Theoretical analysis of spectrum flattening in fiber laser oscillator

Optics Communications 383 (2017) 460–466

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Theoretical analysis of spectrum flattening in fiber laser oscillator Chen Shi a,b,c, Xiaolin Wang a,b,c,n, Pu Zhou a,b,c, Zefeng Wang a,b,c, Xiaojun Xu a,b,c, Qisheng Lu a,b,c a

College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, PR China Hunan Provincial Key Laboratory of High Energy Laser Technology, PR China c Hunan Provincial Collaborative Innovation Center of High Power Fiber Laser, PR China b

art ic l e i nf o

a b s t r a c t

Article history: Received 7 April 2016 Received in revised form 31 August 2016 Accepted 14 September 2016

The flatness of laser spectrum is important in many applications. In this manuscript, a method of acquiring flattened spectrum directly from a fiber oscillator by optimizing the reflective spectrum of Fiber Bragg Gratings (FBG) was demonstrated and optimized result at wavelength around 1064 nm and 1080 nm was presented. An optimization path to alter the reflectivity of FBGs using greedy algorithm was interpreted by analyzing the single-trip gain inside the resonant cavity. Our method has a guiding significance of controlling the output spectrum of laser oscillator using FBGs. & 2016 Elsevier B.V. All rights reserved.

Keywords: Fiber laser Fiber Bragg grating Optimization Greedy algorithm Spectrum flattening

1. Introduction Because of the high slope efficiency and compactness, fiber lasers are widely used in the field of laser marking, material processing, medical, communication and many industrial applications [1,2]. Multi-wavelength laser has already been widely used in many applications in commercial and medical area [3–5]. In multiwavelength system, the flatness of spectrum can greatly affect the characteristics of the whole system. This is because the unequalization of spectrum creates undesirable nonlinear effects, such as self-phase, cross-phase modulation, wave mixing and nonelastic scatterings, which can potentially hamper the performance of the system [6]. In general, spectrum flattening is usually acquired outside a laser resonate cavity. One common method is to use filters, including long-period fiber grating, tunable acousticoptical modulator (AOM), Sagnac loop filter, etc. [6–9]. The flat spectrum can also be achieved by fiber Raman amplifier with multi-pumping and different kinds of algorithms, such as particle swarm optimization (PSO), shooting algorithm and artificial fish school algorithm, has been studied in order to find the optimal pump parameters [10–12]. In this manuscript, we presented a simulation work on acquiring flattened spectrum directly from a single laser oscillator by n Corresponding author at: College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, PR China. E-mail address: [email protected] (X. Wang).

http://dx.doi.org/10.1016/j.optcom.2016.09.029 0030-4018/& 2016 Elsevier B.V. All rights reserved.

optimizing reflective spectrum of output-coupling (OC) FBG. The appropriate reflective spectrum of OC FBG for generation of flattened spectrum was theoretically found out using our method which based on a preliminary model. Our result has two meanings: (1) Our method flattens the spectrum inside the laser cavity, which means higher efficiency compared with filters and can provide a possible solution for building compact sources for Dense Wavelength Division Multiplexing (DWDM) system; (2) Our method provide a way to control the output spectrum from laser oscillator using FBGs. It can also be used to compensate the spectrum distortion caused by gain difference in amplifier. This work might have a guiding significance to FBG design.

2. Configuration and theoretical model The typical configuration of a fiber laser oscillator is schematically shown in Fig. 1. HR FBG stands for high reflectivity fiber Bragg grating while OC FBG stands for output coupling fiber Bragg grating. These two FBGs act as cavity mirrors in fiberized laser oscillators. The fiber between two FBGs is a piece of rare-earthdoped fiber which acts as gain medium inside the cavity and YDDCF is short for ytterbium-doped double-clad fiber. The forward and backward pump are coupled into different sides of resonant cavity using combiners. The output fiber end is angle-cleaved in order to eliminate facet reflection. Both the forward and backward pump laser diodes are wideband and thus divided into P channels with central wavelength λp,

C. Shi et al. / Optics Communications 383 (2017) 460–466

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Fig. 1. Typical configuration of a fiber laser oscillator.

Fig. 2. Simulation configuration of fiber inside cavity.

p ¼1, 2, …, P, and even spacing of Δλp. The signal and amplified spontaneous emission (ASE) is also divided into K channels, the central wavelength of each channel is λk, k¼1, 2, …, K, the wavelength difference between channels border upon each other is Δλk. Suppose that the fiber length inside the resonant cavity is L, obviously, it consists of two passive and one active fiber parts. We separate the whole fiber length into M small pieces as shown in Fig. 2. N0 represents the doping concentration of each part of the fiber. For the purpose of considering the whole length of fiber uniformly, we set N0 as a function of position coordinate z. Based on all the assumptions above, we can obtain the steady-state two level rate equation by simplifying the model given in Ref. [13].

η( z ) =

±

±

±

dPp± dz

∑n Γnλ nσa( λ n)⎡⎣ Pn+( z ) + Pn−( z )⎤⎦ hcA τ

+ ∑n Γnλ n⎡⎣ σa( λ n) + σe( λ n)⎤⎦⎡⎣ Pn+( z ) + Pn−( z )⎤⎦

Fig. 3. Absorption (dashed) and emission (solid) cross section of Yb-ions.

= ΓpN0⎡⎣ σe λ p η − σa λ p ( 1 − η)⎤⎦Pp± − αpPp±

( )

(1)

( )

(2)

dPk± hc 2 = ΓkN0⎡⎣ σe( λk )η − σa( λk )( 1 − η)⎤⎦Pk± − αkPk± + σe( λk )N0η 3 Δλ dz λk (3)

dPR±j dz

⎡ ⎤ = ΓR jN0⎣⎢ σe λ R j η − σa λ R j ( 1 − η) − αR j⎦⎥PR±j

( )

−

+

λ Rj +1

gR , R

λ Rj gR

j − 1, R j

Aeff

j

j +1

Aeff

(P

( )

(P

+ Rj −1

+ Rj +1

)

+ PR−j + 1 PR±j

)

( )

+ PR−j − 1 PR±j + σe λ R j N0η

hc 2 λ R3j

A is core area of active fiber, Гx (x ¼p, k, Rj) are overlapping factors between light and doping area of active fiber (p ¼pump, k¼ signal, Rj ¼Raman). αp, αk and αRj are attenuate coefficient of fiber to pump, signal and Raman Stokes wave, respectively. τ is the upper-level particle lifetime. It is worth mentioning that because the multi-phonon transition is usually strong if the gain medium has a high phonon energy, this two-level model also works for erbium doped fibers [15], which means our model can be easily generalized to 1.55 μm wavelength. The boundary conditions of resonant cavity are + − ⎧ ⎪ Pk ( 0) = Pk ( 0) × RHR( k ) ⎨ − + ⎪ ⎩ Pk ( L ) = Pk ( L ) × ROC ( k )

Δλ (4)

where η represents the upper-level population ratio as a function of coordinate z, and η(z)N0(z), [1  η(z)]N0(z) are the upper-level and ground population at position z, respectively. Pp(z) is the pump power of channel p, Pk(z) is the signal power of channel k and PRj(z) is the power of j-th order Raman Stokes wave (note that j¼0 corresponds to signal, 7 corresponds to forward and backward direction propagations, respectively). gRj,Rj þ 1 is Raman gain between j-th and (jþ 1)-th order Raman Stokes. h is the Planck's constant, c is the light velocity in vacuum, sa and se are absorption and emission cross section of ytterbium ions, respectively. Fig. 3 shows the sa and se of Yb-ions [14].

(5)

where RHR and ROC are reflectivity of HR FBG and OC FBG, respectively. The reflective spectrum of FBGs is also divided into different channels that corresponds to channels of signal powers, Table 1 Key parameters used in simulation. Item

Value(s)

Item

Value(s)

sa se τ A

see Fig. 3 see Fig. 3 8.4  10  4 s 7.85  10  11 m2

αk Гp Гk, ΓRj L

1.5  10  3 6.4  10  3 0.88 10 m

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C. Shi et al. / Optics Communications 383 (2017) 460–466

Fig. 4. Calculated spectrum of fiber oscillator; Inset: corresponding HR and OC FBG reflectivity.

and R(k) means the corresponding reflectivity to channel k of signal power. Obviously, R(k) of different channel could be distinct to each other. The output signal power is thus obtained by

Pout =

∑k Pk+( L)ROC ( k)

(6)

Through solving the differential equations above, we can numerically investigate the changes of output spectrum along with the changes in reflectivity of FBGs in order to construct reflective spectrum for generating flattened output spectrum. Because our simulation is running under a relatively low power level, the

Fig. 6. Calculated single-trip gain inside the cavity. Inset: single-trip gain detail range from 1060–1070 nm.

Raman scattering will not affect the result significantly and we can also safely ignore stimulated Brillioun scattering in such wide spectrum. Some key parameters used in simulation are listed in Table 1.

3. Intracavity gain estimation Before we start optimizing FBGs to generate flattened output spectrum, we have to consider what makes spectrum uneven and

Fig. 5. Changes of magnitude difference between Kk and Ck/P along with variations of P; (a) P¼ 1 nW (b) P ¼10 nW (c) P¼ 100 nW (d) P¼ 1 mW. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).

C. Shi et al. / Optics Communications 383 (2017) 460–466

463

Fig. 7. An output spectrum that is under optimization procedure around 1064 nm.

Table 2 Parameters used in optimization. Item

Value(s)

OPsR OPdBdiff OPBW Ac

5% 3 dB 5 nm 0.4%

rugged. Actually, if we set the reflective spectrum of FBGs to rectangular functions as shown in the inset of Fig. 4, what we get is an uneven output spectrum. The reason is simply because different channel has different gain inside the resonant cavity. In this section, we will discuss how to estimate intracavity gain for different channels. In order to investigate intracavity gain, we have to concentrate on the Eq. (3) which describes the variation of signal power. Through observing Eq. (3), we can find that the derivative of signal power consists of two parts: P(z) related part and P(z) independent part. We define two functions below:

Kk( z ) = ΓkN0⎡⎣ σe( λk )η( z ) − σa( λk )( 1 − η( z ))⎤⎦ − αk

Ck( z ) = σe( λk )N0( z )η( z )

hc 2 λk3

(7)

(8)

(9)

In order to obtain a simple form of Eq. (9) to estimate intracavity gain, we use exponential descendent method to process Eq. (9). We set forward direction as an example. First of all, suppose we already have the value of Pk þ at coordinate zm, we can calculate the value at next position from a recurrence relation:

Pk ( z m + 1) = Pk ( z m) +

∫z

zm+1 m

⎡⎣ K ( z )P +( z ) k k

+ Ck( z )⎤⎦dz

(10)

We create an exponential function to fit function Pk þ (z) at coordinate zm:

F ( z ) = Ae

Bz

⎧ F ( z ) = P +( z ) m k m ⎨ + ⎪ ⎩ F ′( z m) = Kk( z m)Pk ( z m) + Ck( z m)

(12)

Solving equation set (12), we get:

dP ±( z ) ± k = Kk( z )Pk±( z ) + Ck( z ) dz

+

where A and B are indeterminate constant functions which fulfill the relations below: ⎪

Δλk

Then Eq. (3) can be written as:

+

Fig. 8. The greedy algorithm flow chart.

(11)

⎛ ⎧ C (z ) ⎞ −⎜⎜ Kk ( z m)+ k+ m ⎟⎟z m ⎪ + Pk ( z m) ⎠ ⎝ A = P ( z ) e k m ⎪ ⎪ Ck( z m) ⎨ ⎪ B = Kk( z m) + P +( z ) k m ⎪ + ⎪ ⎩ Pk ( z m) ≠ 0

(13)

Put A and B into Eq. (11), then solve the derivation of F(z), we get: ⎡ ⎤ ⎧ ⎡ ⎤ ⎢ K ( z )+ Ck( z m) ⎥( z − z m) ⎪ ⎪ F ′( z ) = P +( z )⎢ K ( z ) + Ck( z m) ⎥e⎢⎣ k m Pk+( z m) ⎥⎦ k m k m ⎨ Pk +( z m) ⎦ ⎣ ⎪ + ⎪ ⎩ Pk ( z m) ≠ 0

(14)

Because Pk þ (z) is fitted by function F(z) at coordinate zm, we can

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C. Shi et al. / Optics Communications 383 (2017) 460–466

Fig. 9. Optimization result around 1064 nm. (a) Optimized OC-FBG reflective spectrum; (b) Output spectrum; (c) changes of OC FBG reflective spectrum during optimization procedure.

Fig. 10. Optimization result around 1080 nm. (a) Optimized OC-FBG reflective spectrum; (b) Output spectrum.

put Eq. (14) back into Eq. (10) to replace the function inside the integral sign:

Pk +( z m + 1) = Pk +( z m) +

∫z

zm+1

F ′( z )dz

m

(15)

Through a simplification of Eq. (15), we get a recurrence expression of Pk þ (z): ⎡ ⎧ C (z ) ⎤ ⎢ Kk ( z m)+ k m ⎥dz ⎪ + + + ⎢⎣ P ⎦ k ( z m) ⎥ ⎨ Pk ( z m + 1) = Pk ( z m)e ⎪ + ⎩ Pk ( z m) ≠ 0

in gain process of signal power. In this case, we ignore Ck(zm)/ Pk þ (zm) in Eq. (16). Finally, we get an extremely simple recurrence relation to describe intracavity gain of each signal channel:

⎧ + ⎪ P (z ) = Pk +( z m)e Kk( z m)dz ⎨ k m+1 ⎪ + ⎩ Pk ( z m) ≠ 0

(17)

The steady state single-trip gain inside the cavity can be expressed as follow: M

(16)

In order to make the recurrence relation more convenient to calculate, we now discuss the magnitude of Kk(zm) and Ck(zm)/ Pk þ (zm). Typically, Pk þ (zm) ranges from 10  9 to 103 W. The variations of Kk(zm) and Pk þ (zm)/Ck(zm) with the changes in Pk þ (zm) are plotted in Fig. 5. In Fig. 5, the blue line (solid) represents value of Kk(zm) while the green line (dashed) represents value of Ck(zm)/Pk þ (zm). As we can see in the figure, with the growth of Pk þ (zm), Ck(zm)/Pk þ (zm) tends approximate to 0. The signal power will higher than 0.1 mW at steady state under normal circumstances, so Kk(zm) is dominate

ln( Gk ) =

∑ m=1

Kk( z m)dz

(18)

Using Eq. (18), we can calculate single-trip gain of each channel when system reached steady state. For instance, suppose a 10 m 10/125 YDDC fiber pumped forwardly by a 25 W multimode 915 nm LD source, the HR and OC FBG reflectivity is 99.9% and 11%, respectively, the single-trip gain of each channel can be calculated and plotted in Fig. 6. The inset figure of Fig. 6 is a detailed gain curve ranges from 1060 to 1070 nm. Compare the inset figure with the spectrum in Fig. 4, we can find that two curves have the same variation trends,

C. Shi et al. / Optics Communications 383 (2017) 460–466

and that indicates intracavity single-trip gain has direct proportion with output intensities of each channel under same optical feedbacks provided by the cavity.

4. Optimization using the greedy algorithm and results When the laser oscillates in the resonant cavity, the single-trip gain and the intensity of optical feedback determinate the ability to extract energy from the cavity of each channel together. We can get the relationship under steady state:

Gk _ round = Gk2ROC

(19)

where Gk_round represents the round-trip gain inside the cavity. As we analyzed in Section 3, Gk is determined by the instinct properties of Yb3 þ ions. So Eq. (19) instructs us to balance the gain of different channels by changing reflectivity spectrum of outputcoupling FBG while we suppose the reflectivity of HR FBG is 99.9% to all wavelengths. Based on the theoretical analysis above, greedy algorithm is employed in order to obtain the flat-top pattern spectrum. Greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum [16]. Because the greedy algorithm is easy to realize and understand and it will do produce the optimum or near-optimum solution in most cases, we choose it as the optimize algorithm while acquiring flat-top spectrum. It usually takes four steps when using the greedy algorithm [17]. First of all, we should build up a mathematical model for a specific problem. For our problem, we are using the steady-state two-level rate equation model to describe the whole oscillator system. The second step is to separate the problem into sub problems in a smaller scale. In our specific problem, our aim is to adjust reflectivity of every channels to make output spectrum flattened. Then the sub problems should be the adjustment of each independent channels. The third step is to solve every sub problem and get every locally optimal choice, and this is the key step in our manuscript. Before we adjust reflectivity of every channel, we should set up several parameters to describe the standards and error tolerance to the optimization procedure. In the first place, we should set a reflectivity baseline to OC-FBG, a targeting bandwidth of output spectrum and a flatness tolerance, and these parameters are denoted by OPsR, OPBW and OPdBdiff, respectively. All reflectivity of channels in the range of OPBW should be set to OPsR in the first time. The baseline is the starting point of optimization. Then, we should rearrange the order of channels according to Gk that calculated by solving Eqs. (1)–(3) using current conditions. We find a channel among all the unoptimized channels that has the maximal Gk to be the next sub problem. We are on the basis of that channels which has higher Gk is more sensitive to the variation of reflectivity and it may affect whole shape of output spectrum and make optimization procedure unstable. Fig. 7 plotted a half-optimized output spectrum that can help us to understand the whole procedure more easily. As we can see in Fig. 7, the targeting spectrum bandwidth is 5 nm and the central wavelength is 1064 nm. The spectrum is obviously divided into two parts: optimized part and unoptimized part. Our next sub problem is to make another channel flattened with optimized channels. As we analyzed above, our optimization sequence is sorted by Gk. Considering about Eq. (19), we can infer that the reflectivity of next channel will be greater or equal to the reflectivity of previous channel. So the optimization starting point of next sub problem is the reflectivity of previous sub problem. Now we have our sub problem and we have an optimization

465

starting point, now we need an optimization direction: the adjust standard. It is very hard to make a wide band spectrum absolutely flat. As shown in Fig. 7, the optimized part of spectrum is also serrated by a small amplitude. We define the arithmetic average of intensities under linear coordinate of all optimized channel as the adjust standard of corresponding sub problem, which is denoted by OPstd. The dashed horizon line in Fig. 7 shows the adjustment standard. Suppose the pending adjust channel intensity under linear coordinate is Ic, then the adjustment amount can be expressed as follow:

⎧⎛ I ⎞ ⎪ ⎜ 1 − c ⎟ × Ac OPstd − Ic > OPdBdiff OPstd ⎠ ⎪⎝ ⎪ Adjn = ⎨ 0 OPstd − Ic ∈ ⎡⎣ −OPdBdiff , OPdBdiff ⎤⎦ ⎪ ⎪ 1 Adjn − 1 Ic − OPstd > OPdBdiff ⎪− ⎩ 10

(20)

The suffix n represents the optimization times of this channel. Parameter Ac is adjustment coefficient in optimization. In this manuscript, Ac is set to 0.4%, and it can be set depend on the wavelength range and required optimization accuracy. From Eq. (20), we can figure out that parameter OPdBdiff quantize the tolerable error when optimizing FBG, the intensities near adjustment standard are all acceptable, and this is the reason why optimized part in Fig. 6 is serrated. Parameters that needed in optimization are listed in Table 2 and they can be modified considering about practical demands. The last step of greedy algorithm is to combine results of sub problems. To sum up, the optimization algorithm flow chart is plotted in Fig. 8. Considering the actual manufacturing technique of FBG, we can insert transition zones on both edges of reflective spectrum, which will not affect the whole shape of output spectrum, to eliminate the kinks that may have difficulties in fabrications. Using the algorithm which we discussed and analyzed above, we present two optimization results under different conditions. The first result is plotted in Fig. 9. The simulation configuration uses only forward pump. A 10 m long 10/125 YDDCF is pumped by a 25 W multimode LD that operates at 915 nm, the central wavelength of FBGs are set to 1064 nm. Fig. 9(a) plots the optimized OC FBG reflective spectrum and the Fig. 9(b) plots the output spectrum of corresponding laser oscillator. In this case, the 5 dB bandwidth is 5 nm. To better show the optimization procedure, Fig. 9(c) shows the variation of reflective spectrum of OC FBG. All sub-figures in Fig. 9(c) is sorted by time from left to right. At the beginning, the shape of OC FBG reflective spectrum is a square function which is set to the baseline. The adjusted channel gradually moves from high gain channel to low gain channel, and Fig. 9(a) shows the final optimized result. Using the same configuration, we shift central wavelengths of FBGs to 1080 nm, we can get the optimization result around 1080 nm which is plotted in Fig. 10. The 5 dB bandwidth of optimized spectrum is 3.4 nm and the 10 dB bandwidth is 5 nm. When recalling Figs. 9(a) and 10(a), we can see that the optimized reflective spectrum is irregular compared with ordinary FBGs. It is commonly known that the Bragg wavelength of FBGs is related to the period Λ and effective index neff of the grating λB ¼ 2neffΛ. This inspires us to resolve complicated reflective spectrum into a combination of different Bragg wavelengths by introducing chirping in FBGs [18]. This indicates that, theoretically, FBGs with arbitrary reflective spectrum shape can be designed and fabricated.

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5. Conclusion In this manuscript, we have successfully demonstrated an optimization method, which based on a preliminary model, to acquire flattened output spectrum directly from a single laser oscillator. Through the analysis of intracavity gain, using the greedy algorithm to implement the optimization of OC-FBG's reflective spectrum, we successfully flattened the output spectrum around 1064 nm and 1080 nm in our simulation, and it can be easily generalized to 1.55 μm wavelength or other wavelength with proper rate equations modification. Compared with more common used methods that flatten the spectrum outside of cavity, our method has high efficiency and simpler structure and it can be used as broadband source in many applications. Our method has a guiding significance of controlling the output spectrum of laser oscillator using FBGs.

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