Comparative Analysis of Optical Ring Resonators with Cross- and Direct-Coupled Configurations

This paper presents a proposed alternate configuration for optical ring resonator (ORR) having a resonance loop made of single-mode optical fiber. Under steady state condition, the responses of the proposed ORR are formulated by considering the output and loop intensities for various conditions of transmission coefficient, power coupling coefficient, and compared the results with the previous reported ORR configuration. The analyses have shown that the main difference between two ORRs is due to the values of power coupling coefficients of the couplers used in their configurations. Another major point, that is to be considered in design of ORR for a maximum output power is that the transmission coefficient and power coupling coefficient in the proposed configuration are not required to be equal, as the case of the previously reported configuration. The analytical results given in this article can be useful for selection of ORR configuration for particular applications, specially in case of nonlinear use of ORR configuration.

A basic structure of an ORR consists of a 2×2 directional coupler where one of the input port is connected to one of the output port, making a closed loop to configure a ring resonator. To operate an ORR at resonance, the loop length could be of the order of few micrometers [10] to tens of meters [32].
Recently, a full steady-state analysis of the ORR is reported where characteristic parameters such as coupling coefficient of the coupler and transmission parameters of the loop fiber affecting the response are investigated [35]. In another report, the dynamic behavior of the ORR, by considering the influences of light-source parameters on the response, is presented [36]. An ORR may be realized mainly by two types of configurations: Cross-coupled and direct-coupled. To the best of our knowledge, in the reported analyses of the ORR in the literatures, the importance and difference of these configurations have not been addressed with application point of views. With respect to the type of high power applications of ORR, variety of configuration may play an important role [37][38][39].
In this paper, we analyze the two ORR configurations and compare their characteristic intensities at the output and in the resonance loop [35]. It is shown that in one configuration the, the resonance loop intensity is about twice the other one.

Intensities of Output and Resonance Loop
The two ORR configurations, which are considered for analysis, are shown in Fig. 1. In Fig. 1(a), a previously reported configuration [35] is shown in which the resonance loop is made by connecting two cross-coupled ports (2) and (3), whereas the resonance loop of our suggested ORR con-figuration in Fig. 1(b) is made by connecting two direct-coupled ports (2) and (4).
For cross-coupled ORR (CCORR) in Fig. 1(a), let us assume that the loop and the coupler are ideal with no polarization coupling and the birefringence in the loop fiber is zero. In Fig. 1(a), if we denote in E , r2 E , r1 E and 0 E the electric fields of light waves at ports (1), (2), (3), and (4), respectively, the complex field amplitudes at ports (3) and (4), under steady state operation, can be expressed as [1,35].
Where κ is the power coupling coefficient of the coupler and 0 γ is its fractional intensity loss. For an ideal condition and based on conservation of energy [35,40], the electric fields r2 E and r1 E are related by: ω is the optical angular frequency, and α is the power transmission coefficient of the loop fiber expressed as   (4) we can obtain the field at port 2 [35]: Substituting Eqs. (1), (4), and (5) in Eq. (2), 0 E results in:

Intensities of Output and Resonance Loop
By some algebraic manipulations of Eqs. (5) and (6), the output intensity at port (4) and the loop intensity at port (2) both normalized to input intensity can be shown, respectively, as:

Condition for Resonance
The condition for resonance in the ORR is obtained by finding the value of c ω at which the loop intensity loop I builds up to a maximum value and the output intensity out I reduces to a minimum value. By inspection of Eq. (8), the condition for resonance is found to be c 2 / 2 q ω τ = π − π , where q is an integer. Equation (8) shows that at resonance, the value of the output intensity out I becomes a minimum.
The additional condition required to make out I to become zero at resonance can be shown as: If the coupler is lossless 0 ( 0) γ = , we have κ = α . Figure  2 shows the theoretical output and loop intensities of the CCORR as a function of c ω τ for different values of ( ) κ =α . We note that for all values of κ , the output response dips to zero, and loop intensity peaks to a maximum value, indicating the resonance condition. As the value of ( ) κ =α decreases, the sharpness of the resonance response decreases as well. A reduction of 0.2 in the value of ( ) κ =α from 0.95 to 0.75 causes the maximum loop intensity to decrease from 19 to 3. We note that in Fig. 2(b) the resonance response is not sharp when the value of ( ) κ =α is small.

Response of CCORR at Resonance
Equation (13) gives the minimum value of out I at c 2 / 2 q ω τ = π − π as a function of κ and α . The normalized output intensity is equal to κ for 0 α = , is zero when κ = α , and is unity at κ = α . According to Eq. (14), under the resonance condition, if κ = α , the theoretical value of loop I tends to be large when κ tends to unity. But when 1 κ = , the power coupling in the coupler of the CCORR will be 100 percent, resulting in a zero loop intensity.

Suggested Configuration of ORR
In this section, we analyze our suggested configuration of an ORR, as presented in Fig. 1(b), which is hereafter termed as DCORR. Principally, this configuration has a response similar to that of the CCORR discussed so far. The difference between the configurations of Fig. 1(b) and of Fig 1(a) is that in former port (4) is used for feedback loop instead of port (3) as in the latter case.
Under steady state operation, as in the previous section, the complex electric field at ports (3) and (4) of Fig. 1(b) can respectively be written as: where all the parameters have their usual meaning. In a similar procedure, the normalized loop and output intensities can be shown as:

Response of DCORR at Resonance
The normalized loop intensity at resonance the condition (1 ) In general, Eq. (21) gives the minimum value of the out put intensity as function of κ and α . The additional condition for the output intensity to become zero at resonance is given by: (22) For 0 0 γ = will be: 1 κ = − α (23) Figure 3 shows the normalized output intensity out I of the DCORR as a function of c ω τ and Fig. 4 illustrates the normalized loop intensity loop I , assuming 1 α = − κ and 0 0 γ = .

Comparison of CCORR and DCORR
If we compare Fig. 1 with Fig. 2, the responses of the CCORR and the DCORR indicate that the behavior is the same in both the cases except that the resonance in the former occurs at c 2 / 2, 0,1, 2,... q q ω τ = π − π = whereas in the latter it occurs at c 2 , 0,1, 2,... q q ω τ = π = . The major difference between two configurations lies in the value of κ , where in the DCORR, when value of κ is lower, the finesse of the resonator will be higher, while in case of the CCORR is reverse. Moreover, we note that at resonance conditions, at whatsoever value it maybe, the loop intensity in CCORR is nearly twice the value in the DCORR.
For practical reasons, the CCORR configuration in Fig.  1(a) is usually employed, since such a configuration can be out I 0 1 (1 ) κ = − α − γ fabricated by using a single length of an optical fiber without a need of any splice joint. To obtain high finesse or to make as large as possible, it is desirable to avoid a splice joint in the loop fiber. To fabricate the configuration of DCORR in Fig. 1(b), two separate lengths of optical fibers would be needed to realize the coupler and a splice joint would be required to make the loop.

Conclusions
This paper suggested a configuration for optical ring resonator having a resonance loop made of single-mode optical fiber. Under steady state condition, the responses of the proposed ORR are formulated by considering the output and loop intensities for various conditions of transmission coefficient α , power coupling coefficient κ , and compared the results with the previously reported ORR configuration.
The analyses have shown that the characteristic behaviors of both the configurations are the same. The main difference between two ORRs is due to the value of power coupling coefficient of the couplers used for their configurations. Another major point is that in CCORR configuration the loop power is nearly double of the value in the DCORR case.
The analytical results given in this article can be useful for selection of ORR configuration for particular applications, such as high power generation and investigation of nonlinear phenomena.