Abstract

Recent advances in power scaling of fiber lasers are hindered by the thermal issues, which deteriorate the beam quality. Anti-Stokes fluorescence cooling has been suggested as a viable method to balance the heat generated by the quantum defect and background absorption. Such radiation-balanced configurations rely on the availability of cooling-grade rare-earth-doped gain materials. Herein, we perform a series of tests on an ytterbium-doped ZrF4BaF2LaF3AlF3NaF (ZBLAN) optical fiber to extract its laser-cooling-related parameters and show that it is a viable laser-cooling medium for radiation balancing. In particular, a detailed laser-induced modulation spectrum test is performed to highlight the transition of this fiber to the cooling regime as a function of the pump laser wavelength. Numerical simulations support the feasibility of a radiation-balanced laser, but they highlight that practical radiation-balanced designs are more demanding on the fiber material properties, especially on the background absorption, than solid-state laser-cooling experiments.

© 2020 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
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2019 (6)

2018 (1)

2016 (1)

S. R. Bowman, “Low quantum defect laser performance,” Opt. Eng. 56, 011104 (2016).
[Crossref]

2014 (2)

2012 (2)

2010 (4)

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. Di Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[Crossref]

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

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46, 1076–1085 (2010).
[Crossref]

X. Zhu and N. Peyghambarian, “High-power ZBLAN glass fiber lasers: review and prospect,” Adv. OptoElectron. 2010, 501956 (2010).
[Crossref]

2008 (1)

2007 (1)

T. Newell, P. Peterson, A. Gavrielides, and M. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007).
[Crossref]

2004 (2)

A. P. Rizzato, C. V. Santilli, S. H. Pulcinelli, Y. Messaddeq, and P. Hammer, “XPS study of the corrosion protection of fluorozirconate glasses dip-coated with SnO2 transparent thin films,” J. Sol-Gel Sci. Technol. 32, 155–160 (2004).
[Crossref]

A. P. Rizzato, C. V. Santilli, S. H. Pulcinelli, Y. Messaddeq, A. F. Craievich, and P. Hammer, “Study on the initial stages of water corrosion of fluorozirconate glasses,” J. Non-Cryst. Solids 348, 38–43 (2004).
[Crossref]

2002 (1)

H. Malik and K. Maqsood, “Effect of distilled water on the optical properties and surface degradation of Zr-Ba based glass,” J. Mater. Sci. 37, 5367–5369 (2002).
[Crossref]

2001 (2)

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, 207–217 (2001).
[Crossref]

A. Rayner, M. Hirsch, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Distributed laser refrigeration,” Appl. Opt. 40, 5423–5429 (2001).
[Crossref]

1999 (2)

T. Gosnell, “Laser cooling of a solid by 65  K starting from room temperature,” Opt. Lett. 24, 1041–1043 (1999).
[Crossref]

S. R. Bowman, “Lasers without internal heat generation,” IEEE J. Quantum Electron. 35, 115–122 (1999).
[Crossref]

1998 (2)

X. Luo, M. D. Eisaman, and T. R. Gosnell, “Laser cooling of a solid by 21  K starting from room temperature,” Opt. Lett. 23, 639–641 (1998).
[Crossref]

I. Kelson and A. A. Hardy, “Strongly pumped fiber lasers,” IEEE J. Quantum Electron. 34, 1570–1577 (1998).
[Crossref]

1997 (1)

C. Mungan, M. Buchwald, B. Edwards, R. Epstein, and T. Gosnell, “Laser cooling of a solid by 16  K starting from room temperature,” Phys. Rev. Lett. 78, 1030–1033 (1997).
[Crossref]

1995 (1)

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).
[Crossref]

1993 (1)

L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11, 1435–1446 (1993).
[Crossref]

1982 (1)

B. Aull and H. Jenssen, “Vibronic interactions in Nd: Yag resulting in nonreciprocity of absorption and stimulated emission cross sections,” IEEE J. Quantum Electron. 18, 925–930 (1982).
[Crossref]

1964 (1)

D. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957 (1964).
[Crossref]

Abaie, B.

E. Mobini, M. Peysokhan, B. Abaie, M. P. Hehlen, and A. Mafi, “Spectroscopic investigation of Yb-doped silica glass for solid-state optical refrigeration,” Phys. Rev. Appl. 11, 014066 (2019).
[Crossref]

M. Peysokhan, E. Mobini, B. Abaie, and A. Mafi, “Method for measuring the resonant absorption coefficient of rare-earth-doped optical fibers,” Appl. Opt. 58, 1841–1846 (2019).
[Crossref]

M. Peysokhan, E. M. Souchelmaei, B. Abaie, and A. Mafi, “A non-destructive method for measuring the absorption coefficient of a doped optical fiber,” Proc. SPIE 10936, 109360K (2019).
[Crossref]

E. Mobini, M. Peysokhan, B. Abaie, and A. Mafi, “Thermal modeling, heat mitigation, and radiative cooling for double-clad fiber amplifiers,” J. Opt. Soc. Am. B 35, 2484–2493 (2018).
[Crossref]

M. Peysokhan, E. Mobini, B. Abaie, and A. Mafi, “A non-destructive method for measuring the absorption coefficient of a Yb-doped fiber,” in Laser Science (Optical Society of America, 2018), paper JW3A–138.

M. Peysokhan, B. Abaie, E. Mobini, S. Rostami, and A. Mafi, “Measuring quantum efficiency and background absorption of an ytterbium-doped ZBLAN fiber,” in CLEO: Applications and Technology (Optical Society of America, 2018), paper JW2A–118.

Albrecht, A. R.

Arora, A.

Asmerom, Y.

Aull, B.

B. Aull and H. Jenssen, “Vibronic interactions in Nd: Yag resulting in nonreciprocity of absorption and stimulated emission cross sections,” IEEE J. Quantum Electron. 18, 925–930 (1982).
[Crossref]

Barty, C.

Beach, R. J.

Bernier, M.

Bigotta, S.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. Di Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[Crossref]

Biswal, S.

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46, 1076–1085 (2010).
[Crossref]

Bowman, S. R.

S. R. Bowman, “Low quantum defect laser performance,” Opt. Eng. 56, 011104 (2016).
[Crossref]

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46, 1076–1085 (2010).
[Crossref]

S. R. Bowman, “Lasers without internal heat generation,” IEEE J. Quantum Electron. 35, 115–122 (1999).
[Crossref]

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, 207–217 (2001).
[Crossref]

Buchwald, M.

C. Mungan, M. Buchwald, B. Edwards, R. Epstein, and T. Gosnell, “Laser cooling of a solid by 16  K starting from room temperature,” Phys. Rev. Lett. 78, 1030–1033 (1997).
[Crossref]

Buchwald, M. I.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).
[Crossref]

Clarkson, W. A.

Codemard, C. A.

M. N. Zervas and C. A. Codemard, “High power fiber lasers: a review,” IEEE J. Sel. Top. Quantum Electron. 20, 219–241 (2014).
[Crossref]

Condon, N. J.

S. R. Bowman, S. P. O’Connor, S. Biswal, N. J. Condon, and A. Rosenberg, “Minimizing heat generation in solid-state lasers,” IEEE J. Quantum Electron. 46, 1076–1085 (2010).
[Crossref]

Cozic, S.

Craievich, A. F.

A. P. Rizzato, C. V. Santilli, S. H. Pulcinelli, Y. Messaddeq, A. F. Craievich, and P. Hammer, “Study on the initial stages of water corrosion of fluorozirconate glasses,” J. Non-Cryst. Solids 348, 38–43 (2004).
[Crossref]

Dajani, I.

Dawson, J. W.

Di Lieto, A.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. Di Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[Crossref]

Digonnet, M. J. F.

Edwards, B.

C. Mungan, M. Buchwald, B. Edwards, R. Epstein, and T. Gosnell, “Laser cooling of a solid by 16  K starting from room temperature,” Phys. Rev. Lett. 78, 1030–1033 (1997).
[Crossref]

Edwards, B. C.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).
[Crossref]

Eidam, T.

Eisaman, M. D.

Epstein, R.

C. Mungan, M. Buchwald, B. Edwards, R. Epstein, and T. Gosnell, “Laser cooling of a solid by 16  K starting from room temperature,” Phys. Rev. Lett. 78, 1030–1033 (1997).
[Crossref]

Epstein, R. I.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).
[Crossref]

Gavrielides, A.

T. Newell, P. Peterson, A. Gavrielides, and M. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007).
[Crossref]

Gosnell, T.

T. Gosnell, “Laser cooling of a solid by 65  K starting from room temperature,” Opt. Lett. 24, 1041–1043 (1999).
[Crossref]

C. Mungan, M. Buchwald, B. Edwards, R. Epstein, and T. Gosnell, “Laser cooling of a solid by 16  K starting from room temperature,” Phys. Rev. Lett. 78, 1030–1033 (1997).
[Crossref]

Gosnell, T. R.

X. Luo, M. D. Eisaman, and T. R. Gosnell, “Laser cooling of a solid by 21  K starting from room temperature,” Opt. Lett. 23, 639–641 (1998).
[Crossref]

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).
[Crossref]

Hammer, P.

A. P. Rizzato, C. V. Santilli, S. H. Pulcinelli, Y. Messaddeq, A. F. Craievich, and P. Hammer, “Study on the initial stages of water corrosion of fluorozirconate glasses,” J. Non-Cryst. Solids 348, 38–43 (2004).
[Crossref]

A. P. Rizzato, C. V. Santilli, S. H. Pulcinelli, Y. Messaddeq, and P. Hammer, “XPS study of the corrosion protection of fluorozirconate glasses dip-coated with SnO2 transparent thin films,” J. Sol-Gel Sci. Technol. 32, 155–160 (2004).
[Crossref]

Hardy, A. A.

I. Kelson and A. A. Hardy, “Strongly pumped fiber lasers,” IEEE J. Quantum Electron. 34, 1570–1577 (1998).
[Crossref]

Heckenberg, N. R.

Heebner, J. E.

Hehlen, M. P.

E. Mobini, M. Peysokhan, B. Abaie, M. P. Hehlen, and A. Mafi, “Spectroscopic investigation of Yb-doped silica glass for solid-state optical refrigeration,” Phys. Rev. Appl. 11, 014066 (2019).
[Crossref]

Hirsch, M.

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, 207–217 (2001).
[Crossref]

Jansen, F.

Jauregui, C.

Jenssen, H.

B. Aull and H. Jenssen, “Vibronic interactions in Nd: Yag resulting in nonreciprocity of absorption and stimulated emission cross sections,” IEEE J. Quantum Electron. 18, 925–930 (1982).
[Crossref]

Kelson, I.

I. Kelson and A. A. Hardy, “Strongly pumped fiber lasers,” IEEE J. Quantum Electron. 34, 1570–1577 (1998).
[Crossref]

Knall, J.

Limpert, J.

Luo, X.

Mafi, A.

E. Mobini, M. Peysokhan, B. Abaie, M. P. Hehlen, and A. Mafi, “Spectroscopic investigation of Yb-doped silica glass for solid-state optical refrigeration,” Phys. Rev. Appl. 11, 014066 (2019).
[Crossref]

M. Peysokhan, E. Mobini, B. Abaie, and A. Mafi, “Method for measuring the resonant absorption coefficient of rare-earth-doped optical fibers,” Appl. Opt. 58, 1841–1846 (2019).
[Crossref]

E. Mobini, M. Peysokhan, and A. Mafi, “Heat mitigation of a core/cladding Yb-doped fiber amplifier using anti-Stokes fluorescence cooling,” J. Opt. Soc. Am. B 36, 2167–2177 (2019).
[Crossref]

M. Peysokhan, E. M. Souchelmaei, B. Abaie, and A. Mafi, “A non-destructive method for measuring the absorption coefficient of a doped optical fiber,” Proc. SPIE 10936, 109360K (2019).
[Crossref]

E. Mobini, M. Peysokhan, B. Abaie, and A. Mafi, “Thermal modeling, heat mitigation, and radiative cooling for double-clad fiber amplifiers,” J. Opt. Soc. Am. B 35, 2484–2493 (2018).
[Crossref]

M. Peysokhan, E. Mobini, B. Abaie, and A. Mafi, “A non-destructive method for measuring the absorption coefficient of a Yb-doped fiber,” in Laser Science (Optical Society of America, 2018), paper JW3A–138.

M. Peysokhan, B. Abaie, E. Mobini, S. Rostami, and A. Mafi, “Measuring quantum efficiency and background absorption of an ytterbium-doped ZBLAN fiber,” in CLEO: Applications and Technology (Optical Society of America, 2018), paper JW2A–118.

Malik, H.

H. Malik and K. Maqsood, “Effect of distilled water on the optical properties and surface degradation of Zr-Ba based glass,” J. Mater. Sci. 37, 5367–5369 (2002).
[Crossref]

Maqsood, K.

H. Malik and K. Maqsood, “Effect of distilled water on the optical properties and surface degradation of Zr-Ba based glass,” J. Mater. Sci. 37, 5367–5369 (2002).
[Crossref]

McCumber, D.

D. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957 (1964).
[Crossref]

Melgaard, S.

Melgaard, S. D.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. Di Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[Crossref]

Meng, J.

Messaddeq, Y.

A. P. Rizzato, C. V. Santilli, S. H. Pulcinelli, Y. Messaddeq, and P. Hammer, “XPS study of the corrosion protection of fluorozirconate glasses dip-coated with SnO2 transparent thin films,” J. Sol-Gel Sci. Technol. 32, 155–160 (2004).
[Crossref]

A. P. Rizzato, C. V. Santilli, S. H. Pulcinelli, Y. Messaddeq, A. F. Craievich, and P. Hammer, “Study on the initial stages of water corrosion of fluorozirconate glasses,” J. Non-Cryst. Solids 348, 38–43 (2004).
[Crossref]

Messerly, M. J.

Mobini, E.

E. Mobini, M. Peysokhan, B. Abaie, M. P. Hehlen, and A. Mafi, “Spectroscopic investigation of Yb-doped silica glass for solid-state optical refrigeration,” Phys. Rev. Appl. 11, 014066 (2019).
[Crossref]

M. Peysokhan, E. Mobini, B. Abaie, and A. Mafi, “Method for measuring the resonant absorption coefficient of rare-earth-doped optical fibers,” Appl. Opt. 58, 1841–1846 (2019).
[Crossref]

E. Mobini, M. Peysokhan, and A. Mafi, “Heat mitigation of a core/cladding Yb-doped fiber amplifier using anti-Stokes fluorescence cooling,” J. Opt. Soc. Am. B 36, 2167–2177 (2019).
[Crossref]

E. Mobini, M. Peysokhan, B. Abaie, and A. Mafi, “Thermal modeling, heat mitigation, and radiative cooling for double-clad fiber amplifiers,” J. Opt. Soc. Am. B 35, 2484–2493 (2018).
[Crossref]

M. Peysokhan, E. Mobini, B. Abaie, and A. Mafi, “A non-destructive method for measuring the absorption coefficient of a Yb-doped fiber,” in Laser Science (Optical Society of America, 2018), paper JW3A–138.

M. Peysokhan, B. Abaie, E. Mobini, S. Rostami, and A. Mafi, “Measuring quantum efficiency and background absorption of an ytterbium-doped ZBLAN fiber,” in CLEO: Applications and Technology (Optical Society of America, 2018), paper JW2A–118.

Mungan, C.

C. Mungan, M. Buchwald, B. Edwards, R. Epstein, and T. Gosnell, “Laser cooling of a solid by 16  K starting from room temperature,” Phys. Rev. Lett. 78, 1030–1033 (1997).
[Crossref]

Mungan, C. E.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).
[Crossref]

Newell, T.

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M. Peysokhan, E. Mobini, B. Abaie, and A. Mafi, “A non-destructive method for measuring the absorption coefficient of a Yb-doped fiber,” in Laser Science (Optical Society of America, 2018), paper JW3A–138.

M. Peysokhan, B. Abaie, E. Mobini, S. Rostami, and A. Mafi, “Measuring quantum efficiency and background absorption of an ytterbium-doped ZBLAN fiber,” in CLEO: Applications and Technology (Optical Society of America, 2018), paper JW2A–118.

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Figures (10)

Fig. 1.
Fig. 1. (a) Experimental setup for the LITMoS test of the Yb-doped ZBLAN fiber. (b) Magnified image of the fiber holder and an illustration of the three sources of heat load: convective, conductive, and radiative.
Fig. 2.
Fig. 2. (a) Blue circles correspond to Δ(pixel) (change in the pixel value of the thermal camera image) at each wavelength, and red asterisks represent the area under the S(λ) curve. (b) Red dots represent the measurement of the cooling efficiency (ηc) for the Yb:ZBLAN fiber at different wavelengths. The solid curve shows a fitting of ηc based on Eq. (1) to the measured values, where the positive region in ηc indicates cooling.
Fig. 3.
Fig. 3. (a) Schematic of the experimental setup that is used for the MACSLA method. OSA stands for optical spectrum analyzer, LPF for long-pass filter, and MMF for multimode fiber. (b) Schematic of the propagation of the pump power in the core of the optical fiber, and collection of the spontaneous emission from the side of the Yb-doped ZBLAN fiber by two multimode passive optical fibers.
Fig. 4.
Fig. 4. (a) Emission power spectral density S(λ), measured by the optical spectrum analyzer, is plotted in arbitrary units. The inset shows the resonant absorption coefficient, which is normalized to its peak value and is calculated by using the McCumber theory. (b) The points indicate the values of r(λ) measured at different wavelengths near the peak of the resonant absorption coefficient.
Fig. 5.
Fig. 5. Schematic of the laser system and propagation of the pump power and signal in the double-cladding fiber laser. Pump power is launched at z=0, and the output signal is calculated at z=L at the power delivery port. R1(λ) and R2(λ) are the distributed Bragg reflectors at z=0 and z=L.
Fig. 6.
Fig. 6. Density plot of the optimum efficiency of the fiber laser for different pump and signal wavelengths, when the laser is pumped with 80 W of input pump power. The inset is a magnification of the density plot over the range of wavelengths, which are most relevant for an RBL system.
Fig. 7.
Fig. 7. (a) Propagation of the forward pump (FW pump), backward pump (BW pump), forward signal (FW signal), backward signal (BW signal), and temperature rise along the ZBLAN fiber for a conventional fiber laser pumped at λp=975nm. (b) Similar graph for the RBL operation pumped at λp=1030nm. Both lasers are optimized for the signal output power of 3 W at λs=1070nm for αb from Table 1. Note that the fiber in the RBL design is considerably longer than in the conventional design.
Fig. 8.
Fig. 8. Similar to Fig. 7, except the ZBLAN fiber is chosen with a 10-fold reduction in the background absorption, i.e., αb=αb/10, while maintaining the parasitic absorption of 0.01 dB/m in the cladding. The reduced value is used for both the conventional laser in (a), pumped with 3.65 W at λp=975nm; and the RBL laser in (b), pumped with 10.2 W at λp=1030nm. Both lasers are optimized for the signal output power of 3 W at λs=1070nm. The RBL design has a substantially reduced temperature performance compared with the conventional laser and the trade-off of a nearly two-fold increase in the required pump power.
Fig. 9.
Fig. 9. (a) Images of the polishing fixture and the ZBLAN doped fiber, which are glued together by the Crystalbond. (b) Initial coarse polishing steps to prepare a flat surface for further polishing. From left to right, the side and facet views of the doped fiber are shown for each step of the coarse polishing. (c) Images of the facet of the ZBLAN fiber under microscope after each fine-polishing step.
Fig. 10.
Fig. 10. Thermal camera images of the laser-pumped ZBLAN fiber. Images in subfigure (a) are for pumping at the 1030 nm wavelength and sequentially improved polishing of the facets. The brighter spots indicate heating, and as the polishing quality is improved, the facet heating is reduced. When cooling-grade polishing is reached, the facets no longer are sources of parasitic heating in subfigure (b), and transition from heating to cooling is clearly observed when the pump wavelength is switched from 975 to 1030 nm wavelength.

Tables (1)

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Table 1. Yb-Doped ZBLAN Fiber Simulation Parameters

Equations (10)

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ηc(λp)=λpλfηqηabs(λp)-1,
ηabs(λp)=αr(λp)αr(λp)+αb.
αr(λ)λ5S(λ)exp(hcλkBT),
r(λ)=ζ-αr(λ)|zB-zA|,
N2(z)N=ΓsσsaλsPs˜(z)+ΓpσpaλpPp˜(z)ΓsσsaeλsPs˜(z)+ΓpσpaeλpPp˜(z)+hcAτ-1,
±dPp±dz=-Γp[σpaN-σpaeN2(z)]Pp±(z)-αbPp±(z),
±dPs±dz=-Γs[σsaN-σsaeN2(z)]Ps±(z)-αbPs±(z),
Ps˜(z)Ps+(z)+Ps-(z),σsaeσsa+σse,Pp˜(z)Pp+(z)+Pp-(z),σpaeσpa+σpe.
dN2dt=αrIPhνp-(Wr+Wnr)N2+(1-ηe)WrN2,
Pnet=(αr+αb)IP-αrIPηq(λp/λf).

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