Consider the double-coupler nonlinear fiber ring resonator as shown in Figure 14.17.

328 14. ElectromagneticWaves and Optical Resonators 3

 : 1 − 

 : 1 − 

L/2 L/2 E

2 E 4 1 E E

T Ein ER E

Figure 14.17: Schematic of a double-coupler fiber ring resonator.

Suppose that ER(t) = √κEin(t) + i√1 − κE4(t);

E1(t) = i√1 − κEin(t) + √κE4(t);

E2(t) = E1(t − tR)eiφ1(t−tR);

φ1(t − tR) =

πr2L

λAeff |E1(t − tR)|2;

E3(t) = √κE2(t);

ET (t) = i√1 − κE2(t);

E4(t) = E3(t − tR)eiφ2(t−tR);

φ2(t − tR) =

πr2L

λAeff |E3(t − tR)|2;

where the fiber loop is of length L, both halves are of length L/2, tR is the time taken for the electric field to complete half a fiber loop, and both couplers split the power in the ratio κ : 1 − κ. Assuming that there are no losses in the fiber, show that ET (t) = −(1−κ)Ein(t−tR)eiφ1(t−tR)+κET (t−2tR)ei(φ1(t−tR)+φ2(t−2tR)).