1. INTRODUCTION

Nonreciprocal optical devices, especially optical diodes that transmit forward propagating light and do not transmit backward propagating light, are crucial for the realization of photonic integrated circuits. Magneto-optical materials have nonreciprocal properties and enable optical isolation, but these are unfortunately incompatible with a silicon-on-insulator platform [1,2]. Another approach to realize the diode operation is to use nonlinear optical effects, such as the optical Kerr effect, free-carrier effect, thermo-optic effect, four-wave mixing, stimulated Brillouin scattering, and so on [3–14]. Nonlinear optical diodes based on Kerr-type nonlinearity may be called “all-optical diodes” because they do not require any externally applied electrical fields. Many kinds of all-optical diodes have been studied and a large nonreciprocal transmission ratio (NTR) of $\sim 30\mathrm{dB}$ was successfully demonstrated by the cascaded ring resonator [9], cascaded L3 photonic crystal cavities [12], and the asymmetric H0 photonic crystal cavity [13]. NTR is one of the figures of merit (FOMs) for optical diodes and is defined as $T_{\mathrm{FW}}/T_{\mathrm{BW}}$, where $T_{\mathrm{FW}}$ and $T_{\mathrm{BW}}$ denote the forward and backward transmissions, respectively. Previous research mainly focused on the magnitude of the NTR. However, the forward transmission during the diode operation should be taken into account, which is also one of the most important characteristics. Although the Fano-like spectrum is often used in the all-optical diode to obtain a large NTR instead of the Lorentzian spectrum, it generally decreases the maximum transmission. In [8,9], since the combination of a notch filter and an add-drop filter shows the Fano-like spectrum, 100% transmission cannot be obtained due to the inclusion of the notch filter. In [9,13], the device is composed of an asymmetric cavity that has different coupling coefficients, which degrades the transmission as described in [15]. Furthermore, in [13], maximum transmission falls about several dB due to the inclusion of a partially transmitting element as well as an asymmetric cavity. For these reasons, the forward transmission tends to decrease during the diode operation. In this study, we focus on the box-like spectrum [16,17] rather than the Fano-like spectrum. The box-like spectrum has a wide bandwidth of 100% transmission and a high contrast of transmission between on-resonance and off-resonance. These behaviors are preferable to obtain a high forward transmission as well as a large NTR. Although the box-like spectrum was demonstrated by cascading ring resonators in [16], we confirmed that it can be also exploited by cascading side-coupled photonic crystal cavities [17], resulting in a small device size and low-power operation. The cascaded side-coupled cavities do not include a notch filter, different coupling coefficients, or any reflection elements. Thus, the use of the box-like spectrum in the all-optical diode is preferable to maintain 100% transmission.

This time, we newly found that by changing the distance between two cavities, both a box-like and a Fano-like spectrum can be generated due to the interaction between two cavities. Although the diode operation in cascaded L3 photonic crystal cavities was reported in [12], the forward transmission falls about 5–10 dB, which does not include insertion loss, because the NTR is mainly discussed and the interaction between two cavities is not considered well. Preliminary results show that 95% transmission and $\mathrm{NTR}>30\mathrm{dB}$ can be achieved by the box-like spectrum [17]; however, we also find that to achieve a higher transmission with a large NTR, careful adjustment of the distance between cavities is necessary. In this paper, it is revealed that the distance between two cavities significantly affects the forward transmission in the diode operation and almost perfect transmission with a large NTR can be achieved.

The paper is organized as follows: in Section 2, the concept for the high-forward-transmission all-optical diode is described with the nonlinear coupled-mode theory (NL-CMT) [18]. The Fano-like spectrum and box-like spectrum are compared, and then we indicate that careful adjustment of the phase difference between two cavities is critically important to obtain a high forward transmission and a large NTR. In Section 3, to explore how high transmission can be realized, we investigate optimum resonant characteristic parameters that give a high forward transmission with a large NTR. To verify our discussion, we perform a rigorous nonlinear analysis based on the finite-element time-domain beam propagation method (NL-FETD-BPM) [19,20].

2. THEORETICAL MODEL AND CONCEPT

Figure 1 shows the schematic of the $n$-cascaded side-coupled cavities, where $a_i$ is the amplitude of the $i$th cavity mode, $s_{\pm i}$ is the amplitudes of the incoming and outgoing lightwaves on the left side of the $i$th cavity, ${\tau }_i$ is the decay rate of the $i$th cavity mode, and $l_i$ is the waveguide length of the $i$th cavity. The decay rate is related to the quality factor $Q_i$, which is defined as

 

Fig. 1. Schematic of $n$-cascaded side-coupled cavities.

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To understand the behavior of the nonlinear characteristics in the cascaded cavities, we begin a discussion with two cascaded cavities composed of the same cavities, which gives a reciprocal operation due to the symmetry in the whole structure. Figures 2(a)–2(c) show the nonlinear transmission spectra for various input power and ${\varphi }_{12}$, where ${\varphi }_{(i)(i+1)}$ is defined as

 

Fig. 2. Nonlinear transmission spectra of two symmetrically cascaded cavities for various input power and ${\varphi }_{12}$ when $Q_{1,2}={10}^4$, $N_{1,2}={10}^{-4}\mathrm{\mu m}/\mathrm{pJ}$, and $n_{1,2}=2.76$, where (a) ${\varphi }_{12}=0.5\pi$, (b) ${\varphi }_{12}=0.375\pi$, and (c) ${\varphi }_{12}=0.25\pi$.

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Until now, we considered a symmetric structure. It gives a reciprocal operation even if the nonlinear effect is taken into account. To obtain the nonreciprocal transmission characteristics, at least two different cavities must be cascaded. We next consider two cascaded cavities composed of different cavities. Here we define the useful parameters

Figure 3(a) shows the transmission spectra of the two asymmetrically cascaded cavities when $\mathrm{\Delta }{\omega }_{12}=0.7$, $\mathrm{\Delta }{Q}_{12}=2.0$ (${\omega }_1=2\pi c/1550\mathrm{nm}$, $Q_1=\mathrm{10,000}$, where $c$ is the speed of light in vacuum), ${\varphi }_{12}=0.375\pi$, $N_{1,2}=8.7\times {10}^{-5}\mathrm{\mu m}/\mathrm{pJ}$, and $n_{1,2}=2.76$. For the forward and backward propagations, different transmission spectra can be obtained for input power of 65 mW/μm because of two asymmetrically deployed cavities. Here we focus on the shapes of the transmission spectra. The linear spectrum has two peaks (${\delta }_1=0.0$ and 0.7) due to the difference between two resonant frequencies ${\omega }_1$ and ${\omega }_2$. In the nonlinear regime, both forward and backward spectra are red-shifted, but there is a difference in spectral deformations for the forward and backward propagations, as shown in Fig. 3(a). When light is injected from the first cavity side, the input lightwave first couples to the first cavity before coupling to the second cavity, and, then, the energy stored in the second cavity becomes low, resulting in a strong red shift of the first cavity, and vice versa. If we set $\mathrm{\Delta }{\omega }_{12}>0$, the second cavity can be oscillated with low power, which, however, is not sufficient for the oscillation of the first cavity. Once the second cavity oscillates, most of the input light reflects and does not go to the first cavity, which means the first cavity cannot oscillate and the red shift of the first cavity becomes very low. Therefore, the backward transmission spectrum has two obvious peaks near ${\delta }_1=0.3$ and 2.2, and a high contrast between the forward and backward transmissions can be seen. Although there is no need of the difference of $Q$-factors (namely, $\mathrm{\Delta }{Q}_{12}$), the bandwidth for the diode operation can be improved by setting to $\mathrm{\Delta }{Q}_{12}>1$ as discussed later.

 

Fig. 3. Nonlinear transmission characteristics of two asymmetrically cascaded cavities when $\mathrm{\Delta }{\omega }_{12}=0.7$, $\mathrm{\Delta }{Q}_{12}=2.0$, (${\omega }_1=2\pi c/1550\mathrm{nm}$, $Q_1=\mathrm{10,000}$, where $c$ is the speed of light in vacuum), ${\varphi }_{12}=0.375\pi$, $N_{1,2}=8.7\times {10}^{-5}\mathrm{\mu m}/\mathrm{pJ}$, and $n_{1,2}=2.76$. (a) Transmission spectra for input power of 0 (linear) and 65 mW/μm (nonlinear), and (b) NTR corresponding to (a).

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Figure 3(b) shows the NTR for the input power of 65 mW/μm corresponding to Fig. 3(a). This graph can be divided to three parts, (i)–(iii). Forward transmission is higher than backward transmission in (i) and (iii), and there is a reverse relationship in (ii). We can see the two peaks of $|\mathrm{NTR}|\sim 90\mathrm{dB}$ in (ii), which corresponds to the red-shifted resonant peaks for the forward propagation. An all-optical diode using such peaks has been reported by Zhang et al. [12]. Note that, even though the NTR exceeds 30 dB in each part, a high transmission of about 100% is maintained in only (iii), which is a great advantage of using the box-like spectrum. As shown in Fig. 3(b), a NTR of 30 dB and 100% transmission can be achieved simultaneously in this condition. As illustrated above, to obtain an almost perfect forward transmission as well as an extremely large NTR, it is critically important to appropriately adjust the structural parameters, including not only the asymmetry of the whole structure, but also the distance between cavities.

3. DESIGN OF HIGH-FORWARD-TRANSMISSION ALL-OPTICAL DIODE

In this section, to explore how high transmission can be realized, we investigate optimum parameters, which give maximum forward transmission with a large NTR ($>30\mathrm{dB}$) in the designed all-optical diode. For simplicity, only the two-cascaded type is investigated and we should treat only three parameters, $\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$, and ${\varphi }_{12}$, because other characteristic parameters are correlated to input power (see Appendix B). Figures 4(a)–4(c) show the forward transmission, backward transmission, and NTR of two asymmetrically cascaded cavities as functions of input power and detuning parameters, where the structural parameters are the same as for Fig. 3. Increasing the input power, different behaviors can be seen between the forward and backward transmissions. The regions enclosed by white lines denote an $|\mathrm{NTR}|>30\mathrm{dB}$. The high transmission satisfying $|\mathrm{NTR}|>30\mathrm{dB}$ is only seen in the vicinity of the input power of 65 mW/μm and ${\delta }_1=2.1$.

 

Fig. 4. (a) Forward transmission, (b) backward transmission, and (c) NTR of two asymmetrically cascaded cavities for various input parameters (input power and detuning parameters), where structural parameters are the same as in Figs. 3(a) and 3(b).

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To investigate the optimum $\mathrm{\Delta }{\omega }_{12}$ and $\mathrm{\Delta }{Q}_{12}$, we define the FOM as

 

Fig. 5. FOM in the two asymmetrically cascaded cavities as a function of $\mathrm{\Delta }{\omega }_{12}$ and $\mathrm{\Delta }{Q}_{12}$ for (a) ${\varphi }_{12}=0.375\pi$ and (b) ${\varphi }_{12}=0.5\pi$.

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Figures 6(a)–6(d) show the nonlinear transmission characteristics of two asymmetrically cascaded cavities for ${\varphi }_{12}=0.375\pi$, where $(\mathrm{\Delta }{\omega }_{12},\mathrm{\Delta }{Q}_{12},P_{\text{in}})=(0.1,1.0,0.266\mathrm{W}/\mathrm{\mu m})$, (0.6, 1.6, 0.108 W/μm), (0.9, 2.0, 0.069 W/μm), and (1.5, 4.0, 0.017 W/μm). For each parameter, a NTR of 30 dB with $>99\%$ forward transmission is obtained. A flat spectral lineshape similarly to Fig. 2(b) is obtained by setting ${\varphi }_{12}=0.375\pi$, which contributes a wider bandwidth achieving a high forward transmission and a large NTR. For $\mathrm{\Delta }{Q}_{12}=1.0$, although a NTR of 30 dB and $T_{\mathrm{FW}}=99\%$ can be obtained, the bandwidth (here, detuning parameter) is very narrow. Increasing $\mathrm{\Delta }{Q}_{12}$ and appropriately setting $\mathrm{\Delta }{\omega }_{12}$, the bandwidth becomes wider. If setting a much larger $\mathrm{\Delta }{Q}_{12}$ such as 4.0, the bandwidth becomes narrow. Therefore, there is an optimum combination of $\mathrm{\Delta }{Q}_{12}$, $\mathrm{\Delta }{\omega }_{12}$, and ${\varphi }_{12}$ to obtain a large FOM and a wider bandwidth.

 

Fig. 6. Nonlinear transmission characteristics of two asymmetrically cascaded cavities when ${\varphi }_{12}=0.375\pi$, $N_{1,2}=8.7\times {10}^{-5}\mathrm{\mu m}/\mathrm{pJ}$, and $n_{1,2}=2.76$, where the parameters ($\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$, $P_{\text{in}}$) are (a) (0.1, 1.0, 0.266 W/μm), (b) (0.6, 1.6, 0.108 W/μm), (c) (0.9, 2.0, 0.069 W/μm), and (d) (1.5, 4.0, 0.017 W/μm), respectively.

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Finally, to confirm the validity of our design method, we perform a rigorous numerical analysis based on NL-FETD-BPM [19,20]. Figure 7 shows the schematics of the two asymmetrically cascaded L3 cavities. It consists of two L3 cavities and a photonic crystal line-defect waveguide. The structural parameters are lattice constant $a=420\mathrm{nm}$, air-hole radius $r=0.29a$, and the effective refractive index of the core is $n_{\text{core}}=2.76$, where $n_{\text{core}}$ corresponds to the refractive index of Si [22]. The number of rows of air-holes separating the $i$th cavity and the waveguide is $d_i$, where we set $d_1=4$ ($Q_1=4795$, ${\omega }_1=2\pi c/1559.828\mathrm{nm}$) and $d_2=5$ ($Q_2=\mathrm{14,641}$, ${\omega }_2=2\pi c/1560.027\mathrm{nm}$), which corresponds to $\mathrm{\Delta }{\omega }_{12}=1.21$ and $\mathrm{\Delta }{Q}_{12}=3.05$. $l$ is a discrete variable defined as $l(p)=(p+0.5((d_2-d_1)\%2))·a$, where $p$ is an integer and % represents the modulus operator. We set $p=20$, resulting in $l=8.61\mathrm{\mu m}$, which corresponds to ${\varphi }_{12}\approx -45.91$ (equivalent to $0.386\pi$). The nonlinear parameter $N_{1,2}$ is evaluated to be $1.06\times {10}^{-4}\mathrm{\mu m}/\mathrm{pJ}$, where we obtain it by performing a NL-FETD-BPM analysis for the normal L3 cavity with a nonlinear refractive index of $6.0\times {10}^{-18}{\mathrm{m}}^2/\mathrm{W}$ [23]. The linear transmission spectrum of these cascaded cavities is shown in Fig. 8, where a solid line and circles denote the results calculated by NL-CMT and NL-FETD-BPM, respectively. The numerical results obtained by NL-FETD-BPM are in excellent agreement with those calculated by NL-CMT. We can confirm almost 100% transmission and a flat spectral lineshape for a wavelength larger than 1560 nm, which is made by careful adjustment of the distance between two cavities.

 

Fig. 7. Schematics of the two asymmetrically cascaded L3 cavities.

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Fig. 8. Linear transmission spectrum in the two cascaded L3 cavities where $a=420\mathrm{nm}$, $r=0.29a$, $d_1=4$, $d_2=5$, and $p=20$. The dashed line and circles denote results calculated by CMT and FETD-BPM, respectively.

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Figure 9 shows the forward and backward transmissions in the two asymmetrically cascaded L3 cavities as functions of input power. Solid lines and circles with dashed lines denote the results calculated by NL-CMT and NL-FETD-BPM, respectively. The numerical results obtained by NL-FETD-BPM and NL-CMT agree well except for a slight difference of threshold power, which is also seen in a previous work [14]. For the forward and backward propagations, a high contrast of transmission can be seen in the vicinity of input power of 0.1 W/μm. Forward and backward transmissions of, respectively, 99.8% and 0.0005% are obtained, and the corresponding NTR is 53 dB. To the best of our knowledge, such a high forward transmission and large NTR in a Kerr-type all-optical diode have never been reported. Noting that, its characteristics can be achieved by carefully designing the cascaded cavities, which includes the optimum parameters of $\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$, and ${\varphi }_{12}$. Figures 10(a) and 10(b) show, respectively, the magnetic field distributions for forward and backward propagations during the diode operation when input power is 0.1 W/μm. Our designed all-optical diode performs almost 100% forward transmission and 0% backward transmission. Figures 11(a) and 11(b) show the time dependence of the forward and backward transmissions calculated by NL-FETD-BPM. In Fig. 11(a), solid and dashed lines denote, respectively, the forward and backward input power (correspond to ${|s_{+1}|}^2$ and ${|s_{-3}|}^2$). Lightwave is injected alternately from the left side and right side. In Fig. 11(b), solid and dashed lines denote, respectively, the normalized output power toward the forward and backward directions, which are normalized by input power [correspond to ${|s_{+3}|}^2/({|s_{+1}|}^2+{|s_{-3}|}^2)$ and ${|s_{-1}|}^2/({|s_{+1}|}^2+{|s_{-3}|}^2)$]. Although a momentary backward transmission arises from an unstable state of the cavity, we can see that the lightwave is constantly output to the right side, regardless of the direction of propagation.

 

Fig. 9. Forward and backward transmissions in the two cascaded L3 cavities as functions of input power where ${\delta }_1=2.1$ (${\omega }_{\text{in}}=2\pi c/1560.169\mathrm{nm}$). The solid lines and circles with dashed lines denote results calculated by NL-CMT and NL-FETD-BPM, respectively.

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Fig. 10. Magnetic field distribution calculated by NL-FETD-BPM during the diode operation in the two cascaded L3 cavities, where input power is 0.1 W/μm and the other parameters are the same as for Fig. 9. (a) Forward and (b) backward propagation.

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Fig. 11. Time dependence of forward and backward transmissions in the two cascaded L3 cavities where the parameters are the same as for Fig. 10. (a) Input power and (b) normalized output power calculated by NL-FETD-BPM.

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4. CONCLUSION

In this paper, we have proposed a high-forward-transmission all-optical diode based on cascaded photonic crystal cavities. It is proved that almost perfect transmission with a large NTR can be achieved by using a well-designed box-like spectrum, which is generated by cascading side-coupled photonic crystal cavities with appropriate distance. We have also shown the dependences of forward transmission, NTR, and bandwidth for the diode operation on $\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$, and ${\varphi }_{12}$. There is an optimum combination of $\mathrm{\Delta }{\omega }_{12}$ and $\mathrm{\Delta }{Q}_{12}$ to obtain a wider bandwidth. The numerical results obtained by a rigorous NL-FETD BPM analysis reveal that a high transmission of 99.8% and a large NTR of 53 dB can be obtained by the two carefully designed cascaded L3 cavities. It verifies our discussion and the effectiveness of the design method.

APPENDIX A

For $n$-cascaded side-coupled photonic crystal cavities, the coupled-mode equations in the $i$th ($i=1,2,\dots ,n$) cavity are expressed as [18]

(A1)$$\frac{d}{dt}{a}_i=(j{\omega }_i-\frac{1}{{\tau }_i})a_i+{\kappa }_i(s_{+i}\pm s_{-(i+1)}),$$

(A2)$$s_{-i}=\exp (-j\beta l_i)(s_{-(i+1)}\mp {\kappa }_i^{*}{a}_i),$$

(A3)$$s_{+(i+1)}=\exp (-j\beta l_i)(s_{+i}-{\kappa }_i^{*}{a}_i),$$

(A4)$${\kappa }_i=\sqrt{\frac{1}{{\tau }_i}}\exp (j{\theta }_i),$$

where we omit the term of intrinsic decay rate for simplicity, and the stationary solution for the monochromatic source $[s_i\propto \exp (j{\omega }_{\text{in}}t)]$ is

(A5)$$a_i=\frac{{\kappa }_i(s_{+i}\pm s_{-(i+1)})}{j({\omega }_{\mathrm{in}}-{\omega }_i)+\frac{1}{{\tau }_i}}.$$

To consider the Kerr-type nonlinearity, we assume the following relationship between the energy stored in the $i$th cavity and the change of refractive index:

(A6)$$\mathrm{\Delta }{n}_{\mathrm{NL}}\equiv N_i{|a_i|}^2.$$

Hence, the change of resonant frequency can be expressed as

$${\omega }_i\to \frac{n_i}{n_i+N_i{|a_i|}^2}{\omega }_i.$$

From Eqs. (A1) and (A6), the coupled-mode equation for a nonlinear cavity is given as

(A7)$$\frac{d}{dt}{a}_i=(j\frac{n_i}{n_i+N_i{|a_i|}^2}{\omega }_i-\frac{1}{{\tau }_i})a_i+{\kappa }_i(s_{+i}\pm s_{-(i+1)}).$$

Assuming $n_i\gg N_i{|a_i|}^2$, we can find the stationary solution of Eq. (A7) as

(A8)$$a_i=\frac{{\kappa }_i(s_{+i}\pm s_{-(i+1)})}{j({\omega }_{\mathrm{in}}-\frac{n_i}{n_i+N_i{|a_i|}^2}{\omega }_i)+\frac{1}{{\tau }_i}}$$

and the transfer matrix as

(A9)$$\left[\begin{array}{c}{s}_{+i}\\ s_{-i}\end{array}\right]=[Q_i][P_i][Q_i]\left[\begin{array}{c}{s}_{+(i+1)}\\ s_{-(i+1)}\end{array}\right],$$

where

(A10)$$[P_i]=\left[\begin{array}{cc}1-j{X}_i& -j{X}_i\\ +j{X}_i& 1+j{X}_i\end{array}\right],$$

(A11)$$[Q_i]=\left[\begin{array}{cc}\exp (+j\frac{\beta }{2}{l}_i)& 0\\ 0& \pm \exp (-j\frac{\beta }{2}{l}_i)\end{array}\right],$$

(A12)$$X_i=\frac{1}{{\delta }_i+\frac{N_i{|a_i|}^2}{n_{\mathrm{i}}}2Q_i}.$$

At the same time, the forward transmission and reflection can be calculated as, respectively,

(A13)$$T=\frac{{|s_{+n}|}^2}{{|s_{+1}|}^2},$$

(A14)$$R=\frac{{|s_{-1}|}^2}{{|s_{+1}|}^2}.$$

APPENDIX B

When we search the optimum resonant characteristic parameters in the two cascaded cavities, only $\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$ and ${\varphi }_{12}$ should be focused. To verify it, we explain the dependencies between variables showing some parameter conversion. First, we rewrite Eqs. (A8), (A2), and (A3) into, respectively,

(B1)$$a_i=\frac{\sqrt{\frac{2Q_i}{{\omega }_i}}(s_{+i}\pm s_{-(i+1)})}{j[(\mathrm{\Delta }{\omega }_{1i}-{\delta }_1)\frac{{\omega }_1\mathrm{\Delta }{Q}_{1i}}{{\omega }_i}+\frac{2Q_i{N}_i{|a_i|}^2}{n_i}]+1}\exp (j\theta ),$$

(B2)$$s_{-i}=\exp (-j\beta l_i)(s_{-(i+1)}\mp \sqrt{\frac{{\omega }_i}{2Q_i}}\exp (-j{\theta }_i)a_i),$$

(B3)$$s_{+(i+1)}=\exp (-j\beta l_i)(s_{+i}-\sqrt{\frac{{\omega }_i}{2Q_i}}\exp (-j{\theta }_i)a_i),$$

where we use the following approximation:

(B4)$$\frac{N_i{|a_i|}^2}{n_i+N_i{|a_i|}^2}\approx \frac{N_i{|a_i|}^2}{n_i}.$$

When ${\delta }_1$, ${\omega }_1$, $\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$, ${\varphi }_{12}$, $N_{1,2}$, and $n_{1,2}$ are fixed, if considering the parameter conversion

$$Q_1\to {\xi }^{-1}{Q}_1,$$

where $\xi$ is an arbitrary positive real number, we find the following change of the solution:

$$(a_i,s_{\pm i})\to ({\xi }^{1/2}{a}_i,\xi s_{\pm i}),$$

where we use the following approximation:

$$1\gg \frac{\mathrm{\Delta }{\omega }_{12}(1-\xi )}{2Q_1-\mathrm{\Delta }{\omega }_{12}}.$$

It indicates that when ${\delta }_1$, ${\omega }_1$, $\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$, ${\varphi }_{12}$, $N_{1,2}$, and $n_{1,2}$ are fixed, even if $Q_1$ is changed, the same forward and backward transmissions are obtained by setting input powers of ${\xi }^2{|s_{+1}|}^2$ and ${\xi }^2{|s_{-3}|}^2$, respectively. Therefore, unless ${\delta }_1$, ${\omega }_1$, $\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$, ${\varphi }_{12}$, $N_{1,2}$, and $n_{1,2}$ are changed, FOM is constant. There are some similar relationships and we have summarized them in Table 1. From the relationships in Table 1, when investigating optimum parameters, we should search only three parameters, $\mathrm{\Delta }{\omega }_{12}$, $\mathrm{\Delta }{Q}_{12}$ and ${\varphi }_{12}$. Furthermore, for $n$-cascaded cavities, we should search only $\mathrm{\Delta }{\omega }_{1i}$, $\mathrm{\Delta }{Q}_{1i}$, and ${\varphi }_{1i}$.

Table 1. Parameter Conversions and Corresponding Solutions

View Table

Funding

Japan Society for the Promotion of Science (JSPS) (17J00378).

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