## Abstract

We perform optimization of all-optical EDFA-based Sagnac – interferometer switch through an analytical model and numerical simulations by solving nonlinear Schrödinger equations. The effects of the performance of EDFA on the bit rate and the switching power are investigated for all-optical switch based on self-phase or cross-phase modulation. The simulated results show that ultra-low switching power (<1mW) all-optical switch for 40 Gb/s data can be realized by properly selecting the length of highly nonlinear photonic crystal fiber and adjusting the performance of EDFA.

© 2007 Optical Society of America

## 1. Introduction

In the present ultrahigh-speed communication networks, where information is encoded in pulse, optical-electrical conversion of information is still necessary at each network node to process the incoming signals [1]. However, when the single-channel bit rate increases beyond the electric bandwidth, all-optical processing must be performed in order to extract the routing information from the incoming packets and switch them to the selected node output. Besides, all-optical switching can offer transparency to data rate and format, fine granularity and flexibility, reducing the packet latent time as well. For this purpose, various optical switches were developed to process the incoming signals, such as optical microelectro- mechanical-systems-based switches [2], electrooptical switches [3], acoustooptical switches [4], and thermal optical switches [5]. These schemes exhibit a common intrinsic limit due to their slow switching time. A promising alterative is represented by all-optical switches, exploiting ultrafast nonlinear effects [6] including self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM) in optical fiber or waveguides, thus allowing a substantial improvement in terms of processing velocity.

Up to now, two types of all-optical switches based on ultrafast nonlinear effects were developed. One is based on the interferometric components, such as nonlinear Sagnac loop mirror [7] and Mach-Zehnder interferometer [8], and the other noninterferometric. Very recently, Berrettini et al. [9] reported all-optical 2 × 2 switch based on XPM-induced polarization rotation. In comparison with the noninterferometric ones, all-optical switches based on interferometric components have many advantages, such as low switching power, high extinction ratio and compactness.

In the case of Sagnac-interferometer switch, since the first report of all-optical switch based on nonlinear Sagnac loop mirror [7], all-optical fiber Sagnac-interferometer switch has attracted much attention due to its low switching power, simplicity and compactness. To reduce the switching power, an erbium-doped fiber amplifier (EDFA) was introduced into the fiber loop to make it more practical [10]. Very recently, Liu et al. [11] demonstrated low switching power (~40 mW) EDFA-based Sagnac-interferometer switch using highly nonlinear photonic crystal fiber. However, the effects of the performance of EDFA, a key component of the device, on the bit rate and the switching power have not yet been investigated so far. In this paper, we have performed optimization of all-optical EDFA-based Sagnac – interferometer switch through an analytical model and numerical simulations by solving nonlinear Schrödinger equations. The effects of the performance of EDFA on the bit rate and the switching power were investigated for all-optical switch based on self-phase or cross-phase modulation. The simulated results showed that ultra-low switching power (<1mW) all-optical switch for 40 Gb/s data could be realized by properly selecting the length of highly nonlinear photonic crystal fiber and adjusting the performance of EDFA.

## 2. Modeling of all-optical EDFA-based Sagnac-interferometer switches

Figure 1 shows the schematic of all-optical Sagnac-interferometer switches based on (a) SPM and (b) XPM by including an EDFA. The fiber Sagnac interferometer is formed by connecting a fiber loop with two output ports of a 3 dB coupler, which can totally reflect any input signal, so called the fiber loop mirror. Such a symmetric device can not act as an optical switch even using the nonlinear fiber loop. It is because the two signal beams divided from the input signal pass through the coupler to get the *π*/2 -phase difference, and propagate oppositely in the same loop with the same phase shift. After they pass through the coupler again to produce another *π*/2 -phase deference, and then they interfere with each other at the two output ports. In the reflected port, two beams have same phase (*π*/2), while at the transmitted port, they have reversed phase (0 and *π*), therefore, all the signal power will be output from the reflected point [6].

If the device can be designed to be asymmetric structures, for example, introducing of an EDFA into the fiber loop; and the self-pump (SPM) or the cross-pump (XPM) power is strong enough to induce an additional *π*-phase difference; the two signals have same phase at reflected point and have reversed phase at reflected point. Therefore, the device can accomplish a complete optical switching: the high-power signal will be transmitted, while the signal at low-power level will be reflected.

In our case, the fiber loop is composed of a bi-directional EDFA (small signal gain G
_{0} = 30 dB at 1550 nm, the saturated output power P_{S} = 35 or 100 mW, the effective length of erbium-doped fiber (EDF) L_{E} = 2.3 m, the group velocity dispersion parameter of EDF β_{2} = -20 ps^{2}/km at 1550 nm, the nonlinear parameter of EDF γ_{E} = 3.78 /(W∙km) at 1550 nm), one meter long single mode fiber (SMF) to connect the EDFA with highly nonlinear photonic crystal fiber together (the group velocity dispersion parameter of SMF β_{2} = -20 ps^{2}/km at 1550 nm, the nonlinear parameter of SMF γ_{S} = 3.78 /(W∙km) at 1550 nm, and L_{S} = 1 m), and a certain length of high nonlinear photonic crystal fibers (HNPCF) (the effective length of HNPCF L_{P} = 1~1000 m range, the group velocity dispersion parameter of HNPCF β_{2} = 0.1 ps^{2}/km at 1550 nm, the nonlinear parameter of HNPCF γ_{P} = 45.38 /(W∙km) at 1550 nm). The envelope of input signals (40 or 10 Gb/s non-return-to-zero bit streams) is assumed to be ${A}_{0}(t)=\sqrt{{P}_{in}}$ exp⌊-*t*
^{2} /(2*T*
_{0}
^{2})⌋, where *P _{in}* is the peak power of input pulse,

*T*

_{0}= 5 ps the half-width at 1 /

*e*-intensity point.

For all-optical Sagnac-interferometer switches based on SPM shown in Fig. 1(a), the XPM interaction between two counterpropagating optical pulses is generally quite weak and can be neglected in the case of ultrashort pulses. Let *A*
_{3} (*z*, *t*) or *A*
_{4} [*z*, *t*) be the envelope of clockwise or anti-clockwise propagating pulse, z the propagating distance of pulse along the fiber loop. So the envelope of the pulse output from the port 3 after amplified by EDFA is ${A}_{3}\left(0,t\right)=\sqrt{\frac{G}{2{A}_{0}\left(t\right)}},$ , and the envelope of the pulse output from the port 4 is ${A}_{4}\left(0,t\right)=\sqrt{\frac{1}{2{A}_{0}\left(t\right)}},$. Since the width of the input pulse is larger than 1 ps, the propagation of the pulse inside the HNPCF can be described by the simplified nonlinear Schrödinger equation [6]

where α is the background loss of HNPCF, β_{2} the GVD parameter of HNPCF, *T* = *t* - *z* / *ν _{g}* (

*ν*the group velocity of the signal), and γ the nonlinear parameter of HNPCF. By neglecting the effect of the GVD on the pulse propagation due to small GVD value of HNPCF, the envelope of the pulse

_{g}*A*

_{3}after propagating from the port 3 to the port 4 is

Similarly, the envelope of the pulse *A*
_{4} after propagating from the port 4 to the port 3 is

where *ϕ*
_{0} = *βL* is the linear phase shift, and *β* the mode-propagation constant.

By solving the transfer matrix of the fiber coupler [6]

we derived the transmission ratio *T _{SPM}* = |

*A*|

_{t}^{2}/|

*A*

_{0}|

^{2}of the nonlinear Sagnac loop mirror

and the switching power of all-optical Sagnac-interferometer switch based on SPM

where *P _{SPM}* is the peak power for switching.

As aforementioned, the EDFA, as a key component of the device, has a large effect on the performance of all-optical switch. In the case of short pulse train as the input signal of EDFA, the gain of EDFA is represented by [1]

where *P _{i}* is the averaged power of input signals for EDFA. By substituting

*P*into Eq. (7), we obtained the required gain for switching

_{SPM}and the final form of the switching power

In addition, the EDFA requires that the ratio

For all-optical Sagnac-interferometer switch based on XPM (Signal peak power at 1550 nm ~ 50 μ[J W, the pumping wavelength ~ 1540 nm, the pump pulse width ~ 5 ps), similarly, we derived the signal transmission ratio *T _{XPM}* = |

*A*|

_{t}^{2}/|

*A*

_{0}|

^{2}of the nonlinear Sagnac loop mirror

(where ${A}_{p0}=\sqrt{{P}_{pump}}$ exp(*t*
^{2} /(2*T*
_{0}
^{2}
_{0})) is the envelope of the pump pulse) and the switching pump power of all-optical Sagnac-interferometer switch based on XPM

By substituting *P _{xpm}* into Eq. (11), we obtained the required gain for switching

and the final form of the switching pump power

Similarly, the EDFA requires that the ratio

We have performed optimization of all-optical EDFA-based Sagnac – interferometer switch through numerical analysis using Eqs. (5)–(15) and simulations by solving nonlinear Schrödinger equation Eq. (1). The simulated results were shown in the following sections.

## 3. Simulated results and discussion

Figure 2(a) shows the switching power and the required gain for switching of all-optical Sagnac-interferometer switch based on SPM as a function of the effective fiber length of HNPCF for EDFA (P_{s} = 35 mW) with different small signal gain values (G_{0}=20, 30, 40, and 50 dB, respectively) and 40 Gb/s data. It is seen that, with increasing the small signal gain value G_{0} of EDFA, the switching power decreases and the required gain for switching increases when the effective fiber length is fixed. The switching power decreases and the required gain for switching increase with increasing the effective fiber length of HNPCF when the small signal gain value of EDFA is fixed. As aforementioned, the EDFA requires that the ratio
$\frac{{P}_{\mathrm{SPM}\_F}{G}_{\mathrm{SPM}}}{5{P}_{s}}\le 1$for 40 Gb/s data. Fig. 2(b) presents the corresponding critical ratio as a function of the effective fiber length for EDFA (P_{s} = 35 mW) with different small signal gain values (G_{0}=20, 30, 40, and 50 dB, respectively) and 40 Gb/s data. As we can see from Fig. 2(b), the effective fiber length of HNPCF must be larger than 800 m to satisfy Eq. (10) for EDFA (P_{s} = 35 mW) with different small signal gain values (G_{0}=20, 30, 40, and 50 dB, respectively) and 40 Gb/s data. It means that the required effective fiber length of HNPCF should be larger than 800m to get switching. Let us move from Fig. 2(b) to Fig. 2(a), the data of Fig. 2(a) does not show that the fiber loop composed of less than 800 m long HNPCF could not give switching, which indicates that Eqs. (8), (9) do not consider the gain saturation effects of EDFA inside the fiber loop. These results show that we must consider the gain saturation effects of EDFA by using Eq. (10) to get switching. Otherwise, we can not get switching if we just use Eqs. (8), (9) to optimize the device. This is the first time to optimize all-optical EDFA-based Sagnac-interferometer switch by considering Eq. (10), to our best knowledge.

Figure 2(c) shows the switching power and the required gain for switching of all-optical Sagnac-interferometer switch based on SPM as a function of the effective fiber length of HNPCF for EDFA (P_{s} = 35 mW) with different small signal gain value (G_{0}=20, 30, 40, and 50 dB, respectively) and 10 Gb/s data. Fig. 2(d) presents the corresponding critical ratio as a function of the effective fiber length for EDFA (P_{s} = 35 mW) with different small signal gain value (G_{0}=20, 30, 40, and 50 dB, respectively) and 40 Gb/s data. In comparison with Fig. 2(a) and Fig. 2(b) for 40 Gb/s data, the switching power become smaller and the required gain for switching larger for 10 Gb/s data, and the required effective fiber length of HNPCF to satisfy Eq. (10) is reduced from 800 m to 200 m. This is because that the averaged power for 10 Gb/s data is substantially smaller than that for 40 Gb/s when the input peak power is same for both cases, which make the suffered gain G for 10 Gb/s larger than that for 40 Gb/s data due to weak gain saturation effects of EDFA, and furthermore the output power after amplified by the EDFA for 10 Gb/s data higher than the case for 40 Gb/s data. Therefore, the switching power becomes smaller with decreasing the bit rate from 40 Gb/s to 10 Gb/s. As a result, the required effective fiber length of HNPCF to satisfy Eq. (10) is reduced from 800 m to 200 m, as shown in Fig. 2(d).

Figure 3(a) and Fig. 3(c) present the switching power and the required gain for switching of all-optical Sagnac-interferometer switch based on SPM including A EDFA with P_{s} = 100 mW as a function of the effective fiber length of HNPCF for 40 Gb/s and 10 Gb/s data, respectively. Fig. 3(b) and Fig. 3(d) show the corresponding critical ratio as a function of the effective fiber length for 40 Gb/s and 10 Gb/s data, respectively. In comparison with Fig.2, the switching power become smaller and the required gain for switching larger, and the required effective fiber length of HNPCF to satisfy Eq. (10) is reduced substantially for both cases due to weak gain saturation effects with enhancing P_{s} from 35 mW to 100 mW.

Figure 4(a) and Fig. 4(c) show the switching pump power and the required gain for switching of all-optical Sagnac-interferometer switch based on XPM as a function of the effective fiber length of HNPCF for 40 Gb/s and 10 Gb/s data, respectively. Fig. 4(b) and Fig. 4(d) show the corresponding critical ratio as a function of the effective fiber length for 40 Gb/s and 10 Gb/s data, respectively. In comparison with the switch based on SPM shown Fig. 3, the switching power for the pump pulse decreases and the required gain for the pump pulse increases, and the required effective length of HNPCF is reduced to below 200 m to satisfy Eq. (15) and make the switching power less than 1 mW for 40 Gb/s data due to large phase shift for XPM. For example, the switching pump power of the device by using 200 m long HNPCF and a EDFA with G_{0} = 30 dB is 0.7 mW for 40 Gb/s data.

To understand the effects of the GVD parameters of the fiber on the performance of all-optical Sagnac-interferometer switch, we performed numerical simulations by solving nonlinear Schrödinger equation. Fig. 5(a) shows the transmission ratio of the nonlinear Sagnac loop mirror against the input signal power for all-optical Saganac-interferometer switch based on SPM composed of a EDFA with G_{0} = 30 dB and P_{s} = 100 mW, a 300 m long HNPCF for 40 Gb/s data. It is seen that the switching power is about 1.16 mW, which agree well with the analytical value show in Fig. 3(a). In addition, the gain saturation effects occur with increasing the input signal power after switching. Fig. 5(b) presents the pulse evolution of the input signal propagating clockwisely inside the fiber loop when the device is switched on, which shows that no substantially pulse widening occurs due to small GVD value (0.1 ps_{2}/km at 1550 nm) [6]. Fig. 5(c) shows the corresponding phase difference evolution of the input pulse inside the fiber loop, which give a phase difference of ~ π when two counter-propagating pulse meet before the 3 dB fiber coupler. Fig. 5(d) gives the corresponding spectral evolution of the input signal inside the fiber loop, which describes a general feature of SPM for phase shifting from 0 to *π* [6].

Figure 6(a) gives the transmission ratio of the nonlinear Sagnac loop mirror against the input signal power for all-optical Saganac-interferometer switch based on XPM composed of a EDFA with G_{0} = 30 dB and P_{s} = 100 mW, a 150 m long HNPCF for 40 Gb/s data. It is seen that the switching pump power is about 1.21 mW, which agree well with the analytical value show in Fig. 4(a). Fig. 6(b) shows the pulse evolution of the input signal propagating clockwisely inside the fiber loop when the device is switched on, which shows that no substantially pulse widening occurs due to small GVD value (0.1 ps^{2}/km at 1550 nm). Fig. 6(c) shows the corresponding phase difference evolution of the input pulse inside the fiber loop, which give a phase difference of ~ *π* when two counter-propagating pulse meet before the 3 dB fiber coupler. Fig. 6(d) gives the corresponding spectral evolution of the input signal inside the fiber loop, which gives a general feature of XPM for phase shifting from 0 to *π*.

## 4. Conclusion

We performed optimization of all-optical EDFA-based Sagnac – interferometer switch through an analytical model and numerical simulations by solving nonlinear Schrödinger equations. The effects of the performance of EDFA on the bit rate and the switching power were investigated for all-optical switch based on self-phase or cross-phase modulation. The simulated results showed that ultra-low switching power (<1mW) all-optical switch for 40 Gb/s data could be realized by properly selecting the length of highly nonlinear photonic crystal fiber and adjusting the performance of EDFA.

## Acknowledgments

This work was supported by the Natural Science Foundation of China (10474017).

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