## Abstract

In this paper, a model is proposed to study the behavior of four-wave mixing assisted multiwavelength erbium doped fiber ring laser based on the theoretical model of the multiple FWM processes and Gile’s theory of erbium-doped fiber. It is demonstrated that the mode competition can be effectively suppressed through FWM. The effect of phase matching, the nonlinear coefficient, the power in the cavity and the length of the nonlinear medium on output spectrum uniformity are also investigated.

©2008 Optical Society of America

## 1. Introduction

Multiwavelength erbium-doped fiber laser (MWEDFL) has received extensive attention for two decades, because of their potential applications in wavelength-division-multiplexed (WDM) communication systems, fiber sensors and optical instrumentations. Erbium-doped fiber (EDF) is a homogenous broadening medium under room temperature, which leads to fierce mode competition and makes it difficult to obtain muiltiwavelength lasing simultaneously. Various techniques have been exploited to overcome this barrier, including frequency shift feedback [1,2], stimulated Brillouin scattering [3,4], nonlinear fiber-loop mirror [5], nonlinear polarization rotation [6], polarization hole burning [7,8], and four-wave mixing (FWM) [9–15] and so on.

FWM has been proposed and demonstrated as a means to suppress the mode competition through transferring the energy among different frequency components by X. Liu [9]. The nonlinear coefficient of PCF can achieve more than 30/W/km while the flat near zero-dispersion region covers more than 100nm. Fierce FWM can generate in a short high nonlinear PCF, which makes it an ideal medium to suppress mode competition. Despite a lot of experiments carried out to study FWM assisted MWEDFL [9–15], there are rare reports on theoretical analysis of FWM assisted MWEDFL. In this paper, a theoretical model is proposed, based on which the behavior of FWM assisted Multiwavelength Erbium-doped Fiber Ring Laser is studied. According to the author’s knowledge, this paper is the first reported complete numerical approach in this field.

## 2. Theoretical model

The theoretical model follows the arrangement shown in Fig.1. The model assumes single direction pumping at 980nm. The backward amplified spontaneous emission (ASE) is prohibited by optical isolator (OI). The mode competition induced by homogenous broadening of the EDF is suppressed by multiple FWM processes in a piece of high nonlinear dispersion shifted fiber (HNDSF) or PCF. The multifrequency band-limited frequency periodic filter is used to provide periodic loss in the spectrum domain to generate multiwavelength lasing. Polarization controller (PC) is used to adjust the polarizations of the beam that enters the nonlinear medium. A 3-dB coupler is used for output. Since a lot of theoretical work has been done on EDF, we will use the EDF model in Ref. [16] directly and focus on the theoretical model of multiple FWM processes.

In a dispersion medium, each frequency components need to be considered separately. Because of the PC, the polarization of every frequency component is assumed to be along *x* direction. Therefore, the total electrical field can be represented as the sums of its various frequency components:

where *A _{n}*(

*z*) is the complex amplitude.

*F*(

_{n}*x*,

*y*) is the normalized transverse field distribution of frequency

*ω*.

_{n}*β*(

*ω*) is the propagation factor. Substituting equations (1) into the general nonlinear optical wave equation, the power of each frequency component after FWM can be described as [17]:

_{n}$$\phantom{\rule{2.5em}{0ex}}+\mathrm{i\gamma}\sum _{k}{A}_{k}^{*}\left(z\right){A}_{k}^{2}\left(z\right)\mathrm{exp}\left(-i\Delta {\beta}_{\mathrm{mnk}}z\right)$$

where *γ* is nonlinear coefficient approximated as independent of frequency, since the frequency shift is slight compared with frequency itself. Δ*β _{ijk}*=

*β*(

*ω*)+

_{i}*β*(

*ω*)-2

_{j}*β*(

*ω*) is the propagation constant difference describing the phase mismatch of the quasi-degenerate FWM induced by material and waveguide dispersion. After Taylor expansion, it can be simplified as:

_{k}where *β*
_{2}(*ω*
_{0}) and *β*
_{3}(*ω*
_{0}) are second and third derivatives of the propagation constant at frequency *ω*
_{0}. In most situation, only *β*
_{2}(*ω*
_{0}) needs to be considered except in the region near *ω*
_{0} where the contribution of *β*
_{3}(*ω*
_{0}) is comparable with *β*
_{2}(*ω*
_{0}). Higher order derivatives of the propagation constant do little contribution to Δ*β _{ijk}*, and therefore can be ignored. Ω is the frequency shift.

In fiber, quasi- degeneration FWM is more likely to happen than non-degeneration FWM, because the phase matching condition of quasi-degeneration is more easy to satisfy [13,17]. Thus in Eq. (2), non-degeneration FWM is ignored. For certain frequency component, it can acted as pump, Stokes or anti-Stokes wave in different FWM process, which are represented as the second and third terms on the right hand side of Eq.(2). The second term must satisfy 2*ω _{n}*=

*ω*+

_{m}*ω*and the third must satisfy

_{k}*ω*+

_{n}*ω*=2

_{m}*ω*. The first term describes Kerr effect. Letting

_{k}*A*(

_{n}*z*)=

*B*(

_{n}*z*)exp(

*iϕ*), the equation can be separated into:

_{n}$$\phantom{\rule{2.2em}{0ex}}+\gamma {B}_{n}^{-1}\left(z\right)\sum _{k}{B}_{m}\left(z\right){B}_{k}^{2}\left(z\right)\mathrm{cos}{\theta}_{\mathrm{mnk}}$$

The solution to Eq. (4),(5) can be represented in the form of elliptic functions [17], but it is more reasonable to solve it numerically. Because they are stiff in character, the equations are integrated by Gear’s method. To make the discussion more explicit, the roles of components shown in Fig. 1 are treated as operators obeying the operator algebra. Then iteration process for iteration times *i*>1 can be simply described as:

When *i*=1, *P*
_{1}(*ω*)=0. *EDF* denotes the theoretical mode in Ref.[16]. *FWM* represents the effect of FWM described in previous passage. *F* is the transmission spectrum of filter and *L* is the total loss of the cavity.

## 3. Simulation results and analysis

In the numerical model, a 40m typical Lucent Technology EDF HP980 is considered. A 1.2 km HNDSF is inserted into the cavity, whose nonlinear coefficient is 11/W/km, the dispersion at 1550nm is zero and dispersion slope is 0.03ps/nm^{2}/km. A 3-dB coupler is used for output. The total loss of the cavity is approximated to be 10dB. The period filter is simulated by the superposition of Gaussian band pass filters with central frequencies shifted by 0.8nm and full width at half maximum (FWHM) of 0.15 nm, covering 1540~1560nm.

The whole spectrum region from 1450nm to 1650nm has been subdivided into 2000 slots of Δ*λ*=0.1*nm* and substituted into the EDF model. However, it is unrealistic to consider all the frequency slots in the FWM model because the possible mixing products of 2000 frequency components are horrible. To reduce the calculation work, only the 26 modes obtained through the filter are considered. This simplification is reasonable because the power of other frequency components is so small that their contribution to the FWM process can be ignored.

To begin with, we will demonstrate the multifrequency operation is impossible without FWM effect. Fig.2 (a) and (b) show the output power spectrum when the 1.2km HNDSF is absent or not. Without HNDSF, one strong single lasing line appears at 1555.2nm, shown in Fig. 2(a), the wavelength of highest gain. With HNDSF, energy will transfer from the higher power frequency components to the lower ones during FWM process and therefore the mode competition is suppressed effectively. As a result, a lot of lines appear in the spectrum. Three strong lasing are built up at 1557.6nm, 1558.4nm and 1559.2nm with power difference < 3dB, as shown in Fig. 2(b). Due to the phase mismatching induced by the dispersion character of the HNDSF, the flatness of the spectrum is weak. Lines in the region >1550nm are developed better. This result is in good coincidence with the experiment reported in Ref. [11].

In order to obtain more lasing lines, the critical issue of phase mismatching needs to be examined. The phase mismatching can be evaluated as Δ*δ*=Δ*δ _{M}*+Δ

*δ*+Δ

_{W}*δ*, where Δ

_{NL}*δ*, Δ

_{M}*δ*and Δ

_{W}*δ*represents the phase mismatching induced by material dispersion, waveguide dispersion and nonlinear effects, respectively. In single mode fiber, the contribution of Δ

_{NL}*δ*can be ignored except in the region near the zero dispersion wavelength, when it becomes comparable with Δ

_{W}*δ*for identified polarized waves. To control Δ

_{M}*δ*into a tolerate value, the following methods can be adopted:

*a*. reducing Δ

*δ*and Δ

_{M}*δ*by using small frequency shifts.

_{NL}*b*. operating near the zero dispersion wavelength so that Δ

*δ*almost cancels Δ

_{W}*δ*and Δ

_{M}*δ*.

_{NL}*c*. operating in the abnormal group velocity dispersion (GVD) regime so that Δ

*δ*is negative and can be canceled by Δ

_{M}*δ*and Δ

_{W}*δ*[17]. For the MWEDFL, only the third method is applicable. That is why many reported FWM assisted MWEDFL operates in the region >1550nm [11].

_{NL}Here we change the filter band to 1550~1570nm, where is in the abnormal GVD region of the HNDSF considered in the simulation. The result is shown in Fig. 3. Comparing Fig.3(a) with Fig. 2(b), the flatness of the output spectrum is improved apparently. From 1556.4nm to 1566.8, there are 14 lines with power difference <3-dB while only 3 lines in Fig.2 (b). However, the lasing lines near 1550nm and 1570nm are not well developed, because they are involved in fewer FWM processes than those near 1560nm and little energy would be transferred to them unless FWM effect is fierce enough.

The FWM effect can be enhanced by maintaining higher power in the cavity, using fiber with larger *γ* and increasing the length *L* of the HNDSF [11]. High cavity power can be obtained through increasing pump power, reducing the loss of the cavity or coupling more power back into the cavity. Here, we will only study the effect of pump power. Similar results can be obtained by using the other methods. In Fig. 3, the output spectrum is plotted for *L*=1.2km, *γ*=11/W/km and the pump power is 100mW, 200mW, 300mW and 400mW, respectively. When the pump power is 200mW, the output spectrum is flattened, as shown in Fig.3 (b). The number of wavelengths in 3-dB bandwidth exceeds twenty. However, when pump power increases to 300mW and 400mW, the flatness of the output spectrum is abated, because too much power is transferred to wavelength near 1550nm and 1570nm. The number of the wavelengths in 3-dB bandwidth reduces to 14 and 9, respectively, as shown in Fig.3(c) and (d). Similar phenomenon can be observed through increasing the nonlinear coefficient *γ*. The output spectrums for *P*=100mW, *L*=1.2km *γ*=20/W/km and 30/W/km are plotted in Fig.4. When *γ*=20/W/km, 17 lasing lines in 3-dB bandwidth are obtained, as shown in Fig.4 (a), while the number reduces to 14 when *γ*=30/W/km. Thus the pump power and the nonlinear coefficient need to be optimized in order to get satisfying output spectrum.

Another interesting phenomenon appears when the length of the nonlinear medium increases. The relationships between the standard derivation of the output spectrum and the length of the medium when pump power is 100mW, 300mW and 500mW are plotted in Fig.5. In the beginning, the standard derivation reduces as the length of HNDSF increases, but later, the standard derivation begins to fluctuate. This phenomenon becomes eminent when the pump power is high enough. Take the pump power of 500 mW for example. When the length of HNDSF is 200m, the standard derivation comes to the lowest point. Beyond this length, the standard derivation begins to fluctuate. This phenomenon is believed to be caused by the periodic character of FWM. The power of each frequency component varies at different periods. Because of this character, it is difficult to obtain output spectrum with high uniformity, because any change to the FWM related parameters will equally affects the behavior of each frequency component.

This problem can be solved through introducing nonuniform effect into the cavity. In the reported experiment, the PC can play this role [11,12]. Adjusting PC can change the polarization and loss slightly and therefore affected the FWM efficiency of each frequency component unequally. However, the effect of PC can’t be described numerically, so it isn’t considered in our model. Another way is to use specially designed filter to compensate the gain difference, as shown in Ref. [11].

## 4. Conclusion

In this paper, a model is proposed to study the behavior of FWM assisted MWEDFL based on Gile’s homogeneously broadened EDF model and the multiple FWM process theory. When a piece of HNDSF is inserted into the cavity, the homogenous broadening is efficiently suppressed and multiwavelength lasing is achieved. To improve the performance of the FWM assisted MWEDFL, we studied the effect of phase mismatching on the multifrequencies oscillations and the result shows that the laser should operates in the abnormal GVD region. Other factors such as the power in the cavity, the nonlinear coefficient and the length of the HNDSF are also investigated and the simulation demonstrates that an increase in these factors can improve the flatness of the output spectrum.

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