Statistical Properties of a Ring Laser with Injected Signal and Backscattering

Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 87 92 c International Academic Publishers Vol. 35, No. 1, January 15, 2001 Statistical Properties of a Ring Laser with Injected Signal and Backscattering LENG Feng and ZHU Shi-Qun Department of Physics, College of Sciences, Suzhou University, Suzhou 215006, China (Received November 1, 1999; Revised April 14, 2000) Abstract The statistical properties of a homogeneously broadened ring laser with an injected signal are investigated and the normalized two-mode intensity auto- and cross-correlation functions are calculated by a full saturation laser theory with backscattering. The theoretical predictions are in good agreement with the experimental measurements. Further investigation reveals that the backscattering can reduce the fluctuations in the system while the full saturation effect plays a major role when the laser is operated above threshold. It is also quite important to notice that the injected signal can drive the weak mode from incoherent light to coherent light. PACS numbers: 42.55.Lt, 05.40.+j, 42.60.Mi Key words: homogeneously broadened ring laser, injected signal, full saturation laser theory, backscattering 1 Introduction The statistical properties of a bidirectional two-mode ring laser have attracted much attention both experimentally and theoretically in recent years. [1 10] Some of these investigations were concerned on the applications of ring laser system as a gyroscope, [1 3] the intensity correlation and mode competition effects in a ring laser. [4 7] Others were concerned on the possibility of controlling the output of a laser by injection of the light signal from another laser. [8 10] In most of the previous analyses, the traditional third-order laser model is employed and only cubic nonlinearities are contained. It is necessary to model a bidirectional two-mode ring laser using a theory with full saturation effects in the presence of backscattering especially when the injected signal is included in the system. In this paper, the full saturation effects and backscattering are taken into account in a homogeneously broadened two-mode ring laser with injected signal. In Sec. 2, the equation of motion in a laser field with injected signal is presented and the expression for the steady-state two-mode intensity distribution function is derived. The mean, auto- and cross-correlation functions are calculated. In Sec. 3, the theoretical predictions are compared with experimental measurements of Cheng and Mandel. [10] A good agreement is obtained. In Sec. 4, the backscattering and full saturation effects are discussed. A discussion of the laser theory concludes the paper in the last section. 2 Equation of Motion The complex electric fields E 1 and E 2 of a bidirectional ring laser with an injected signal follow the Langevin equation when the full saturation effects and backscattering are taken into account, de 2 dt = K 2 E 2 + FE 2 1 + ( E 2 2 + ξ E 1 2 )/F + R 2 E 1 + q 2 (t), (1) where K 1 and K 2 are the cavity decay rates for the two modes, F is the gain parameter with a 1 = F K 1 and a 2 = F K 2, ξ is the mode coupling constant, it takes the value 2 for a homogeneously broadened two-mode ring laser and 1 for an inhomogeneously broadened ring laser resonant at line center, E 3 is the electric field of the injected signal, η is the constant that relates the injected signal to the output laser field, R 1 and R 2 are complex backscattering coefficients, q 1 (t) and q 2 (t) are additive quantum noises, their first and second moments obey q 1 (t) = q 2 (t) = 0, q i (t)q j (t ) = 2Pδ ij δ(t t ), (i,j = 1,2), (2) here P is the strength of additive noise. A simple binomial expansion of the denominator in Eq. (1) leads to the traditional third-order laser theory discussed in Ref. [10] with R 1 = R 2 = 0. By writing E 1 = x 1 + ix 2,E 2 = x 3 + ix 4,E 3 = x 5 + ix 6,R 1 = r 1 + ir 2,R 2 = r 3 + ir 4,η = η 1 + iη 2, the corresponding Fokker Planck equation can be written as [11] Q(E 1,E 2,t) t = + P 2 4 i=1 with the drift coefficients x i [h i (E 1,E 2 )Q(E 1,E 2,t)] 4 2 x 2 i=1 i [Q(E 1,E 2,t)] (3) de 1 dt = K 1 E 1 + FE 1 1 + ( E 1 2 + ξ E 2 2 )/F + R 1 E 2 + ηe 3 + q 1 (t), h 1 = K 1 x 1 + Fx 1 1 + [(x 2 1 + x2 2 ) + ξ(x2 3 + x2 4 )]/F + r 1 x 3 r 2 x 4 + η 1 x 5 η 2 x 6, The project supported by National Natural Science Foundation of China (Grant No. 19874046) and Natural Science Foundation of Jiangsu Education Commission of China

88 LENG Feng and ZHU Shi-Qun Vol. 35 h 2 = K 1 x 2 + Fx 2 1 + [(x 2 1 + x2 2 ) + ξ(x2 3 + x2 4 )]/F + r 1 x 4 + r 2 x 3 + η 1 x 6 + η 2 x 5, h 3 = K 2 x 3 + + r 3 x 1 r 4 x 2, h 4 = K 2 x 4 + Fx 3 1 + [(x 2 3 + x2 4 ) + ξ(x2 1 + x2 2 )]/F Fx 4 1 + [(x 2 3 + x2 4 ) + ξ(x2 1 + x2 2 )]/F + r 3 x 2 + r 4 x 1. (4) In the case of in-phase backcoupling with R 1 = R 2, the two-dimensional steady-state intensity distribution function can be calculated analytically and given by Q ss (I 1,I 2 ) = N s [1 + (I 1 + ξi 2 )/F] β1 [1 + (I 2 + ξi 1 )/F] β2 I 0 (2R I 1 I 2 /P) I 0 ( η [ I 1 I 3 )exp 1 ] P (K 1I 1 + K 2 I 2 ), (5) where N s is the normalization constant, I 1 = E 1 2, I 2 = E 2 2, I 3 = E 3 2, R 1 = R 2 = R, β 1 = F 2 I 1 /P(I 1 + ξi 2 ), β 2 = F 2 I 2 /P(I 2 + ξi 1 ), and I 0 (z) is the zero-order Bessel function which is defined by [12] (z/2) 2k I 0 (z) = (k!) 2. (6) k=0 The expectation values I1 n I2 m of the laser intensities can be calculated from Eq. (5) and given by I1 n I2 m = Q ss (I 1,I 2 )I1 n I2 m di 1 di 2, (n,m = 0,1,2,3, ). (7) The normalized auto- and cross-correlation functions of the intensities are given by and λ 11 (0) = I2 1 1, (8) I 1 2 λ 22 (0) = I2 2 1, (9) I 2 2 λ 12 (0) = I 1I 2 1. (10) I 1 I 2 In this paper, only homogeneously broadened ring laser system with ξ = 2 is discussed. 3 Comparison of Theory and Experiment It is necessary to compare theoretical results and experimental measurements to check the accuracy of the theory. The one-dimensional steady-state intensity probability density of I 1 can be obtained by integrating the variable I 2 and given by Q ss (I 1 ) = 0 Q ss (I 1,I 2 )di 2 = exp[ U(I 1 )], (11) where U(I 1 ) is the so-called pseudopotential and is plotted in Fig. 1 for various values of the injected signal η 2 I 3. The corresponding one-dimensional steady-state probability density obtained from Eq. (11) and the experimental measurements of Cheng and Mandel [10] are shown in Fig. 2. It is clear that the one-dimensional steady-state probability density Q ss (I 1 ) shows a two-peak structure. One peak is near the zero point which corresponds to pseudopotential curves that I 1 is before the saddle point I 1s, in this case mode 1 seems to be off. While another peak is near the point I 1 10 that corresponds to the case that mode 1 is on. So the probability Q off and Q on can be calculated by the integration I1s Q off = Q ss (I 1 )di 1, Q on = Q ss (I 1 )di 1. (12) 0 I 1s The curves of Q off and Q on are shown in Fig. 3 together with the measured data from Ref. [10]. The theoretical predictions of λ 12 (0) from Eq. (5) and the experimental data from Ref. [10] are shown in Fig. 4. From Figs 2 4, it is seen that a good agreement between the full saturation laser theory and experiment is obtained. Though they are not shown in Figs 2 4, the theoretical curves from the third-order laser theory always give lower values than the measured ones. The deviation increases with increasing values of the laser intensity I 1. It is very clear that the full saturation theory can provide correct description of the laser behavior especially when the laser is operated well above the threshold. Fig. 1 Typical curves of the steady-state pseudopotential for the full saturation laser model as a function of intensity I 1 with backscattering. The parameters are chosen as K 1 = 90.1, K 2 = 89.9, a = 0.2, R = 0.1 F = 100. The dotted, dashed and solid lines correspond to η 2 I 3 = 0.0, 3.58 and 9.0, respectively. The behaviors of the two laser intensities I 1 and I 2 are shown in Fig. 5 which is calculated by Monte Carlo simulations of Eq. (1), where Gaussian random numbers are generated to simulate the random noise terms in the Langevin equation at each integration step. The parameters are chosen to be the same as those in the experimental measurements of Ref. [10]. It is seen that the numerical simulations can reproduce the intensity time series of I 1 and I 2 recorded in Ref. [10].

No. 1 Statistical Properties of a Ring Laser with Injected Signal and Backscattering 89 Fig. 2 The one-dimensional steady-state probability density obtained from Eq. (11) for the full saturation theory with injected signal and backscattering. The parameters are chosen as K 1 = 90.1, K 2 = 89.9, a = 0.2, F = 100, R = 0.1, η 2 I 3 = 0.0 (a), 0.58 (b), 1.02 (c), 2.02 (d), 3.58 (e), 9.0 (f). The smooth curves are theoretical predictions and the zigzag lines are the experimental data from Ref. [10]. Fig. 3 The probabilities Q off and Q on for I 1 as functions of injected signal intensity η 2 I 3 with the parameters K 1 = 90.1, K 2 = 89.9, a = 0.2, R = 0.1, F = 100. The solid lines are the theoretical predictions from Eq. (12); the circles are the experimental data from Ref. [10]. Fig. 4 The normalized cross-correlation function of the light intensities obtained from Eq. (10). The parameters are chosen as K 1 = 90.1, K 2 = 89.9, a = 0.2, R = 0.1, F = 100. The solid line is the theoretical prediction from Eq. (12); the circles are the experimental data from Ref. [10].

90 LENG Feng and ZHU Shi-Qun Vol. 35 the coherent light (Fig. 6c) while λ 22 (0) also shows thermal light properties with anomalously larger fluctuations above the threshold (with a 10 30) but approaches zero far above the threshold (with a > 35) (Fig. 7c). These mean that the injected signal can drive weak mode I 1 from incoherent light to coherent light and it can also change the strong mode I 2 from coherent light to incoherent light. This depends on which mode it drives. Fig. 5 The sample of the stochastic trajectories of the light intensities I 1 and I 2 calculated from Eqs (1) by the Monto Carlo simulations. It is almost the duplicate of the curves recorded in Ref. [10]. 4 Backscattering and Saturation Effects The backscattering effects on the statistical fluctuations of a ring laser system with injected signal can be discussed from the auto-correlation functions λ 11 (0) and λ 22 (0) of the intensities I 1 and I 2. The auto-correlation functions λ 11 (0) and λ 22 (0) of the laser intensities I 1 and I 2 are plotted in Figs 6 and 7 respectively for different values of backscattering R. It is seen that the height of the auto-correlation functions λ 11 (0) and λ 22 (0) decreases with increasing backscattering R. This means that the backscattering can reduce the intensity fluctuations in homogeneously broadened ring laser system with mode coupling constant ξ = 2. It is very clear that the injected signal plays an important role in statistical properties of the ring laser system. For weak injected signal, λ 11 (0) of the weak mode I 1 is similar to that of thermal light with anomalously large fluctuation above the threshold (with a around 15 25) but approaches zero far above the threshold (with a > 30) (Fig. 6a) while λ 22 (0) of the strong mode I 2 shows coherent property (Fig. 7a). For intermediate injected signal, λ 11 (0) shows thermal light property with a dip above the threshold (Fig. 6b) while λ 22 (0) is similar to that of thermal light with anomalously large fluctuation above the threshold (with a around 15 30) but approaches zero far above the threshold (with a > 35) (Fig. 7b). For relatively strong injected signal, λ 11 (0) tends to zero with increasing pump parameter a which is the same as that of Fig. 6 The auto-correlation function λ 11(0) for different values of injected signals and backscattering. The parameters are chosen as K 1 = 90.1, K 2 = 89.9, a = 0.2. The solid, dashed and dotted lines are for R = 0.0, 3.0 and 4.0, respectively. (a) η 2 I 3 = 0.0; (b) η 2 I 3 = 1.02; (c) η 2 I 3 = 3.58. The full saturation effects on the statistical properties of a ring laser with injected signal can be investigated from Q ss (I 1 ), λ 11 (0) and λ 22 (0). The one-dimensional steadystate probability Q ss (I 1 ), the auto-correlation functions λ 11 (0) and λ 22 (0) of laser intensities I 1 and I 2 are plotted in Figs 8 10 respectively for different values of cavity decay constants K 1 and K 2. From Fig. 8, one can see that the second peak in Q ss (I 1 ) increases with increasing val-

No. 1 Statistical Properties of a Ring Laser with Injected Signal and Backscattering 91 ues of K 1 and K 2 and it also shifts to smaller values of I 1. There are almost no significant changes of the two-peak structure in Q ss (I 1 ) as K 1 and K 2 are increased. When K 1 and K 2 are increased further, the curve of Q ss (I 1 ) from the full saturation theory approaches that from the third-order laser theory. statistical fluctuations of either weak mode I 1 or strong mode I 2 increase or decrease depends on the properties of that mode. For coherent light, the increase of cavity decay constants K 1 and K 2 can increase the fluctuations of λ 11 (0) and λ 22 (0) (Figs 9c and 10a). For incoherent light, the increase of K 1 and K 2 can reduce the fluctuations in λ 11 (0) and λ 22 (0) (Figs 9a and 10c). For partially coherent light, the increase of K 1 and K 2 can either reduce (Fig. 9b) or increase (Fig. 10b) the fluctuations in the system. Fig. 7 The same as Fig. 6 except for the auto-correlation function λ 22(0). From Figs 9 and 10, it can be seen that for a laser operated below the threshold, there is almost no change in λ 11 (0) and λ 22 (0) even though the cavity decay constants K 1 and K 2 are changed a lot. When the laser is operated above the threshold, the full saturation effect plays a major role in λ 11 (0) and λ 22 (0). For weak and intermediate injected signals, λ 11 (0) of the weak mode I 1 decreases with increasing values of K 1 and K 2 (Figs 9a and 9b) while λ 22 (0) of the strong mode I 2 increases with increasing values of K 1 and K 2 (Figs 10a and 10b). For relatively strong injected signal, λ 11 (0) increases with increasing K 1 and K 2 (Fig. 9c) while λ 22 (0) decreases with increasing K 1 and K 2 (Fig. 10c). These show whether the Fig. 8 The one-dimensional steady-state probability density function obtained from Eq. (11) for full saturation theory with injected signal and backscattering. The parameters are chosen as R = 0.1, a = 0.2. The solid, dashed and dotted lines are for K 1 = 90.1, K 2 = 89.9; K 1 = 50.1, K 2 = 49.9; and K 1 = 20.1, K 2 = 19.9, respectively. (a) η 2 I 3 = 0.0; (b) η 2 I 3 = 1.02; (c) η 2 I 3 = 3.58. Each graph in Figs 9 and 10 shows the similar structure as corresponding one in Figs 6 and 7. This shows the important role played by the injected signal again.

92 LENG Feng and ZHU Shi-Qun Vol. 35 Fig. 9 The auto-correlation function λ 11(0) for different values of injected signals and backscattering. The parameters are the same as those in Fig. 8. 5 Discussion The statistical fluctuations of homogeneously broadened two-mode ring laser (ξ = 2) with an injected signal are investigated. The backscattering and full saturation effects are analyzed through the one-dimensional probability distribution function Q ss (I 1 ), the normalized autocorrelation functions λ 11 (0) and λ 22 (0) of the intensities I 1 and I 2. It is clear that the backscattering can reduce the fluctuations in the ring laser system. The full saturation effects play a major role when the laser is operated Fig. 10 The same as Fig. 9 except for the autocorrelation function λ 22(0). far above the threshold. It is very important to notice the effect of injected signal. When the injected signal is imposed on the weak mode, it can drive the weak mode from incoherent light to coherent light. Meanwhile, it can also change the strong mode from coherent light to incoherent light. Acknowledgment It is a pleasure to thank GAO Wei-Jian for many valuable suggestions in numerical calculations. References [1] R.J. Spreeuw, R. Centeno Neelen, N.J. Van Druten, E.R. Elied and J.P. Woerdman, Phys. Rev. A42 (1990) 4315. [2] T.H. Chyba, Phys. Rev. A40 (1989) 6327; Opt. Commun. 76 (1990) 395. [3] F.C. Cheng, Opt. Commun. 82 (1991) 45. [4] L. Pesquera, R. Blanco and M.A. Rodriguez, Phys. Rev. A39 (1989) 5777. [5] M. San Miguel, L. Pesquera, M.A. Rodriguez and A. Hernandez-Machado, Phys. Rev. A35 (1987) 208. [6] Shi-Qun ZHU, Phys. Rev. A50 (1994) 1710. [7] X. ZHOU, W. GAO and S. ZHU, Phys. Lett. A213 (1996) 43. [8] I. Littler, S. Baue, K. Bergmann, G. Vemuri and R. Roy, Phys. Rev. A41 (1990) 4131. [9] P. Jung, G. Vemuri and R. Roy, Opt. Commun. 78 (1990) 58. [10] F.C. Cheng and L. Mandel, J. Opt. Soc. Am. B8 (1991) 1681. [11] H. Risken, The Fokker Planck Equation, Springer, Berlin (1984). [12] Table of Integrals, Series and Products, eds I.S. Gradshteyn and I.M. Ryzhik, Academic Press, New York (1980).