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Optical bistability and differential amplification in a tapered fiber coupler — microsphere resonator with nonlinear medium
Optics & Laser Technology 32 (2000) 245–249
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Optical bistability and di erential ampli cation in a tapered ber coupler — microsphere resonator with nonlinear medium Ying Lu, Ji-You Wang, Xiao-Xuan Xu, Shi-Hong Pan, Cun-Zhou Zhang∗ Department of Physics, Nankai University, Tianjin 300071, People’s Republic of China Received 22 March 2000; accepted 8 June 2000
Abstract We propose and study a coupling system composed of the tapered ber and the microsphere with nonlinear medium. The optical bistability and di erential ampli cation caused by optical Kerr e ect is demonstrated using a geometrical method. We show that their operation characteristics can be controlled by parameters: linear phase detuning, attenuation coecient and real amplitude coupling c 2000 Elsevier Science Ltd. All rights reserved. coecient. PACS: 42.65.Pc; 42.60Da Keywords: Microsphere; Optical bistability; Optical di erential ampli cation
1. Introduction Optical bistability and di erential ampli cation have attracted considerable interest owing to their potential applications as optical memories, logic gates, optical thresholding and optically controlled photonic switches for high-speed all-optical processing and computing [1]. In particular, the dielectric microspheres with nonlinear medium are attractive candidates for these nonlinear e ects, because they act as optical resonator with high-quality factors and can con ne large electromagnetic elds in small volumes, which locally enhance the elds by orders of magnitude [2]. Furthermore, in comparison with the double-coupler ring resonator [3], double-ring resonator [4], ber loop resonator [5] and F–P cavity [6] whose bistability is theoretically or experimentally demonstrated at large size, optical microsphere resonators are preferred for use in integrated optical systems. This results from the fact that the threshold power in microsphere resonators for optical bistability and di erential ampli cation can be lower at smaller size owing to higher Q’s maintained which in turn can provide larger optical feedback. Optical bistability in a dielectric microsphere was both theoretically and experimentally and demonstrated [7]. The ∗
Corresponding author. E-mail address: [email protected] (C.-Z. Zhang).
results suggest that to obtain and utilize the bistability at low threshold one requires ecient excitation of a single near-resonant mode in the microsphere. This may be achieved by using near- eld coupler devices which can couple light into and out of the microsphere beyond the critical angle. Experimental results on ecient coupler devices in the linear range [8,9] suggest the eciency of tapered ber coupler as one of the best. Furthermore, tapered ber couplers are easily used with ber and tapered ber-microsphere system can serve as versatile units for integrated optics. In this paper, we propose a coupling system consisting of the tapered ber and the microsphere with nonlinear medium, and demonstrate the optical bistability and the di erential ampli cation of the output power caused by the Kerr e ect using a geometrical method. The analytic expressions for the critical conditions and the other essential parameters for the optical bistability and the di erential ampli cation are derived using near-resonance. We show how the various parameters of the system in uence both the operation characteristics of optical bistability and the di erential ampli cation. 2. The model and analysis The geometry of the coupling system of a tapered ber and a microsphere with nonlinear medium is shown in
c 2000 Elsevier Science Ltd. All rights reserved. 0030-3992/00/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 5 2 - 9
246
Y. Lu et al. / Optics & Laser Technology 32 (2000) 245–249
Fig. 1. The geometry of the coupling system of the tapered ber-microsphere with nonlinear medium.
Fig. 1, where the power carried by the fundamental cladding-guided mode in the tapered ber is coupled into equatorial modes near the surface of the microsphere. Azimuthal propagation of these modes is like closed waves circulating in the sphere. To obtain simpli ed expressions for bistability we use the single-mode assumption, i.e., only consider a single near-resonant equatorial mode, which is applicable when the incident frequency is close enough (within a few linewidths) to the resonance frequency of the mode. This results from the fact that under such circumstances the electric eld in the microsphere is dominated by the eld of the single resonant mode whose resonance characteristic makes bistability possible and the contributions from other nonresonant modes to bistability can be ignored [7]. In this section, we brie y review the linear output power function of the coupling system, extend it to account for the nonlinear problem, and then derive the critical conditions for the optical bistability, and the optical di erential ampli cation. Finally, we derive the analytic expressions for some essential parameters in the optical bistability device. 2.1. Linear problem Because the coupling zone is small, the coupling is considered as lumped at the reference plane shown in Fig. 1. The coupling equations for the eld transfer are p (1a) E3 = 1 − t 2 E1 + itE2 ; E4 = itE1 +
p 1 − t 2 E2 ;
(1b)
where Ei is the complex eld amplitude (at the ith port) such that |Ei |2 =Pi , the power entering or exiting that port, and t is the real amplitude coupling coecient related to the usual coupling coecient of the coupling mode formulation, which is given in Ref. [10]. It is clear that E2 and E4 are related by − L exp(i )E4 ; (2) E2 = exp 2 where represents attenuation coecient of the microsphere, L = 2a is the circumference of the sphere, = ÿL is the total phase shift acquired by the wave during one
round trip, and ÿ is the propagation constant of the mode to be excited, which is approximately given by [8,11] l ÿ ≈ k0 x nl n 1=3 p 0 ≈ k0 n0 + x−2=3 x−1 nl n + 2 2 (n0 − 1)1=2 nl # 1=3 1 2 2 −4=3 + n x nl ; (3) 6 n0 where n0 TE mode; p= 1 TM mode; n0 n denotes the nth zero of the Airy function, x nl is the resonance size parameter, n is the radial mode number, l is angular mode number, k0 is the wavenumber of the free space, and n0 is the refractive index. Eqs. (1) and (2) can be solved to give 2 E3 =1− ; (4) E1 1 + F sin2 ( =2) where is peak-coupling eciency given by [1 − exp(− L)]t 2 √ [1 − exp(− L=2) 1 − t 2 ]2 and the parameter F is de ned by =
F=
4[exp(− L)(1 − t 2 )]1=2 √ : [1 − exp(− L=2) 1 − t 2 ]2
(5)
(6)
2.2. Nonlinear problem It is well known that the optical Kerr e ect in the nonlinear microsphere makes refractive index of the microsphere depend on the internal circulating light intensity, that is n = n0 + n2 I:
(7)
Here n0 is its linear value, n2 the nonlinear refractive index coecient (we only consider the case n2 ¿0), and I is the instantaneous internal circulating optical intensity. Accordingly, ÿ and in turn are functions of the circulating light intensity in the microsphere. As in Ref. [12], we can state the nonlinear problem in terms of linear output power function (4) provided that nonlinear round-trip phase shift is expressed as 1 − e− L |E4 |2 = ÿL + R ! 2 R 1 − e− L E3 = ÿL + − 1 |E1 |2 ; (8) 1 − e L E1 where R represents the nonlinear phase shift coecient. According to Ref. [5], we derive R in our system, r Z 1 0 n0 |es |4 ds; (9) R ≈ n2 k0 2 0 S
Y. Lu et al. / Optics & Laser Technology 32 (2000) 245–249
where 0 and 0 are the permittivity and permibility of free space, respectively, and es is the normalized power circulating eigenmode eld in the microsphere. Eqs. (4) and (8) described the input–output characteristics of the system completely. The characteristics depends on the parameters such as linear phase shift (ÿL = 2l + , being linear phase detuning), t; ; L and n2 . It is worth pointing that optical bistability clearly results from the second term of Eq. (8) associated with the input power, which implies that the smaller (higher Q) the smaller is the threshold power required for the optical bistability. Because dielectric sphere with a few hundred of micrometers can maintain high Q (¿109 ) while the other analogous cavity resonator of similar size, such as FP cavity, have much smaller Q, optical bistability device made of microsphere can operate at lower input power. 2.3. Critical conditions for bistability and di erential ampliÿcation From Eqs. (4) and (8) by taking the approximation 1 − = L, we can get the equations for the input power e and output power, 2 L 2 + RL|E4 | 2 1 + F sin |E4 |2 ; (10) |E1 | = 2 − L
L |E3 | = 2
1 − + F sin
2
+ RL|E4 |2 2
2
|E4 | :
(11)
Substituting this value in Eqs. (10) and (14) implies r 16 3 2 |E1 |c = : (16) 9R F It is clear that in order to achieve optical bistability, |E1 |2 ¿ |E1 |2c and ¡ c are required, while for |E1 |2 = |E1 |2c and = c , the system operates as an optical differential ampli er with in nite gain (g = d|E3 |2 =d|E1 |2 ). This also means that the linear phase shift must always be smaller than the resonance phase shift of the mode. Owing to their importance in practical applications, we approximately derive the analytic expressions for the switch-on power, the switch-o power, and bistability domain from Eqs. (10), (11), and (14), respectively, p (−2 − 2 − 12=F) |E1 |2on = 3R !# " p − 2 − 12=F 2 ; (17a) × 1 + F sin 6 |E3 |2H =
+ RL|E4 |2 1 + F sin 2 RL |E4 |2 F sin( + RL|E4 |2 ) = 0: (12) + 2 Clearly, in order to get analytical expressions, it is necessary to use the approximation given below, which is applicable for the small size microsphere and low input power, sin( + RL|E4 |2 ) ≈ + RL|E4 |2 : Then, we can get p −2 ± 2 − 12=F 2 : |E4 | = 3RL 2
(13)
(14)
If |E4 | has two di erent real and positive roots, the bistability can be obtained, while with having one real and positive root, the di erential amplication with in nity gain can be achieved. The critical condition is thus given by r √ 12 3 ≈− ; (15) c = − F 2 √ where = 4= F is the half-width of resonance.
p (−2 − 2 − 12=F) 3R !# " p − 2 − 12=F 2 ; × 1 − + F sin 6
|E1 |2o =
To nd the conditions for bistability and di erential ampli cation, we calculate d|E1 |2 =d|E3 |2 from Eqs. (10) and (11) and set it equal to zero, which yields the equation 2
247
|E3 |2L =
p (−2 + 2 − 12=F) 3R !# " p + 2 − 12=F 2 ; × 1 + F sin 6
p (−2 + 2 − 12=F) 3R !# " p + 2 − 12=F 2 ; × 1 − + F sin 6
(17b)
(17c)
(17d)
where |E3 |2H is the output power for higher critical transition point of the high-power state and |E3 |2L is the output power for the lower critical transition point of the low-power state. The domain of the bistability is given by |E1 |2 = |E1 |2on − |E1 |2o ;
(18a)
|E3 |2 = |E3 |2H − |E3 |2L :
(18b)
Using the above equations, we can choose di erent values of the system parameters depending on di erent applications. 3. Results and discussion We consider microsphere with a = 50 m, = 10−71 m, n0 = 1:44, and n2 = 2:3 × 10−22 m2 =V2 at ≈ 1:55 m.
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Y. Lu et al. / Optics & Laser Technology 32 (2000) 245–249
Fig. 2. 1 − |E3 |2 =|E1 |2 versus the nonlinear phase detuning in number of half-widths ( − 2l)= described by Eqs. (4)√and (8) with, t 2 = L = 0:86, = −4 (¿ c ), and = c = −( 3=2) and |E1 |2 = |E1 |2c = 0:0014 mW. The later is plotted for several input powers |E1 |2 = 0 (curve a), 0.0046 mW (curve b), 0.008 mW (curve c), 0.019 mW (curve d), 0.09 mW (curve e), and |E1 |2c = 0:0014 mW (curve f).
Fig. 3. The dependence of the switch-on and switch-o power of the bistability on parameter || for (1) t 2 = L = 1, (a) |E1 |2o , (b) |E1 |2on , (2) t 2 = L = 2(¿ 1), (a) |E1 |2o , (b) |E1 |2on , and (3) t 2 = L = 0:1(¡ 1), (a) |E1 |2o , (b) |E1 |2on , respectively.
Fig. 4. The dependence of the domain of the bistability on parameter || for (1) t 2 = L=1, (a) |E1 |2 , (b) |E3 |2 , (2) t 2 = L=2(¿ 1), (a) |E1 |2 , (b) |E3 |2 , (3) t 2 = L = 0:1(¡ 1), (a) |E1 |2 , (b) |E3 |2 , respectively.
Figs. 3 and 4 show the dependence of the switch-on and switch-o power, and the domain of bistability on || for t 2 = L = 1; 2, and 0.1, respectively. It can be clearly observed that the switching powers and domain of the bistability decrease monotonically with || for the xed t 2 = L, while having the smallest value at t 2 = L ≈1 for xed ||. From the analysis, we deduce that to obtain optical bistability and optical di erential ampli cation with lower input power, smaller ||, smaller (higher Q) and t 2 = L ≈1 at small size are required. The smaller || means that the input light frequency must be very near to the eigenfrequency of the mode and the propagation constant of mode of the coupler and that of microsphere mode required to be matched. This can be achieved with a tapered ber for which optimal parameters have been studied in Ref. [10]. Acknowledgements
Fig. 2 shows the relations between 1 − |E3 |2 =|E1 |2 and the nonlinear phase detuning in number of half-widths ( − 2l)= as described by Eqs. (4) and (8) with = −4 (¿ c ) and t 2 = L = 0:86. The latter is plotted for several input powers. It is clear that the intersection points of two curves described by Eqs. (4) and (8) are the solution of 1 − |E3 |2 =|E1 |2 . When the input power exceeds 4:6 W, two stable output powers exist at the same input powers. We observe that when input power is gradually increased from 0 to 90 W, 1 − |E3 |2 =|E1 |2 follows the path 1–2–3–30 – 4, but as the input power is decreased from 90 W to 0, 1 − |E3 |2 =|E1 |2 follows the path 4 – 30 –B–20 –1 (point B is not a stable state). The resulting hysteresis loop corresponds to the optical bistability. √ Fig. 2 also shows the case = c = −( 3=2) and |E1 |2 = |E1 |2c = 0:0014 mW, in which the intersect point is the in ection point of the curve described by Eq. (4) and the optical di erential gain becomes in nite. Thererfore the system can contribute to di erential ampli cation.
This work was projection 69678015 supported by National Natural Science Foundation of China. References [1] Saleh BEA, Teich MC. Fundmentals of photonics. New York: Wiley, 1991 [Chapter 21.3]. [2] Barber PW, Chang RK. Optical e ects associated with small particles. Singapore: World Scienti c, 1988. [3] Ja YH. Multiple bistability in an optical- ber double-ring resonator utilizing the Kerr e ect. IEEE J Quantum Electron 1994;30(2):329– 33. [4] Fraile-Pelaez FJ, Capmany J, Muriel MA. Transmission bistability in a double-coupler ber ring resonator. Opt Lett 1991;16(12):907– 9. [5] Crosignani B, Daino B, Di Porto P, Wabnitz S. Optical multistability in a ber-optic passive-loop resonator. Opt Commun 1986;59(4):309–12. [6] Marburger JH, Felber FS. Theory of a lossless nonlinear Fabry– Perot interferometer. Phys Rev A 1978;17(1):335–42.
Y. Lu et al. / Optics & Laser Technology 32 (2000) 245–249 [7] Mazumder Md M, Hill SC, Chowdhury DQ, Chang RK. Dispersive optical bistability in a dielectric sphere. J Opt Soc Am B 1995;12(2):297–309. [8] Knight JC, Cheung JC, Jacques F, Birks TA. Phase-matched excitation of whispering-gallery-mode resonances by a ber-taper. Opt Lett 1997;22(15):1129–31. [9] Ilchenko VS, Yao XS, Maleki L. Pigtailing the high-Q microsphere cavity: A simple ber-coupler for optical whispering-gallery. Opt Lett 1999;24(11):723–5.
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[10] Lu Ying, Wang Ji-You, Xu Xiao-Xuan, Pan Shi-Hong, Zhang Cun-Zhou. Optimal conditions of coupling between the propagating mode in a tapered ber and the given WG mode in a high-Q microsphere. Optik, submitted for publication. [11] Lock JA. Improved gaussian beam-scattering algorithm. Appl Opt 1995;34(3):559–70. [12] Capmany J, Fraile-Pelaez FJ, Muriel MA. Optical bistability and di erential ampli cation in nonlinear ber resonators. IEEE J Quantum Electron 1994;30(11):2578–88.
www.elsevier.com/locate/optlastec
Optical bistability and di erential ampli cation in a tapered ber coupler — microsphere resonator with nonlinear medium Ying Lu, Ji-You Wang, Xiao-Xuan Xu, Shi-Hong Pan, Cun-Zhou Zhang∗ Department of Physics, Nankai University, Tianjin 300071, People’s Republic of China Received 22 March 2000; accepted 8 June 2000
Abstract We propose and study a coupling system composed of the tapered ber and the microsphere with nonlinear medium. The optical bistability and di erential ampli cation caused by optical Kerr e ect is demonstrated using a geometrical method. We show that their operation characteristics can be controlled by parameters: linear phase detuning, attenuation coecient and real amplitude coupling c 2000 Elsevier Science Ltd. All rights reserved. coecient. PACS: 42.65.Pc; 42.60Da Keywords: Microsphere; Optical bistability; Optical di erential ampli cation
1. Introduction Optical bistability and di erential ampli cation have attracted considerable interest owing to their potential applications as optical memories, logic gates, optical thresholding and optically controlled photonic switches for high-speed all-optical processing and computing [1]. In particular, the dielectric microspheres with nonlinear medium are attractive candidates for these nonlinear e ects, because they act as optical resonator with high-quality factors and can con ne large electromagnetic elds in small volumes, which locally enhance the elds by orders of magnitude [2]. Furthermore, in comparison with the double-coupler ring resonator [3], double-ring resonator [4], ber loop resonator [5] and F–P cavity [6] whose bistability is theoretically or experimentally demonstrated at large size, optical microsphere resonators are preferred for use in integrated optical systems. This results from the fact that the threshold power in microsphere resonators for optical bistability and di erential ampli cation can be lower at smaller size owing to higher Q’s maintained which in turn can provide larger optical feedback. Optical bistability in a dielectric microsphere was both theoretically and experimentally and demonstrated [7]. The ∗
Corresponding author. E-mail address: [email protected] (C.-Z. Zhang).
results suggest that to obtain and utilize the bistability at low threshold one requires ecient excitation of a single near-resonant mode in the microsphere. This may be achieved by using near- eld coupler devices which can couple light into and out of the microsphere beyond the critical angle. Experimental results on ecient coupler devices in the linear range [8,9] suggest the eciency of tapered ber coupler as one of the best. Furthermore, tapered ber couplers are easily used with ber and tapered ber-microsphere system can serve as versatile units for integrated optics. In this paper, we propose a coupling system consisting of the tapered ber and the microsphere with nonlinear medium, and demonstrate the optical bistability and the di erential ampli cation of the output power caused by the Kerr e ect using a geometrical method. The analytic expressions for the critical conditions and the other essential parameters for the optical bistability and the di erential ampli cation are derived using near-resonance. We show how the various parameters of the system in uence both the operation characteristics of optical bistability and the di erential ampli cation. 2. The model and analysis The geometry of the coupling system of a tapered ber and a microsphere with nonlinear medium is shown in
c 2000 Elsevier Science Ltd. All rights reserved. 0030-3992/00/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 5 2 - 9
246
Y. Lu et al. / Optics & Laser Technology 32 (2000) 245–249
Fig. 1. The geometry of the coupling system of the tapered ber-microsphere with nonlinear medium.
Fig. 1, where the power carried by the fundamental cladding-guided mode in the tapered ber is coupled into equatorial modes near the surface of the microsphere. Azimuthal propagation of these modes is like closed waves circulating in the sphere. To obtain simpli ed expressions for bistability we use the single-mode assumption, i.e., only consider a single near-resonant equatorial mode, which is applicable when the incident frequency is close enough (within a few linewidths) to the resonance frequency of the mode. This results from the fact that under such circumstances the electric eld in the microsphere is dominated by the eld of the single resonant mode whose resonance characteristic makes bistability possible and the contributions from other nonresonant modes to bistability can be ignored [7]. In this section, we brie y review the linear output power function of the coupling system, extend it to account for the nonlinear problem, and then derive the critical conditions for the optical bistability, and the optical di erential ampli cation. Finally, we derive the analytic expressions for some essential parameters in the optical bistability device. 2.1. Linear problem Because the coupling zone is small, the coupling is considered as lumped at the reference plane shown in Fig. 1. The coupling equations for the eld transfer are p (1a) E3 = 1 − t 2 E1 + itE2 ; E4 = itE1 +
p 1 − t 2 E2 ;
(1b)
where Ei is the complex eld amplitude (at the ith port) such that |Ei |2 =Pi , the power entering or exiting that port, and t is the real amplitude coupling coecient related to the usual coupling coecient of the coupling mode formulation, which is given in Ref. [10]. It is clear that E2 and E4 are related by − L exp(i )E4 ; (2) E2 = exp 2 where represents attenuation coecient of the microsphere, L = 2a is the circumference of the sphere, = ÿL is the total phase shift acquired by the wave during one
round trip, and ÿ is the propagation constant of the mode to be excited, which is approximately given by [8,11] l ÿ ≈ k0 x nl n 1=3 p 0 ≈ k0 n0 + x−2=3 x−1 nl n + 2 2 (n0 − 1)1=2 nl # 1=3 1 2 2 −4=3 + n x nl ; (3) 6 n0 where n0 TE mode; p= 1 TM mode; n0 n denotes the nth zero of the Airy function, x nl is the resonance size parameter, n is the radial mode number, l is angular mode number, k0 is the wavenumber of the free space, and n0 is the refractive index. Eqs. (1) and (2) can be solved to give 2 E3 =1− ; (4) E1 1 + F sin2 ( =2) where is peak-coupling eciency given by [1 − exp(− L)]t 2 √ [1 − exp(− L=2) 1 − t 2 ]2 and the parameter F is de ned by =
F=
4[exp(− L)(1 − t 2 )]1=2 √ : [1 − exp(− L=2) 1 − t 2 ]2
(5)
(6)
2.2. Nonlinear problem It is well known that the optical Kerr e ect in the nonlinear microsphere makes refractive index of the microsphere depend on the internal circulating light intensity, that is n = n0 + n2 I:
(7)
Here n0 is its linear value, n2 the nonlinear refractive index coecient (we only consider the case n2 ¿0), and I is the instantaneous internal circulating optical intensity. Accordingly, ÿ and in turn are functions of the circulating light intensity in the microsphere. As in Ref. [12], we can state the nonlinear problem in terms of linear output power function (4) provided that nonlinear round-trip phase shift is expressed as 1 − e− L |E4 |2 = ÿL + R ! 2 R 1 − e− L E3 = ÿL + − 1 |E1 |2 ; (8) 1 − e L E1 where R represents the nonlinear phase shift coecient. According to Ref. [5], we derive R in our system, r Z 1 0 n0 |es |4 ds; (9) R ≈ n2 k0 2 0 S
Y. Lu et al. / Optics & Laser Technology 32 (2000) 245–249
where 0 and 0 are the permittivity and permibility of free space, respectively, and es is the normalized power circulating eigenmode eld in the microsphere. Eqs. (4) and (8) described the input–output characteristics of the system completely. The characteristics depends on the parameters such as linear phase shift (ÿL = 2l + , being linear phase detuning), t; ; L and n2 . It is worth pointing that optical bistability clearly results from the second term of Eq. (8) associated with the input power, which implies that the smaller (higher Q) the smaller is the threshold power required for the optical bistability. Because dielectric sphere with a few hundred of micrometers can maintain high Q (¿109 ) while the other analogous cavity resonator of similar size, such as FP cavity, have much smaller Q, optical bistability device made of microsphere can operate at lower input power. 2.3. Critical conditions for bistability and di erential ampliÿcation From Eqs. (4) and (8) by taking the approximation 1 − = L, we can get the equations for the input power e and output power, 2 L 2 + RL|E4 | 2 1 + F sin |E4 |2 ; (10) |E1 | = 2 − L
L |E3 | = 2
1 − + F sin
2
+ RL|E4 |2 2
2
|E4 | :
(11)
Substituting this value in Eqs. (10) and (14) implies r 16 3 2 |E1 |c = : (16) 9R F It is clear that in order to achieve optical bistability, |E1 |2 ¿ |E1 |2c and ¡ c are required, while for |E1 |2 = |E1 |2c and = c , the system operates as an optical differential ampli er with in nite gain (g = d|E3 |2 =d|E1 |2 ). This also means that the linear phase shift must always be smaller than the resonance phase shift of the mode. Owing to their importance in practical applications, we approximately derive the analytic expressions for the switch-on power, the switch-o power, and bistability domain from Eqs. (10), (11), and (14), respectively, p (−2 − 2 − 12=F) |E1 |2on = 3R !# " p − 2 − 12=F 2 ; (17a) × 1 + F sin 6 |E3 |2H =
+ RL|E4 |2 1 + F sin 2 RL |E4 |2 F sin( + RL|E4 |2 ) = 0: (12) + 2 Clearly, in order to get analytical expressions, it is necessary to use the approximation given below, which is applicable for the small size microsphere and low input power, sin( + RL|E4 |2 ) ≈ + RL|E4 |2 : Then, we can get p −2 ± 2 − 12=F 2 : |E4 | = 3RL 2
(13)
(14)
If |E4 | has two di erent real and positive roots, the bistability can be obtained, while with having one real and positive root, the di erential amplication with in nity gain can be achieved. The critical condition is thus given by r √ 12 3 ≈− ; (15) c = − F 2 √ where = 4= F is the half-width of resonance.
p (−2 − 2 − 12=F) 3R !# " p − 2 − 12=F 2 ; × 1 − + F sin 6
|E1 |2o =
To nd the conditions for bistability and di erential ampli cation, we calculate d|E1 |2 =d|E3 |2 from Eqs. (10) and (11) and set it equal to zero, which yields the equation 2
247
|E3 |2L =
p (−2 + 2 − 12=F) 3R !# " p + 2 − 12=F 2 ; × 1 + F sin 6
p (−2 + 2 − 12=F) 3R !# " p + 2 − 12=F 2 ; × 1 − + F sin 6
(17b)
(17c)
(17d)
where |E3 |2H is the output power for higher critical transition point of the high-power state and |E3 |2L is the output power for the lower critical transition point of the low-power state. The domain of the bistability is given by |E1 |2 = |E1 |2on − |E1 |2o ;
(18a)
|E3 |2 = |E3 |2H − |E3 |2L :
(18b)
Using the above equations, we can choose di erent values of the system parameters depending on di erent applications. 3. Results and discussion We consider microsphere with a = 50 m, = 10−71 m, n0 = 1:44, and n2 = 2:3 × 10−22 m2 =V2 at ≈ 1:55 m.
248
Y. Lu et al. / Optics & Laser Technology 32 (2000) 245–249
Fig. 2. 1 − |E3 |2 =|E1 |2 versus the nonlinear phase detuning in number of half-widths ( − 2l)= described by Eqs. (4)√and (8) with, t 2 = L = 0:86, = −4 (¿ c ), and = c = −( 3=2) and |E1 |2 = |E1 |2c = 0:0014 mW. The later is plotted for several input powers |E1 |2 = 0 (curve a), 0.0046 mW (curve b), 0.008 mW (curve c), 0.019 mW (curve d), 0.09 mW (curve e), and |E1 |2c = 0:0014 mW (curve f).
Fig. 3. The dependence of the switch-on and switch-o power of the bistability on parameter || for (1) t 2 = L = 1, (a) |E1 |2o , (b) |E1 |2on , (2) t 2 = L = 2(¿ 1), (a) |E1 |2o , (b) |E1 |2on , and (3) t 2 = L = 0:1(¡ 1), (a) |E1 |2o , (b) |E1 |2on , respectively.
Fig. 4. The dependence of the domain of the bistability on parameter || for (1) t 2 = L=1, (a) |E1 |2 , (b) |E3 |2 , (2) t 2 = L=2(¿ 1), (a) |E1 |2 , (b) |E3 |2 , (3) t 2 = L = 0:1(¡ 1), (a) |E1 |2 , (b) |E3 |2 , respectively.
Figs. 3 and 4 show the dependence of the switch-on and switch-o power, and the domain of bistability on || for t 2 = L = 1; 2, and 0.1, respectively. It can be clearly observed that the switching powers and domain of the bistability decrease monotonically with || for the xed t 2 = L, while having the smallest value at t 2 = L ≈1 for xed ||. From the analysis, we deduce that to obtain optical bistability and optical di erential ampli cation with lower input power, smaller ||, smaller (higher Q) and t 2 = L ≈1 at small size are required. The smaller || means that the input light frequency must be very near to the eigenfrequency of the mode and the propagation constant of mode of the coupler and that of microsphere mode required to be matched. This can be achieved with a tapered ber for which optimal parameters have been studied in Ref. [10]. Acknowledgements
Fig. 2 shows the relations between 1 − |E3 |2 =|E1 |2 and the nonlinear phase detuning in number of half-widths ( − 2l)= as described by Eqs. (4) and (8) with = −4 (¿ c ) and t 2 = L = 0:86. The latter is plotted for several input powers. It is clear that the intersection points of two curves described by Eqs. (4) and (8) are the solution of 1 − |E3 |2 =|E1 |2 . When the input power exceeds 4:6 W, two stable output powers exist at the same input powers. We observe that when input power is gradually increased from 0 to 90 W, 1 − |E3 |2 =|E1 |2 follows the path 1–2–3–30 – 4, but as the input power is decreased from 90 W to 0, 1 − |E3 |2 =|E1 |2 follows the path 4 – 30 –B–20 –1 (point B is not a stable state). The resulting hysteresis loop corresponds to the optical bistability. √ Fig. 2 also shows the case = c = −( 3=2) and |E1 |2 = |E1 |2c = 0:0014 mW, in which the intersect point is the in ection point of the curve described by Eq. (4) and the optical di erential gain becomes in nite. Thererfore the system can contribute to di erential ampli cation.
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