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Analysis of modal characteristics and thermal effects on photonic crystal fiber laser
Optical Fiber Technology xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optical Fiber Technology www.elsevier.com/locate/yofte
Analysis of modal characteristics and thermal effects on photonic crystal fiber laser S.H. Aref ⇑, J. Yekta Research Photonics Laboratory, Department of Physics, University of Qom, 3716146611 Qom, Iran
a r t i c l e
i n f o
Article history: Received 26 October 2013 Revised 11 March 2014 Available online xxxx Keywords: Index-guiding photonic crystal fiber Double clad fiber lasers Plane wave expansion method Thermo optic effects
a b s t r a c t In this paper, thermo-optics effects on fiber lasers based on index-guiding photonic crystal fibers (PCFs) are studied. The modal characteristics of the PCF lasers are discussed in terms of the thermal effects. Modal analysis of PCF has been done with help of plane wave expansion and supercell lattice methods. The thermal analysis is performed by obtaining the effective index of fundamental mode and core confinement factor (CCF) when the fiber laser was pumped. The results show that PCF laser structure has negligible modal sensitivity to thermal load in comparison to conventional fiber laser. Analyzing the mode performance of the fiber laser under thermal load shows that the assumed PCF laser at 80 W/m pump power show 500° rise in temperature and 0.34% propagation constant variation in fundamental mode index and 0.04% CCF variation in fundamental mode. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In recent years, high power fiber lasers have found in large industrial and scientific applications. Continuous-wave output powers well above the kilowatt range with excellent beam quality have been achieved [1]. In conventional photonic crystal fibers, the cladding structure is formed by embedding a number of air channels which form square-lattice or triangular-lattice structures around the core and run along the fiber. Beside the triangular-lattice structures as a common PCF, the propagation characteristics of the PCFs with square-lattice structures have been recently studied and reported [2,3]. The first simple PCF laser was reported in 2000 [4]. Fiber lasers based on index-guiding PCF offer the laser designer novel degrees of freedom not available in the conventional fiber lasers. For instance, they can operate in single-mode, they reach high numerical aperture values for the pump core up to 0.9, they have high power-scaling factor and finally their all-glass structure make the high-power operation possible [5,6]. The air-cladding in the PCF lasers is used to achieve high numerical aperture for the pump and also it prevents the contact between radiation and coating materials. The main disadvantage of these structures seems to be that the air-cladding acts as a thermal insulation layer and it interrupts the heat dissipation from the inner to the outer cladding [6]. Since thermal effects are crucial for efficient operation of fiber lasers, extensive studies have been done so far [6–11]. ⇑ Corresponding author. Fax: +98 (251) 2812129. E-mail address: [email protected] (S.H. Aref).
Limpert et al., modeled the thermo-optical behavior of air-clad ytterbium-doped large-mode-area PCF lasers. The authors drew the conclusion that the air-clad PCF lasers are likely to be scalable to power levels of several kWs [6]. Cheng et al. studied the temperature distribution and thermo-optical properties of PCF lasers [7]. They reported that the air-filling factor of PCF is a key parameter to the thermal performance of the PCF lasers and the fiber core temperature is a critical limitation to the power scaling [7]. In this work, the heat generated in the index-guiding PCF laser as well as thermo-optical effect on some modal properties of laser is studied. To the best of our knowledge, the modal characteristics of laser signal in PCF lasers at different pump power resulting heat power were not investigated. We consider a square lattice based structure due to simplicity of analyzing, simulating and also the result will be clearly applicable to other common structure like. Here, as a main objective we try to analyze some modal characteristics of the PCF laser under thermal load. Therefore, in this paper, as a result of PCF laser and thermal analysis, the variation of the effective index of fundamental laser modes, core confinement factor (CCF) of fundamental mode are investigated. The plane wave expansion and supercell lattice method are used to calculate the modal characteristics of the assumed PCF structure. 2. Radial temperature distribution and PCF laser principle The porous cladding of PCFs which containing a regular arrangement of air holes leads to a special heat conduction properties, which commonly used step index fibers do not posses.
http://dx.doi.org/10.1016/j.yofte.2014.04.010 1068-5200/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
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S.H. Aref, J. Yekta / Optical Fiber Technology xxx (2014) xxx–xxx
An assumed PCF cross section used in fiber lasers has been shown in Fig. 1. The figure schematically shows the core, photonic crystal cladding as an inner cladding (first cladding), air-cladding, outer cladding (second cladding) and coating region. The figure also shows the PCF structure as a square lattice and also the unit cell used in the calculation and their geometrical parameters. The gain medium of a PCF laser can be fabricated by introducing a rare-earth ion doping into the core of PCF. 2.1. Radial temperature distribution In the PCF core, heat is mainly generated due to the quantum defects between the pump and laser photons. The heat is transferred to the surface of the fiber by thermal conduction through the core, the photonic crystal cladding, the air-cladding, the outer cladding, and the coating material. Heat dissipation from the coating to the ambient air also involves convection and conduction mechanisms. In a cylindrical coordinate system the heat transfer equation in the core and the outer layers can be expressed as the following [12]
1 @ rk @T@r r @r @T 1 @ rki @r r @r
¼ Q
0 6 r 6 a
¼0
a 6 r 6 a4
ð1Þ
where k is the thermal conductivity of the core and Q is the heat gP
flow density given by Q ¼ pap2 with P p as the pump power per unit length and g as the heat fraction. In Eq. (1) the heat flow density is equal to zero and ki (i = 1,. . .,4) are the thermal conductivity factors of the cladding ai (as shown in Fig. 1). Considering appropriate boundary conditions, one can solve the differential equations of Eq. (1) to obtain a relation describing temperature distribution in the core and claddings as
Q r2 Q a2 1 lna4 1 1 1 1 lna3 þ lna2 T ðrÞ ¼ T c þ þ þ k3 k4 k2 k3 4k 2 a4 h k4 1 1 lna 1 lna1 0 6 r 6 a þ þ k1 2k k1 k2 4 Q a2 r Q a2 X Q a2 an T i ðrÞ ¼ T c ln þ ln ai1 6 r 6 ai ; 1 6 i 6 4 ai1 2a4 h n¼1 2kn an1 4ki ð2Þ where T c and h are coolant temperature and heat transfer coefficient of air, respectively. To determine the effective thermal
conductivity of each unit cell (as shown in Fig. 1), we employ the series–parallel connection concept which is well known in the analysis of electric resistance [7]. The unit cell is divided into three layers, and the effective thermal conductivity is estimated by the weighted summation of these layers
keff ¼
SA SB SC kA þ kB þ kC SA þ SB þ SC SA þ SB þ SC SA þ SB þ SC
ð3Þ
here SðA;B;CÞ is the area occupied by the fused silica and kðA;B;CÞ are thermal conductivities corresponding to layers A, B, C, respectively. The effective thermal conductivity of photonic crystal cladding is given by
keff ¼ ð1 aÞksi þ
4akair ksi paksi þ ð4 paÞkair
ð4Þ
where ksi , kair , and a represent the thermal conductivities of the silica and air and air-filling factor (a = d/D; d is hole diameter and D is pitch), respectively. Also based on the series–parallel model, the thermal conductivity of the air-cladding, k2, is given by [7]
k2 ¼
wb wair ksi þ kair wb þ wair wb þ wair
ð5Þ
where wb and wair are the thicknesses of silica bridges and air gaps, respectively. Since, the Dnstress is usually smaller than Dntemp by three orders of magnitude, it could be ignored in our calculations [7]. Thus the index change in the core is obtained as
Dncore ¼
dn Q a2 dT 4k
ð6Þ
where dn=dT ¼ 1 105 /°C is the thermal coefficient of refractive index change [7]. 2.2. PCF laser principle A quasi analytical model based on a set of rate equations for strongly pumped fiber lasers was developed by Kelson et al. [13] and Hardy et al. [14]. Here we use their model at steady state for a Yb+3-doped PCF laser which is regarded as a quasi-three-level system emitting at 1090 nm. A typical PCF laser consists of a doped index-guiding PCF section of length L, with a Bragg reflector on either side, as shown in Fig. 2.
Fig. 1. An assumed PCF cross section, the square lattice photonic crystal, and the unit cell used in the calculation and their geometrical parameters.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
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S.H. Aref, J. Yekta / Optical Fiber Technology xxx (2014) xxx–xxx
The strongly pumped double-clad fiber lasers have two important features, namely N 2 ðzÞ << N and ras << res from which we benefit in our approximated approach. The parameters N 2 and N are the upper level Yb+3 population density and Yb+3 dopant concentration in the core and ras and res are the absorption and emission cross sections of the laser, respectively. The rate equations include stimulated and spontaneous emission and absorption at the signal wavelength as well as the excited state absorption (ESA) and stimulated emission at the pump wavelength. Corresponding to Fig. 2 the forward (P þ p ) and backward (P p ) pump powers at wavelength kp are launched into the first cladding at z = 0 and propagate along the fiber. The fraction of the pump power actually coupled to the core is represented by the power filling factor, Cp A=S where A is the area of the core cross section and S is the area of the first multimode cladding. Also, P þ s ðzÞ, P s ðzÞ are the signal powers propagating in the positive and negative z direction, respectively. The solution of the rate equations under steady state condition for the propagation of the pump and the signal power in the PCF laser can be described by the following set of coupled differential equations as [13] ½Pþ p ðzÞþP p ðzÞrap Cp
½Pþ s ðzÞþP s ðzÞras Cs
þ N2 ðzÞ hm A hms A ¼ ½Pþ ðzÞþP ðzÞðr þrp ÞC ap ep p ½P þ p p N 1 s ðzÞþP s ðzÞðras þþres ÞCs þ þ hmp A
s
ð7Þ
hms A
dP p ðzÞ dz
¼ Cp ½rap N ðrap þ rep ÞN2 ðzÞPp ðzÞ ap Pp ðzÞ
dP s ðzÞ
¼ Cs ½ðras þ res ÞN2 ðzÞ ras NPs ðzÞ as Ps ðzÞ
dz
ð8Þ
where rap and rep are the emission and absorption cross section of the pump light, respectively. Cs represents the contribution of laser power in the core, ms and mp are the laser and pump frequency, respectively. Also, s is the spontaneous lifetime, h is Planck’s constant. ap and as represent scattering loss coefficients of pump light and laser. Finally, the output power P out can be written as [15]
Pout ¼ Pþs ðLÞ Ps ðLÞ ¼ ð1 R2 ÞPþs ðLÞ
ð9Þ
The details of the analysis and calculations can be found in [13–15]. 3. Modal computations and heat dissipation In order to investigate the mode variations due to thermo-optical effects in PCF lasers, one should solve the Maxwell equations for PCF structure by standard methods like finite difference time domain, finite element, and or plane wave expansion (PWE) [16]. Here we develop a MATLAB code based on PWE to find the eigen value and mode field of the PCF structure. We employed also a so-called supercell enhancement to the PWE method. In this enhancement, PCF structure is considered as a periodic supercell, which contains a crystal structure and its defects [17,18]. Also with a reasonable approximation the ARPACK algorithm [19] is used. The latter is a package based on Arnoldi’s method [20] that make it possible to determine eigen values of a large matrix. In PCF lasers, the pump power absorbed by dopant is partially converted into fluorescence via radiative relaxation processes and partially to heat via non-radiative relaxations. Pump power absorption and quantum defects (i.e. the energy difference between laser photons and pump photons) cause a thermal load in PCF lasers. This can drastically disturb the core mode properties especially in the high power regimes [21]. Here, we use the core confinement factor (CCF) for a mode to describe the mode performance [21,22]
R a R 2p
0 0 Cmn ¼ R 1 R 2p 0
jHmn ðr; /Þj2 rdrd/
0
jHmn ðr; /Þj2 rdrd/
:
ð10Þ
where Hmn ðr; /Þ is the magnetic field profile of the propagating mode, m and n are mode indices and refer to angular and radial components in cylindrical coordination system, respectively.
Fig. 2. Schematic of PCF laser structure.
Fig. 3. (a) 2D and (b) 3D contour plot of fundamental mode distributions of a 13 13 2D square lattice PCF with a 5 5 central defect as a core and lattice constant (pitch) 3 lm and air hole diameter to pitch ratio 0.4.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
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S.H. Aref, J. Yekta / Optical Fiber Technology xxx (2014) xxx–xxx
Perceptible change of CCF can also have some effects on the beam quality factor (M2 factor) of the laser beam, and it might cause deviation of laser beam profile from their standard shape. 3.1. Modal computations To calculate the CCF for PCF lasers, the wave propagation equation is solved numerically and the core mode distribution and propagation constants are obtained. Since the z-component of wave vector (kz ) is non-zero, the wave equation should be solved as a 2D vector problem and it is not possible to consider the equation as a 2D scalar problem or TE and TM mode separation. Using the Bloch’s theorem and PWE method, we can solve the 2D vector problem to obtain the propagation constant and eigen-vectors of transversal component of magnetic field (Hx and Hy ) [23,24]. By performing numerical simulations based on the PWE method, the eigenvalue problem has been solved. To achieve a reasonable approximation we employed the ARPACK algorithm which is mentioned above. The Fig. 3 shows 2D and 3D contour plot of normalized pointing vector for fundamental mode as the initial mode distribution of laser power at 1090 nm. As shown in Fig. 1, the structure is a 13 13 2D square lattice PCF with a 5 5 central defect as a core and lattice constant (pitch) 3 lm and air hole diameter to pitch ratio 0.4. 3.2. Heat dissipation To obtain the heat dissipation of PCF, the differential heat transfer equations are analytically solved. We found out that the obtained results are consistent with other reports [6–8] which commonly use software packages that implement the finite element method. The solutions presented in Eq. (2) shows the temperature distribution in the core and claddings of PCF at different pump powers. In these calculations the first cladding of PCF was considered as a 2D square lattice as shown in Fig. 1 and the fiber is cooled in air. The thermal and geometrical parameters of PCF that will be used in our calculation are compiled in Table 1. 4. Results and discussion The laser signal can be calculated through Eqs. (7)–(9) and all assumed and required parameters of PCF laser doped with Yb+3 are presented in Table 2. The output of laser power at the end of fiber at different pump power was calculated. The computed
Fig. 4. The variation of laser output power versus pump power.
results presented in Fig. 4 shows the calculated laser output power for the pump power increasing up to 80 W/m and fiber length of 2 m. The results show that the threshold, slop efficiency and laser output power for assumed PCF laser are 2.6 W, %81 and 132 W, respectively. The change of the refractive index related to the temperature rise consists of two parts; first the temperature-induced index change and the second the thermal-stress-induced index change. The PCF are made of fused silica and as a result thermal-stressinduced index change is usually three orders of magnitude smaller than temperature-induced index change, so this term can be ignored [7]. The radial temperature distribution of the PCF laser at different air-filling factor and pump power of 80 W/m is shown in Fig. 5. As shown in this figure, with 80 W/m pump power the core temperature increases to 500 °C and correspondingly the refractive index changes by 0.005 in this region. Fig. 6 shows the effect of increasing thermal load on operation of fundamental mode. This figure shows the change of effective refractive index (neff ¼ b=k) of fundamental mode versus a broad
Table 1 The thermal and geometrical parameters of PCF used in the modeling. Parameter
Value
Parameter
Value
Thermal conductivity of fused silica (ksi) Thermal conductivity of coating material (acrylate) (k4) Thermal conductivity of air (kair) Heat transfer coefficient of air Coolant temperature (Tc) Length of fiber (L) Air hole diameter to pitch ratio (a)
1.37 W/(mK) 0.2 W/(mK) 2.58 102 W/(mK) 17 W/(m2K) 288 K 2m 0.4
Silica bridges width to air gaps ratio in air cladding Core diameter Inner cladding diameter Air cladding thickness Outer cladding diameter Coating thickness Pitch
0.1 15 lm 200 lm 30 lm 560 lm 20 lm 3 lm
Table 2 Parameters of PCF laser doped with Yb+3 [11]. Parameters
Value
Parameters
Value
Parameters
Value
ks kp
1090 nm 920 nm 6 1025 m2 2.5 1026 m2 1.4 1027 m2
res
2 1025 m2 103 s 1.6 1026 m3 1 109 m2 3 103 m1
as
5 103 m1 0.055 0.8 0.98 0.04
rap rep ras
T N A
ap
Up Us R1 R2
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
S.H. Aref, J. Yekta / Optical Fiber Technology xxx (2014) xxx–xxx
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Fig. 5. Radial temperature distribution of the PCF laser for fiber length of 2 m and pump power of 80 W/m at different air-filling factors.
wavelength range around the laser wavelength for temperature corresponding with two thermal load one at threshold and other at maximum pump power. In this calculation the dispersion property of silica has been regarded. This shows effects of thermal load on fundamental mode propagation of PCF laser. Fig. 7a shows the comparison of CCF variation of the fundamental mode at two air-filling factors against the pump power per length at the laser wavelength (k = 1090 nm). Also the Fig. 7b indicates the comparison of CCF variation of fundamental mode at two pump power per length (threshold and maximum pump power) versus the air-filling factors at laser wavelengths. The results show that the CCF of the fundamental mode of the laser decreases with increasing the thermal load and is associated with 0.04% variation at 1.3–80 W/m pump power. Compare the results with [22] where Fig. 7. The comparison of CCF variation of fundamental mode (a) at two air-filling factors versus the pump power per length and (b) at two pump power versus the air-filling factors at laser wavelengths.
Fig. 6. Effects of thermal load on fundamental mode propagation of PCF laser Effective refractive index change of fundamental mode versus the laser wavelength. The solid line shows the case of core temperature difference with air is zero and dash line shows the case of core temperature differences with air is about 500 °C.
Fig. 8. Full width at half maximum (FWHM) variation of fundamental mode versus the pump power.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
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S.H. Aref, J. Yekta / Optical Fiber Technology xxx (2014) xxx–xxx
the use of conventional octagonal double clad fiber laser shows that for 29 W thermal loads the CCF variation of fundamental mode in octagonal double clad fiber laser and PCF laser are 0.03% and 0.04%, respectively. It should be noted that at this thermal load the core temperature of PCF laser has a core temperature of 320 degree more than the fiber laser at Ref. [22]. These results make us face the fact that the structure of the PCF laser is such that more heat is generated but the CCF of fundamental mode due to inherent light propagation properties without altering remains. This fact should be seriously taken into account in the design of PCF laser structures, especially in high power regime. To understand the dependency of field distribution on pump power, the parameter of full width at half maximum (FWHM) for fundamental mode was calculated. Fig. 8 shows the variation of FWHM of fundamental mode versus the pump power. Slight increase in the FWHM towards the pump power is the results of a little decrease of CCF for this mode at Fig. 7. 5. Conclusion The thermal analysis of PCF laser and variation of some modal characteristics in connection with the thermal effects were presented. Analyzing of the modal properties of PCF was performed with the help of plane wave expansion and supercell lattice methods. The effective index of fundamental laser mode and core confinement factor (CCF) of PCF is calculated in the presence of pumping. The characterization was done for different pump power and air-filling factor. The results show that the assumed PCF laser at 80 W/m pump power show 500 °C temperature rise and 0.34% propagation constant variation in the fundamental mode, respectively. Meanwhile, the results show that PCF laser structure has reasonable modal sensitivity to thermal load in comparison to conventional fiber laser besides the fact that at the nearly same conditions and 29 W thermal loads the PCF laser has a core temperature of 320 degree more than the conventional fiber laser.
Acknowledgment The authors would like to acknowledge Dr. Mozafari faculty member of Qom University for discussion. References [1] G. Bonati, H. Voelckel, T. Gabler, U. Krause, et al., Photonics West, San Jose, Late Breaking Developments, Session 5709–2a (2005). [2] S. Kim, C.S. Kee, C.G. Lee, Opt. Express 17 (2009) 7952–7957. [3] L. Wang, D. Yang, Opt. Express 15 (2007) 8892–8897. [4] W.J. Wadsworth, J.C. Knight, W.H. Reeves, P.S. Russell, J. Arriaga, Electron. Lett. 36 (2000) 1452–1454. [5] W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, P.St.J. Russell, Opt. Express 11 (2003) 48–53. [6] J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, Opt. Express 11 (2003) 2982–2990. [7] X. Cheng, J. Xu, Opt. Eng. 12 (2006) 124204-1–124204-5. [8] Jian feng Li, Kailiang Duan, Optik 121 (2010) 1243–1250. [9] J. Limpert, O. Schmidt, J. Rothhardt, et al., Opt. Express 7 (2006) 2715–2720. [10] H. Su, Y. Li, K. Lu, K. Wang, Q. Guo, Proc. SPIE 6823 (2007) 682318-1–682318-7. [11] J. Boullet, Y. Zaouter, R. Desmarchelier, et al., Opt. Express 22 (2008) 17891– 17902. [12] D.C. Brown, H.J. Hoffman, IEEE J. Quantum Electron. 37 (2001) 207–217. [13] I. Kelson, A. Hardy, IEEE J. Quantum Electron. 34 (1998) 1570–1577. [14] A. Hardy, R. Oron, IEEE J. Quantum Electron. 33 (1997) 307–313. [15] Limin Xiao, Ping Yan, Mali Gong, et al., Opt. Commun. 230 (2004) 401–410. [16] Igor A. Sukhoivanov, Igor V. Guryevg, Photonic Crystals Physics and Practical Modeling, Springer, 2009. [17] J. Arriaga, J.C. Knight, P.St.J. Russell, Physica D 189 (2004) 100–106. [18] G.J. Pearce, Plane-wave methods for modelling photonic crystal fibre (PhD thesis), University of Bath, 2006. [19] http://www.caam.rice.edu/software/ARPACK. [20] W.E. Arnoldi, Q. Appl. Math. 9 (1954) 17–29. [21] T.D. Visser, H. Blok, B. Demeulenaere, D. Lenstra, IEEE J. Quantum Electron. 33 (1997) 1763–1766. [22] M. Sabaeian, H. Nadgaran, M. De Sario, et al., Opt. Mater. 31 (2009) 1300–1305. [23] Richard Norton, Numerical Computation of Band Gaps in Photonic Crystal Fibers (PhD thesis), University of Bath, 2008. [24] T.M. Monro, D.J. Richardson, N.G.R. Broderick, P.J. Bennett, J. Lightwave Technol. 1 (1999) 1093–1102.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
Contents lists available at ScienceDirect
Optical Fiber Technology www.elsevier.com/locate/yofte
Analysis of modal characteristics and thermal effects on photonic crystal fiber laser S.H. Aref ⇑, J. Yekta Research Photonics Laboratory, Department of Physics, University of Qom, 3716146611 Qom, Iran
a r t i c l e
i n f o
Article history: Received 26 October 2013 Revised 11 March 2014 Available online xxxx Keywords: Index-guiding photonic crystal fiber Double clad fiber lasers Plane wave expansion method Thermo optic effects
a b s t r a c t In this paper, thermo-optics effects on fiber lasers based on index-guiding photonic crystal fibers (PCFs) are studied. The modal characteristics of the PCF lasers are discussed in terms of the thermal effects. Modal analysis of PCF has been done with help of plane wave expansion and supercell lattice methods. The thermal analysis is performed by obtaining the effective index of fundamental mode and core confinement factor (CCF) when the fiber laser was pumped. The results show that PCF laser structure has negligible modal sensitivity to thermal load in comparison to conventional fiber laser. Analyzing the mode performance of the fiber laser under thermal load shows that the assumed PCF laser at 80 W/m pump power show 500° rise in temperature and 0.34% propagation constant variation in fundamental mode index and 0.04% CCF variation in fundamental mode. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In recent years, high power fiber lasers have found in large industrial and scientific applications. Continuous-wave output powers well above the kilowatt range with excellent beam quality have been achieved [1]. In conventional photonic crystal fibers, the cladding structure is formed by embedding a number of air channels which form square-lattice or triangular-lattice structures around the core and run along the fiber. Beside the triangular-lattice structures as a common PCF, the propagation characteristics of the PCFs with square-lattice structures have been recently studied and reported [2,3]. The first simple PCF laser was reported in 2000 [4]. Fiber lasers based on index-guiding PCF offer the laser designer novel degrees of freedom not available in the conventional fiber lasers. For instance, they can operate in single-mode, they reach high numerical aperture values for the pump core up to 0.9, they have high power-scaling factor and finally their all-glass structure make the high-power operation possible [5,6]. The air-cladding in the PCF lasers is used to achieve high numerical aperture for the pump and also it prevents the contact between radiation and coating materials. The main disadvantage of these structures seems to be that the air-cladding acts as a thermal insulation layer and it interrupts the heat dissipation from the inner to the outer cladding [6]. Since thermal effects are crucial for efficient operation of fiber lasers, extensive studies have been done so far [6–11]. ⇑ Corresponding author. Fax: +98 (251) 2812129. E-mail address: [email protected] (S.H. Aref).
Limpert et al., modeled the thermo-optical behavior of air-clad ytterbium-doped large-mode-area PCF lasers. The authors drew the conclusion that the air-clad PCF lasers are likely to be scalable to power levels of several kWs [6]. Cheng et al. studied the temperature distribution and thermo-optical properties of PCF lasers [7]. They reported that the air-filling factor of PCF is a key parameter to the thermal performance of the PCF lasers and the fiber core temperature is a critical limitation to the power scaling [7]. In this work, the heat generated in the index-guiding PCF laser as well as thermo-optical effect on some modal properties of laser is studied. To the best of our knowledge, the modal characteristics of laser signal in PCF lasers at different pump power resulting heat power were not investigated. We consider a square lattice based structure due to simplicity of analyzing, simulating and also the result will be clearly applicable to other common structure like. Here, as a main objective we try to analyze some modal characteristics of the PCF laser under thermal load. Therefore, in this paper, as a result of PCF laser and thermal analysis, the variation of the effective index of fundamental laser modes, core confinement factor (CCF) of fundamental mode are investigated. The plane wave expansion and supercell lattice method are used to calculate the modal characteristics of the assumed PCF structure. 2. Radial temperature distribution and PCF laser principle The porous cladding of PCFs which containing a regular arrangement of air holes leads to a special heat conduction properties, which commonly used step index fibers do not posses.
http://dx.doi.org/10.1016/j.yofte.2014.04.010 1068-5200/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
2
S.H. Aref, J. Yekta / Optical Fiber Technology xxx (2014) xxx–xxx
An assumed PCF cross section used in fiber lasers has been shown in Fig. 1. The figure schematically shows the core, photonic crystal cladding as an inner cladding (first cladding), air-cladding, outer cladding (second cladding) and coating region. The figure also shows the PCF structure as a square lattice and also the unit cell used in the calculation and their geometrical parameters. The gain medium of a PCF laser can be fabricated by introducing a rare-earth ion doping into the core of PCF. 2.1. Radial temperature distribution In the PCF core, heat is mainly generated due to the quantum defects between the pump and laser photons. The heat is transferred to the surface of the fiber by thermal conduction through the core, the photonic crystal cladding, the air-cladding, the outer cladding, and the coating material. Heat dissipation from the coating to the ambient air also involves convection and conduction mechanisms. In a cylindrical coordinate system the heat transfer equation in the core and the outer layers can be expressed as the following [12]
1 @ rk @T@r r @r @T 1 @ rki @r r @r
¼ Q
0 6 r 6 a
¼0
a 6 r 6 a4
ð1Þ
where k is the thermal conductivity of the core and Q is the heat gP
flow density given by Q ¼ pap2 with P p as the pump power per unit length and g as the heat fraction. In Eq. (1) the heat flow density is equal to zero and ki (i = 1,. . .,4) are the thermal conductivity factors of the cladding ai (as shown in Fig. 1). Considering appropriate boundary conditions, one can solve the differential equations of Eq. (1) to obtain a relation describing temperature distribution in the core and claddings as
Q r2 Q a2 1 lna4 1 1 1 1 lna3 þ lna2 T ðrÞ ¼ T c þ þ þ k3 k4 k2 k3 4k 2 a4 h k4 1 1 lna 1 lna1 0 6 r 6 a þ þ k1 2k k1 k2 4 Q a2 r Q a2 X Q a2 an T i ðrÞ ¼ T c ln þ ln ai1 6 r 6 ai ; 1 6 i 6 4 ai1 2a4 h n¼1 2kn an1 4ki ð2Þ where T c and h are coolant temperature and heat transfer coefficient of air, respectively. To determine the effective thermal
conductivity of each unit cell (as shown in Fig. 1), we employ the series–parallel connection concept which is well known in the analysis of electric resistance [7]. The unit cell is divided into three layers, and the effective thermal conductivity is estimated by the weighted summation of these layers
keff ¼
SA SB SC kA þ kB þ kC SA þ SB þ SC SA þ SB þ SC SA þ SB þ SC
ð3Þ
here SðA;B;CÞ is the area occupied by the fused silica and kðA;B;CÞ are thermal conductivities corresponding to layers A, B, C, respectively. The effective thermal conductivity of photonic crystal cladding is given by
keff ¼ ð1 aÞksi þ
4akair ksi paksi þ ð4 paÞkair
ð4Þ
where ksi , kair , and a represent the thermal conductivities of the silica and air and air-filling factor (a = d/D; d is hole diameter and D is pitch), respectively. Also based on the series–parallel model, the thermal conductivity of the air-cladding, k2, is given by [7]
k2 ¼
wb wair ksi þ kair wb þ wair wb þ wair
ð5Þ
where wb and wair are the thicknesses of silica bridges and air gaps, respectively. Since, the Dnstress is usually smaller than Dntemp by three orders of magnitude, it could be ignored in our calculations [7]. Thus the index change in the core is obtained as
Dncore ¼
dn Q a2 dT 4k
ð6Þ
where dn=dT ¼ 1 105 /°C is the thermal coefficient of refractive index change [7]. 2.2. PCF laser principle A quasi analytical model based on a set of rate equations for strongly pumped fiber lasers was developed by Kelson et al. [13] and Hardy et al. [14]. Here we use their model at steady state for a Yb+3-doped PCF laser which is regarded as a quasi-three-level system emitting at 1090 nm. A typical PCF laser consists of a doped index-guiding PCF section of length L, with a Bragg reflector on either side, as shown in Fig. 2.
Fig. 1. An assumed PCF cross section, the square lattice photonic crystal, and the unit cell used in the calculation and their geometrical parameters.
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The strongly pumped double-clad fiber lasers have two important features, namely N 2 ðzÞ << N and ras << res from which we benefit in our approximated approach. The parameters N 2 and N are the upper level Yb+3 population density and Yb+3 dopant concentration in the core and ras and res are the absorption and emission cross sections of the laser, respectively. The rate equations include stimulated and spontaneous emission and absorption at the signal wavelength as well as the excited state absorption (ESA) and stimulated emission at the pump wavelength. Corresponding to Fig. 2 the forward (P þ p ) and backward (P p ) pump powers at wavelength kp are launched into the first cladding at z = 0 and propagate along the fiber. The fraction of the pump power actually coupled to the core is represented by the power filling factor, Cp A=S where A is the area of the core cross section and S is the area of the first multimode cladding. Also, P þ s ðzÞ, P s ðzÞ are the signal powers propagating in the positive and negative z direction, respectively. The solution of the rate equations under steady state condition for the propagation of the pump and the signal power in the PCF laser can be described by the following set of coupled differential equations as [13] ½Pþ p ðzÞþP p ðzÞrap Cp
½Pþ s ðzÞþP s ðzÞras Cs
þ N2 ðzÞ hm A hms A ¼ ½Pþ ðzÞþP ðzÞðr þrp ÞC ap ep p ½P þ p p N 1 s ðzÞþP s ðzÞðras þþres ÞCs þ þ hmp A
s
ð7Þ
hms A
dP p ðzÞ dz
¼ Cp ½rap N ðrap þ rep ÞN2 ðzÞPp ðzÞ ap Pp ðzÞ
dP s ðzÞ
¼ Cs ½ðras þ res ÞN2 ðzÞ ras NPs ðzÞ as Ps ðzÞ
dz
ð8Þ
where rap and rep are the emission and absorption cross section of the pump light, respectively. Cs represents the contribution of laser power in the core, ms and mp are the laser and pump frequency, respectively. Also, s is the spontaneous lifetime, h is Planck’s constant. ap and as represent scattering loss coefficients of pump light and laser. Finally, the output power P out can be written as [15]
Pout ¼ Pþs ðLÞ Ps ðLÞ ¼ ð1 R2 ÞPþs ðLÞ
ð9Þ
The details of the analysis and calculations can be found in [13–15]. 3. Modal computations and heat dissipation In order to investigate the mode variations due to thermo-optical effects in PCF lasers, one should solve the Maxwell equations for PCF structure by standard methods like finite difference time domain, finite element, and or plane wave expansion (PWE) [16]. Here we develop a MATLAB code based on PWE to find the eigen value and mode field of the PCF structure. We employed also a so-called supercell enhancement to the PWE method. In this enhancement, PCF structure is considered as a periodic supercell, which contains a crystal structure and its defects [17,18]. Also with a reasonable approximation the ARPACK algorithm [19] is used. The latter is a package based on Arnoldi’s method [20] that make it possible to determine eigen values of a large matrix. In PCF lasers, the pump power absorbed by dopant is partially converted into fluorescence via radiative relaxation processes and partially to heat via non-radiative relaxations. Pump power absorption and quantum defects (i.e. the energy difference between laser photons and pump photons) cause a thermal load in PCF lasers. This can drastically disturb the core mode properties especially in the high power regimes [21]. Here, we use the core confinement factor (CCF) for a mode to describe the mode performance [21,22]
R a R 2p
0 0 Cmn ¼ R 1 R 2p 0
jHmn ðr; /Þj2 rdrd/
0
jHmn ðr; /Þj2 rdrd/
:
ð10Þ
where Hmn ðr; /Þ is the magnetic field profile of the propagating mode, m and n are mode indices and refer to angular and radial components in cylindrical coordination system, respectively.
Fig. 2. Schematic of PCF laser structure.
Fig. 3. (a) 2D and (b) 3D contour plot of fundamental mode distributions of a 13 13 2D square lattice PCF with a 5 5 central defect as a core and lattice constant (pitch) 3 lm and air hole diameter to pitch ratio 0.4.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
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Perceptible change of CCF can also have some effects on the beam quality factor (M2 factor) of the laser beam, and it might cause deviation of laser beam profile from their standard shape. 3.1. Modal computations To calculate the CCF for PCF lasers, the wave propagation equation is solved numerically and the core mode distribution and propagation constants are obtained. Since the z-component of wave vector (kz ) is non-zero, the wave equation should be solved as a 2D vector problem and it is not possible to consider the equation as a 2D scalar problem or TE and TM mode separation. Using the Bloch’s theorem and PWE method, we can solve the 2D vector problem to obtain the propagation constant and eigen-vectors of transversal component of magnetic field (Hx and Hy ) [23,24]. By performing numerical simulations based on the PWE method, the eigenvalue problem has been solved. To achieve a reasonable approximation we employed the ARPACK algorithm which is mentioned above. The Fig. 3 shows 2D and 3D contour plot of normalized pointing vector for fundamental mode as the initial mode distribution of laser power at 1090 nm. As shown in Fig. 1, the structure is a 13 13 2D square lattice PCF with a 5 5 central defect as a core and lattice constant (pitch) 3 lm and air hole diameter to pitch ratio 0.4. 3.2. Heat dissipation To obtain the heat dissipation of PCF, the differential heat transfer equations are analytically solved. We found out that the obtained results are consistent with other reports [6–8] which commonly use software packages that implement the finite element method. The solutions presented in Eq. (2) shows the temperature distribution in the core and claddings of PCF at different pump powers. In these calculations the first cladding of PCF was considered as a 2D square lattice as shown in Fig. 1 and the fiber is cooled in air. The thermal and geometrical parameters of PCF that will be used in our calculation are compiled in Table 1. 4. Results and discussion The laser signal can be calculated through Eqs. (7)–(9) and all assumed and required parameters of PCF laser doped with Yb+3 are presented in Table 2. The output of laser power at the end of fiber at different pump power was calculated. The computed
Fig. 4. The variation of laser output power versus pump power.
results presented in Fig. 4 shows the calculated laser output power for the pump power increasing up to 80 W/m and fiber length of 2 m. The results show that the threshold, slop efficiency and laser output power for assumed PCF laser are 2.6 W, %81 and 132 W, respectively. The change of the refractive index related to the temperature rise consists of two parts; first the temperature-induced index change and the second the thermal-stress-induced index change. The PCF are made of fused silica and as a result thermal-stressinduced index change is usually three orders of magnitude smaller than temperature-induced index change, so this term can be ignored [7]. The radial temperature distribution of the PCF laser at different air-filling factor and pump power of 80 W/m is shown in Fig. 5. As shown in this figure, with 80 W/m pump power the core temperature increases to 500 °C and correspondingly the refractive index changes by 0.005 in this region. Fig. 6 shows the effect of increasing thermal load on operation of fundamental mode. This figure shows the change of effective refractive index (neff ¼ b=k) of fundamental mode versus a broad
Table 1 The thermal and geometrical parameters of PCF used in the modeling. Parameter
Value
Parameter
Value
Thermal conductivity of fused silica (ksi) Thermal conductivity of coating material (acrylate) (k4) Thermal conductivity of air (kair) Heat transfer coefficient of air Coolant temperature (Tc) Length of fiber (L) Air hole diameter to pitch ratio (a)
1.37 W/(mK) 0.2 W/(mK) 2.58 102 W/(mK) 17 W/(m2K) 288 K 2m 0.4
Silica bridges width to air gaps ratio in air cladding Core diameter Inner cladding diameter Air cladding thickness Outer cladding diameter Coating thickness Pitch
0.1 15 lm 200 lm 30 lm 560 lm 20 lm 3 lm
Table 2 Parameters of PCF laser doped with Yb+3 [11]. Parameters
Value
Parameters
Value
Parameters
Value
ks kp
1090 nm 920 nm 6 1025 m2 2.5 1026 m2 1.4 1027 m2
res
2 1025 m2 103 s 1.6 1026 m3 1 109 m2 3 103 m1
as
5 103 m1 0.055 0.8 0.98 0.04
rap rep ras
T N A
ap
Up Us R1 R2
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S.H. Aref, J. Yekta / Optical Fiber Technology xxx (2014) xxx–xxx
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Fig. 5. Radial temperature distribution of the PCF laser for fiber length of 2 m and pump power of 80 W/m at different air-filling factors.
wavelength range around the laser wavelength for temperature corresponding with two thermal load one at threshold and other at maximum pump power. In this calculation the dispersion property of silica has been regarded. This shows effects of thermal load on fundamental mode propagation of PCF laser. Fig. 7a shows the comparison of CCF variation of the fundamental mode at two air-filling factors against the pump power per length at the laser wavelength (k = 1090 nm). Also the Fig. 7b indicates the comparison of CCF variation of fundamental mode at two pump power per length (threshold and maximum pump power) versus the air-filling factors at laser wavelengths. The results show that the CCF of the fundamental mode of the laser decreases with increasing the thermal load and is associated with 0.04% variation at 1.3–80 W/m pump power. Compare the results with [22] where Fig. 7. The comparison of CCF variation of fundamental mode (a) at two air-filling factors versus the pump power per length and (b) at two pump power versus the air-filling factors at laser wavelengths.
Fig. 6. Effects of thermal load on fundamental mode propagation of PCF laser Effective refractive index change of fundamental mode versus the laser wavelength. The solid line shows the case of core temperature difference with air is zero and dash line shows the case of core temperature differences with air is about 500 °C.
Fig. 8. Full width at half maximum (FWHM) variation of fundamental mode versus the pump power.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010
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the use of conventional octagonal double clad fiber laser shows that for 29 W thermal loads the CCF variation of fundamental mode in octagonal double clad fiber laser and PCF laser are 0.03% and 0.04%, respectively. It should be noted that at this thermal load the core temperature of PCF laser has a core temperature of 320 degree more than the fiber laser at Ref. [22]. These results make us face the fact that the structure of the PCF laser is such that more heat is generated but the CCF of fundamental mode due to inherent light propagation properties without altering remains. This fact should be seriously taken into account in the design of PCF laser structures, especially in high power regime. To understand the dependency of field distribution on pump power, the parameter of full width at half maximum (FWHM) for fundamental mode was calculated. Fig. 8 shows the variation of FWHM of fundamental mode versus the pump power. Slight increase in the FWHM towards the pump power is the results of a little decrease of CCF for this mode at Fig. 7. 5. Conclusion The thermal analysis of PCF laser and variation of some modal characteristics in connection with the thermal effects were presented. Analyzing of the modal properties of PCF was performed with the help of plane wave expansion and supercell lattice methods. The effective index of fundamental laser mode and core confinement factor (CCF) of PCF is calculated in the presence of pumping. The characterization was done for different pump power and air-filling factor. The results show that the assumed PCF laser at 80 W/m pump power show 500 °C temperature rise and 0.34% propagation constant variation in the fundamental mode, respectively. Meanwhile, the results show that PCF laser structure has reasonable modal sensitivity to thermal load in comparison to conventional fiber laser besides the fact that at the nearly same conditions and 29 W thermal loads the PCF laser has a core temperature of 320 degree more than the conventional fiber laser.
Acknowledgment The authors would like to acknowledge Dr. Mozafari faculty member of Qom University for discussion. References [1] G. Bonati, H. Voelckel, T. Gabler, U. Krause, et al., Photonics West, San Jose, Late Breaking Developments, Session 5709–2a (2005). [2] S. Kim, C.S. Kee, C.G. Lee, Opt. Express 17 (2009) 7952–7957. [3] L. Wang, D. Yang, Opt. Express 15 (2007) 8892–8897. [4] W.J. Wadsworth, J.C. Knight, W.H. Reeves, P.S. Russell, J. Arriaga, Electron. Lett. 36 (2000) 1452–1454. [5] W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, P.St.J. Russell, Opt. Express 11 (2003) 48–53. [6] J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, Opt. Express 11 (2003) 2982–2990. [7] X. Cheng, J. Xu, Opt. Eng. 12 (2006) 124204-1–124204-5. [8] Jian feng Li, Kailiang Duan, Optik 121 (2010) 1243–1250. [9] J. Limpert, O. Schmidt, J. Rothhardt, et al., Opt. Express 7 (2006) 2715–2720. [10] H. Su, Y. Li, K. Lu, K. Wang, Q. Guo, Proc. SPIE 6823 (2007) 682318-1–682318-7. [11] J. Boullet, Y. Zaouter, R. Desmarchelier, et al., Opt. Express 22 (2008) 17891– 17902. [12] D.C. Brown, H.J. Hoffman, IEEE J. Quantum Electron. 37 (2001) 207–217. [13] I. Kelson, A. Hardy, IEEE J. Quantum Electron. 34 (1998) 1570–1577. [14] A. Hardy, R. Oron, IEEE J. Quantum Electron. 33 (1997) 307–313. [15] Limin Xiao, Ping Yan, Mali Gong, et al., Opt. Commun. 230 (2004) 401–410. [16] Igor A. Sukhoivanov, Igor V. Guryevg, Photonic Crystals Physics and Practical Modeling, Springer, 2009. [17] J. Arriaga, J.C. Knight, P.St.J. Russell, Physica D 189 (2004) 100–106. [18] G.J. Pearce, Plane-wave methods for modelling photonic crystal fibre (PhD thesis), University of Bath, 2006. [19] http://www.caam.rice.edu/software/ARPACK. [20] W.E. Arnoldi, Q. Appl. Math. 9 (1954) 17–29. [21] T.D. Visser, H. Blok, B. Demeulenaere, D. Lenstra, IEEE J. Quantum Electron. 33 (1997) 1763–1766. [22] M. Sabaeian, H. Nadgaran, M. De Sario, et al., Opt. Mater. 31 (2009) 1300–1305. [23] Richard Norton, Numerical Computation of Band Gaps in Photonic Crystal Fibers (PhD thesis), University of Bath, 2008. [24] T.M. Monro, D.J. Richardson, N.G.R. Broderick, P.J. Bennett, J. Lightwave Technol. 1 (1999) 1093–1102.
Please cite this article in press as: S.H. Aref, J. Yekta, Analysis of modal characteristics and thermal effects on photonic crystal fiber laser, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.010