Analytical formulation of high-power Yb-doped double-cladding fiber laser
AAnalytical formulation of high-power Yb-doped double-cladding fiber laser ∗ Mostafa Peysokhan , , ∗ , Esmaeil Mobini , , and Arash Mafi , Department of Physics & Astronomy, University of New Mexico, Albuquerque, NM 87131, USA Center for High Technology Materials, University of New Mexico, Albuquerque, NM 87106, USA
Here a detailed formalism to achieve an analytical solution of a lossy high power Yb-doped silicafiber laser is introduced. The solutions for the lossless fiber laser is initially attained in detail. Next,the solution for the lossy fiber laser is obtained based on the lossless fiber laser solution. To examinethe solutions for both lossless and lossy fiber laser two sets of values are compared with the exactnumerical solutions and the results are in a good agreement. Furthermore, steps and procedures forachieving the final solutions are explained clearly and precisely.
I. INTRODUCTION
Over the past five decades following the first demon-stration of glass fiber lasers by Snitzer [1], fiber lasershave excelled in all performance attributes. The oper-ating wavelengths cover a broad range, from ultravioletto mid-infrared [2–8], and high-power Yb-doped double-cladding fiber lasers (YDCFL) are one of the primarysources of high-power radiation for industrial and di-rected energy applications [9, 10]. Fiber lasers can deliverpowers on the order of a few kilowatts (kW) [11–15] andare promising candidates for coherent beam combiningto achieve even higher powers [16–18].Fiber lasers are typically designed using extensive nu-merical solutions and optimizations. It is often thecase that a broad range of parameters over a multi-dimensional design space needs to be covered to find theoptimal parameters for the best performance. The avail-ability of an analytical solution would simplify the designproblem significantly; moreover, it can provide a moreintuitive design platform compared with brute-force nu-merical solutions. There are a few published papers inthe literature that present analytical solutions to the fiberlaser equations [19–25]. However, either the parasiticbackground absorption is absent in the formulation, orthe results are not presented in a fully analytical closedform. In particular, it is quite essential to include theparasitic background absorption because it is one of themain detriments in modern high-power laser designs, andsignificant effort is put into reducing its contribution tothe heating problem in fiber lasers [26–28].In this paper, we present a complete analytical solutionof the YDCFL considering the parasitic background ab-sorption. We treat the parasitic background absorptionas a perturbation. We first obtain an analytical solu-tion to the pump and signal propagation problem in theabsence of loss and then treat the parasitic backgroundabsorption using the first-order perturbation theory. Weverify the accuracy of the solutions by comparing themwith the full numerical solutions and show that our ana-lytical treatment accurately models the full laser design ∗ [email protected] problem in the regimes of interest to high-power fiberlaser design. II. LASER EQUATIONS
The basic schematic of the laser that we study in thispaper is shown in Fig. 1. The laser system includes anactive region (Yb-doped fiber) with a length of L and aBragg reflector ( R ) at the pumping port ( Z = 0) andanother Bragg reflector ( R ) at the signal port ( Z = L ).The rate equations for the Yb ions are based on a quasi-three-level system [6]. We assume that the pump poweris strong enough to saturate the gain, and the signal issufficiently strong to suppress the spontaneous emissions.Both of these assumptions are quite reasonable in high-power fiber laser designs. FIG. 1. Schematic of the laser system and propagation ofthe pump power and signal in the double-cladding fiber laser.Pump power is launched at z = 0 and the output signal iscalculated at z = L at the power delivery port. R and R are the distributed Bragg reflectors at z = 0 and z = L . For continuous wave (CW) lasers, the upper manifoldpopulations is given by [19] N ( z ) N = Γ p σ ap (cid:102) P p ( z ) hν p A + Γ s σ as (cid:102) P s ( z ) hν s A Γ p σ aep (cid:102) P p ( z ) hν p A + 1 τ + Γ s σ aes (cid:102) P s ( z ) hν s A , (1)where for simplicity of expression we have used the fol- a r X i v : . [ phy s i c s . op ti c s ] F e b lowing definitions: (cid:102) P p ( z ) := P + p ( z ) + P − p ( z ) , σ aep := σ ap + σ ep , (2) (cid:102) P s ( z ) := P + s ( z ) + P − s ( z ) , σ aes := σ as + σ es . (3)Equation 1 describes the variation of the upper-level pop-ulation density of the Yb +3 ions along the fiber throughits dependance on the z -varying pump and signal pow-ers. The differential equations for forward pump propa-gation, P + p ( z ), and backward pump propagation, P − p ( z ),are given by+ dP + p ( z ) dz = Γ p ( σ aep N ( z ) − σ ap N ) P + p ( z ) − α p P + p ( z ) , (4) − dP − p ( z ) dz = Γ p ( σ aep N ( z ) − σ ap N ) P − p ( z ) − α p P − p ( z ) . (5)The rate equations for forward signal propagation, P + s ( z ), and backward signal propagation, P − s ( z ), are alsogiven by+ dP + s ( z ) dz = Γ s ( σ aes N ( z ) − σ as N ) P + s ( z ) − α s P + s ( z ) , (6) − dP − s ( z ) dz = Γ s ( σ aes N ( z ) − σ as N ) P − s ( z ) − α s P − s ( z ) . (7) N is the total Yb +3 concentration, which is assumed tobe constant along the fiber laser, ν s ( ν p ) is the signal(pump) frequency, σ as ( σ ap ) is the absorption cross sectionat the signal (pump) wavelength, σ es ( σ ep ) is the emissioncross section at signal (pump) wavelength, τ is the up-per manifold lifetime, A is the cross-sectional area of theoptical fiber core, and h is the Planck constant. The frac-tion of pump power that is coupled to the doped core ofthe gain fiber is represented by Γ p . In a double-claddingconfiguration, where the pump mode is fully scrambled,Γ p can be approximated as the ratio of the doped corearea to the area of the inner cladding. Γ s is the fractionof the signal power that overlaps the doped core area.Because the core of the optical fiber is single-mode, thepower filling factor can be easily approximated using ananalytical formula presented in Ref. [29] based on thenearly Gaussian profile of the LP mode in a step-indexfiber. III. ANALYTICAL SOLUTION
As already highlighted, the proposed solution appliedto the high power laser regime, where the signal powercirculating in the cavity is much larger than the satu-ration power of the fiber, i.e. (cid:102) P s (cid:29) P sat s and P sat s = Ahν s /σ aes τ . Because the total signal power is largeenough to saturate the gain at each z location, from Eq. 1 combined with the fact that σ es (cid:29) σ as , we have N ( z ) (cid:28) N [19, 20]. For simplicity, we assume thatthe pump reflection from the second mirror is negligi-ble; however, the extension to the more general case canbe readily implemented. Taking these assumptions intoaccount, Eq. 4 can be written as+ dP + p ( z ) dz ≈ − (Γ p σ ap N + α p ) P + p ( z ) , (8)The solution to Eq. 8 can be expressed as P + p ( z ) = P + p (0) e − αz , (9)where we have defined α according to α := (Γ p σ ap N + α p ) . (10)We will use Eq. 8 to account for the propagation ofthe pump power. We next consider the derivation of therelevant equation for the propagation of the forward- andbackward-moving signal power. Equations 6 through 7are subject to the following boundary conditions at thelocation of the mirrors P + s (0) = R P − s (0) , (11) P − s ( L ) = R P + s ( L ) . (12)Using Eq. 6 and, 7, we can readily show that dP + s ( z ) P + s ( z ) + dP − s ( z ) P − s ( z ) = 0 , (13)which can be used to show that the z -derivative of P + s ( z ) P − s ( z ) vanishes, therefore, it is constant along thefiber. In other words, P + s (0) P − s (0) = P + s ( L ) P − s ( L ) = R P − s (0) . (14)In Eq. 1, when we multiply the sides of the fractions,one of the terms is in the form of N ( z ) /τ ; in the Ap-pendix, we have shown that the variation of N with z is very slow and N ( z ) /τ can be reliably replaced by N ( z ) /τ , where N ( z ) is the z -averaged value of N ( z ).In Eq. 1, if we replace the N ( z ) /τ with N ( z ) /τ , whilekeeping the z -dependence of other N ( z ) terms, we arriveat τhν p A (cid:0) Γ p ( σ aep N ( z ) − σ ap N ) (cid:1)(cid:102) P p + (15) τhν s A (cid:0) Γ s ( σ aes N ( z ) − σ as N ) (cid:1) (cid:102) P s + N = 0 . Using the pump and signal propagation equations di-rectly, we obtain( dP + p dz − dP − p dz ) + α p (cid:102) P p = Γ p ( σ aep N ( z ) − σ ap N ) (cid:102) P p , (16)( dP + s dz − dP − s dz ) + α s (cid:102) P s = Γ s ( σ aes N ( z ) − σ as N ) (cid:102) P s . (17)Inserting Eq. 16, and Eq. 17 into Eq. 15 results in τhν s A (cid:16) ( dP + s dz − dP − s dz ) + α s (cid:102) P s (cid:17) + (18) τhν p A (cid:16) ( dP + p dz − dP − p dz ) + α p (cid:102) P p (cid:17) + N = 0 . Using Eq. 9 and the fact that P − p ( z ) = 0, Eq. 18 can besimplified as (cid:0) dP + s dz − dP − s dz (cid:1) + α s (cid:0) P + s + P − s (cid:1) + ν s ν p e − αz P + p (0) (cid:0) α p − α (cid:1) + hν s AN τ = 0 . (19)In order to solve Eq. 19, we first assume that α s ≈ α s , we can integrateEq. 19 and obtain P + s ( z ) − P + s (0) − P − s ( z ) + P − s (0) (20)+ ν s ν p P + p (0)( α p − α )( e − αz − − α ) + hν s AN zτ = 0 . Using the boundary conditions (Eq. 11) and implement-ing Eq. 14, Eq. 20 can be written in following form: P + s ( z ) + (cid:16) (1 − R ) P − s (0) − ν s ν p P + p (0)(1 − α p α )(1 − e − αz )(21)+ hν s AN zτ (cid:17) P + s ( z ) − R P − s (0) = 0 . Equation 21 is a quadratic polynomial equation in P + s ( z )and can be readily solved for P + s ( z ). We can write thisequation formally as X + 2 bX − c = 0 , X = P + s ( z ) , (22)where b := (1 − R ) P − s (0)2 − ν s ν p P + p (0)(1 − α p α )(1 − e − αz )(23)+ hν s AN z τ ,c := R P − s (0) . The relevant solution to the quadratic polynomial equa-tion that will be on interest to the problem is expressedas X ≡ P + s ( z ) = − b + (cid:112) b + c. (24)On the other hand, using P + s ( z ) P − s ( z ) = R P − s (0) , wecan show that P + s ( z ) P − s ( z ) = c . If we define X (cid:48) ≡ P − s ( z ), we then have X (cid:48) = c/X , which results in thefollowing quadratic polynomila equation for X (cid:48) : X (cid:48) − bX (cid:48) − c = 0 . (25)The relevant solution of Eq. 25 is given by X (cid:48) ≡ P − s ( z ) = b + (cid:112) b + c. (26)Equations 24 and 26 provide the forward- and backward-propagating signal powers; these solutions in combina-tion with Eq. 1 and Eq. 9 present the complete solutionto the laser problem. However, we still need to find thevalue of P − s (0) that shows up in b parameter of Eq. 23based on the design parameters of the laser, which iswhat we will do next.As explained before, we would now like to find P − s (0).Let’s put the two solutions that we found together P + s ( z ) = − b + (cid:113) b + R P − s (0) ,P − s ( z ) = + b + (cid:113) b + R P − s (0) . (27)We would like to evaluate these equations at z = L , not-ing that b is also a function of z , so we use b L = b ( z = L )in these equations. If we apply the following boundaryconditions P + s ( L ) P − s ( L ) = P + s (0) P − s (0) ,P − s ( L ) = R P + s ( L ) ,P + s (0) = R P − s (0) , (28)which result in (cid:40) R P + s ( L ) = R P − s (0) ,P + s ( L ) = (cid:113) R R − P − s (0) , (29)then we obtain the following quadratic polynomial equa-tion for P + s ( L ): P + s ( L ) + 2 b L P + s ( L ) − c = 0 . (30)For simplicity the part of b ( L ) that contains P − s (0) canbe separated by the following definition: b ( L ) ≡ (1 − R ) P − s (0)2 + f ( L ) , (31) f ( L ) ≡ − ν s ν p P + p (0)(1 − α p α )(1 − e − αL ) + hν s AN L τ . (32)By inserting P + s ( L ) = (cid:112) R /R P − s (0) into Eq. 30 thefollowing equation can be obtained: R R P − s (0) + 2 (cid:18) − R P − s (0) + f ( L ) (cid:19) (cid:114) R R P − s (0)(33) − R P + s (0) = 0 . Dividing both side of the Eq. 33 by P − s (0) results in:( R R + (1 − R ) (cid:114) R R − R ) P − s (0) + 2 f ( L ) (cid:112) R R = 0 , ⇒ ( (cid:112) R R + ( R − − (cid:114) R R ) P − s (0) = 2 f ( L ) , ⇒ ( √ R R + (cid:112) R R − √ R − √ R P − s (0) = 2 (cid:112) R f ( L ) . (34)Finally P − s (0) can be expressed as: P − s (0) = √ R (cid:2) √ R (1 − R ) + √ R (1 − R ) (cid:3) (35) × (cid:20) ( ν s ν p ) P + p (0)(1 − α p α )(1 − e − αL ) − hν s AN Lτ (cid:21) . In the Appendix, Eq. 48, we have an expression for N L that can be substituted in Eq. 35 to obtain P − s (0) = √ R (cid:2) √ R (1 − R ) + √ R (1 − R ) (cid:3) (36) × (cid:34) ( ν s ν p ) P + p (0)(1 − α p α )(1 − e − αL ) − hν s Aτ ( ln √ R R + (Γ s σ as N + α s ) L Γ s ( σ es + σ as ) ) (cid:35) . All the parameters used in Eq. 36 are the known laserparameters. Equation 36 shows how P − s (0) depends onthe laser parameters. It is interesting to note that evenwithout considering the complete solution of the signalpropagation through the laser, one can see how the valueof P − s (0) depends the mirror reflectivities R and R , orhow the length of the fiber, L , affects the value of P − s (0).By plugging the value of P − s (0) from Eq. 36 in Eq. 27,we obtain the full z -dependence of P + s and P − s . How-ever, for the optimization of a laser’s performance usingthe output signal, we can just use the value of P − s (0) inEqs. 11, and 14 and calculate P + s ( L ) and P − s ( L ) directly.We can now compare our analytical solutions for P + p ( z ), P + s ( z ), and P − s ( z ) with direct numerical simu-lations. These constitute the full characterization of thelaser. The values of the parameters that are used for thelaser system are given in Table. I. TABLE I.
YDCFL parameters
Symbol Parameter Value 1 Value 2 A Core area 5 . × − m . × − m Γ s Signal power filling factor 0.82 0.82Γ p Pump power filling factor 1 . × − . × − N Y b +3 concentration 6 . × m − . × m − τ Radiative lifetime 1.0 ms 1.0 ms σ ap Pump absorption cross section 6 . × − m − . × − m − σ ep Pump emission cross section 2 . × − m − . × − m − σ as Signal absorption cross section 1 . × − m − . × − m − σ es Signal emission cross section 2 . × − m − . × − m − α p Background absorption 6 × − m − × − m − λ p Pump wavelength 920 nm nmλ s Signal wavelength 1090 nm nmR First reflector 99% 30% R Second reflector 4% 40% L Fiber length 35 m mP + p (0) Pump power 40 W W Figure 2 shows a detailed comparison of the analyticalversus numerical solution and the agreement is excellent.Figure 2a corresponds to the set of parameters labeled asValue 1 in Table I, while Fig. 2b corresponds to Value 2.
FIG. 2. a) Comparison of the propagation of the analyticalforward pump (Anal. FW pump), analytical forward signal(Anal. FW signal), and analytical backward signal (Anal.BW signal) with their exact numerical counterpart solutionswhich are the exact numerical forward pump (Num. FWpump), exact numerical forward signal (Num. FW signal),and exact numerical backward signal (Num. BW signal) forthe set of Value 1 which are represented in Table I. b) Asimilar graph for the set of Value 2.
As it is shown in Fig. 2, the analytical solution andthe exact numerical solution are in very good agreement;however, the model assumes that the parasitic attenua-tion of the signal α s is negligible. While in most lasersystems this assumption can be acceptable, here we ex-tend the analytical formalism to include α s using first-order perturbation theory. If we include the α s term, wecan write (see Eq. 19): P + s ( z ) − P + s (0) − P − s ( z ) + P − s (0) + ν s ν p P + p (0)( α p − α ) (cid:16) e − αz − − α (cid:17) + hν s AN zτ + α s (cid:90) z ( P + s ( z ) + P − s ( z )) dz = 0 . (37)We next define δ as δ = ( α s (cid:90) z ( P + s ( z ) + P − s ( z )) dz, (38)which is a small perturbation to the main Eq. 23.This perturbation in effect changes the value of the b -parameter in Eqs. 22 and 25 to b + δ , which modifiesEq. 27 to the new form of P + sδ ( z ) = − ( b + δ ) + (cid:113) ( b + δ ) + R P − sδ (0) ,P − sδ ( z ) = +( b + δ ) + (cid:113) ( b + δ ) + R P − sδ (0) . (39)To find an expression for δ in the first-order perturbationtheory, we need to use P + s ( z ) and P − s ( z ) from Eq. 27(corresponding to α s = 0) in Eq. 38. We obtain δ = α s (cid:90) z (cid:113) b ( z ) + R P − s (0) dz. (40)In general, b is a complicated function of z and the inte-gral in Eq. 41 cannot be simplified analytically. To pro-ceed further analytically, we can implement the midpointrule (rectangle rule) for the integration and approximate δ as δ ( z ) ≈ α s (cid:113) b ( z/ + R P − s (0) z, (41)where b is evaluated at the midpoint z/
2. We next insertthe value of b ( L ) from Eq. 41 in Eq. 39 and obtain P + sδ ( L ).Note that we use P − s (0) under the square root in Eq. 39to comply with the first order perturbation. We then ap-ply the boundary condition P − sδ (0) = (cid:112) R /R P + sδ ( L ) andcalculate the first-order improved P − sδ (0), which is thenused in Eq. 39 to obtain the full z -dependence of P + sδ and P − sδ . This procedure results in a rather accurate accountof the forward- and backward- moving signal propaga-tion. Figure 3a corresponds to the set of parameters la-beled as Value 1 in Table I, while Fig. 3b corresponds toValue 2. The signal attenuation parameter is taken to beequal to that of the pump, i.e. α s = α p , in either case. FIG. 3. a) Results of the analytical solution of a lossy fiberlasers which is a comparison of the propagation of the ana-lytical forward pump (Anal. FW pump), analytical forwardsignal (Anal. FW signal), and analytical backward signal(Anal. BW signal) with their exact numerical counterpartsolutions which are the exact numerical forward pump (Num.FW pump), exact numerical forward signal (Num. FW sig-nal), and exact numerical backward signal (Num. BW signal)for the set of Value 1 which are represented in Table I. b) Asimilar graph for the set of Value 2.
IV. CONCLUSION
In summary, we have presented a consistent fully an-alytical solution for the propagation of signal and pumpin YDCFL for both lossy and lossless cases. The resultsare in excellent agreement with the direct numerical so-lution. This study is important because calculating thepump and signal power in a fiber laser using a numericalsolution involves iterative solutions of coupled differen-tial equations and can be time consuming. An analyticalsolution is especially beneficial for multi-parameter opti-mization in designing a fiber laser. The presence of ananalytical solution can significantly speed up optimiza-tion algorithms that targets such metrics as the laseroutput power, efficiency, heat generation or temperature,or a combination of these parameters. The optimizationparameters may include the choice of the fiber geome-try especially the length, fiber dopant concentration, andmirror reflectivities. However, when addressing the ther-mal issues such as in a radiation balanced laser (RBL)design [30, 31], pump and signal wavelengths are alsoincluded as design parameters, making direct numericaloptimization a very time-consuming task [32]. This studywill pave the way to make it possible to find optimallydesigned lasers over a large design parameter space.
V. APPENDIX: AVERAGE OF N ( z ) OVER THELENGTH
Equation 1 can be written in the form of (cid:102) P p ( z )( τ Γ p hν p A ) (cid:16) σ ap − (cid:0) N ( z ) /N (cid:1) σ aep (cid:17) (42)= N ( z ) N + (cid:102) P s ( τ Γ s hν s A ) (cid:16)(cid:0) N ( z ) /N (cid:1) σ aes − σ as (cid:17) , We also define Π( z ) and Σ( z ) asΠ( z ) := (cid:102) P p ( z )( τ Γ p hν p A ) (cid:16) σ ap − (cid:0) N ( z ) /N (cid:1) σ aep (cid:17) , (43)Σ( z ) := (cid:102) P s ( z )( τ Γ s hν s A ) (cid:16)(cid:0) N ( z ) /N (cid:1) σ aes − σ as (cid:17) . Therefore, we can write Eq. 42 as Π( z ) − Σ( z ) = N ( z ) /N . As shown in Fig. 4, while Π( z ) and Σ( z )vary considerably along the z -coordinate, the variationof N ( z ) /N along the length of the fiber is very small.Therefore, it is reasonable to use, instead of N ( z ) /N , N /N which is its average value over the length of thefiber. It should be noted that this substitution is justapplied to the first term of right-hand side of Eq. 42 andnot to the N ( z ) /N term present in Π( z ) and Σ( z ). FIG. 4. Comparison of the Π( z ), Σ( z ), and N ( z ) /N alongthe doped fiber for the Value 1 that are listed in Table I The average value of N ( z ) over the fiber length can be expressed as N = 1 L (cid:90) L N ( z ) dz. (44)The best way to represent N is by utilizing the totalgain of the fiber laser system. The total gain of the fiberlaser for each round trip is given by G = (cid:90) L dP + s ( z ) P + s ( z ) + (cid:90) L dP − s ( z ) P − s ( z )= (cid:90) L (cid:16) Γ s (cid:0) σ aes N ( z ) − σ as N (cid:1) − α s (cid:17) dz − (cid:90) L (cid:16) Γ s (cid:0) σ aes N ( z ) − σ as N (cid:1) − α s (cid:17) dz = 2Γ s σ aes (cid:90) L N ( z ) dz − s σ as N + α ) L, (45)where G is the total gain of the fiber laser. From the finalresult of Eq. 45, the following equation can be derived:1 L (cid:90) L N ( z ) dz = N = G L + (Γ s σ as N + α s )Γ s σ aes . (46)The gain parameter, G should be evaluated based onlaser characters. In each round trip the following rela-tions are R R e G = 1 ⇒ G L = 1 L ln 1 √ R R . (47)By combining the Eq. 45 and Eq. 47 the following ex-pression can be obtained for the average value of N ( z ): N = L ln √ R R + (Γ s σ as N + α s )Γ s ( σ es + σ as ) , (48)where N depends only on the known laser parameters. ACKNOWLEDGMENTS
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