1. Introduction

Optical bistability effect has attracted considerable academic interest because it provides a basis for various optical devices, such as optical logic gates, optical memories, and optical switches [1, 2]. Many experimental schemes have been carried out to obtain optical bistability, and several relevant theories were proposed to explain them [3, 4]. Yuri Mitnick et al. discovered optical bistability in cavities with EDFA, which can be explained by the bidirectional pump-beam interference [3]. Optical bistability can also be observed in a dual-pump EDF ring laser without isolator [4]. Recently, Jung Mi Oh and Donghan Lee reported a strong optical bistability phenomenon in a unidirectional erbium doped fiber ring laser (EDFRL) operating in L-band, and gave an explanation to it using the gain characteristic curves [5].

In this paper, optical bistability is obtained in a single fiber ring laser using EDF as gain medium and factors affecting the region of the bistability are proposed and discussed. To explain the bistability phenomenon of the EDF theoretically, the propagation equations should be modified by including the useless pump loss and ASE. This modification leads to the lowering of the small-signal gain. When the signal becomes strong, the gain decreases because of the gain saturation effect. As a result, the gain coefficient reaches a maximum point with the increase of the input signal power. Optical bistability in our experimental scheme can be explained well according to this unique gain characteristic curve.

2. Experiment

The experimental setup is shown in Fig. 1(a). The fiber ring laser contains a fiber filter with 3-dB bandwidth of 1.6nm and central wavelength of 1556nm, an isolator, a coupler and a 36 meters long EDF with 100-ppm erbium-ion concentration. This system is pumped by a 980-nm laser diode through a 980/1550 WDM coupler. The fiber coupler is used to extract 10% of the laser power in the ring cavity for detection. The isolator makes the laser unidirectional. The laser obtained near the threshold in the experiment is steady over time, and its bandwidth is approximately 0.1nm.

 

Fig. 1. (a) Schematic diagram of the single fiber ring laser; (b) Optical bistability observed in the proposed scheme.

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The optical bistability phenomenon (as shown in Fig. 1(b)) can be observed in the proposed scheme. When the power of 980nm LD is turned up to 74.2mW (turn-up point), the 1556nm laser appears in the ring cavity. The intensity of laser becomes increasingly strong as the pump power being gradually turned up. In contrary to the turning up process, the laser will not disappear unless the pump power is tuned down to 55.8mW (turn-down point). The whole process indicates that optical bistability exists even in a single fiber ring laser employing EDF. The power range of this bistability region is given by: 74.2 − 55.8 = 18.4mW.

This range of bistability region decreases to 3.9mW with the turn-up point 40.6mW and the turn-down point 36.7mW, by replacing the EDF with a 25 meters long one. Compared to the experimental results of the 100-ppm EDF, the bistability region obtained by employing a 36 meters long 400-ppm EDF has a range of 39.2mW, with the turn-up point and the turndown point at 126.3mW and 87.1mW, respectively. The above experimental results can be concluded as: the bistability region is related to erbium-ion concentration and the length of the EDF.

If the pump light is not strong enough, the amplifying and absorbing regions both exist in the EDF. Useless pump loss and ASE can result in different gain and saturation behavior in the two regions. The gain characteristic curve of the 36 meters long EDF (100-ppm) is measured at pump power 55.8mW and 74.2mW respectively. Figure 2(a) shows the sketch map of the experiment for this measurement. Its result is illustrated in Fig. 2(b).

 

Fig. 2. (a) Experimental schematic for the gain character curve of the EDF; (b) Experimental gain characteristic curve under pump of 74.2mW and 55.8mW.

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Using the gain characteristic curves under different pump power, we can explain the bistability phenomenon observed in the above experiments [5]. The line parallel to the x-axis (Fig. 2(b)) denotes the total loss of the fiber ring. At the beginning, there is no laser in the cavity. The gain characteristic curve moves up slowly as we carefully turning up the pump light. Accordingly, the gain characteristic will enter the left end of the curve because the signal power is very weak in the ring cavity at this time. This small-signal gain determines the laser threshold. The turn-up point of the bistability appears when the pump light is turned up to 74.2mW, at which the small-signal gain becomes equal to the total cavity loss. Afterwards the signal light power will continue to grow under the positive gain. This process can be demonstrated by a point moving along the gain characteristic curve from A to B shown in Fig. 2(b). On the other side, the turn-down point of the bistability is the pump power at which the peak gain equals the total cavity loss. Laser will not disappear until the pump light is turned down to 55.8mW (Point C in Fig. 2(b)) at which the gain is not strong enough to counteract the cavity loss. As the pump light being turned down gradually, the point which determines the power of the signal light moves from B to B’ and eventually reaches C, the peak point of the curve under 55.8mW pump light. The whole course, which starts at A and ends at C, can be defined as the bistability phenomenon of EDF. This bistability is formed mainly because of the difference of the upper and lower threshold values.

3. Theoretical analysis

To calculate the features of the optical bistability, we need to at first determine the useless loss of the pump light. The theoretical maximum of the pump light’s energy transfer rate in the EDF can be calculated as 63.0% by assuming that 980nm photons can totally transfer to 1556nm photons (R_photon =100%). Thus:

In fact, this value is very hard to achieve in the real experiments due to various attenuation factors. Assume that in our experiment the transfer rate is about 40%. This means about (63.0−40)/63.0 = 36.5% of the pump light’s energy is useless. We use α_cp to denote the attenuation coefficient of useless pump loss. It is our approximation that α_cp is a constant throughout the fiber amplifier.

Active spontaneous emission (ASE), another factor which affects pump transferring in the EDF, should also be thought about. In our calculation its influence to the signal light can be ignored because the bandwidth of ASE measured is 30nm, which significantly surpasses the bandwidth of the signal light, 0.1nm. In the following calculations, the EDF is described by the three-level system. Under the assumption that the difference between the absorption cross section and the emission cross section of signal light can be ignored, the pump (P^o_p), signal (P^o_s), forward ASE (P^+o_a) and backward ASE (P^−o_a) power can be described by the following rate equation [6]:

Notice that N_t = N ₁ + N ₂, where N ₁ and N ₂ are the population densities of the ground level and the pump level, respectively, and N_t is the total Er³⁺ density. σ, v, a, Γ are the emission cross section, light frequency, fiber core area and light-to-core overlap, respectively. The footnotes p and s denote the pump and signal light. The steady state solution of Eq. (2) is:

where P_p =P^o_p/P^sat_p, P_s = P^o_s/P^sat_s, P ⁺ _a = P ^+o _a/P^sat_s, P ⁻ _a = P ^−o _a/P^sat_s. Pump power, signal power, forward and backward ASE power are normalized by P^sat_p and P^sat_s, which are defined as P^sat_p = a_p hv_p/(Γ_pσ_p T ₁) and P^sat_s = a_s hv_s/(Γ_sσ_sT ₁).

Using the result of Eq. (3) and including the useless pump loss, the convective equations describing the spatial development of the pump, signal and ASE in the fiber can be expressed as follows [6-8]:

where α_p = Γ_pσ_p N_t, α_s = Γ_sσ_s N_t, k = 2hv_sΔv/P^sat_s. The loss of pump is divided into two parts. In the right-hand side of Eq. (4), the first term relates to the pump light power which transfers into the power of signal light and ASE, and the second term describes the effect of useless pump loss. According to the transfer rate mentioned at the beginning of this section, α_cp = 0.365 /(1 − 0.365)α_p = 0.575α_p.

4. Numerical calculation

Equations (4-7) can be solved numerically. The typical fiber parameters of the EDF used in the calculation are listed in Table.1 [7, 9].

Table 1. Typical Fiber Parameters

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The gain characteristic curves are sketched numerically using Eq. (4-7) in different cases.

 

Fig. 3. (a) The EDF is 36m long, and with no useless pump loss or ASE; (b) The EDF is 36m long, α_p =0.21, α_s =0.28 , α_cp =0.12 and the cavity loss is 5dB.

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Fig. 4. (a) The EDF is 36m long, α =0.21, α_s =0.28 , α_cp =0.15 , and the cavity loss is 5dB; (b) The EDF is 45m long, α_p =0.21, α_s =0.28, α_cp =0.12, and the cavity loss is 5dB;

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As shown in Fig. 3(a), the gain characteristic curve of the EDF without useless pump loss and ASE has a maximum when the signal is small. It is obvious that this curve cannot bring about bistability phenomenon. The useless pump loss and ASE can both lower the small-signal gain of the EDF (Fig. 3(b)), and can thus result in the bistability phenomenon whose cause has already been demonstrated in section 2. The useless loss coefficient (α_cp) can influence the bistability region seriously. The bistability range is 8.86 with the parameter α_cp =0.12 (Fig. 3(b)). This range increases to 20.94 as α_cp = 0.15 (Fig. 4(a)).

Comparing Fig. 3(b) with Fig. 4(b), it is concluded that a wider bistability region can be obtained with a longer EDF. The bistability range is 8.86 with a 36 meters long EDF, while it changes to 25.06 using a 45 meters long EDF with the same α_cp. According to our calculation, the bistability phenomenon almost vanishes when the EDF is shorter than 15m.

 

Fig. 5. (a) The EDF is 36m long, α_p =0.42, α_s =0.56, α_cp =0.12, and the cavity loss is 5dB; (b) The EDF is 36m long, α_p =0.21, α_s=0.28, α_cp =0.12, and the cavity loss is 10dB.

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Fig. 6. (a) The EDF is 36m long, α_p =0.21, α_s =0.28, and α_cp =0.12; (b) The EDF is 36m long, α_p =0.42, α_s =0.56, and α_cp=0.12.

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The bistability range also depends on α_p and α_s which increase accordingly with the increasing of the erbium-ion doping concentration of the EDF. A larger bistability range 19.04 can be obtained when α_p and α_s double (Fig. 5(a)). This indicates that a higher doping concentration of erbium-ion can bring about wider bistability range.

Cavity loss is another factor which can affect the range of the bistability region. By comparing Fig. 3(b) and Fig. 5(b), we can conclude that the bistability range changes along with the changing of the cavity loss: 8.86 at 5dB and 6.92 at 10dB. While useless pump loss and ASE cannot be controlled precisely for a certain EDF, one can adjust the range of the bistability region by tuning the value of the cavity loss.

Useless pump loss, which should be controlled as small as possible in the common application, is the main reason of the bistability phenomenon. By comparing the two curves shown in Fig. 6(a), it is clear that ASE has a much weaker influence to the gain characteristic curve of the EDF compared to that induced by the useless pump loss. The comparison between Fig. 6(a) and Fig. 6(b) shows that ASE has different influence to gain characteristic curves of EDFs with different α_p and α_s.

5. Conclusion

Optical bistability phenomenon is observed in a single EDF ring laser. Gain characteristic is measured at related pump powers. The gain characteristic curves obtained in our experiment which have maximums can explain the optical bistability phenomenon well. The effect of useless pump loss and ASE are included in the convective equations to explain the phenomenon theoretically. Through calculation, we find that the bistability phenomenon mainly depends on useless pump loss of the EDF. It is also concluded that the length, transfer rate and the erbium-ion concentration of the EDF, and the cavity loss can all influence the range of the bistability region. By verifying the reasons leading to optical bistability and factors related to its range in the single EDF ring laser amplifier, we can find methods to control the bistability to meet various needs in real applications.