Comparison of fiber lasers based on distributed side-coupled cladding-pumped fibers and double-cladding fibers Zhihe Huang, Jianqiu Cao, Shaofeng Guo,* Jinbao Chen, and Xiaojun Xu College of Optoelectric Science and Engineering, National University of Defense Technology, Changsha 410073, China *Corresponding author: [email protected] Received 5 November 2013; revised 20 February 2014; accepted 26 February 2014; posted 28 February 2014 (Doc. ID 200753); published 31 March 2014

We compare both analytically and numerically the distributed side-coupled cladding-pumped (DSCCP) fiber lasers and double cladding fiber (DCF) lasers. We show that, through optimization of the coupling and absorbing coefficients, the optical-to-optical efficiency of DSCCP fiber lasers can be made as high as that of DCF lasers. At the same time, DSCCP fiber lasers are better than the DCF lasers in terms of thermal management. © 2014 Optical Society of America OCIS codes: (060.2320) Fiber optics amplifiers and oscillators; (060.2280) Fiber design and fabrication; (140.3510) Lasers, fiber. http://dx.doi.org/10.1364/AO.53.002187

1. Introduction

High-power fiber lasers and amplifiers offer many advantages in terms of high optical efficiency, degree of integration, beam quality, reliability, and spatial compactness [1–3]. With the development of rareearth-doped double cladding fibers (DCFs) and highpower laser diodes, multikilowatt high-power fiber lasers have been demonstrated [3–6]. The DCF lasers usually employ an end-pump scheme, and use combiners to inject the pump light. However, there is one main problem in the scheme. The pump light is injected directly into the cladding of the active fiber, which may cause serious thermal deposit on the ends of the fiber and impact the safe running of the combiner and active fiber [7]. To avoid this problem, another kind of pump scheme based on multiport distributed side-coupled cladding-pumped (DSCCP) fibers (also called GTWave fibers) [8–12] has been developed. In this scheme, the pump light is absorbed gradually along the fiber, and the heat could be dissipated all along 1559-128X/14/102187-09$15.00/0 © 2014 Optical Society of America

the fiber. As a result, the thermal management can be easier. Based on this pump scheme, a 1.1 kW highpower single mode fiber laser has been achieved [11]. Such a pump scheme appears to be quite promising, but up to now there have been no comprehensive investigations on DSCCP fiber lasers, especially comparisons with DCF lasers. In this work, we provide an analytical model of the DSCCP fiber lasers, and compare their performances with those of DCF lasers in terms of both optical-tooptical efficiency and thermal property. This work is arranged as follows. In Section 2, the model of the DSCCP fiber laser is provided, and then the opticalto-optical efficiency and thermal characteristics of DSCCP fiber lasers and DCF lasers are compared based on the model. In Section 3, the differences between the DSCCP fiber lasers and DCF lasers are discussed through three numerical examples. Finally, we draw the conclusions in Section 4. 2. Model and Theory

There are two cores in the DSCCP fiber structure. One is the pump core [core 1 in Fig. 1(a)], through which the pump light is injected. The other is the signal core [core 2 in Fig. 1(a)], which is doped with 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS

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Fig. 1. (a) Cross section of a DSCCP fiber: (1) pump core; (2) signal core; (3) signal cladding; and (4) outer cladding. (b) Side view of DSCCP fiber lasers. HR, high reflective grating; LR, low reflective grating.

rare-earth ions. The lasing light is produced and propagates in the signal core. Outside the signal core, there is an inner cladding [cladding 3 in Fig. 1(a), named “signal cladding” here], which is in optical contact with the pump core. The pump light is coupled between the pump core and signal cladding. The pump core and signal cladding are coated together with the same outer cladding [cladding 4 in Fig. 1(a)]. Figure 1(b) shows the side view of DSCCP fiber lasers. The pump light can be injected through both ends of the pump core, and the signal cladding is connected to high reflective grating and low reflective grating, which forms an oscillation cavity. A.

Modeling and Quasi-analytic Analysis

The model of DCF lasers has been established in [13] and been widely used in [14–16]:

the pump light power in the signal fiber, and P s z; λ is the signal power. The plus and minus superscripts represent propagation along the positive or negative z directions, respectively. vp is the pump frequency, A is the cross-section area of the core, and α represents scattering loss. Other parameters in Eqs. (1)–(3) are h, Planck’s constant; c, the speed of light in vacuum; and P0 λ, the contribution of the spontaneous emission into the propagating laser. σ es , σ as , σ ep , and σ ap are the emission and absorption cross sections of the signal light and pump light, respectively. In the DSCCP scheme, the fiber consists of the pump core and signal cladding, which are optically coupled with each other, so the coupling of pump light between the pump core and signal cladding should be considered in the model of DSCCP fiber

− Γp σ ap P Γs R p zPp z − σ as λP  hcA s z; λ  Ps z; λλdλ N 2 z hνp A  Γ σ σ P zP− z ; R Γs p ap ep p p 1 − N  hcA σ as λ  σ es λP s z; λ  Ps z; λλdλ  τ hνp A

dP p z∕dz  −Γp fσ ap N − N 2 z − σ ep N 2 zg  · P p z − αz; λp Pp z;



(2)

(4) (3)

where the upper lasing level population density is given by N 2 z with spontaneous lifetime of τ, N is the rare-earth ions dopant concentration, and λ is the wavelength of the signal light. The pump power filling factor Γp is approximately the ratio between the area of the core and that of the signal fiber cladding, and Γs is the signal power filling factor. P p z is 2188

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lasers [17]. Thus, in order to describe the coupling of pump light between the pump core and signal cladding, the pump light propagation equation [i.e., Eq. (2)] should be modified as    dP pp z∕dz  −αz; λp Ppp z − k1 Ppp z  k2 Pp z;

dP s z; λ  Γs fσ as λ  σ es λN 2 z − Nσ as λg dz  · P s z; λ − αz; λPs z; λ  Γs σ es λN 2 zP0 λ;

(1)

 dP p z∕dz  −Γp fσ ap N − N 2 z − σ ep N 2 zg · Pp z   − αz; λp P p z  k1 Ppp z − k2 Pp z;

(5) where P pp z is the pump light power in the pump core. k1 is the averaged coupling coefficient of the pump light from the pump core to signal cladding, while k2 is that from the signal cladding to the pump

core [17]. The averaged coupling coefficient is mainly affected by the parameters (e.g., the radii and numerical aperture) of pump core and signal cladding, and the incident condition of the pump light [18]. As the pump core can support a large number of modes, the optical field is so complicated that the averaged coupling coefficient is generally evaluated by the ray trace theory [18,19]. The averaged coupling coefficient is generally less than 20 m−1 . Then, the model of DSCCP fiber laser is based on Eqs. (1), (3)–(5). First, we will give an analytic study on the model of DSCCP fiber laser. On the assumption of N 2 ≪ N [15], we can neglect N 2 in Eq. (5). With the bidirectional pumping scheme, i.e., the pump light is injected through both ends of pump cladding simultaneously [see Fig. 1(b)], Eqs. (4) and (5) can be analytically solved to (

P p z  P−p z 

the cladding area of DCF is the sum of that of the pump core and signal cladding in DSCCP fiber. Here, we assume the area of pump core is equal to that of the signal cladding. Then the pump power filling factor Γp of DCF is half of that of the signal cladding in the DSCCP fiber. Thus, the absorbing coefficient γ of DCF will be half of that in the DSCCP fiber. Then, from Eq. (2), we can get the total pump light distribution of DCF as −αγ∕2z PDCF z  P  P−PP Le−αγ∕2L−z : (9) PP 0e

By replacing PDCF z with Pp z in Eq. (8), we can get the power of the signal light in the DCF laser. It can be found that the difference of the pump light distributions of DSCCP and DCF will lead to different characteristics of fiber lasers. On the basis of this

 1−2α−k −k −γρz  1 k1 P pp 0 1 2 − e2−2α−k1 −k2 −γ−ρz e2 ρ ; 1 k1 P−pp L  1−2α−k1 −k2 −γρL−z 2 2−2α−k1 −k2 −γ−ρL−z − e e ρ

i h 1 8 1 P pp 0 −12αk1 k2 γz < P e2ρz k2 − k1  γ  ρ  e−2ρz k1 − k2 − γ  ρ pp z  2ρ e 2 i h : P−pp z  P−pp L e−122αk1 k2 γL−z e12ρL−z k2 − k1  γ  ρ  e−12ρL−z k1 − k2 − γ  ρ 2ρ

with the abbreviation γ  σ ap NΓp and ρ  k21  2k1 k2  k22  2k2 γ − 2k1 γ  γ 2 1∕2 . For the strong pumping conditions, the amplified spontaneous emission can be negligible. Furthermore, the lasing wavelength is usually adapted as σ es λ ≫ σ as λ to get a high efficiency of output power. Then, if the scattering loss of signal light is small enough to be neglected, the laser power at both ends can be analytically solved as p 8 Psat R1  > p p   P L  > s > R1 1−R2  R2 1−R1  > > > h  i > p R > > < · ln R1 R2  vpγvPssat 0L Pp zdz − NΓs σ as L ; p > R2 > − 0  p Psat p   > P > s > R1 1−R2  R2 1−R1  > > h p i > R > : · ln R1 R2  vpγvPssat 0L Pp zdz − NΓs σ as L 8 where R1 and R2 are the power reflectivity at the signal wavelength of the high and low Bragg reflectors, respectively, Psat  hvs A∕Γs σ es τ, and Pp z  − P p z  Pp z. As the pump light is coupling between the pump core and signal cladding of DSCCP fibers, the pump light exists in both the pump core and signal cladding. In order to compare to the DCF model with the same core and cladding areas, we assume that

6

7

model, we can make a comparison between the output properties of the DCF and DSCCP fiber lasers. First, we adopt the optical-to-optical efficiency to evaluate the output properties of fiber lasers. In order to compare the optical-to-optical efficiencies of DSCCP fiber and DCF lasers, we define the relative efficiency as the optical-to-optical efficiency ratio of DSCCP fiber to DCF lasers. The optical-tooptical efficiency can be given as P s L1 − R2 ∕ − P pp 0  Ppp L. Then the plot of the 3D relation of the relative efficiency as a function of coupling coefficient k and absorbing coefficient γ is shown in Fig. 2. Here, we assume that the two coupling coefficients are equal (k1  k2  k). The parameters used in our calculations, unless stated otherwise, are given in Table 1. From Fig. 2, we know that the optical-to-optical efficiency of DSCCP fiber lasers is always smaller than DCF lasers. But when the coupling coefficient and absorbing coefficient is big enough, the difference between the optical-to-optical efficiency of DSCCP fiber and DCF is very small. When the coupling coefficient k is larger than 0.5 m−1 , the optical-to-optical efficiency of DSCCP fiber is almost as high as DCF. When the coupling coefficient k is smaller than 0.2 m−1 , the optical-to-optical efficiency may be much lower than DCF fiber laser. Figure 2 shows that the absorbing coefficient γ only has a negligible effect on the relative efficiency, especially when the absorbing coefficient γ is larger than 1 m−1 . The results indicate 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS

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B. Thermal Analysis

According to the laser model above, we can get the thermal source equation as [20] Qz  fα  γ − Γp σ ep  σ ap N 2 zηp · Pp z  α · Ps zg∕πr21 ;

Fig. 2. 3D plot of the relative efficiency as a function of coupling coefficient k and absorbing coefficient γk1  k2  k.

that the different lasing performance of the DCF and DSCCP fiber lasers should be mainly induced by the coupling coefficient, rather than the absorbing coefficient. To make the optical-to-optical efficiency of DSCCP fiber laser as high as 90% of that of DCF, the following equation has to be satisfied: k1  k2  γ∕k1 γ ≤ L∕2.3;

(10)

as long as the pump absorption γL is large enough (see Appendix A). The relation in Eq. (10) indicates distinctly that small k2 , big k1 , and γ will be suitable for high-efficiency lasing of the DSCCP fiber lasers. This relation can be useful for the design of DSCCP fiber lasers. When it is satisfied, the DSCCP fiber could perform as well as DCF on the optical-to-optical efficiency of fiber lasers. Table 1.

Parameter Value Used for the Examples

Parameter λp λs τ σ ap σ ep σ as σ es αp αs r1 d Aclad − P pp 0, Ppp L Γs N L R1 R2 Tc g κ γ

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Value 976 nm 1064 nm 0.84 ms 2.5 × 10−24 m2 2.5 × 10−24 m2 1.4 × 10−27 m2 2.0 × 10−25 m2 0.005 m−1 0.003 m−1 5 μm 400 μm 2.45 × 10−8 m2 1000 W 0.82 1 × 1026 m−3 10 m 99.9% 10% 25°C 2000 W m−2 K −1 1.38 W m−1 K −1 1.6 m−1

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(11)

− where Ps z  P s z  Ps z, ηp is the quantum defect, and r1 is the radius of the fiber core. As the term Pp z of DSCCP fiber is different from that of DCF, the thermal effect is quite different. Here, we note that the heat is mainly produced in the active core of fiber and the thermal effects are generally induced by the high core temperature. Thus, the core temperature can be considered as the parameter evaluating the thermal effect, i.e., the higher the core temperature is, the more serious the thermal effect is. We utilize the core temperature to evaluate the thermal difference between the DCF and DSCCP fiber lasers. In order to estimate the core temperature, the twolayer model which includes the thermal conduction in the core and cladding can be utilized [18]:

8 h i 2 ∂ > r ∂T 1∂rr;z  ∂ T∂z1 r;z < r∂r  − Qz 2 κ ; 0 ≤ r ≤ r1  ; h i > : ∂ r ∂T 2 r;z  ∂2 T 2 r;z  0; r ≤ r ≤ d 2 1 r∂r ∂r ∂z

12

where T 1 and T 2 are the temperature of the fiber core and cladding, respectively, and d is the radius of the cladding. Here, in order to compare the core temperatures of DSCCP fiber and DCF with the same cooling ability of fiber core, we neglect the eccentricity of the active core of DSCCP fiber. This will also bring convenience to the mathematical deduction. We assume that the adiabatic slow-varying approximation conditions can be applied to temperature distribution in the z direction: ∂2 Tr; z∕ ∂z2  0. As the temperatures and their derivatives must be continuous across the boundary, the solutions of thermal conductive equations are [18] 8 2 Qzr2 Qzr2 Qzr2 > < T 1 r;zT s  2dg 1  2κ 1 ln rd  4κ 1 − Qzr 4κ ;0≤r≤r1  1

> : T r;zT  Qzr21  Qzr21 ln d ; r ≤r≤d 2 s 1 2dg 2κ r 13 where T s is the temperature of the surroundings, κ is the thermal conductive coefficient, and g is the thermal convection coefficient. Note that the thermal effect, especially the thermal damage, is generally induced by the maximum temperature in the fiber core. The control of the maximum temperature of the fiber core is very important to the safe running of the fiber lasers. Thus, in the following part, we will make a comparison between the maximum core temperatures of the two sorts of fiber lasers. To get the value and location of the maximum temperature of the DSCCP fiber, we

;

take the differential of Eq. (13) by z. We can get the approximate location corresponding to the maximum temperature (see Appendix B),  zm 

1∕ρ · lnρ  k1  k2  γ∕k1  k2  γ − ρ L − 1∕ρ · lnρ  k1  k2  γ∕k1  k2  γ − ρ

:

(14) We replace z with zm in Eq. (13), and apparently the maximum temperature is at the fiber core center,  r21 r2 d r2  1 ln  1 Qzm : (15) 2dg 2κ r1 4κ

 T m 0; zm   T s 

As to the DCF laser, the maximum temperature in the fiber core is located at the two pumping ends where the pump light is injected. Thus, the maximum temperature of the DCF laser is located at zm  0 and L. Considering that there are two maximum temperatures, we used the larger one to calculate the maximum temperature ratio. Then we can give out the maximum temperature ratio of the DSCCP and DCF fibers as a function of coupling coefficient k and absorbing coefficient γ in Fig. 3. Figure 3(a) shows the plot of the maximum temperature ratio as a function of absorbing coefficient γ at different coupling coefficients k. It can be found that the temperature ratio is no more than 1, which

Fig. 3. (a) Plot of the maximum temperature ratio as a function of absorbing coefficient γ. (b) Plot of the maximum temperature ratio as a function of coupling coefficient kk1  k2  k.

means that the maximum core temperature of DSCCP fiber laser is no more than that of the DCF laser. Furthermore, it also shows that the temperature ratio decreases with the increasing absorbing coefficient γ. It is indicated that the thermal effect of DSCCP fiber be alleviated compared with DCF, and the alleviation become more and more obvious when the absorbing coefficient γ becomes large. It means that the DSCCP fiber laser will face less serious thermal effects than the DCF laser, especially when the active fiber is highly doped. Besides, it can be found that the decrease of the temperature ratio becomes less obvious when the coupling coefficient k is large. The decrease becomes small when the coupling coefficient k is larger than 6 m−1 . It means that the increase of the coupling coefficient k is not beneficial to the improvement of thermal management of DSCCP fiber laser. A similar conclusion can also be found in Fig. 3(b), which shows that the temperature ratio increases monotonously with the coupling coefficient k. Figure 3(b) also indicates that the temperature ratio should decrease monotonously with the increasing absorbing coefficient γ. Especially when the absorbing coefficient γ is small enough [e.g., γ  0.01 m−1 in Fig. 3(b)], the DSCCP fiber laser will have the same maximum core temperature as the DCF laser. The contour plot of the maximum temperature ratio as a function of coupling coefficient k and absorbing coefficient γ is shown in Fig. 4. From Fig. 4, it can be found that the maximum temperature ratio increases with the coupling coefficient k but decreases with the absorbing coefficient γ. So when the coupling coefficient k is not too large and absorbing coefficient γ is large enough, the maximum temperature of the fiber core of the DSCCP can be very low compared with DCF. The figure shows the decrease of the maximum temperature ratio and the corresponding relations of the coupling coefficient k

Fig. 4. Contour plot of the maximum temperature ratio as a function of coupling coefficient k and absorbing coefficient γk1  k2  k. 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS

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Table 2.

Region 1 2 3

Three Regions of the Coupling Coefficient

k-Value

Relative Efficiency

Maximum Temperature Ratio

k > 2γ − 0.8 m−1  2γ − 0.8 m−1  > k > 1∕4.35 m−2 ∕γ 1∕4.35 m−2 ∕γ > k

>90% >90% 70%

Comparison of fiber lasers based on distributed side-coupled cladding-pumped fibers and double-cladding fibers.

We compare both analytically and numerically the distributed side-coupled cladding-pumped (DSCCP) fiber lasers and double cladding fiber (DCF) lasers...
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