorption occurs in the EIA-like resonance. The dispersion is abnormal and large fast-light occurs in this EIA-like resonance, as are shown in Figs. 7(e) and (f), respectively.

For *T*3, when *Q _{c}* → ∞, Eq. 14 is also an EIT equation. Figure 8 provides the transmission, phase shift and group delay for

*T*3 when mode

*a*

_{1}is in over-coupled case (

*Q*

_{i1}>

*Q*). As direct coupling increases, a dip appears in the resonance notch first. When 1/

_{c}*Q*

^{2}

*= (1/*

_{c}*Q*- 1/

_{w}*Q*

_{i1})/

*Q*

_{i2},

*T*(

*ω*

_{0}) = 0, indicating that zero transmission is achieved with the aid of direct coupling. As the direct coupling increases further, the dip disappears and EIT-like resonance appears. Normal dispersion and large slow-light can be observed in the EIT-like resonance of

*T*3, which are the same as that of

*S*6. The dispersions are normal in the two splittings, where slow-light occurs. This is different from

*S*6, where abnormal dispersion and fast-light occur in the two splittings. For under-coupled case (

*Q*

_{i1}<

*Q*), the transmission, phase shift and group delay are similar to those of

_{w}*S*6 system.

## 4. Two indirectly coupled cavity modes

In this part, we analyze two cavity modes coupled through waveguide. For the SW cavity system S8 shown in Fig. 9(a), the two cavities are connected by a waveguide, which induces a phase shift of *ϕ* (we neglect the waveguide dispersion here for simplicity) and thus indirect coupling between the two modes *a*_{1} and *a*_{2}. We assume the two modes are degenerate. The resonant frequency, intrinsic quality factor and waveguide coupling quality factor are denoted by *ω*_{0}, *Q _{i}* and

*Q*, respectively. According to the CMT, the equations for the evolution of the cavity modes

_{w}*a*

_{1}and

*a*

_{2}in time can be given as follows:

Therefore, the transfer functions for the transmitted wave and reflected wave can be expressed:

where *γ*_{0} = −1/[2*Q _{w}* (

*jδ*+1/(2

*Q*)+1/(2

_{i}*Q*))].

_{w}*γ*is actually the transfer function of the reflected port for the single side-coupled SW cavity, as has been pointed before. Therefore, the two cavities can be regarded as two reflecting mirrors of a Fabry-Perot etalon. The two mirrors have an amplitude reflectance of

*γ*and transmission of 1+

*γ*. Eqs.(17) and (18) can be regarded as transfer functions for the transmission and reflectance of the Fabry-Perot etalon, respectively. From Eq.(17) for

*S*8, we can see that the resonant frequency is modified due to introduction of the waveguide. Eq.(17) can be rewritten as:

Table 3 provides the mode-splitting characteristics for *S*8 for different *ϕ* introduced by the waveguide. We consider three cases here: (1) when *ϕ*=*mπ* (*m* is an integer), *t*_{S8} = (−1)* ^{m}*(

*j*2

*δ*+ 1 /

*Q*)/(

_{i}*j*2

*δ*+ 1/

*Q*+2/

_{i}*Q*), which is the expression for

_{w}*S*1, except that

*Q*is modified to

_{w}*Q*/2. In this case, the two modes

_{w}*a*

_{1}and

*a*

_{2}can be regarded as one mode with a waveguide coupling quality factor of

*Q*/2; (2) when

_{w}*ϕ*=(

*m*+ 1/2)

*π*, the indirect coupling due to the waveguide leads to mode splitting and the frequencies of the two split modes are

*ω*

_{0}±

*ω*

_{0}sin

*ϕ*/(2

*Q*). However, the waveguide coupling quality factors for the two modes are still

_{w}*Q*. Therefore, the transmission spectrum keeps symmetric. Note that the maximum resonant frequency separation between the two split modes is

_{w}*ω*

_{0}/

*Q*. (3) when

_{w}*ϕ*≠

*mπ*/2, the frequencies of the two split modes are

*ω*

_{0}±

*ω*

_{0}sin

*ϕ*/(2

*Q*). Not only the resonant frequency is modified, but also the waveguide coupling quality factors (

_{w}*Q*) are changed, which is different from the mode-splitting in the direct coupling case. For the mode with a frequency of

_{w}*ω*

_{0}+

*ω*

_{0}sin

*ϕ*/(2

*Q*), the waveguide coupling quality factor is modified to

_{w}*Q*/(1 -cos

_{w}*ϕ*). For the mode with a frequency of

*ω*

_{0}-

*ω*

_{0}sin

*ϕ*/(2

*Q*), the waveguide coupling quality factor is modified to

_{w}*Q*/(1 + cos

_{w}*ϕ*). Figure 11 plots the transmission, phase shift and group delay when

*ϕ*= 0.785 rad. We can see from Fig. 10 that the transmission ia asymmetrical in this case due to different waveguide coupling quality factors and resonant frequencies of the two split modes. This is actually a kind of Fano-transmission which can be potentially used for lowering power threshold in bistable optical devices and for sensing applications [10].

If the two modes are non-degenerate (*ω*_{1} and *ω*_{2}), the EIT-like and EIA-like transmission can be obtained, which have been demonstrated in both TW ring cavity and SW PhC cavity [23-25]. Here we just compare this kind of EIT-like transmission with the EIT-like transmission between two degenerate modes shown for the *S*6 system. When 1 /*τ _{w}* ≫

*ω*

_{1}−

*ω*

_{2}≫ 1 /

*τ*and

_{i}*ϕ*= −

*mπ*(

*m*is an integer), the transmission

*T*and reflection

*R*(=

*abs*(

*r*

^{2})) are denoted by

*T*

_{S8}=1−

*δ*

^{2}/

*Q*

_{w}^{2}/[

*δ*

^{2}/

*Q*

_{w}^{2}+(

*δ*

^{2}−(Δ

*ω*/(2

*ω*

_{0}))

^{2})

^{2}] and

*R*

_{S8}= 1 −

*T*

_{s8}, respectively.

*ω*

_{0}= (

*ω*

_{1}+

*ω*

_{2})/2 and Δ

*ω*= ∣

*ω*

_{1}−

*ω*

_{2}∣ and

*Q*→ ∞ is assumed. By comparing with the EIT equation given in Section 3.2, we can see that there are actually the EIT-like resonance and EIA-like resonance for

_{i}*ω*

_{0}if we regard

*ω*

_{0}

*δ*→ Δ,

*ω*

_{0}/

*Q*→ Γ, Δ

_{w}*ω*→ Ω

_{2}.

*T*(

*ω*

_{0}) = 1 and

*T*(

*ω*

_{1}) =

*T*(

*ω*

_{2}) = 0. The dashed line in Fig. 9(a) describes the path for EIT-like resonance mode and thus the phase shift for each round is 2

*ϕ*+ arg(

*γ*

_{1})+ arg(

*γ*

_{2}). The maximum transmission in the EIT-like resonance occurs at

*ϕ*(

*ω*

_{0}) = −

*mπ*. The transmission, phase shift and group delay characteristics are similar to

*S*6 system, which are demonstrated in Figs. 7(a)–(c).

For *S*9 system, we can use the CMT to obtain the transfer functions of transmitted port and reflected port. We can also treat the whole system as a Fabry-Perot etalon. The two cavities works as two mirrors. According to the transfer functions for *S*2, the amplitude transmission for the two mirrors are −*γ*_{1} and −*γ*_{2} and the amplitude reflectance are −*γ*_{1} −1 and -*γ*_{2} −1, respectively. Therefore, we can easily obtain the transfer functions for *S*9 system:

Here we just consider the case that the two modes are degenerate (*γ*_{1} = *γ*_{2}). Eq. (20) can be rewritten as:

It can be seen from Eq. (22) that the two degenerate resonances are split into two: *ω*_{0} − *ω*_{0}cot(*ϕ*/2)/4*Q _{w}* and

*ω*

_{0}+

*ω*

_{0}tan(

*ϕ*/2)/4

*Q*. The intrinsic quality factor and waveguide coupling quality factor of the two modes are

_{w}*Q*and 2

_{i}*Q*, respectively. Therefore, the resonant frequency separation is quite sensitive to

_{w}*ϕ*and can be changed just by adjusting

*ϕ*. This is a distinct feature for

*S*9 compared with

*S*8. Figure 11plots the transmission, phase shift and group delay for

*S*9 when

*ϕ*= −0.2 rad and

*ϕ*= 1.57 rad. We can draw two conclusions from Fig. 11 and Eq. (22):

(1) When 0 < sin^{2}
*ϕ* < *Q _{i}*

^{2}/(

*Q*+

_{w}*Q*)

_{i}^{2}, there is a splitting in the transmission and the frequencies of the two splittings in the transmission are ${\omega}_{0}-{\omega}_{0}\left(\mathrm{cos}\mathrm{\varphi \pm}\sqrt{1-{\left(1+{Q}_{w}/{Q}_{i}\right)}^{2}{\mathrm{sin}}^{2}\varphi}\right)/\left({4Q}_{w}\mathrm{sin}\varphi \right)$. When

*Q*>>

_{i}*Q*, the two frequencies are

_{w}*ω*

_{0}and

*ω*

_{0}−

*ω*

_{0}/(2

*Q*tan

_{w}*ϕ*), respectively, and the peak transmission achieves unity. From Figs. 11(a)-(c), we can see that the dispersion and delay characteristics for each splitting are similar to those of

*S*2.

(2) When *Q _{i}*

^{2}/(

*Q*+

_{w}*Q*)

_{i}^{2}≤ sin

^{2}≤ 1, there is no splitting in the transmission although the mode-splitting still exists. The transmission maximum/minimum occurs at

*ω*

_{0}−

*ω*

_{0}cot

*ϕ*/ (4

*Q*). From Figs. 11(d)-(e), it is actually a second-order peak filter for the transmitted port and notch filter for the reflected port.

_{w}For two TW cavity modes indirectly coupled by two waveguides (*T*4) shown in Fig. 9(d), through CMT analysis in [24], its transfer functions are the same as those of two SW cavity modes directly coupled by one waveguide (*S*8), namely *t*_{T4} = *t*_{S8} and *d*_{T4} = *r*_{S8}. Since the resonance mode in *S*8 is standing wave and both forward and backward modes are stimulated simultaneously, the transmission in the waveguide is bi-directional, as is shown in Fig. 9(a); therefore only one waveguide is enough to couple the two standing modes in the two SW cavities. However, for cascaded TW cavity system *T*4, the resonance mode is traveling wave and thus two waveguides are needed to form a closed path for the coupling. The propagation path for the EIT-like resonance mode is shown in Fig. 9(c).

## 5. Two cavity modes with both indirect and direct couplings

In this section, we consider the case that both direct coupling and indirect coupling through waveguide are introduced. The resonant frequencies of the two cavity modes are *ω*_{1} and *ω*_{2}, respectively. The quality factors for the two cavity modes are the same. For the system (*S*10) shown in Fig. 12(a), the transfer functions are given by:

Here we just consider the case of the two degenerate modes (*γ*_{1} = *γ*_{2}). Both direct coupling and indirect coupling through waveguide can lead to mode-splitting and lift the degeneracy of the two resonance modes. For the mode splitting induced by direct coupling, the frequency difference between the two split modes is inversely proportional to the *Q _{c}*. For the mode splitting induced by indirect coupling, the frequency difference between the two split modes greatly depends on the phase shift

*ϕ*. From Eq. (23), we can see that these two effects can offset each other so that the original resonant frequency is retained if sin

*ϕ*= −

*Q*/

_{w}*Q*. However, we must point out that although the resonant frequencies for these two modes are the same, the waveguide coupling quality factors are modified to

_{c}*Q*/(1 + 2

_{w}*Q*cos

_{w}*ϕ*) and

*Q*/(1 − 2

_{w}*Q*cos

_{w}*ϕ*), respectively. Figure 13 provides the transmission, phase shift and group delay for

*S*10 system when sin

*ϕ*= −

*Q*/

_{w}*Q*is satisfied. In this case, it turns to be a second order notch filter for the transmission and peak filter for the reflectance.

_{c}For two TW cavity modes coupled by one waveguide together with direct coupling (*T*5) shown in Fig. 12(b), we can deduce the transfer function as follows:

Here we just consider the case that the two modes are degenerate (*γ*_{1} = *γ*_{2}). From Eq.(25), we can see that when *Q _{w}* ≤ 2

*Q*and sin

_{c}*ϕ*= −1, the resonant frequencies for the two split modes are still

*ω*0. The waveguide coupling quality factors are modified to ${Q}_{w}{Q}_{c}/\left({Q}_{c}+{2Q}_{w}\sqrt{{Q}_{w}\left({2Q}_{c}-{Q}_{w}\right)}\right)\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}{Q}_{w}{Q}_{c}/\left({Q}_{c}-{2Q}_{w}\sqrt{{Q}_{w}\left({2Q}_{c}-{Q}_{w}\right)}\right),\phantom{\rule{.2em}{0ex}}$ respectively. Figure 14 provides the transmission, phase shift and group delay for

*T*5 system when the mode-splitting due to direct coupling is offset by indirect coupling through waveguide in resonant frequency. Both the over-coupling (

*Q*>

_{i}*Q*) and the under-coupling (

_{w}*Q*<

_{i}*Q*) cases are provided. In both cases, they are actually second order filter.

_{w}For two TW coupled cavity modes with both direct coupling and indirect coupling through two waveguides (*T*6) shown in Fig. 12(d), the transfer functions are completely the same as those of *S*10 system, namely *t*_{T6} = *t*_{S10} and *d*_{T6} = *r*_{S10}.

## 6. Conclusion

In the paper, the mode-splitting in coupled SW and TW cavities are analyzed based on the temporal CMT. Both the direct coupling and indirect coupling through waveguide can lead to the mode-splitting. For two directly coupled identical cavity modes with symmetrical waveguide coupling configuration (*S*3-*S*5 and *T*2), the resonant frequency is split and the separation between the two resonant frequencies is proportional to the coupling strength. For *T*2 in the over-coupling case, a zero transmission can be obtained by appropriately choosing the coupling. For two directly coupled identical cavity modes with asymmetrical waveguide positioning (*S*6, *S*7 and *T*3), the two split modes can be different either in resonant frequency or in waveguide coupling quality factor, depending on the strength of coupling. In special cases, the EIT-like and EIA-like transmission spectra can be obtained in these configurations. For two coupled cavity modes through indirect waveguide coupling, both the resonant frequency and waveguide coupling quality factor can be different for the two split modes and thus asymmetrical Fano-transmission can be obtained (*S*8 and *T*4). The separation between the two split modes can be easily controlled, in the case for *S*9, by changing the phase shift introduced by the waveguide. The direct and indirect coupling mechanisms can be concurrently deployed in cavity systems to retain original resonant frequency but the waveguide coupling quality factors are still modified (*S*10, *T*5 and *T*6). This research will be useful for the design of cavity-based devices for integration in nanophotonics, either to mitigate the crosstalk due to the coupling between cavities in densely packed photonics chips or to optimize the coupling between cavities for designing new functional photonics devices.

## Acknowledgments

This work is supported by the Swedish Foundation for Strategic Research (SSF) and the Swedish Research Council (VR).

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