# Optics Communications

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3 F. Li et al. / Optics Communications 284 (11) independent. The output power at the i-th wavelength, P i III for an input power of P i II is therefore given by dp i dz = α ð P iþp i ð2þ where α(p i )N. Similar to the definition of the gain, the transmission L i at the i-th wavelength of the intensity-dependent loss is defined as L i (P i II )=P i III /P i II. The nonlinear loss coefficient can be further approximated to first order in the input optical power as αðp i Þ = α + α 1 P i ð3þ where α and α 1 are the small signal loss coefficient and the first order intensity-dependent loss coefficient respectively. Eqs. (2) and (3) can be combined with the gain model in Eq. (1) to form a single master equation. Here however, we will use a discrete model in the simulations. Eqs. (2) and (3) can be solved directly giving P III i = L i P II i = α L α + α 1 ð1 L ÞPi II P II i where L is the small signal loss, i.e. the loss when α 1 =. From Eq. (4), the output power of the loss element (P III i ) increases monotonically with the input laser light (P II i ). If α 1, as the input power increases, loss increases and the output power approaches the constant value of α L /(α 1 α 1 L ), i.e. for very high input power, the output is always the same value. Thus the intensity dependent loss given by Eqs. (2) and (3) behaves like a saturable transmitter, i.e., it preferentially transmits low-intensity light but absorbs high-intensity light in contrast to the property of a saturable absorber. Fig. 2 shows the output power versus the input power of the loss element when the transmission function of the loss element is constant, saturable, and periodic. From Fig. 2, the transmission of a constant loss is independent of the input power so the output power is proportional to the input power. For the saturable transmitter, the output power increases monotonically when the input power increases and approaches a constant value at high input power. Another curve shown in Fig. 2 as a solid line is of periodic loss which will be discussed in Section 4. To determine the dynamics of the multiwavelength CW laser with homogeneous gain and saturable transmitter, we iterate Eqs. (1) and (4) together around the ring cavity of Fig. 1. For the normalized gain profile in Eq. (1), we have chosen a Gaussian function, i.e. h i f i = exp ðλ i λ c Þ 2 = Δ 2 ð5þ ð4þ and λ i = λ c + iδλ where λ c is the center wavelength and Δ is the bandwidth of the gain profile. δλ is the wavelength separation of the comb filter which is assumed to be.8 nm and i=, ±1, ±2,..., ±N, and 2Nδλ is the spectrum range used in the simulation. We use a simple Gaussian gain profile instead of a more complex gain model based on emission and absorption cross sections because we want to delineate the effects of an intensity dependent loss and the gain properties on the laser dynamics. For example, in erbium doped fiber amplifiers (EDFA), when the gain saturates, the fraction of upper level inversion will decrease. The gain profile will be flattened and shifted to longer wavelength because of the offset of the absorption and emission cross sections. The change in the gain profile will then affect laser dynamics. We note that a Gaussian gain profile is more typical of inhomogeneous gain broadening while a Lorentzian profile is more typical of homogeneous gain broadening. Since the details of the gain profile do not determine whether multiwavelength CW lasing is possible for a homogeneous gain, we have chosen a Gaussian profile in order to investigate the relationship between the laser output spectrum and the gain profile by using the family of super-gaussian profiles later. We have also carried out simulation using Lorentzian gain profiles and found that the results obtained are qualitatively similar to that using Gaussian gain profiles shown. In the simulations that follow, the saturation power is assumed to be P sat =.1 W. For the normalized gain profile, we have chosen the parameters λ c =155 nm, and Δ= nm. The small signal gain for the center wavelength G s, max = db. For the intensity-dependent loss, we have chosen, L =.9 and α 1 /α = W 1. All these parameter choices are consistent with physically realizable systems and operating standards. Fig. 3 shows the variation of the optical power at different wavelengths in the multiwavelength laser as a function of iteration number. The zero-th wavelength is at the center wavelength of the gain band and the i-th wavelength is at wavelength iδλ away from the center wavelength on either side. Note that the gain profile is symmetric with respect to the center wavelength. Initially all the wavelengths satisfying the lasing condition that the cavity gain is larger than the cavity loss, i.e. L G i N1, will grow. As the power in each wavelength grows, the loss each wavelength experiences will also increase because of Eqs. (2) and (3). The gain G i begins to decrease because of gain saturation. As a result, some of the wavelengths will begin to have their gain value less than the loss, i.e. L i G i b1. The power in these wavelengths will begin to decay, e.g. the 13-th wavelength in ð6þ Output Power (W) Constant loss Input Power (W) Power (dbm) No. of iterations Fig. 2. The output power versus the input power curve for constant loss (dotted lines), saturable transmitter (dashed lines) and periodic loss (solid lines). Fig. 3. Variation of the optical powers with the number of iterations at different wavelengths in the multiwavelength laser with a saturable transmitter.

4 23 F. Li et al. / Optics Communications 284 (11) Power (W) No. of iterations 5 wavelength. All other wavelengths will have L i G i b1 so that their steady state output power levels are zero. For a given loss element, the output power, the number of lasing wavelengths and spectrum are determined by the gain characteristics such as the small signal gain G, the saturation power P sat and the gain profile f i. Fig. 6(a) and (b) shows the number and the total power of a λ (nm) Fig. 4. The evolution of the output power profile of a multiwavelength laser with a saturable transmitter. The wavelength spacing is.8 nm. Every symbol represents one wavelength. In this simulation, there are 25 wavelengths at steady state. Fig. 3. However, since the loss now depends on the power at each wavelength, the loss in more than one wavelength can now be adjusted to balance the gain at that wavelength by varying the power in that wavelength. Thus the stable lasing condition of cavity gain equals loss can be satisfied in more than one wavelength, e.g. the -th to 12-th modes in Fig. 3. Multiple wavelength lasing is therefore possible. Fig. 4 shows the evolution of the output power spectrum of a multiwavelength laser with a saturable transmitter. Initially 51 wavelengths grow because their small signal gains are larger than the small signal loss. The power of the center wavelength will take most power at first but then drop due to the homogeneous gain saturation. The power in other wavelengths will increase slowly. Eventually 25 wavelengths survive at steady state. Thus the homogeneous gain medium can support multiwavelength CW operation in the presence of the saturable transmitter loss. We observed that the output power is different at different wavelengths because of the wavelength dependence of the gain. Fig. 5 shows the steady state gain (solid lines), loss (dashed lines) and the product of gain and loss (dotted lines) for the simulation parameters used in Fig. 3. From Fig. 5, we observed that the wavelengths at higher power will not only have higher gain (solid lines) but also experience higher losses (dashed lines) such that the gain loss balance condition of L i G i =1 is satisfied for more than one Wavelengths b P total (W) c 1 P sat G (db) 1.1 P sat G (db) 1.3 Gi Gain / Loss Li Gi*Li P total (W) 1.1 G o db db db 4 db 5 db 6 db λ (nm) Fig. 5. The steady state gain (solid line), loss (dashed line) and the product gain and loss (dotted line) for the simulation parameters used in Fig. 3. 1E P sat (W) Fig. 6. (a) Variation of the number of lasing channels with the small signal gain of the center wavelength for different values of P sat (in Watt). (b) Variation of the total power with the small signal gain of the center wavelength for different values of P sat (in Watt). (c) Variation of the total power with the saturation power of the gain medium for different values of small signal gain.

5 stable lasing CW wavelengths, respectively as a function of the gain G for different values of saturation power. The normalized gain profile f i is chosen to be a Gaussian function as before. From Fig. 6(a), we observed that the number of lasing wavelengths increases when G increases. Specifically, when G increases the small signal gain profile g f i increases, thus more modes can grow initially. When G reaches a threshold value, the number of stable lasing wavelengths will increase as shown in Fig. 6(a). The number of stable lasing wavelengths increases by two at every step because we have used a symmetric normalized gain profile. We note that the final number of stable lasing wavelengths depends on the nonlinear balance of the gain and loss in all the wavelengths that are reached simultaneously. Thus sufficient energy must be added to the cavity before new wavelengths can be supported in steady state. We also observed that for the same saturation power, it takes increasingly larger G to increase the stable lasing wavelengths by two, i.e. the step size in Fig. 6(a) increases as G increases. This is because the gain is Gaussian. For the same separation between wavelengths defined by the comb filter, it takes increasingly larger values of G to compensate the rapidly decreasing normalized gain profile f i. Finally, we observed that for the same G, the number of stable lasing wavelengths increases when saturation power P sat increases. When P sat increases, more power can be built up in the laser cavity before the gain is saturated, thus more stable wavelengths can be supported. Fig. 6(b) shows that the total power increases but the rate of increase decreases as G increases. There are however no discrete jumps in the total power as shown in Fig. 6(b). Thus when G increases, the number of lasing wavelengths initially remains the same but the total output power of the final stable lasing wavelengths increases. The extra energy input to the cavity at a larger G value increases the lasing power. When G continues to increase, a threshold is reached such that more wavelengths can be supported. Fig. 6(c) shows the total output power of the laser as a function of the saturation power P sat for different small signal gains G. We found that the variation of total power P total with P sat obeys a power law for all the values of G simulated. Next we studied the dependence of the output laser power spectrum on the gain profile. We found that for the saturable transmitter given in Eqs. (2) and (3), the output power spectrum profiles are similar to the gain profiles for small to moderate values of the saturation power P sat. Fig. 7(a) shows the output power spectra and their curve fitting results for different values of P sat and different order supergaussian functions f i =exp[ (λ i λ c ) 2n /Δ 2n ] chosen as the gain profiles where n is the order of the supergaussian function. Note that the Gaussian function is given by n=1. The symbols are the simulated output power spectral profiles and the lines are the curve fitting results by using the same order of supergaussian functions as that used for gain profiles. The curve fittings are performed on the higher power part of the output spectrum because the output power spectra drop to zero when the gain is less than the loss, but the supergaussian functions used for the gain profiles approach but never equal to zero. Thus the supergaussian functions will not be able to fit the whole of the laser output spectra including the wavelengths with zero output. The deviation between the output spectra and the supergaussian functions will become more significant near the edge of the output spectra. The results from fitting the whole non-zero part of the spectra would be skewed by this difference. We therefore only fit the top half of the spectrum between the peak and the half maximum power points. The simulated results show that the output power spectra resemble closely the gain profiles for small to moderate values of P sat. Fig. 7(b) shows the quality of the curve fitting results for the power output spectra of the gain profiles using the supergaussian function at different orders. The vertical axis is 1 R 2 [24] which gives the difference between the fitted curve and the upper half of the spectra. A zero value of 1 R 2 means that the fitted curve matches exactly the output power spectrum. We F. Li et al. / Optics Communications 284 (11) a Power (W) b 1-R E-3 1E-4 1E-5 Order, P sat 1, dbm 1, dbm 2, dbm 5, 25 dbm, dbm Order P sat (dbm) 2331 Fig. 7. (a) The output spectra and curve-fitted spectra with gain profiles using different order supergaussian functions and P sat. The symbols are the output spectra and the lines are the curve-fitting results. (b) Goodness of the curve fitting results for different order supergaussian gain profiles at different P sat. observed that the curve fitting has R 2 N.99 for all values of P sat b4 dbm. The resemblance increases with the order of the supergaussian functions used. For high P sat (N4 dbm), the resemblance between the laser output power spectrum and the gain profile decreases when P sat increases. We also observed discrete jumps on the 1 R 2 curves for all the supergaussian functions studied. These jumps coincide with the increase in the number of output wavelength channels in the upper half of the output spectra. When P sat increases, the power in all the wavelengths, hence the total output power, increases as shown in Fig. 6(c). The curve fitting results improve indicating that the higher power part of the output power spectra resembles the gain profile closer than the lower power part. When P sat increases further, new wavelengths enter the curve fitting region, i.e. the upper half of the output spectra, from the lower half of the spectra. Since these new wavelengths are located at the lower power part of the curve fitting region, the curve fitting performance decreases. Thus these jumps are the results of fitting a fixed portion of the laser output power profiles. If we choose to curve fit a fixed number of output wavelengths instead, these jumps will be removed. However, the portion of output power curve represented by the same number of output wavelengths is different for different values of P sat. Refs. [17,18] reported that a very flat output power spectrum is possible for lasers with homogeneous gain and intensity dependent loss. The authors of the article suggested that it was not clear whether the ultra-flat output spectrum was the result of the gain dynamics or

6 2332 F. Li et al. / Optics Communications 284 (11) the intensity-dependent loss property. However, Fig. 7 definitively shows that the output laser spectrum depends on the gain profile at lower or moderate power, thus implying that a flat gain profile will also lead to a flat laser output spectrum. For EDFA, the homogeneous gain used in Refs. [17,18], because of the imbalance of the absorption and emission cross sections of the erbium ions in the fiber, the gain coefficient will change with the pump power and signal power. If the population density of excited state N 2 is close to the population density of ground state N 1, there will be a wide flat region between 155 and 16 nm on the gain coefficient profile [16]. But the gain profile will not be flat if the required population inversion ratio is not maintained. Furthermore, the measured laser output spectrum has sharp edges in Refs. [17,18] but the gain profiles have very gentle edges near the flat region. In the next section, we will show that a laser with homogeneous loss and intensity dependent loss can have an ultraflat multiwavelength output spectrum even if the gain is not flat. 4. Periodic intensity-dependent loss In Section 3, we show that a laser with homogeneous gain broadening can have multiwavelength CW output if a saturable transmitter is present in the cavity. The saturable transmitter has a monotonic output versus input power relationship. In this section, we will study the laser behavior when the response of the intensitydependent loss is not monotonic, but a simple sinusoidal function given by T i ðp i Þ = a + b cosðγp i + ϕþ ð7þ where a, b, γ, and ϕ are constants, a+b 1 and a b. It is difficult to combine the loss function in Eq. (7) and the gain model in Eq. (1) to form a master equation without making drastic approximations of the loss function. However, it is straight forward to use a discrete model in which the gain and loss elements are treated independently. The full sinusoidal behaviors of the loss function can therefore be retained. It should be noted that Eq. (7) describes the transmission characteristics of a NOLM for both single and multiple wavelength operations. In our analysis, we consider the NOLM without birefringence since our simulations are based on continuous wave assumption. Specifically, when a signal propagates in a NOLM, the signals at other wavelengths will affect the phase of the signal via cross phase modulation (XPM). Thus the co-propagating and counter-propagating signals at other wavelengths will affect the phase by the same factor. So the signals propagating clockwise and counterclockwise will see the same phase changes due to XPM. Because the transmission will only be affected by the phase difference of the clockwise and counterclockwise signals at the same wavelength, the XPM induced phase changes by other channels will be cancelled, implying that the transmission of the NOLM only depends on individual channels and can be considered as inhomogeneous. For simplicity, we assume in the simulations that a=.5, b=.4 and γ=1 in Section 4. The output power versus the input power curve of periodic loss with ϕ= is shown in Fig. 2 as a solid line. Different to the monotonic response of the saturable transmitter, the output power of periodic loss varies periodically when the input power increases. The characteristics of the periodic loss at low input power depend on the choice of the initial phase ϕ in Eq. (7). At low input power, the transmission decreases with the input power for ϕbπ but increases with the input power for π ϕb2π. Thus at low input power the intensity-dependent loss behaves like a saturable transmitter for ϕbπ but like a saturable absorber for π ϕb2π. Following the discussion in Section 3, multiple CW output will be possible if the initial phase ϕ in Eq. (7) is chosen such that ϕbπ.an important characteristic of the periodic loss is that it has a turning point that corresponds to a peak on the input output curve differing from the saturable transmitter. In the following simulations, for comparison the gain profile is chosen to be a Gaussian function as defined in Eqs. (5) and (6). The small signal gain parameter is chosen to be G = db Ultra-flat output power spectrum Fig. 8(a) shows the output spectra with periodic loss and homogeneous gain for different saturation powers. The phase shift ϕ= in simulations. The laser with periodic loss and homogeneous gain can give multiwavelength CW output similar to the laser with a saturable transmitter. While the saturation power increases, the number of the lasing channels and output power increases similar to that of the saturable transmitter. However, we observed that the output power spectrum is flattened as the saturation power increases. Fig. 8(b) shows the output power spectra of the laser with periodic loss (squares) and saturable transmitter (circles) at saturation power P sat =3. W. The spectrum of the laser with periodic loss is very flat while that of the saturable transmitter resembles that of the gain profile. It is also observed that the power at the center wavelength for the periodic loss is slightly lower than that of its adjacent channels. We simulate the laser behavior using the same gain profiles and saturation power as we have done in Section 3 for the saturable transmitter. We observed that while the output power spectra of the saturable transmitter are similar to the gain profiles for the cases studied, the spectra of the periodic loss can differ from the gain profiles significantly. The flat output power spectrum generated by the laser with periodic loss is due to the existence of a peak in the input output power curve in the periodic loss. Fig. 8(c) shows the operation points of the lasing wavelengths on the input output power curve of the periodic loss and the saturable transmitter when the laser is at steady state. The solid and dashed lines represent the input output power curves of the periodic loss and the saturable transmitter respectively. The squares and circles represent the operation points of the lasing wavelengths for a laser with periodic loss and P sat =.1 and 3. W respectively. The operation points of the lasing wavelengths for the saturable transmitter (triangles) are also plotted for comparison. For the periodic loss at small saturation power (Fig. 8(c)), the operation points of the lasing wavelengths spread out near the low power part of the input output power curve. Thus the output power profile is not flat. Because we used a Gaussian gain profile, the center wavelength receives the highest gain and hence has the largest output power. Thus the leading operation point on the input output power curve, i.e. the one with the highest input power, corresponds to that of the center wavelength. The operation points of the rest of the wavelengths are ordered according to the separation of these wavelengths from the center wavelength. Wavelengths closer to the center wavelength will have their operation points closer to that of the center wavelength. This ordering of the operation points on the input output power curve of the loss element will not be altered by changing the saturation power. When the saturation power increases, the total power emitted by the laser increases. The ordered operation points of the lasing wavelengths migrate up the input output power curve towards the peak. Since loss increases with power, when the power of a wavelength increases, the effective gain it experiences decreases. The migration of the operation point of a wavelength up the input output power curve therefore slows down as its power increases resulting in a clustering of the operation points in the front of the ordered group. Because of the clustering, the output powers of the operation points in the front become closer to each other, hence the central portion of the output power spectrum becomes flattened. The clustering effect increases as the operation points approach the peak of the input output power curve. Very flat output spectrum can be obtained if the saturation power is chosen such that most of the operation points cluster around the peak. However, it should be noted that two wavelengths which are not symmetrically located with respect to the center wavelength cannot have the same operation

7 F. Li et al. / Optics Communications 284 (11) a Power (W) b P sat.5 W W.1 W.2 W 3. W wavelength can therefore be lower than that of the other operation points, i.e. its adjacent wavelengths, as shown in Fig. 8(a) and (b), even though it has a high small signal gain coefficient. The input output power curve of periodic loss can be used to determine the maximum output power at a wavelength. From Fig. 8(c), the maximum output power of a wavelength is.8 W which is the output power at the peak of the curve. For the saturable transmitter in Eq. (4), although the loss increases with power, the output power increases monotonically with the input power and the input output power curve does not contain a peak. As the saturation power increases, the operation points of the lasing wavelengths migrate up the input output power curve. Clustering of the operation points also occurs and the output power spectrum is flattened. The extent of the spectral flattening depends on the rate of decrease of the slope of the input output power curve Gain variations So far we have used a simple Gaussian function to model the gain profile. In this section, we consider gain profiles with multiple gain peaks and show that the intensity-dependent loss can still give very uniform multiwavelength output spectra. As an illustration, we use a gain profile with three gain peaks given by Power (W) c Output power (W) with P sat =.1 W with P sat = 3. W with P sat = 3. W Input power (W) Fig. 8. (a) The output spectra with periodic loss at different saturation powers. (b) The output spectra with periodic loss and saturable transmitter at P sat =3. W. (c) The operation points of the periodic loss and saturable transmitter at P sat =.1 and 3 W. point in the input output power curve in Fig. 8(c) because they experience different gains as a result of the gain profile used. When the saturation power increases further, the leading operation point, i.e. the center wavelength, will move pass the peak of the input output power curve. The output power of the center f i = e ð λ i λ c Þ 2 = Δ 2 + e ð λ i λ c1 Þ 2 = Δ 2 + e ð λ i λ c2 Þ 2 = Δ 2 ð8þ where λ c, λ c1, λ c2 and Δ are constants. In the simulations, we have chosen λ c =155 nm, λ c1 =15 nm, and λ c2 =157 nm. The small signal gain fluctuation is defined as the ratio of the maximum gain to the minimum of the local minimum gain of the small signal gain profile. The output power fluctuation is similarly defined. From Eq. (8), the small signal gain fluctuation can be varied by varying Δ. For example the gain fluctuation will be db if Δ=9.66 nm as shown in Fig. 9(a) (squares). We simulated the steady state laser output for intensity dependent loss using the saturable transmitter and the periodic loss. For the saturable transmitter, we assumed L =.9 and α 1 /α = W 1 in the simulations as in Section 3. For the periodic loss, we assumed a=.5, b=.4, γ=1 and ϕ=. The gain fluctuation is chosen to be db. Fig. 9 shows the output power spectra with the saturable transmitter and periodic loss. Fig. 9(a), (b) and (c) corresponds to saturation powers of.5, 1 and W respectively. The solid lines with squares represent the small signal gain profile. The dashed lines with circles and the dotted lines with triangles represent results for the saturable transmitter and periodic loss respectively. The output power spectra are normalized with respect to the peak power and shown in db on the vertical axis to the right hand sides of the figures. We observed that the laser operates in multiple CW wavelengths with both saturable transmitters and periodic loss. The output wavelengths cover the whole gain spectrum. When P sat =.5 W, the output power spectrum for the saturable transmitter has a fluctuation of 2.74 db while that for periodic loss is 5.96 db. When P sat is increased to 1 W, the output power fluctuations of the saturable transmitter and periodic loss decrease to 2.27 db and 2.14 db respectively. For large P sat of W, the output power fluctuation of the saturable transmitter is 1.33 db while that of the periodic loss is only.12 db. Thus the output power fluctuation decreases when the saturation power increases for both the saturable transmitter and the periodic loss. However, the periodic loss can have a much more uniform output power spectrum. The change in the output power fluctuation of the saturable transmitter and periodic loss with the saturation power can be explained by inspecting the operation points of the laser on the respective input output power curves as shown in Fig.. The dashed and solid lines are the input output power curves for the saturable transmitter and the periodic loss respectively. Squares, circles and

8 2334 F. Li et al. / Optics Communications 284 (11) a Small Signal Gain (db) b Small Signal Gain (db) c Small Signal Gain (db) 4 Gain Gain Gain Fig. 9. The output spectra with saturable transmitter (circles) and periodic loss (triangles). The small signal gain profile (squares) is also plotted for comparison. The saturation powers are.5,, and W in (a), (b) and (c) respectively. triangles represent the operation points with P sat =.5,, and 1 W respectively. Since the laser operates in many wavelengths, for clarity we only plot the operation points corresponding to wavelengths with the maximum (λ max ) and the minimum (λ min ) input powers for each choice of saturation power. All other operation points between the Relative Power (db) Relative Power (db) Relative Power (db) Output Power (W) W W 1 W Input Power (W) Fig.. The operation points on the input output power curves of periodic loss (solid line) and saturable transmitter (dashed line) for different saturation powers. The gain of the laser contains multiple gain peaks given in Eq. (8). Squares, circles and triangles represent the operation points with P sat =.5,, and 1 W respectively. We only plot the operation points of the wavelengths with the maximum and the minimum input powers for each choice of saturation power. wavelengths λ min and λ max lie within the pair of symbols plotted. In dbs, the output power fluctuations are RP db =log P max P min where P max and P min are the maximum and minimum output powers between the wavelengths λ min and λ max. From Fig., the slope of the input output power curve of the periodic loss is larger than that of the saturable transmitter at low power, thus the operation points in the laser with periodic loss spread out more and hence have a larger output power fluctuation. When the saturation power increases, the operation points move up the input output power curves and the spread of the operation points of the periodic loss decreases. At P sat = W, the laser with periodic loss operates near the peak of the input output power curve, hence the output power fluctuation is much smaller than that of the saturable transmitter. Fig. 11 shows the variations of the output power fluctuation with saturation power for different small signal gain profiles and intensitydependent loss. The curves with open symbols represent the gain fluctuations of periodic loss for the small signal gain fluctuation from 5 Output power fluctuations (db) 1 db with 5 db.1 db 15 db db 25 db P sat (W) Fig. 11. The output power fluctuations for saturable transmitter (solid symbols) and periodic loss (open symbols). The different curves marked with open symbols represent different small signal gain fluctuations from 5 to 25 db. ð9þ

9 F. Li et al. / Optics Communications 284 (11) to 25 db as shown in Fig. 11. The gain fluctuation is varied by varying the parameter Δ in Eq. (8). The curve with solid circles represents the results of saturable transmitter. From Fig. 11, the output power fluctuation with periodic transmitter is less than 2 db if the small signal gain fluctuation and the saturation power are 5 db and 7.8 W respectively. Note that the output power fluctuation will not go to zero for a given gain fluctuation because two wavelengths cannot be colocated on the same operation point of the input output power curve if they experience a different small signal gain. Thus unless the gain profile is uniform, the output power spectrum will not be absolutely flat. For the periodic loss at a given small signal gain fluctuation, the output power fluctuation has a minimum corresponding to the laser working near the peak of the input output power curve. When the operation points of the wavelengths close to the center wavelength move beyond the peak of the input output power curve, the output power fluctuations begin to increase because the output power of these wavelengths decreases when the input power increases beyond the peak on the input output power curve. Fig. 11 also shows that when the small signal gain fluctuation increases, the output power fluctuation also increases. We also note that larger saturation power is required for the laser to operate on the flat output spectrum regime. When the gain fluctuation increases beyond db, the laser can no longer give uniform output power spectrum even when the saturation power is very large. For example, at a gain fluctuation of 25 db, the laser becomes unstable at large saturation power. For comparison, the output power fluctuation of a laser with the saturable transmitter decreases gradually when the saturation power increases but never reaches a minimum as shown in Fig Bias point of the periodic loss As discussed in the beginning of this section, the characteristics of the periodic loss in Eq. (7) depend on the phase bias ϕ. The intensitydependent loss behaves like a saturable transmitter for ϕbπ and a saturable absorber for π ϕb2π. The phase bias ϕ can be varied by adjusting the polarization controllers with a nonlinear optical loop mirror. In this subsection, we will study the effect of changing the phase bias ϕ of the transmission of the periodic loss on the laser output characteristics. To study the effect of the phase bias ϕ in the intensity-dependent loss, we vary ϕ in Eq. (7) from π/2 to π/2. For the saturable gain, we will use the simple Gaussian function Eqs. (5) and (6) for the gain profile. The small signal gain at the center wavelength is db and the saturation power is 5 W. For the periodic loss, we used the same parameters as that used for Fig. 8 except for the phase bias ϕ. Fig. 12 shows the laser output spectra for different phase biases. We observe that when the phase bias varies from π/2 to π/2, the average output power level of the lasing wavelengths drops. The spectra for phase bias ϕb have very sharp edges when compared to those for ϕn. When ϕb the intensity-dependent loss behaves like a saturable absorber for P i b ϕ /γ, i.e. the transmission of a wavelength increases when the power of the wavelength increases. Because of the positive feedback effect of the saturable absorber, the wavelengths with gain larger than loss will grow faster than those in the case for intensityindependent loss. Thus the wavelengths with higher power can suppress the growth of wavelengths with lower power by gain saturation more effectively. The process continues until the power in a wavelength is larger than ϕ /γ when the characteristics of the intensity-dependent loss change from being a saturable absorber to a saturable transmitter. Consequently the loss in that wavelength will begin to increase with its power. The loss of the saturable transmitter then dynamically balances the gain in different wavelengths to enable the laser to operate in multiple wavelengths at steady state. Since no wavelengths can operate at P i b ϕ /γ, the steady state requirement is P i N ϕ /γ which thus produces the sharp edges of the output power spectra. Fig. 13 shows the operation points of the wavelengths on the Power (W) Fig. 12. The output power spectra with periodic loss. The phase bias varies from π/2 to π/2. The gain profile is a Gaussian function given by Eqs. (5) and (6). The small signal gain at the center wavelength is db and the saturation power is 5 W. The parameters for the periodic loss are the same as that used for Fig. 8 except for the phase bias ϕ. input output power curves for ϕ= π/6 (circles) and π/6 (squares). In the case ϕ= π/6, there are no operation points located at bp i b ϕ /γ. For ϕ=π/6, the operation points begin from near P i =. There are no threshold powers. Fig. 14 shows the variation of power of the center wavelength (squares) and the number of wavelengths at steady states (circles) versus the phase bias. We observe that the power of the center wavelength decreases when the phase bias varies from π/2 to π/2 but the number of wavelengths has a maximum near ϕ= and decreases when ϕ increases from zero. For ϕb, the threshold ϕ /γ increases when the magnitude of the phase bias ϕ increases. The steady state wavelengths' power will therefore also increase. For the same saturation power the number of wavelengths at steady state then decreases. For ϕ N, the transmission decreases when ϕ increases, thus the loss experienced by each wavelength increases. The steady state wavelengths' power therefore also decreases. For the same saturation power the number of wavelengths at steady state will also decrease. Thus one can control the number of wavelengths and the power of the wavelengths by adjusting the phase bias of the transmission of the periodic loss. Output power (W) Input power (W) Fig. 13. The operation points of lasers on the input output power of the periodic loss for the phase bias of π/6 (circles) and π/6 (squares). The parameters for the gain and loss are the same as that in Fig. 12.

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