All-optical logical gates based on pump-induced resonant nonlinearity in an erbium-doped fiber coupler Qiliang Li,1 Zhen Zhang,1,* Dongqiang Li,1 Mengyun Zhu,1 Xianghong Tang,1,2 and Shuqin Li3 1
Institute of Communication and Information Systems, College of Communication, Hangzhou Dianzi University, 310018, China
2
College of Information Engineering, Hangzhou Dianzi University, 310018, China
3
Department of Electronic Engineering, The Sino-British College, University of Shanghai Science and Technology, 200031, China *Corresponding author: [email protected] Received 18 August 2014; revised 20 October 2014; accepted 27 October 2014; posted 31 October 2014 (Doc. ID 221133); published 24 November 2014
In this paper, we theoretically investigate all-optical logical gates based on the pump-induced resonant nonlinearity in an erbium-doped fiber coupler. The resonant nonlinearity yielded by the optical transitions between the 4 I15∕2 states and 4 I13∕2 states in Er3 induces the refractive index to change, which leads to switching between two output ports. First, we do a study on the switching performance, and calculate the extinction ratio (Xratio) of the device. Second, using the Xratio, we obtain the truth tables of the device. The results reveal that compared with other undoped nonlinear couplers, the erbium-doped fiber coupler can drop the switching threshold power. We also obtain different logic gates and logic operations in the cases of the same phase and different phase of two initial signals by changing the pump power. © 2014 Optical Society of America OCIS codes: (060.1810) Buffers, couplers, routers, switches, and multiplexers; (190.4370) Nonlinear optics, fibers; (230.3750) Optical logic devices. http://dx.doi.org/10.1364/AO.53.008036
1. Introduction
The presence of a nonlinear phase shift in optically pumped Er3 -doped fibers has been the object of increasing interest [1–5]. Pumping an Er3 -doped SiO2 fiber at 980 nm, Fleming and Whitley measured phase shifts of light for wavelengths close to the resonant wavelength of 1530 nm, where the amplification is maximum, and the phase shifts were measured using a fiber Mach–Zehnder interferometer [6]. In 1992, Chu and Wu determined that the pump induces phase shifts indirectly from the switching characteristics of a twin core fiber coupler. They 1559-128X/14/348036-07$15.00/0 © 2014 Optical Society of America 8036
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investigated the switching in a nonlinear twin-core erbium-doped fiber coupler at a slow speed of response and showed that switching action takes place at a power of a fraction of 1 mW [7]. Thirstrup et al. presented a systematic investigation of the dependence of pump-induced refractive index changes in erbium-doped fibers on wavelength, pump power, and Er3 concentration. The model based on measured spectra of the absorption and emission cross sections yields good agreement to experimental results and proves that the fiber with the highest doping concentration yields the largest change of pump-induced refractive index [8]. For the pump-induced modulation of the refractive index for a signal at 1550 nm and a pump at 980 nm, when strong pump light at 980 nm strikes the
erbium-doped fiber, the erbium ion yields an optical transition between the ground states (4 I15∕2 ) and the metastable states (4 I13∕2 ) by absorbing the photon of the pump light. Because of the population inversion, the signal nearby 1.55 nm will be amplified [9]. The propagation constant is changed because of the gain and the resonant nonlinearity caused by the pump. Both the change of refractive index and gain coefficient in the coupling region are regarded as the real and imaginary parts of the variation of the complex propagation constant [10]. All-optical logical operations using the optical Kerr effect as a switching mechanism have been intensively studied in theory and experiment [11–16], but the threshold power of the pump is very large [17]. To avoid damage to the fiber by the pump light, the erbium-doped fiber coupler is chosen to drop the threshold power [18,19]. The resonant nonlinearity in rare-earth-doped fibers, which is not the same as the Kerr nonlinearity induced by third-order susceptibility, is yielded mainly by the resonant transition of the rare-earth elements from the excited state to the ground state. This type of resonant nonlinearity is usually many orders of magnitude larger than that provided by the Kerr effect [20]. In this paper, we theoretically investigate the operation of all-optical logical gates based on the pump-induced resonant nonlinearity in an erbiumdoped fiber coupler. As shown in Fig. 1, the arms of the coupler are made of two identical erbiumdoped fibers, I and II, and the coupling length of the coupler satisfies L π∕2κ, where κ is the coupling parameters. A signal and a pump light in port A are input coupled into channel I by a wavelength division multiplexer (WDM). Because the wavelength of the pump light is different from that of the signal light, the pump light can remain in channel I; thus, one arm of the coupler is pumped to provide gain while the other arm is the absence of gain. As a result, the propagation constant of channel I is changed because of the gain and the resonant nonlinearity caused by the pump. Both the change of refractive index and gain coefficient in the coupling region are regarded as the real and imaginary parts of the variation of the complex propagation constant in channel I. Consequently, the modification of the phase difference, between channel I and channel II, leads to switching of the output signal in ports C and D. Compared with undoped couplers, the erbium-doped fiber coupler requires a relatively
Fig. 1. Schematic of an erbium-doped fiber coupler based on pump-induced resonant nonlinearity.
low threshold power and may be more appropriate to all-optical switch applications. This paper is organized as follows: in Section 2, we introduce the theoretical model based on pumpinduced resonant nonlinearity and solve the coupled equations. Section 3 presents analysis of the switching characteristics and various logical functions based on pump-induced resonant nonlinearity. Section 4 is the conclusion. 2. Theoretical Model
We know that strong pump light through the erbiumdoped fiber can cause an erbium ion energy level transition [20]. The resonant nonlinearity, yielded by the optical transitions between 4 I15∕2 states and 4 I13∕2 states in Er3 , leads to the refractive index change [8]. The resonant transition of Er3 from the excited state to the ground state can also cause complex propagation constant change, in which the real part is related to the change of refractive index n, and the imaginary part corresponds to the change of gain coefficient g [10]. According to the resonant nonlinearity model that is mentioned above, the coupling mode equations are expressed as dA1 2π Δg iκ 12 A2 iβ1 A1 i ΔnA1 A ; dz λ 2 1
(1)
dA2 iκ21 A1 iβ2 A2 : dz
(2)
Here, we neglect the time-related items. β1 and β2 are the propagation constants of channel 1 and channel 2, respectively. κ12 and κ 21 are the coupling parameters and λ is the wavelength of the signal light. Δn and Δg are the refractive index change and the gain coefficient change, respectively, and they are functions of the pump power. On the basis of the three-level model of Er3 ions, the refractive index change and the gain coefficient change due to pumping can be expressed as [8,10,21] Δn
Nλ3 ξ p pth ln ; 2 2 p exp−αp l pth 16π n0 τ2 αp l
(3)
Δg
2p exp−αp l δ00 N; p exp−αp l pth a
(4)
where l is the coupling length of the fiber, τ2 is the lifetime of the metastable state 4 I13∕2 , N is the total doping concentration of Er3 ions, pth is the saturation power of the pump light, n0 is the refractive index, p is the input pump power, αp is the absorption coefficient at the pump wavelength, and αp δp N, with δp being the absorption section of the pump light. δ00a is the stimulated absorption cross section, and ξ is the line-shape function for the ion transition. 1 December 2014 / Vol. 53, No. 34 / APPLIED OPTICS
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We make conversion as follows: A1 B1 expiβ1 β2 z∕2, A2 B2 expiβ1 β2 z∕2, and we can obtain simpler equations as follows:
XratiodB 10 log10 X ij :
(12)
3. Results and Discussion
dB2 iκB1 iξ2 B2 : dz
(5)
(6)
In these equations, because the two fiber cores are the same, we make the coupling parameters κ 12 κ 21 κ, Δβ β1 − β2 ∕2, ξ1 Δβ 2πΔn∕λ− iΔg∕2, and ξ2 −Δβ. According to Eqs. (5) and (6), we can suppose the initial conditions B1 0 B10 and B2 0 B20 , and we get the following results: B1 ze
ik1 z
k1 κ B10 cosk2 zi sink2 z ; B B k2 10 k2 20 (7)
k κ B2 z eik1 z B20 cosk2 z − i 1 B20 − B10 sink2 z ; k2 k2 (8) where k1 πΔn∕λ − iΔg∕4, k2 p 2 2 1∕2 2πΔn∕λ − iΔg∕2 4κ , and when solving the equations, we make an approximation as Δβ 0. As we mentioned above, Δβ1 2πΔn∕λ− iΔg∕2. The real part is related to the refractive index change Δn and the imaginary part is related to the gain coefficient change Δg. We define the transmission T of the nonlinear coupler as follows: T1
T2
jB1 j2 ; jB1 j2 jB2 j2
(9)
jB2 j2 : jB1 j2 jB2 j2
(10)
jBi j2 ; jBj j2
(11)
where i, j 1; 2, and the Xratio in decibel units is as follows: 8038
The data used in the simulation are as shown in Table 1. The curves of the refractive index change Δn and the gain coefficient change Δg with increase of the pump power are calculated with Eqs. (3) and (4), respectively. Here the wavelength of pump light is taken as 980 nm. Figures 2 and 3 show transmission T as a function of the pump power. The initial conditions are given by B1 0 1 mW and B2 0 0 mW, and B1 0 1 mW and B2 0 1 mW, respectively. In Fig. 2 we have a low signal incident in port A, while we add a pump in port A. We can notice that most of the signal light is transferred to fiber core II. The transmission T 2 is up to 0.9617, while the transmission T 1 is 0.0383 when p 40 mW. In Fig. 3 we input two low power signal incidents in port A and port B, respectively. Pump light with power p is also injected into fiber core I and directly Table 1.
Fixed Parameters of Our Proposed Device
Parameters Signal wavelength Refractive index Total doping concentration of Er3 ions Coupling coefficient Coupling length Line-shape function for the ion transition Lifetime of the metastable state 4 I13∕2 Saturation power of the pump Absorption section of the pump light Stimulated absorption cross section
Symbol
Value
λ n0 N
1.55 × 10−6 m 1.49 1.5 × 1026 m−3
κ l ξ
5 m−1 0.314 m 10−11 ms
τ2 pth δp δ00a
10 ms 1 mW 2.58 × 10−25 m2 3 × 10−25 m2
1
Last, we use the extinction ratio (Xratio) to judge the switch. The Xratio of an on–off switch is the ratio of the output power in the on state to the output power in the off state. This ratio should be as high as possible. It is expressed by X ij
A. Switching Characteristics with the Same Initial Phase and Logical Gates
APPLIED OPTICS / Vol. 53, No. 34 / 1 December 2014
T1
0.8 transmission
dB1 iκB2 iξ1 B1 ; dz
T2 0.6
0.4
0.2
0
0
10
20 P(mW)
30
40
Fig. 2. Transmission T as a function of pump power when B10 1 mW and B20 0.
5
0.65
0.55
Xratio(dB)
transmission
0.6
T1
0.5
T2
0.45
X
0
X
12 21
0.4 0.35 0
10
20 P(mW)
30
-5
40
0
10
20 P(mW)
30
40
Fig. 3. Transmission T as a function of pump power when B10 B20 1 mW.
Fig. 5. Xratio level as a function of pump power with B10 B20 1 mW.
output from fiber core I with power p, because the pump light with wavelength 980 nm is not coupled into fiber II. Therefore, it has little effect on the signal in fiber core II. The pump light induces variation of the propagation constant in fiber I, so the output power in port C is always larger than the output power in port D. From Fig. 3 we can notice that transmission T 2 reduces gradually with the increase of pump power. When the pump power is 0, transmission T 1 and T 2 are both equal to 0.5. When p 40 mW, transmission T 1 is up to 0.6897 and transmission T 2 is 0.3103. Thus, we can realize different logic gates by choosing different combinations of the input and the pump power. Figures 4 and 5 show the Xratio level as a function of the pump power, which we can obtain from Eqs. (11) and (12). The powers of the initial signals at two input ports are given by B1 0 1 mW, B2 0 0 mW, and B1 0 1 mW, B2 0 1 mW, respectively. In Fig. 4 we add a pump in port A and the pump light passes through fiber core I only. This induces variation of the propagation constant in fiber I; fiber core II undergoes no change. We can see that most of
the signal light is coupled into fiber core II. We notice that the extinction ratio (Xratio) is the same when signal inputs in port B. When the pump power is 25 mW, the Xratio21 is up to 15.16 dB and when the pump power is 40 mW, the Xratio21 is 13.99 dB. Figure 5 shows the plot of the Xratio as a function of pump power when the incident signals enter in two ports and pump light is also injected into fiber core I. As mentioned above, the pump light is not coupled into fiber II and it has little effect on the signal in fiber core II. As shown in Fig. 5, we can notice that Xratio12 will increase with the increase of pump power. When the pump power is 25 mW, Xratio12 is 3.035 dB, and when p 40 mW, Xratio12 is up to the highest value of 3.467 dB. We next study the logical function of the erbiumdoped fiber coupler. Table 2 shows the optical logic truths when p 25 mW and have been selected to provide the best average values of Xratio for the configuration (0,1), (1,0), and (1,1). For the (0,1) input configuration in Table 2, we can notice that Xratio12 is 15.16 dB and Xratio21 is −15.16 dB. For the (1,0) input configuration, Xratio12 is −15.16 dB and Xratio21 is 15.16 dB. For the (1,1) input configuration, Xratio12 is 3.035 dB and Xratio21 is −3.035 dB. We can obtain the arithmetic operation ¯ From Table 2, we realize logical operation D A · B. with pump power on the milliwatt magnitude. Table 3 shows the optical logical truths when p 40 mW. We can notice that, for the (0,1) input configuration, Xratio12 is 14 dB and Xratio21 is −14 dB. For the (1,0) input configuration, Xratio12 is −13.99 dB and Xratio21 is 13.99 dB. For the (1,1)
50 X
Xratio(dB)
X
12 21
0
Table 2.
Port A -50
0
10
20 P(mW)
30
40
Fig. 4. Xratio level as a function of pump power with B10 1 mW and B20 0.
0 1 1 0
Optical Logic Truth Table with p 25 mW
Port B
X 12 (dB)
Port C
X 21 (dB)
Port D
1 0 1 0
15.16 −15.16 3.035
1 0 1 0
−15.16 15.16 −3.035
0 1 0 0 ¯ DA·B
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Optical Logic Truth Table with p 40 mW
Table 3.
Port A
Port B
X 12 (dB)
Port C
X 21 (dB)
Port D
1 0 1 0
14 −13.99 3.467
1 0 1 0
−14 13.99 −3.467
0 1 0 0 ¯ DA·B
0 1 1 0
input configuration, Xratio12 is 3.467 dB and Xratio21 is −3.467 dB. We can also realize the arithmetic ¯ From Table 2 and Table 3, we operation D A · B. can conclude that the logical gates do not change with the increase of pump power. B. Switching Characteristics with Different Initial Phase and Logical Gates
Here we explore the situation that the input channels are excited, considering a phase difference between two input signals in the erbium-doped fiber coupler. We add a dephasing value (Δϕ) into the initial input B10 , so Eqs. (7) and (8) are changed as follows: B1 z
eik1 z
B10 eiΔϕ cosk2 z
k1 κ iΔϕ B e B20 sink2 z ; i k2 10 k2
B2 z
eik1 z −i
(13)
B20 cosk2 z
k1 κ B20 − B10 eiΔϕ sink2 z : k2 k2
(14)
We input signals of low power in ports A and B, and the pump light is also injected into port A. Figures 6 and 7 show the three dimensional plots of Xratioij level. As we mentioned before, it is a function of p and Δϕ, and, from Figs. 6 and 7, we can more clearly study the effect of pump power and dephasing
Fig. 7. Xratio level as a function of the pump and Δϕ.
value on the Xratio. From Fig. 6, we notice that the highest point is Xratio12 3.509 dB and the lowest point is Xratio12 −3.515 dB when Δϕ 1.96π and p 40 mW, and Δϕ 0.96π and p 40 mW, respectively. Figures 8 and 9 are the curves of the Xratio versus the initial phase difference, when B10 1 mW and B20 0 mW, and B10 B20 1 mW, respectively, and the pump power is 40 mW. In Fig. 8 we can notice that the Xratio will be invariable when we have a low signal incident in only one port. From Fig. 9 we notice that the Xratio fluctuates with the dephasing value Δϕ when we input two low signal incidents in ports A and B. When Δϕ 0.96π, Xratio12 reduces to the minimum value −3.515 dB, and when Δϕ 1.96π, Xratio12 increases to the maximum value 3.515 dB. In this section we will study the influence of different initial phases on the erbium-doped fiber coupler. The dephasing value (Δϕ 0.96π) has been selected to provide the best values of Xratio for the configurations (0,1), (1,0), and (1,1), so it is appropriate to judge the logic gates. 20
Xratio(dB)
10
X 0
X
12 21
-10
-20
0
2
4
6
phase Fig. 6. Xratio level as a function of the pump and Δϕ. 8040
APPLIED OPTICS / Vol. 53, No. 34 / 1 December 2014
Fig. 8. Xratio level as a function of Δϕ when B10 1 mW and B20 0.
10 X X
Xratio(dB)
5
changing the initial phase of the two input signals, we obtain the logical operation C A · B in port C; here, the initial phase difference is Δϕ 0.96π. We use the pump to yield resonant nonlinearity so that the switching threshold power can drop. At the same time, we can also obtain the various logical operations by adjusting the initial phase of two input signals.
12 21
0
We are grateful to the reviewers for their helpful comments. This work was partially supported by the Natural Science Foundation of Zhejiang Province under grant Y1110078 and the National Natural Science Foundation of China (NSFC) under grant 10904028.
-5
-10
0
2
4 phase
6
Fig. 9. Xratio level as a function of Δϕ when B10 B20 1 mW.
Optical Logic Gate with Δϕ 0.96π, p 40 mW
Table 4.
Port A 0 1 1 0
Port B
X 12 (dB)
Port C
X 21 (dB)
Port D
1 0 1 0
14 −13.99 −3.515
1 0 0 0 ¯ ·B CA
−14 13.99 3.515
0 1 1 0
As is described above, we choose Δϕ 0.96π in Table 4. Xratio12 is around −3.515 dB compared to 3.467 dB in Table 3 for the input configuration (1, 1). Then we can notice that the dephasing value (Δϕ) has realized an exchange between ports C and D. The input configurations (0, 1) and (1, 0) present an Xratio12 around 14 dB and −13.99 dB, which is the same as Xratio12 in Table 3. For the configurations of Table 4, we can get different logical gates. ¯ · B in We obtain the arithmetic operations C A the output of port C. To sum up, we can also realize switching with the change of the initial phase under low pump power. 4. Conclusion
In conclusion, we have proposed the design of alloptical logical gates based on pump-induced propagation-constant change. Based on the model of resonant nonlinearity, we solve the coupled equations by using an analytical method. In our scheme, the signals with low power input from port A and port B, respectively, and the pump light is also coupled into port A by a WDM. We also study the switching characteristics and the curve of the Xratio of the system. We find that compared with other undoped couplers, the doped-erbium fiber coupler can drop the switching threshold power. To realize various logical operations, the Xratio of the on–off switch should be as high as possible. When the initial phases of two input signals are the same, we obtain the logical operation D A · B in port D by changing the pump power. When
References 1. R. I. Laming, S. B. Poole, and E. J. Tarbox, “Pump excitedstate absorption in erbium-doped fibers,” Opt. Lett. 13, 1084–1086 (1988). 2. R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991). 3. Y. L. Xue, P. L. Chu, and W. Zhang, “Resonance-enhanced refractive index and its dynamics in rare-earth-doped fibers,” J. Opt. Soc. Am. B 10, 1840–1847 (1993). 4. Y. Imai and N. Matsuda, “Nonlinear refractive index of Er3+-doped fiber and its application to nonlinear fiber coupler,” Opt. Rev. 1, 97–99 (1994). 5. J. W. Arkwright, G. R. Atkins, J. Canning, P. L. Chu, M. Janos, M. G. Sceats, and B. Wu, “Resonantly enhanced nonlinearities in rare-earth-doped fibers and waveguides,” Proc. SPIE 2841, 172–182 (1996). 6. S. C. Fleming and T. J. Whitley, “Measurement of pump induced refractive index change in erbium doped fiber amplifier,” Electron. Lett. 27, 1959–1961 (1991). 7. P. L. Chu and B. Wu, “Optical switching in twin-core erbiumdoped fibers,” Opt. Lett. 17, 255–257 (1992). 8. C. Thirstrup, Y. Shi, and B. Pálsdóttir, “Pump-Induced refractive index modulation and dispersions in Er3+-doped fibers,” J. Lightwave Technol. 14, 732–738 (1996). 9. G. P. Agrawal, Applications of Nonlinear Fiber Optics, 4th ed. (Elsevier, 2009). 10. C. Li, G. Xu, L. Ma, N. Dou, and H. Gu, “An erbium-doped fibre nonlinear coupler with coupling ratios controlled by pump power,” J. Opt. A: Pure Appl. Opt. 7, 540–543 (2005). 11. S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987). 12. S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 13, 904–906 (1988). 13. C. C. Yang, “All-optical ultrafast logic gates that use asymmetric nonlinear directional couplers,” Opt. Lett. 16, 1641–1643 (1991). 14. W. Samir, C. Pask, and S. J. Garth, “Signal switching by a control beam in a nonlinear coupler,” J. Opt. Soc. Am. B 11, 2193–2205 (1994). 15. W. B. Fraga, J. W. M. Menezes, M. G. da Silva, C. S. Sobrinho, and A. S. B. Sombra, “All optical logic gates based on an asymmetric nonlinear directional coupler,” Opt. Commun. 262, 32–37 (2006). 16. Q. Li, A. Zhang, and X. Hua, “Numerical simulation of solitons switching and propagating in asymmetric directional couplers,” Opt. Commun. 285, 118–123 (2012). 17. Q. Li and H. Yuan, “All-optical logic gates based on crossphase modulation in an asymmetric coupler,” Opt. Commun. 319, 90–94 (2014). 1 December 2014 / Vol. 53, No. 34 / APPLIED OPTICS
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18. R. H. Pantell, R. W. Sadowski, M. J. Digonnet, and H. J. Shaw, “Laser-diode-pumped nonlinear switch in erbium-doped fiber,” Opt. Lett. 17, 1026–1028 (1992). 19. R. H. Pantell, M. J. F. Digonnet, R. W. Sadowski, and H. J. Shaw, “Analysis of nonlinear optical switching in an erbium-doped fiber,” J. Lightwave Technol. 11, 1416–1424 (1993).
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20. P. L. Chu, “Nonlinear effects in rare-earth-doped fibers and waveguides,” in Lasers and Electro-Optics Society Annual Meeting (IEEE, 1997), pp. 371–372. 21. E. Desurvire, “Study of the complex atomic susceptibility of erbium-doped fiber amplifiers,” J. Lightwave Technol. 8, 1517–1527 (1990).
Institute of Communication and Information Systems, College of Communication, Hangzhou Dianzi University, 310018, China
2
College of Information Engineering, Hangzhou Dianzi University, 310018, China
3
Department of Electronic Engineering, The Sino-British College, University of Shanghai Science and Technology, 200031, China *Corresponding author: [email protected] Received 18 August 2014; revised 20 October 2014; accepted 27 October 2014; posted 31 October 2014 (Doc. ID 221133); published 24 November 2014
In this paper, we theoretically investigate all-optical logical gates based on the pump-induced resonant nonlinearity in an erbium-doped fiber coupler. The resonant nonlinearity yielded by the optical transitions between the 4 I15∕2 states and 4 I13∕2 states in Er3 induces the refractive index to change, which leads to switching between two output ports. First, we do a study on the switching performance, and calculate the extinction ratio (Xratio) of the device. Second, using the Xratio, we obtain the truth tables of the device. The results reveal that compared with other undoped nonlinear couplers, the erbium-doped fiber coupler can drop the switching threshold power. We also obtain different logic gates and logic operations in the cases of the same phase and different phase of two initial signals by changing the pump power. © 2014 Optical Society of America OCIS codes: (060.1810) Buffers, couplers, routers, switches, and multiplexers; (190.4370) Nonlinear optics, fibers; (230.3750) Optical logic devices. http://dx.doi.org/10.1364/AO.53.008036
1. Introduction
The presence of a nonlinear phase shift in optically pumped Er3 -doped fibers has been the object of increasing interest [1–5]. Pumping an Er3 -doped SiO2 fiber at 980 nm, Fleming and Whitley measured phase shifts of light for wavelengths close to the resonant wavelength of 1530 nm, where the amplification is maximum, and the phase shifts were measured using a fiber Mach–Zehnder interferometer [6]. In 1992, Chu and Wu determined that the pump induces phase shifts indirectly from the switching characteristics of a twin core fiber coupler. They 1559-128X/14/348036-07$15.00/0 © 2014 Optical Society of America 8036
APPLIED OPTICS / Vol. 53, No. 34 / 1 December 2014
investigated the switching in a nonlinear twin-core erbium-doped fiber coupler at a slow speed of response and showed that switching action takes place at a power of a fraction of 1 mW [7]. Thirstrup et al. presented a systematic investigation of the dependence of pump-induced refractive index changes in erbium-doped fibers on wavelength, pump power, and Er3 concentration. The model based on measured spectra of the absorption and emission cross sections yields good agreement to experimental results and proves that the fiber with the highest doping concentration yields the largest change of pump-induced refractive index [8]. For the pump-induced modulation of the refractive index for a signal at 1550 nm and a pump at 980 nm, when strong pump light at 980 nm strikes the
erbium-doped fiber, the erbium ion yields an optical transition between the ground states (4 I15∕2 ) and the metastable states (4 I13∕2 ) by absorbing the photon of the pump light. Because of the population inversion, the signal nearby 1.55 nm will be amplified [9]. The propagation constant is changed because of the gain and the resonant nonlinearity caused by the pump. Both the change of refractive index and gain coefficient in the coupling region are regarded as the real and imaginary parts of the variation of the complex propagation constant [10]. All-optical logical operations using the optical Kerr effect as a switching mechanism have been intensively studied in theory and experiment [11–16], but the threshold power of the pump is very large [17]. To avoid damage to the fiber by the pump light, the erbium-doped fiber coupler is chosen to drop the threshold power [18,19]. The resonant nonlinearity in rare-earth-doped fibers, which is not the same as the Kerr nonlinearity induced by third-order susceptibility, is yielded mainly by the resonant transition of the rare-earth elements from the excited state to the ground state. This type of resonant nonlinearity is usually many orders of magnitude larger than that provided by the Kerr effect [20]. In this paper, we theoretically investigate the operation of all-optical logical gates based on the pump-induced resonant nonlinearity in an erbiumdoped fiber coupler. As shown in Fig. 1, the arms of the coupler are made of two identical erbiumdoped fibers, I and II, and the coupling length of the coupler satisfies L π∕2κ, where κ is the coupling parameters. A signal and a pump light in port A are input coupled into channel I by a wavelength division multiplexer (WDM). Because the wavelength of the pump light is different from that of the signal light, the pump light can remain in channel I; thus, one arm of the coupler is pumped to provide gain while the other arm is the absence of gain. As a result, the propagation constant of channel I is changed because of the gain and the resonant nonlinearity caused by the pump. Both the change of refractive index and gain coefficient in the coupling region are regarded as the real and imaginary parts of the variation of the complex propagation constant in channel I. Consequently, the modification of the phase difference, between channel I and channel II, leads to switching of the output signal in ports C and D. Compared with undoped couplers, the erbium-doped fiber coupler requires a relatively
Fig. 1. Schematic of an erbium-doped fiber coupler based on pump-induced resonant nonlinearity.
low threshold power and may be more appropriate to all-optical switch applications. This paper is organized as follows: in Section 2, we introduce the theoretical model based on pumpinduced resonant nonlinearity and solve the coupled equations. Section 3 presents analysis of the switching characteristics and various logical functions based on pump-induced resonant nonlinearity. Section 4 is the conclusion. 2. Theoretical Model
We know that strong pump light through the erbiumdoped fiber can cause an erbium ion energy level transition [20]. The resonant nonlinearity, yielded by the optical transitions between 4 I15∕2 states and 4 I13∕2 states in Er3 , leads to the refractive index change [8]. The resonant transition of Er3 from the excited state to the ground state can also cause complex propagation constant change, in which the real part is related to the change of refractive index n, and the imaginary part corresponds to the change of gain coefficient g [10]. According to the resonant nonlinearity model that is mentioned above, the coupling mode equations are expressed as dA1 2π Δg iκ 12 A2 iβ1 A1 i ΔnA1 A ; dz λ 2 1
(1)
dA2 iκ21 A1 iβ2 A2 : dz
(2)
Here, we neglect the time-related items. β1 and β2 are the propagation constants of channel 1 and channel 2, respectively. κ12 and κ 21 are the coupling parameters and λ is the wavelength of the signal light. Δn and Δg are the refractive index change and the gain coefficient change, respectively, and they are functions of the pump power. On the basis of the three-level model of Er3 ions, the refractive index change and the gain coefficient change due to pumping can be expressed as [8,10,21] Δn
Nλ3 ξ p pth ln ; 2 2 p exp−αp l pth 16π n0 τ2 αp l
(3)
Δg
2p exp−αp l δ00 N; p exp−αp l pth a
(4)
where l is the coupling length of the fiber, τ2 is the lifetime of the metastable state 4 I13∕2 , N is the total doping concentration of Er3 ions, pth is the saturation power of the pump light, n0 is the refractive index, p is the input pump power, αp is the absorption coefficient at the pump wavelength, and αp δp N, with δp being the absorption section of the pump light. δ00a is the stimulated absorption cross section, and ξ is the line-shape function for the ion transition. 1 December 2014 / Vol. 53, No. 34 / APPLIED OPTICS
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We make conversion as follows: A1 B1 expiβ1 β2 z∕2, A2 B2 expiβ1 β2 z∕2, and we can obtain simpler equations as follows:
XratiodB 10 log10 X ij :
(12)
3. Results and Discussion
dB2 iκB1 iξ2 B2 : dz
(5)
(6)
In these equations, because the two fiber cores are the same, we make the coupling parameters κ 12 κ 21 κ, Δβ β1 − β2 ∕2, ξ1 Δβ 2πΔn∕λ− iΔg∕2, and ξ2 −Δβ. According to Eqs. (5) and (6), we can suppose the initial conditions B1 0 B10 and B2 0 B20 , and we get the following results: B1 ze
ik1 z
k1 κ B10 cosk2 zi sink2 z ; B B k2 10 k2 20 (7)
k κ B2 z eik1 z B20 cosk2 z − i 1 B20 − B10 sink2 z ; k2 k2 (8) where k1 πΔn∕λ − iΔg∕4, k2 p 2 2 1∕2 2πΔn∕λ − iΔg∕2 4κ , and when solving the equations, we make an approximation as Δβ 0. As we mentioned above, Δβ1 2πΔn∕λ− iΔg∕2. The real part is related to the refractive index change Δn and the imaginary part is related to the gain coefficient change Δg. We define the transmission T of the nonlinear coupler as follows: T1
T2
jB1 j2 ; jB1 j2 jB2 j2
(9)
jB2 j2 : jB1 j2 jB2 j2
(10)
jBi j2 ; jBj j2
(11)
where i, j 1; 2, and the Xratio in decibel units is as follows: 8038
The data used in the simulation are as shown in Table 1. The curves of the refractive index change Δn and the gain coefficient change Δg with increase of the pump power are calculated with Eqs. (3) and (4), respectively. Here the wavelength of pump light is taken as 980 nm. Figures 2 and 3 show transmission T as a function of the pump power. The initial conditions are given by B1 0 1 mW and B2 0 0 mW, and B1 0 1 mW and B2 0 1 mW, respectively. In Fig. 2 we have a low signal incident in port A, while we add a pump in port A. We can notice that most of the signal light is transferred to fiber core II. The transmission T 2 is up to 0.9617, while the transmission T 1 is 0.0383 when p 40 mW. In Fig. 3 we input two low power signal incidents in port A and port B, respectively. Pump light with power p is also injected into fiber core I and directly Table 1.
Fixed Parameters of Our Proposed Device
Parameters Signal wavelength Refractive index Total doping concentration of Er3 ions Coupling coefficient Coupling length Line-shape function for the ion transition Lifetime of the metastable state 4 I13∕2 Saturation power of the pump Absorption section of the pump light Stimulated absorption cross section
Symbol
Value
λ n0 N
1.55 × 10−6 m 1.49 1.5 × 1026 m−3
κ l ξ
5 m−1 0.314 m 10−11 ms
τ2 pth δp δ00a
10 ms 1 mW 2.58 × 10−25 m2 3 × 10−25 m2
1
Last, we use the extinction ratio (Xratio) to judge the switch. The Xratio of an on–off switch is the ratio of the output power in the on state to the output power in the off state. This ratio should be as high as possible. It is expressed by X ij
A. Switching Characteristics with the Same Initial Phase and Logical Gates
APPLIED OPTICS / Vol. 53, No. 34 / 1 December 2014
T1
0.8 transmission
dB1 iκB2 iξ1 B1 ; dz
T2 0.6
0.4
0.2
0
0
10
20 P(mW)
30
40
Fig. 2. Transmission T as a function of pump power when B10 1 mW and B20 0.
5
0.65
0.55
Xratio(dB)
transmission
0.6
T1
0.5
T2
0.45
X
0
X
12 21
0.4 0.35 0
10
20 P(mW)
30
-5
40
0
10
20 P(mW)
30
40
Fig. 3. Transmission T as a function of pump power when B10 B20 1 mW.
Fig. 5. Xratio level as a function of pump power with B10 B20 1 mW.
output from fiber core I with power p, because the pump light with wavelength 980 nm is not coupled into fiber II. Therefore, it has little effect on the signal in fiber core II. The pump light induces variation of the propagation constant in fiber I, so the output power in port C is always larger than the output power in port D. From Fig. 3 we can notice that transmission T 2 reduces gradually with the increase of pump power. When the pump power is 0, transmission T 1 and T 2 are both equal to 0.5. When p 40 mW, transmission T 1 is up to 0.6897 and transmission T 2 is 0.3103. Thus, we can realize different logic gates by choosing different combinations of the input and the pump power. Figures 4 and 5 show the Xratio level as a function of the pump power, which we can obtain from Eqs. (11) and (12). The powers of the initial signals at two input ports are given by B1 0 1 mW, B2 0 0 mW, and B1 0 1 mW, B2 0 1 mW, respectively. In Fig. 4 we add a pump in port A and the pump light passes through fiber core I only. This induces variation of the propagation constant in fiber I; fiber core II undergoes no change. We can see that most of
the signal light is coupled into fiber core II. We notice that the extinction ratio (Xratio) is the same when signal inputs in port B. When the pump power is 25 mW, the Xratio21 is up to 15.16 dB and when the pump power is 40 mW, the Xratio21 is 13.99 dB. Figure 5 shows the plot of the Xratio as a function of pump power when the incident signals enter in two ports and pump light is also injected into fiber core I. As mentioned above, the pump light is not coupled into fiber II and it has little effect on the signal in fiber core II. As shown in Fig. 5, we can notice that Xratio12 will increase with the increase of pump power. When the pump power is 25 mW, Xratio12 is 3.035 dB, and when p 40 mW, Xratio12 is up to the highest value of 3.467 dB. We next study the logical function of the erbiumdoped fiber coupler. Table 2 shows the optical logic truths when p 25 mW and have been selected to provide the best average values of Xratio for the configuration (0,1), (1,0), and (1,1). For the (0,1) input configuration in Table 2, we can notice that Xratio12 is 15.16 dB and Xratio21 is −15.16 dB. For the (1,0) input configuration, Xratio12 is −15.16 dB and Xratio21 is 15.16 dB. For the (1,1) input configuration, Xratio12 is 3.035 dB and Xratio21 is −3.035 dB. We can obtain the arithmetic operation ¯ From Table 2, we realize logical operation D A · B. with pump power on the milliwatt magnitude. Table 3 shows the optical logical truths when p 40 mW. We can notice that, for the (0,1) input configuration, Xratio12 is 14 dB and Xratio21 is −14 dB. For the (1,0) input configuration, Xratio12 is −13.99 dB and Xratio21 is 13.99 dB. For the (1,1)
50 X
Xratio(dB)
X
12 21
0
Table 2.
Port A -50
0
10
20 P(mW)
30
40
Fig. 4. Xratio level as a function of pump power with B10 1 mW and B20 0.
0 1 1 0
Optical Logic Truth Table with p 25 mW
Port B
X 12 (dB)
Port C
X 21 (dB)
Port D
1 0 1 0
15.16 −15.16 3.035
1 0 1 0
−15.16 15.16 −3.035
0 1 0 0 ¯ DA·B
1 December 2014 / Vol. 53, No. 34 / APPLIED OPTICS
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Optical Logic Truth Table with p 40 mW
Table 3.
Port A
Port B
X 12 (dB)
Port C
X 21 (dB)
Port D
1 0 1 0
14 −13.99 3.467
1 0 1 0
−14 13.99 −3.467
0 1 0 0 ¯ DA·B
0 1 1 0
input configuration, Xratio12 is 3.467 dB and Xratio21 is −3.467 dB. We can also realize the arithmetic ¯ From Table 2 and Table 3, we operation D A · B. can conclude that the logical gates do not change with the increase of pump power. B. Switching Characteristics with Different Initial Phase and Logical Gates
Here we explore the situation that the input channels are excited, considering a phase difference between two input signals in the erbium-doped fiber coupler. We add a dephasing value (Δϕ) into the initial input B10 , so Eqs. (7) and (8) are changed as follows: B1 z
eik1 z
B10 eiΔϕ cosk2 z
k1 κ iΔϕ B e B20 sink2 z ; i k2 10 k2
B2 z
eik1 z −i
(13)
B20 cosk2 z
k1 κ B20 − B10 eiΔϕ sink2 z : k2 k2
(14)
We input signals of low power in ports A and B, and the pump light is also injected into port A. Figures 6 and 7 show the three dimensional plots of Xratioij level. As we mentioned before, it is a function of p and Δϕ, and, from Figs. 6 and 7, we can more clearly study the effect of pump power and dephasing
Fig. 7. Xratio level as a function of the pump and Δϕ.
value on the Xratio. From Fig. 6, we notice that the highest point is Xratio12 3.509 dB and the lowest point is Xratio12 −3.515 dB when Δϕ 1.96π and p 40 mW, and Δϕ 0.96π and p 40 mW, respectively. Figures 8 and 9 are the curves of the Xratio versus the initial phase difference, when B10 1 mW and B20 0 mW, and B10 B20 1 mW, respectively, and the pump power is 40 mW. In Fig. 8 we can notice that the Xratio will be invariable when we have a low signal incident in only one port. From Fig. 9 we notice that the Xratio fluctuates with the dephasing value Δϕ when we input two low signal incidents in ports A and B. When Δϕ 0.96π, Xratio12 reduces to the minimum value −3.515 dB, and when Δϕ 1.96π, Xratio12 increases to the maximum value 3.515 dB. In this section we will study the influence of different initial phases on the erbium-doped fiber coupler. The dephasing value (Δϕ 0.96π) has been selected to provide the best values of Xratio for the configurations (0,1), (1,0), and (1,1), so it is appropriate to judge the logic gates. 20
Xratio(dB)
10
X 0
X
12 21
-10
-20
0
2
4
6
phase Fig. 6. Xratio level as a function of the pump and Δϕ. 8040
APPLIED OPTICS / Vol. 53, No. 34 / 1 December 2014
Fig. 8. Xratio level as a function of Δϕ when B10 1 mW and B20 0.
10 X X
Xratio(dB)
5
changing the initial phase of the two input signals, we obtain the logical operation C A · B in port C; here, the initial phase difference is Δϕ 0.96π. We use the pump to yield resonant nonlinearity so that the switching threshold power can drop. At the same time, we can also obtain the various logical operations by adjusting the initial phase of two input signals.
12 21
0
We are grateful to the reviewers for their helpful comments. This work was partially supported by the Natural Science Foundation of Zhejiang Province under grant Y1110078 and the National Natural Science Foundation of China (NSFC) under grant 10904028.
-5
-10
0
2
4 phase
6
Fig. 9. Xratio level as a function of Δϕ when B10 B20 1 mW.
Optical Logic Gate with Δϕ 0.96π, p 40 mW
Table 4.
Port A 0 1 1 0
Port B
X 12 (dB)
Port C
X 21 (dB)
Port D
1 0 1 0
14 −13.99 −3.515
1 0 0 0 ¯ ·B CA
−14 13.99 3.515
0 1 1 0
As is described above, we choose Δϕ 0.96π in Table 4. Xratio12 is around −3.515 dB compared to 3.467 dB in Table 3 for the input configuration (1, 1). Then we can notice that the dephasing value (Δϕ) has realized an exchange between ports C and D. The input configurations (0, 1) and (1, 0) present an Xratio12 around 14 dB and −13.99 dB, which is the same as Xratio12 in Table 3. For the configurations of Table 4, we can get different logical gates. ¯ · B in We obtain the arithmetic operations C A the output of port C. To sum up, we can also realize switching with the change of the initial phase under low pump power. 4. Conclusion
In conclusion, we have proposed the design of alloptical logical gates based on pump-induced propagation-constant change. Based on the model of resonant nonlinearity, we solve the coupled equations by using an analytical method. In our scheme, the signals with low power input from port A and port B, respectively, and the pump light is also coupled into port A by a WDM. We also study the switching characteristics and the curve of the Xratio of the system. We find that compared with other undoped couplers, the doped-erbium fiber coupler can drop the switching threshold power. To realize various logical operations, the Xratio of the on–off switch should be as high as possible. When the initial phases of two input signals are the same, we obtain the logical operation D A · B in port D by changing the pump power. When
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