Abstract

Taking the improved SRS threshold formula into consideration, the power scaling of tandem-pumped Yb-doped silica fiber lasers and amplifiers is analyzed by new models. The results show that the power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers is primarily limited by optical damage, SRS and thermal lens, while the pump brightness induced limitation is almost removed. It is also found that tandem-pumped Yb-doped fiber lasers and amplifiers, based upon state-of-art fiber technology, have the potential to achieve a power limit of 70.7 kW with a core diameter of 63.4 μm, and in the case of a strict single-mode fiber, the power limit is about 13.3 kW with a core numerical aperture of 0.03.

© 2011 OSA

1. Introduction

Fiber lasers and amplifiers have been widely used in communication, scientific research and industrial applications such as marking, cutting and welding. In recent years, the output power of fiber lasers and amplifiers has shown a dramatic increase [18]. The reported maximum extractable output power of conventional diode-pumped Yb-doped fiber sources has been scaled to multi-Kilowatt level [1,3]. Moreover, IPG photonics corp. has demonstrated 10 kW single-mode continuous-wave (CW) fiber lasers by using tandem-pumping approach [7]. It is firmly believed much higher output power produced by fiber lasers and amplifiers may be obtained in the near future. However, the physical limits imposed on the fibers may constrain the ultimate output power of fiber lasers and amplifiers, including thermal fracture, melting of the core, thermal lens, nonlinear effect, optical damage and finite pump brightness [9,10].

According to Dawson’s analysis, concerning conventional diode-pumped Yb-doped silica fiber lasers and amplifiers with the broad signal bandwidth, pump brightness, thermal lens, and stimulated Raman Scattering (SRS) effect become the main physical limits relevant to power scaling, and the diffraction-limited power limit is 36.85 kW [10]. Actually tandem-pumping, in which fiber lasers are used to pump another fiber laser or amplifier, may be an increasingly attractive technique to increase the output power of Yb-doped fiber lasers and amplifiers as result of its high-brightness pumping source [6,11]. All fiber sources generating an output power of 3 kW or above have employed tandem-pumping at present [6]. IPG’s 10 kW single-mode Yb-doped fiber laser with a wavelength of 1070 nm is also tandem-pumped by Yb-doped fiber lasers at 1018 nm [7]. However, to the best of our knowledge, up to now the power limit of tandem-pumped fiber lasers and amplifiers has not yet been studied. Since tandem-pumping has the potential to scale fiber lasers and amplifiers to much higher output power, it is necessary and important to analyze the power scaling of tandem-pumped fiber lasers and amplifiers.

In the present paper, we will firstly review physical limits on tandem-pumped fiber lasers and amplifiers. We also use the improved SRS threshold formula to compute SRS-limited output power and derive new expressions for analyzing the power limit of tandem-pumped fiber lasers and amplifiers. Then we will analyze and calculate the power scaling of tandem-pumped Yb-doped silica fiber lasers and amplifiers, and especially focus on computing the power limit of tandem-pumped Yb-doped strict single-mode fiber sources.

2. Models and theoretical analysis

In previous study [9,10], the power scaling of fiber lasers and amplifiers was limited by the combination of thermal fracture, melting of the core, thermal lens, nonlinear effect, optical damage and finite pump brightness. Here we give a summary of Dawson’s analysis and put forward our own views from the perspective of tandem-pumping. It has to be pointed out that we use the improved SRS threshold formula to compute SRS-limited output power, which has been proved to be more accurate compared with the traditional study [12].

2.1. Thermal effect

Thermal fracture, melting of the core and thermal lens are taken into consideration to study the thermal effect on fibers. The maximum extractable powers at the limits of thermal fracture, melting of the core and thermal lens are given by [9,13]

PTF=4ηlaserπRmLηheat(1-a22b2)
PMC=4ηlaserπk(TmTc)Lηheat(1+2kbh+2ln(ba))
PTL=ηlaserπkλ2L2ηheatdndTa2
where the subscript TF, MC and TL refer to thermal fracture, melting of the core and thermal lens. L is the length of fiber, a is the core radius and b is cladding radius. ηlaser is optical-optical conversion efficiency while ηheat is the faction of pump light converted to heat. In Eq. (1), Rm denotes the rupture modulus of the silica glass; in Eq. (2), k is the thermal conductivity, and h is the convective film coefficient. Tm and Tc represent melt temperature and coolant temperature, respectively; in Eq. (3), λ is the wavelength of laser (1.080 μm for Yb-doped silica fiber sources), and dn/dT denotes the change of index with the core temperature.

Most parameters described above are the same as that in diode pump except ηlaser and ηheat. Tandem-pumping has a distinct advantage of lower quantum defect owing to the in-band pumping architecture. It is reported that tandem-pumping has achieved a quantum defect of less than 5% [6]. However, given that non-radiative decay mechanisms may contribute to heat deposited in the fiber, it is not reasonable to make ηlaser equal to quantum defect. Therefore, we simultaneously consider quantum defect and the heat contributions made by non-radiative decay mechanisms, and make ηheat equal to 5%. It is also reported that tandem-pumping can obtain an optical-optical conversion efficiency of 83% [14]. As a consequence, we take ηlaser = 83% for examples in our calculations.

2.2. SRS effect

The primary nonlinear effect imposed on fiber lasers and amplifiers with a broad bandwidth is stimulated Raman scattering (SRS). In [9], they used Smith’s classical SRS threshold formula to analyze the SRS limit on the power scaling of fiber lasers and amplifiers, which reads as [15]

PSRS16AeffgRLeff

However, this expression has several practical constraints and it is proved to be inaccurate for calculating SRS threshold in high-power CW double-clad fiber amplifiers [16]. For example, the definition of SRS threshold will heavily deplete the signal power in the high-power case (the SRS threshold in Eq. (4) was defined as the input signal power at which the output power of signal and Raman scattering are equal), and the factor of 16 just applies to fiber lasers with an input power of about 100 mW (the factor will increase with the rising input power). Furthermore, other practical considerations are not included in Eq. (4) [12,16]. Therefore, a more accurate and suitable SRS threshold formula should be provided, which is expressed as [12]

PSRS=20.3lnβ+ln(AeffgRLeff)gRLeffAeff
where β is the ratio between Raman power and signal at the output of fiber at the threshold (0.01 is convenient for high power fiber lasers [12]), and gR is peak Raman gain coefficient. Leff and Aeff in Eq. (5) respectively denote effective length of fiber and effective area of mode, which can be written as [9]
Aeff=Γ2πa2LeffGLln(G)
with Γ being the ratio between mode radius and core radius, L denoting the actual fiber length and G representing power gain factor of Raman signal in the fiber amplifiers.

Note that Eq. (5) was originally used to calculate SRS threshold of passive fiber. However, the output power of the fiber amplifiers can be obtained by multiplying Eq. (5) by the power gain factor of G. Therefore, the new formula used to calculate SRS threshold of tandem-pumped fiber lasers and amplifiers is given by

PSRS=[20.3lnβ+ln(Γ2πa2gRGL/ln(G))]gRL/ln(G)×(Γ2πa2)

2.3. Optical damage

Optical damage is often observed in pulsed fiber lasers and amplifiers as it is easy for them to achieve megawatt-level peak power [17]. On the contrary, the output power from conventional CW lasers and amplifiers is limited within multi-kW, so the phenomenon of optical damage hardly occurs. However, the tandem-pumped fiber lasers and amplifiers may undergo optical damage due to their potential high output power. The optical-damage limited power is given by [9]

POD=Γ2πa2Idamage
where Idamage represents the maximum irradiance tolerated in the fiber.

2.4. Pump limitation

The ideal pump source should meet the following two requirements: firstly, it should have a high output power which is used to be converted to laser power; secondly, it should have a good beam quality in order to be easily coupled into fiber and absorbed by the rare-earth-doped fiber. Generally speaking, it is a thorny problem for conventional diode pump sources to meet the two requirements simultaneously. However, tandem-pumping technique, where fiber laser is used to pump another one, would be an effective solution to achieve high output power and simultaneously maintain good beam quality.

It may be predicted that pump brightness limitation is minor for tandem-pumped fiber lasers and amplifiers, even completely removed in some particular situation. Therefore, it may be logical to neglect the influence of pump brightness on the power limit of tandem-pumped fiber lasers and amplifiers. However, in order to ensure the structural integrity of the models and validate the theoretical analysis, the pump limitation is still taken into consideration. The output power limited by finite pump brightness is given by [9]

PPB=ηlaserIpump(πb2)(πNA2)
where the subscript PB refers to pump brightness. b and NA denote cladding radius and numerical aperture of cladding, respectively. The relation between core radius a and cladding radius b can be written as b = a × (αcore × L/A)1/2 [9], where L is fiber length, A is small signal pump absorption and αcore is peak core’s absorption coefficient. Besides, Ipump in Eq. (9) represents tandem-pumping brightness, which is assumed to be hundredfold as high as that of the brightest diode pump source [6]. The conventional diode-pumping source with a pump brightness of 0.1 W/(μm2Sr) is reported [10], so it is possible for tandem-pumping source to obtain 10 W/(μm2Sr) pump brightness. Here it needs to explain that detailed deduction of Eq. (9) and Eq. (3) can be found in [9], while Eq. (1) and Eq. (2) can be seen in [13].

3. Results and discussion

In the section, we study the power scaling of tandem-pumped fiber lasers and amplifiers based on the models proposed in section 2 and the new expressions derived in this section. It is noted that only the Yb-doped fibers are reported to be used in tandem-pumping at present. Therefore, we focus on the analysis of the power scaling of tandem-pumped Yb-doped silica fiber lasers and amplifiers. It is worth to note that most parameters mentioned in section 2 are either physical constants or hardly changed, which can be found in [9,10]. Moreover, a majority of parameters of tandem-pumping share identical value with that of diode pump except ηheat , ηlaser and Ipump. These three parameters with different values have already summarized in section 2.

As shown in section 2, the output power of fiber lasers and amplifiers is limited by thermal fracture, melting of the core, thermal lens, SRS, optical damage and pump brightness. Accordingly, there are six limited output powers (PTF, PMC, PTL, PSRS, POD and PPB). In order to confirm the fiber lasers and amplifiers to work safely, the maximum extractable power (also namely the power limit) of the fiber lasers and amplifiers is the minimal one among these six limited output powers. We calculate the six limited output powers (PTF, PMC, PTL, PSRS, POD and PPB) when the core diameter varies from 0 to 100 μm and the fiber length changes among 30 m. It is obvious that the lowest limit is the actual power limit for any given point (d, L) since the fiber lasers and amplifiers need to simultaneously meet all the six limits. The corresponding physical limits of the fiber lasers and amplifiers will vary with different regions, as shown in Fig. 1 . It is found from Fig. 1 that optical damage, SRS and thermal lens become the primary physical limits of tandem-pumped Yb-doped fiber lasers and amplifiers, while pump brightness induced limitation is almost removed. This is distinctly different from the physical limits of diode-pumped fiber sources. It is reported that diode-pumped fiber lasers and amplifiers are primarily limited by pump brightness, SRS and thermal lens [9,10].

 

Fig. 1 Dependence of the physical limits of tandem-pumped Yb-doped fiberlasers and amplifiers on the diameter and length of fiber. In cyan region the limit is optical damage, in red region the limit is thermal lens, in blue region the limit is SRS, and in yellow region the limit is pump brightness.

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The boundary line of Fig. 1 between the SRS and thermal-lens limited regions which determines the power limit is of great interest. By equating Eq. (3) and Eq. (7), we found

(20.3lnβ+ln(Γ2πln(G)gRG))×a4L2+ln(a2L)×a4L2gRΓ2πln(G)×ηlaserπkλ22ηheatdndT=0
We assume that
P=20.3lnβ+ln(Γ2πln(G)gRG),Q=gRΓ2πln(G)×ηlaserπkλ22ηheatdndT
Then Eq. (10) can be written as
P×a4L2+ln(a2L)×a4L2Q=0
We can solve and express the boundary as follows
LSRSLens=a2exp{12×lambertw[2×Q×exp(2×P)]P}=a25.03×1011
where lambertw represents Lambert’s W function. Lambert’s W solves the equation w × exp(w) = x for w as a function of x. By inserting Eq. (13) into Eq. (3), the ultimate output power limited by the SRS and thermal lens at the boundary can be written as
PSRSLens=ηlaserπkλ22ηheatdndTexp{12×lambertw[2×Q×exp(2×P)]P}=70.7kW
Equation (14) is the new derived expression for computing the power limit of Yb-doped fiber lasers and amplifiers after using the improved SRS threshold formula. It is calculated that the power limit of tandem-pumped fiber lasers and amplifiers is 70.7 kW. Similarly, using Eq. (14), it is found that the maximum extractable output power of conventional diode-pumped fiber lasers and amplifiers is only 49.7 kW. The results imply that tandem-pumping technique can achieve a big increase of 42% in the output power of Yb-doped fiber lasers and amplifiers.

Likewise, the boundary line between the SRS and optical-damage limited regions can be obtained by making Eq. (7) equal to Eq. (8), yielding

LSRSDamage=a2exp{lambertw[gRIdamageln(G)×a2×exp(P)]P}
The third, fourth and fifth boundary lines also can be calculated in the similar way, which read as
LDamageLens=2ηheatIdamageΓ2dndTηlaserkλ2×a4
LDamagePump=IdamageΓ2AηlaserIpumpπNA2αcore=0.325m
dPL=(8kλ2AηheatIpumpπNA2αcoredndT)14=22.6μm
where dPL refers to the boundary line between the pump and thermal-lens limited region, which represents a straight line perpendicular to d-axis. Besides, LDamage-Pump refers to the boundary line between the pump and optical-damage limited region, which is a straight line parallel to d-axis.

According to the expression of boundary lines Eq. (13), Eq. (15), Eq. (16), Eq. (17) and Eq. (18), all the five boundary lines of the different limited regions are plotted and given in Fig. 2 . It is found that the analytic boundary (Fig. 2) is exactly the same as the numerical results (Fig. 1).

 

Fig. 2 Boundary lines of physical limits of tandem-pumped Yb-doped fiber lasers and amplifiers.

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Here what we care about most is the power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers. By respectively substituting Eq. (16) into Eq. (3), Eq. (17) into Eq. (9), the relevant limited powers at the boundary can be given by

PDamageLens=ηlaserπkλ2LDamageLens2ηheatdndTa2
PDamagePump=ηlaserIpumpπ2(NA)2αcoreALDamagePumpa2

On the basis of Eq. (14), Eq. (19), and Eq. (20), the expression of the power limit can be summarized as

P={PPumpDamagePDamageLensPSRSLens0ddPLdPLddDSLddDSL

In Eq. (21), dPL is the core diameter given by Eq. (18) at which the pump and thermal-lens limited regions overlap while dDSL is the core diameter at which the optical damage, SRS and thermal-lens limited regions intersect. By setting Eq. (13) equal to Eq. (15), dDSL can be written as

dDSL=2gRIdamageexp(P)exp{12×lambertw[2×Q×exp(2×P)]}×ln(G)×lambertw[2×Q×exp(2×P)]gRIdamageexp(P)=63.4μm

Shown in Fig. 3 is the dependence of the power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers on the core diameter. The result reveals that the power limit of tandem-pumped Yb-doped fiber lasers and amplifiers gradually increases from 0 to 70.7 kW with the rising core diameter. However, when the core diameter exceeds 63.4 μm, the power limit does not increase any longer. The stable output power when the core diameter is larger than 63.4 μm may be resulted from the combined effect of the SRS limit and thermal-lens limit. Actually, it can be understood by Eq. (14) where the ultimate output power limited by the SRS and thermal lens at the boundary is independent on core diameter. Consequently, even if the core diameter can increase arbitrarily, the power limit is still restricted within 70.7 kW.

 

Fig. 3 Dependence of the power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers on the core diameter.

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It is reported that IPG has achieved 10 kW single-mode CW fiber laser by means of tandem-pumping technique [7]. They use Yb-doped large mode area fibers with a core diameter of 30 μm as active fibers [18]. It can be seen from Fig. 3 that the fiber laser with a core diameter of 30 μm may achieve a power limit of 15.9 kW if the corresponding parameters are the same as these adopted in our models.

It is believed that high power single-mode fiber lasers are of great interest due to their good beam quality. Strict single-mode tandem-pumped fiber lasers and amplifiers are extremely promising as they may maintain good beam quality while simultaneously achieving high output power. It is well known that strict single-mode fiber laser must meet the following requirement:

V=2πa(NAcore)λ<2.405
where NAcore is core numerical aperture. Figure 4 illustrates the strict single-mode power limit of tandem-pumped Yb-doped silica fiber lasers and amplifiers. As shown in Fig. 4, the strict single-mode power limit of tandem-pumped Yb-doped fiber lasers and amplifiers reduces little by little when the core numerical aperture increases from 0.03 to 0.19. It is also found that the strict single-mode power limit is up to 13.3 kW at the core numerical aperture of 0.03. However, it should be noted that it may be not easy for conventional fibers to achieve a core numerical aperture of 0.03.

 

Fig. 4 Dependence of the strict single-mode power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers on the core numerical aperture.

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We also replace Eq. (5) with the traditional SRS threshold formula (Eq. (4)) in the models, and analyze the physical limits and power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers using the same parameters. The results are computed and plotted in Fig. 5 and Fig. 6 . It is seen from Fig. 5 that the SRS-limited region is larger than that shown in Fig. 1, which reveals that the traditional SRS threshold formula overestimates the SRS limit on the maximum extractable power of tandem-pumped Yb-doped fiber lasers and amplifiers. Figure 6 indicates that if we use the traditional SRS threshold formula to compute the power limit of tandem-pumped Yb-doped fiber lasers and amplifiers, the maximum extractable output power is just 51.3 kW, significantly lower than that calculated by using the improved SRS formula in the models (70.7 kW). This again shows evidence of the overestimation of SRS limit on the maximum extractable power of tandem-pumped Yb-doped fiber lasers and amplifiers when using the traditional SRS threshold formula in the models.

 

Fig. 5 Dependence of the physical limits on the core diameter and length of fiber when using the traditional SRS threshold formula in the models.

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Fig. 6 Dependence of the power scaling on the core diameter and length of fiber when using the traditional SRS threshold formula in the models.

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4. Conclusion

In conclusion, the power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers is analyzed. The improved SRS threshold formula is used to compute SRS-limited output power and new expressions for analyzing the power scaling of fiber lasers and amplifiers are derived. The presented analysis reveals that the power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers is primarily limited by optical damage, SRS and thermal lens, while pump induced limitation is almost removed. It is found that, based on state-of-art fiber technology, tandem-pumped Yb-doped fiber lasers and amplifiers have the potential to achieve a power limit of 70.7 kW with a core diameter of 63.4 μm, and in the case of a strict single-mode fiber, the power limit is about 13.3 kW with a core numerical aperture of 0.03. It is worthy noting that novel fiber designs and fabrication with the improved nonlinear management, thermal and optical management may push the output power of tandem-pumped Yb-doped fiber lasers and amplifiers beyond this limit.

References and links

1. Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004). [CrossRef]   [PubMed]  

2. J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007). [CrossRef]  

3. Y. Jeong, A. J. Boyland, J. K. Sahu, S. Chung, J. Nilsson, and D. N. Payne, “Multi-kilowatt single-mode ytterbium-doped large-core fiber laser,” J. Opt. Soc. Korea 13(4), 416–422 (2009). [CrossRef]  

4. T. Ehrenreich, R. Leveille, I. Majid, and K. Tankala, “1 kW, all-glass Tm: fiber lasers,” presented at SPIE Photonics West 2010: Fiber Lasers VII: Technology, Systems and Applications, January 28, 2010.

5. A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010). [CrossRef]   [PubMed]  

6. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]  

7. V. Fomin, M. Abramov, A. Ferin, A. Abramov, D. Mochalov, N. Platonov, and V. Gapontsev, “10 kW single-mode fiber laser,” presented at 5th International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, June 28-July 1, 2010.

8. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011). [CrossRef]   [PubMed]  

9. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008). [CrossRef]   [PubMed]  

10. J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

11. C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in Ytterbium-Doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010). [CrossRef]  

12. C. Jauregui, J. Limpert, and A. Tünnermann, “On the Raman threshold of passive large mode area fibers,” Proc. SPIE 7914, 791408, 791408-6 (2011). [CrossRef]  

13. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001). [CrossRef]  

14. H. Xiao, J. Leng, W. Wu, P. Zhou, X. Xu, and G. Zhao, “High efficiency tandem-pumped fiber amplifier,” Acta Phys. Sin (to be published).

15. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef]   [PubMed]  

16. C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009). [CrossRef]   [PubMed]  

17. A. V. Smith, B. T. Do, G. R. Hadley, and R. L. Farrow, “Optical damage limits to pulse energy from fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 153–158 (2009). [CrossRef]  

18. J. Hecht, “Fiber laser ramp up the power,” Laser Focus World 12, 52–57 (2009).

References

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  1. Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004).
    [Crossref] [PubMed]
  2. J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
    [Crossref]
  3. Y. Jeong, A. J. Boyland, J. K. Sahu, S. Chung, J. Nilsson, and D. N. Payne, “Multi-kilowatt single-mode ytterbium-doped large-core fiber laser,” J. Opt. Soc. Korea 13(4), 416–422 (2009).
    [Crossref]
  4. T. Ehrenreich, R. Leveille, I. Majid, and K. Tankala, “1 kW, all-glass Tm: fiber lasers,” presented at SPIE Photonics West 2010: Fiber Lasers VII: Technology, Systems and Applications, January 28, 2010.
  5. A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010).
    [Crossref] [PubMed]
  6. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
    [Crossref]
  7. V. Fomin, M. Abramov, A. Ferin, A. Abramov, D. Mochalov, N. Platonov, and V. Gapontsev, “10 kW single-mode fiber laser,” presented at 5th International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, June 28-July 1, 2010.
  8. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
    [Crossref] [PubMed]
  9. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
    [Crossref] [PubMed]
  10. J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).
  11. C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in Ytterbium-Doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
    [Crossref]
  12. C. Jauregui, J. Limpert, and A. Tünnermann, “On the Raman threshold of passive large mode area fibers,” Proc. SPIE 7914, 791408, 791408-6 (2011).
    [Crossref]
  13. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
    [Crossref]
  14. H. Xiao, J. Leng, W. Wu, P. Zhou, X. Xu, and G. Zhao, “High efficiency tandem-pumped fiber amplifier,” Acta Phys. Sin (to be published).
  15. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972).
    [Crossref] [PubMed]
  16. C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009).
    [Crossref] [PubMed]
  17. A. V. Smith, B. T. Do, G. R. Hadley, and R. L. Farrow, “Optical damage limits to pulse energy from fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 153–158 (2009).
    [Crossref]
  18. J. Hecht, “Fiber laser ramp up the power,” Laser Focus World 12, 52–57 (2009).

2011 (2)

2010 (4)

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in Ytterbium-Doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
[Crossref]

A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010).
[Crossref] [PubMed]

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
[Crossref]

2009 (4)

2008 (1)

2007 (1)

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

2004 (1)

2001 (1)

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

1972 (1)

Barty, C. P. J.

Bartya, C. P. J.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

Beach, R. J.

Beacha, R. J.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

Boyland, A. J.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Bullingtona, A. L.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

Chung, S.

Clarkson, W. A.

Codemard, C. A.

C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in Ytterbium-Doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
[Crossref]

Dawson, J. W.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[Crossref] [PubMed]

Do, B. T.

A. V. Smith, B. T. Do, G. R. Hadley, and R. L. Farrow, “Optical damage limits to pulse energy from fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 153–158 (2009).
[Crossref]

Dong, X.

Du, W.

Dubinskiib, M.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

Eberhardt, R.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

Farrow, R. L.

A. V. Smith, B. T. Do, G. R. Hadley, and R. L. Farrow, “Optical damage limits to pulse energy from fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 153–158 (2009).
[Crossref]

Hadley, G. R.

A. V. Smith, B. T. Do, G. R. Hadley, and R. L. Farrow, “Optical damage limits to pulse energy from fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 153–158 (2009).
[Crossref]

Hecht, J.

J. Hecht, “Fiber laser ramp up the power,” Laser Focus World 12, 52–57 (2009).

Heebner, J. E.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[Crossref] [PubMed]

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

Jauregui, C.

C. Jauregui, J. Limpert, and A. Tünnermann, “On the Raman threshold of passive large mode area fibers,” Proc. SPIE 7914, 791408, 791408-6 (2011).
[Crossref]

C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009).
[Crossref] [PubMed]

Jeong, Y.

Klingebiel, S.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

Leng, J.

Limpert, J.

C. Jauregui, J. Limpert, and A. Tünnermann, “On the Raman threshold of passive large mode area fibers,” Proc. SPIE 7914, 791408, 791408-6 (2011).
[Crossref]

A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010).
[Crossref] [PubMed]

C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009).
[Crossref] [PubMed]

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

Liu, Z.

Ma, Y.

Messerly, M. J.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[Crossref] [PubMed]

Nilsson, J.

Pax, P. H.

Paxa, P. H.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

Payne, D. N.

Peschel, T.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

Richardson, D. J.

Röser, F.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

Sahu, J. K.

Schreiber, T.

A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010).
[Crossref] [PubMed]

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

Shverdin, M. Y.

Si, L.

Siders, C. W.

Sidersa, C. W.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

Smith, A. V.

A. V. Smith, B. T. Do, G. R. Hadley, and R. L. Farrow, “Optical damage limits to pulse energy from fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 153–158 (2009).
[Crossref]

Smith, R. G.

Sridharan, A. K.

Sridharana, A. K.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

Stappaerts, E. A.

Tünnermann, A.

C. Jauregui, J. Limpert, and A. Tünnermann, “On the Raman threshold of passive large mode area fibers,” Proc. SPIE 7914, 791408, 791408-6 (2011).
[Crossref]

A. Tünnermann, T. Schreiber, and J. Limpert, “Fiber lasers and amplifiers: an ultrafast performance evolution,” Appl. Opt. 49(25), F71–F78 (2010).
[Crossref] [PubMed]

C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009).
[Crossref] [PubMed]

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

Wang, X.

Wirth, C.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

Xiao, H.

Xu, X.

Zhao, Y.

Zhou, P.

Zhu, J.

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

C. A. Codemard, J. K. Sahu, and J. Nilsson, “Tandem cladding-pumping for control of excess gain in Ytterbium-Doped fiber amplifiers,” IEEE J. Quantum Electron. 46(12), 1860–1869 (2010).
[Crossref]

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

A. V. Smith, B. T. Do, G. R. Hadley, and R. L. Farrow, “Optical damage limits to pulse energy from fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 153–158 (2009).
[Crossref]

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 537–545 (2007).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Opt. Soc. Korea (1)

Laser Focus World (1)

J. Hecht, “Fiber laser ramp up the power,” Laser Focus World 12, 52–57 (2009).

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (2)

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Paxa, A. K. Sridharana, A. L. Bullingtona, R. J. Beacha, C. W. Sidersa, C. P. J. Bartya, and M. Dubinskiib, “Power scaling analysis of fiber lasers and amplifiers based on nonsilica materials,” Proc. SPIE 7886, 1–12 (2010).

C. Jauregui, J. Limpert, and A. Tünnermann, “On the Raman threshold of passive large mode area fibers,” Proc. SPIE 7914, 791408, 791408-6 (2011).
[Crossref]

Other (3)

H. Xiao, J. Leng, W. Wu, P. Zhou, X. Xu, and G. Zhao, “High efficiency tandem-pumped fiber amplifier,” Acta Phys. Sin (to be published).

T. Ehrenreich, R. Leveille, I. Majid, and K. Tankala, “1 kW, all-glass Tm: fiber lasers,” presented at SPIE Photonics West 2010: Fiber Lasers VII: Technology, Systems and Applications, January 28, 2010.

V. Fomin, M. Abramov, A. Ferin, A. Abramov, D. Mochalov, N. Platonov, and V. Gapontsev, “10 kW single-mode fiber laser,” presented at 5th International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, June 28-July 1, 2010.

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Figures (6)

Fig. 1
Fig. 1

Dependence of the physical limits of tandem-pumped Yb-doped fiberlasers and amplifiers on the diameter and length of fiber. In cyan region the limit is optical damage, in red region the limit is thermal lens, in blue region the limit is SRS, and in yellow region the limit is pump brightness.

Fig. 2
Fig. 2

Boundary lines of physical limits of tandem-pumped Yb-doped fiber lasers and amplifiers.

Fig. 3
Fig. 3

Dependence of the power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers on the core diameter.

Fig. 4
Fig. 4

Dependence of the strict single-mode power scaling of tandem-pumped Yb-doped fiber lasers and amplifiers on the core numerical aperture.

Fig. 5
Fig. 5

Dependence of the physical limits on the core diameter and length of fiber when using the traditional SRS threshold formula in the models.

Fig. 6
Fig. 6

Dependence of the power scaling on the core diameter and length of fiber when using the traditional SRS threshold formula in the models.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

P T F = 4 η l a s e r π R m L η h e a t ( 1 - a 2 2 b 2 )
P M C = 4 η l a s e r π k ( T m T c ) L η h e a t ( 1 + 2 k b h + 2 ln ( b a ) )
P T L = η l a s e r π k λ 2 L 2 η h e a t d n d T a 2
P S R S 16 A e f f g R L e f f
P S R S = 20.3 ln β + ln ( A e f f g R L e f f ) g R L e f f A e f f
A e f f = Γ 2 π a 2 L e f f G L ln ( G )
P S R S = [ 20.3 ln β + ln ( Γ 2 π a 2 g R G L / ln ( G ) ) ] g R L / ln ( G ) × ( Γ 2 π a 2 )
P O D = Γ 2 π a 2 I d a m a g e
P P B = η l a s e r I p u m p ( π b 2 ) ( π N A 2 )
( 20.3 ln β + ln ( Γ 2 π ln ( G ) g R G ) ) × a 4 L 2 + ln ( a 2 L ) × a 4 L 2 g R Γ 2 π ln ( G ) × η l a s e r π k λ 2 2 η h e a t d n d T = 0
P = 20.3 ln β + ln ( Γ 2 π ln ( G ) g R G ) , Q = g R Γ 2 π ln ( G ) × η l a s e r π k λ 2 2 η h e a t d n d T
P × a 4 L 2 + ln ( a 2 L ) × a 4 L 2 Q = 0
L S R S L e n s = a 2 exp { 1 2 × l a m b e r t w [ 2 × Q × exp ( 2 × P ) ] P } = a 2 5.03 × 10 11
P S R S L e n s = η l a s e r π k λ 2 2 η h e a t d n d T exp { 1 2 × l a m b e r t w [ 2 × Q × exp ( 2 × P ) ] P } = 70.7 k W
L S R S D a m a g e = a 2 exp { l a m b e r t w [ g R I d a m a g e ln ( G ) × a 2 × exp ( P ) ] P }
L D a m a g e L e n s = 2 η h e a t I d a m a g e Γ 2 d n d T η l a s e r k λ 2 × a 4
L D a m a g e P u m p = I d a m a g e Γ 2 A η l a s e r I p u m p π N A 2 α c o r e = 0.325 m
d P L = ( 8 k λ 2 A η h e a t I p u m p π N A 2 α c o r e d n d T ) 1 4 = 22.6 μ m
P D a m a g e L e n s = η l a s e r π k λ 2 L D a m a g e L e n s 2 η h e a t d n d T a 2
P D a m a g e P u m p = η l a s e r I p u m p π 2 ( N A ) 2 α c o r e A L D a m a g e P u m p a 2
P = { P P u m p D a m a g e P D a m a g e L e n s P S R S L e n s 0 d d P L d P L d d D S L d d D S L
d D S L = 2 g R I d a m a g e exp ( P ) exp { 1 2 × l a m b e r t w [ 2 × Q × exp ( 2 × P ) ] } × ln ( G ) × l a m b e r t w [ 2 × Q × exp ( 2 × P ) ] g R I d a m a g e exp ( P ) = 63.4 μ m
V = 2 π a ( N A c o r e ) λ < 2.405

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