Corrections To And Comments On" Dynamic Behavior Of Re Ection ...

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997

Correspondence Corrections to and Comments on “Dynamic Behavior of Reflection Optical Bistability in a Nonlinear Fiber Ring Resonator” Kazuhiko Ogusu, Alan L. Steele, John E. Hoad, and Stephen Lynch In the above paper1 on a transient analysis based on the multiplebeam interference method [1], some errors have been discovered. In this note, we detail corrections and discuss the validity of the multiple-beam interference method by comparison with another numerical method. Initially, we correct the mistakes in the analysis. The problem lies in the calculation of the total nonlinear phase shift of the partial wave entering the ring resonator at time t 0 mR and leaving it at time t after the mth circulation. In the text, the nonlinear phase change suffered during each circulation was calculated using the intracavity field at the exit (port 3, see Fig. 1(a) of the above paper1 ) of the ring resonator. The integration of the last circulation (corresponding to time interval from t 0 R to t) was carelessly omitted. As a corrected formula becomes implicit, i.e., the phase change suffered during the last circulation is expressed in terms of the output field, itself to be determined, this formulation is not advantageous. A more adequate procedure is to use the intracavity field at the input (port 4) of the resonator, in which case the output field is computed by using previously determined values. In such a formulation, Eq. (10) on p. 1538 should be replaced by

1(t) = k0

(a)

L n0 n2

2 20 jEring (t; z)j dz 2 = k0 n0 n2 j (1 0 )(1 0 )Eout (t) 0 (1 0 )Ein (t)j 20(1 0 ) 1 0 exp( 0 2 L ) 2 (10 ) 2 0

0

and similarly, Eq. (15) on p. 1539 should be

1(t) = k0

L n0 n2

20 jEring (t; z)j dz 2 = k0 n0n2 jj (1 0 )Eout (t) + (1 0 )Ein (t)j 20(1 0 )(1 0 ) 1 0 exp( 0 2L) : 2 (15 ) 2 2

0

0

Next, before presenting numerical results calculated using corrected Eq. (100 ), we describe another method to confirm the validity of the multiple-beam interference method [1]. Here we examine the direct-coupled ring resonator [Fig. 1(a) of the original paper1 ). The dynamics of such a fiber resonator can be analyzed by using the Ikeda theory [2]–[4]. The group-velocity dispersion effects of the Manuscript received July 17, 1997. K. Ogusu is with the Department of Electrical and Electronic Engineering, Shizuoka University, Hamamatsu, 432 Japan. A. L. Steel, J. E. Hoad, and S. Lynch are with the Department of Mathematics and Physics, The Manchester Metropolitan University, Manchester, M1 5GD, U.K. Publisher Item Identifier S 0018-9197(97)07823-8. 1 IEEE J. Quantum Electron., vol. 32, pp. 1537–1543, Sept. 1996.

(b)

=

Fig. 1. Nonlinear pulse response of the nonlinear ring resonator with L 4 cm,  0.1, and 0 0:1 for two values of fractional coupler intensity loss: (a) = 0 and (b) = 0.15. (a) and (b) are revised graphs of Figs. 2(b) and 6(c) in the original paper, respectively.

=

1 =0

ring fiber are neglected in the following formulation since the length of the ring treated here is short. When the relaxation time of the nonlinearity is much shorter than the cavity round-trip time R , the coupled differential-difference equations describing the dynamics of the cavity are reduced to a difference equation. The following analysis is rigorous in the limit of instantaneous relaxation of the nonlinearity. For a slowly time-varying input field, Ein , the intracavity field, Ering (t; 0), at position 4 and output field, Eout , at position 2 are described in the adiabatic limit by the following equations:

Ering (t; 0) = 0j (1 0 )Ein (t) + (1 0 )(1 0 ) 2 Ering (t 0 R ; 0) exp[0j (0 + N )] Eout (t) = (1 0 )(1 0 )Ein (t) 0 j (1 0 ) 2 Ering (t 0 R ; 0) exp[0j (0 + N )]

(1) (2)

with

0 = k0 n0 L

2 N = k0 n0 n2 jEring (t 0 R ; 0)j

0018–9197/97$10.00  1997 IEEE

20

1

0 exp(02L) 2

(3) (4)

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997

where parameters , , etc., are the same as those used in the original paper. The parameters 0 and N are the linear and nonlinear phase changes per round trip, respectively. Note that (2) is simply an iteration of the intracavity field, Ering (t; 0), with respect to the cavity round-trip time, R . Finally, we show the numerical results calculated using (100 ). Although the results in Figs. 2, 5, 6, 7, and 8 of the original paper are generally affected by the present correction, we present two typical results when the coupler has no loss, = 0, and a moderate loss of = 0.15. Fig. 1(a) and (b) are new graphs replacing Figs. 2(b) and 6(d) of the original paper, respectively. Fig. 1(a) shows that the output oscillation reported in the original paper is significantly damped. It seems that the numerical error brought about by the above-mentioned mistake induced the strong oscillation through a competition of accumulation and release of the intracavity power. Fig. 1(b) shows that the discrepancy between the two results before and after the correction is small when the coupler has a loss of

= 0.15. Therefore, we consider that the discussion of the results in Figs. 7 and 8 of the original paper is valid. Regarding the

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comparison between the multiple-beam interference method and the Ikeda approach, we have found both methods to be in very good agreement, i.e., indistinguishable when plotted. This shows that the multiple-beam interference method is as valid as the Ikeda approach for this application, although it should be noted that it is more computationally intensive. REFERENCES [1] T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A, vol. 19, pp. 1169–1176, 1979. [2] K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun., vol. 30, pp. 257–261, 1979. [3] K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: Chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett., vol. 45, pp. 709–712, 1980. [4] A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fiber loop mirror with feedback,” Opt. Commun., vol. 137, pp. 136–142, 1997.