Abstract

This article presents the fundamental principles of operational performance of a continuous wave (cw) thin-disk laser with multiple disks in one resonator. Based on the model of an end-pumped Yb:YAG thin-disk laser with nonuniform temperature distribution, the effect of the multiple disks in one resonator is considered. The analytic expressions are derived to analyze the laser output intensity, laser intensity in the resonator, threshold intensity, and the optical efficiency of a thin-disk laser with multiple disks arranged in series. The dependence of output coupler reflectivity and the number of thin disks on various parameters are investigated, which are useful to determine the optimal output coupler reflectivity of the thin-disk lasers and control the laser intensity in the resonator.

© 2012 Optical Society of America

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References

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  1. A. Giesen, “Thin-disk solid-state lasers,” Proc. SPIE 5620, 112–127 (2004).
    [CrossRef]
  2. A. Giesen, “Results and scaling laws of thin disk lasers,” Proc. SPIE 5332, 212–227 (2004).
    [CrossRef]
  3. D. Havrilla and M. Holzer, “High power disk lasers-advances and applications,” Proc. SPIE 7912, 79120F (2011).
    [CrossRef]
  4. R. J. Beach, “CW theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123, 385–393 (1995).
    [CrossRef]
  5. G. L. Bourdet, “Theoretical investigation of quasi-three-level longitudinally pumped continuous wave lasers,” Appl. Opt. 39, 966–971 (2000).
    [CrossRef]
  6. K. Contag, S. Erhard, and A. Giesen, “Calculations of optimum design parameters for Yb:YAG thin disk lasers,” in Advanced Solid State Lasers, Technical Digest (CD) (Optical Society of America, 2000), pp. 124–130.
  7. D. A. Copeland, “Optical extraction model and optimal outcoupling for a CW quasi-three level thin disk laser,” Proc. SPIE 7912, 79120D (2011).
    [CrossRef]
  8. D. C. Brown and V. A. Vitali, “Yb:YAG kinetics model including saturation and power conservation,” IEEE J. Quantum Electron. 47, 3–12 (2011).
    [CrossRef]
  9. H. Yu, G. Bourdet, and S. Ferre, “Comprehensive modeling of the temperature-related laser performances of the amplifiers of the LUCIA laser,” Appl. Opt. 44, 6413 (2005).
    [CrossRef]
  10. A. K. Jafari and M. Aas, “Continuous-wave theory of Yb:YAG end-pumped thin-disk lasers,” Appl. Opt. 48, 106–113 (2009).
    [CrossRef]
  11. C. Lim and Y. Izawa, “Modeling of end-pumped cw quasi-three-level lasers,” IEEE J. Quantum Electron. 38, 306–311 (2002).
    [CrossRef]
  12. G. Zhu, X. Zhu, C. Zhu, J. Shang, and H. Wang, “Modeling of end-pumped Yb:YAG thin-disk lasers with nonuniform temperature distribution,” Appl. Opt. 51, 2521–2531 (2012).
    [CrossRef]
  13. Q. Liu, X. Fu, M. Gong, and L. Huang, “Effects of the temperature dependence of the absorption coefficients in edge-pumped Yb:YAG slab lasers,” J. Opt. Soc. Am. B 24, 2081–2089 (2007).
    [CrossRef]
  14. B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
    [CrossRef]
  15. M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009).
    [CrossRef]
  16. C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1 kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
    [CrossRef]
  17. J. Mende, E. Schmid, J. Speiser, G. Spindler, and A. Giesen, “Thin-disk laser-power scaling to kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
    [CrossRef]

2012

2011

D. Havrilla and M. Holzer, “High power disk lasers-advances and applications,” Proc. SPIE 7912, 79120F (2011).
[CrossRef]

D. A. Copeland, “Optical extraction model and optimal outcoupling for a CW quasi-three level thin disk laser,” Proc. SPIE 7912, 79120D (2011).
[CrossRef]

D. C. Brown and V. A. Vitali, “Yb:YAG kinetics model including saturation and power conservation,” IEEE J. Quantum Electron. 47, 3–12 (2011).
[CrossRef]

2009

A. K. Jafari and M. Aas, “Continuous-wave theory of Yb:YAG end-pumped thin-disk lasers,” Appl. Opt. 48, 106–113 (2009).
[CrossRef]

M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009).
[CrossRef]

J. Mende, E. Schmid, J. Speiser, G. Spindler, and A. Giesen, “Thin-disk laser-power scaling to kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
[CrossRef]

2007

2005

H. Yu, G. Bourdet, and S. Ferre, “Comprehensive modeling of the temperature-related laser performances of the amplifiers of the LUCIA laser,” Appl. Opt. 44, 6413 (2005).
[CrossRef]

2004

A. Giesen, “Thin-disk solid-state lasers,” Proc. SPIE 5620, 112–127 (2004).
[CrossRef]

A. Giesen, “Results and scaling laws of thin disk lasers,” Proc. SPIE 5332, 212–227 (2004).
[CrossRef]

2003

B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
[CrossRef]

2002

C. Lim and Y. Izawa, “Modeling of end-pumped cw quasi-three-level lasers,” IEEE J. Quantum Electron. 38, 306–311 (2002).
[CrossRef]

2000

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1 kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

G. L. Bourdet, “Theoretical investigation of quasi-three-level longitudinally pumped continuous wave lasers,” Appl. Opt. 39, 966–971 (2000).
[CrossRef]

1995

R. J. Beach, “CW theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123, 385–393 (1995).
[CrossRef]

Aas, M.

Bass, M.

B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
[CrossRef]

Beach, R. J.

R. J. Beach, “CW theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123, 385–393 (1995).
[CrossRef]

Bourdet, G.

H. Yu, G. Bourdet, and S. Ferre, “Comprehensive modeling of the temperature-related laser performances of the amplifiers of the LUCIA laser,” Appl. Opt. 44, 6413 (2005).
[CrossRef]

Bourdet, G. L.

Brown, D. C.

D. C. Brown and V. A. Vitali, “Yb:YAG kinetics model including saturation and power conservation,” IEEE J. Quantum Electron. 47, 3–12 (2011).
[CrossRef]

Chen, B.

B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
[CrossRef]

Chen, Y.

B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
[CrossRef]

Contag, K.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1 kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

K. Contag, S. Erhard, and A. Giesen, “Calculations of optimum design parameters for Yb:YAG thin disk lasers,” in Advanced Solid State Lasers, Technical Digest (CD) (Optical Society of America, 2000), pp. 124–130.

Copeland, D. A.

D. A. Copeland, “Optical extraction model and optimal outcoupling for a CW quasi-three level thin disk laser,” Proc. SPIE 7912, 79120D (2011).
[CrossRef]

Dong, J.

B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
[CrossRef]

Erhard, S.

K. Contag, S. Erhard, and A. Giesen, “Calculations of optimum design parameters for Yb:YAG thin disk lasers,” in Advanced Solid State Lasers, Technical Digest (CD) (Optical Society of America, 2000), pp. 124–130.

Ferre, S.

H. Yu, G. Bourdet, and S. Ferre, “Comprehensive modeling of the temperature-related laser performances of the amplifiers of the LUCIA laser,” Appl. Opt. 44, 6413 (2005).
[CrossRef]

Fu, X.

Giesen, A.

J. Mende, E. Schmid, J. Speiser, G. Spindler, and A. Giesen, “Thin-disk laser-power scaling to kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
[CrossRef]

A. Giesen, “Thin-disk solid-state lasers,” Proc. SPIE 5620, 112–127 (2004).
[CrossRef]

A. Giesen, “Results and scaling laws of thin disk lasers,” Proc. SPIE 5332, 212–227 (2004).
[CrossRef]

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1 kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

K. Contag, S. Erhard, and A. Giesen, “Calculations of optimum design parameters for Yb:YAG thin disk lasers,” in Advanced Solid State Lasers, Technical Digest (CD) (Optical Society of America, 2000), pp. 124–130.

Golpaygani, A. H.

M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009).
[CrossRef]

Gong, M.

Havrilla, D.

D. Havrilla and M. Holzer, “High power disk lasers-advances and applications,” Proc. SPIE 7912, 79120F (2011).
[CrossRef]

Holzer, M.

D. Havrilla and M. Holzer, “High power disk lasers-advances and applications,” Proc. SPIE 7912, 79120F (2011).
[CrossRef]

Huang, L.

Hugel, H.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1 kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Izawa, Y.

C. Lim and Y. Izawa, “Modeling of end-pumped cw quasi-three-level lasers,” IEEE J. Quantum Electron. 38, 306–311 (2002).
[CrossRef]

Jafari, A. K.

Kar, A.

B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
[CrossRef]

Larionov, M.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1 kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Lim, C.

C. Lim and Y. Izawa, “Modeling of end-pumped cw quasi-three-level lasers,” IEEE J. Quantum Electron. 38, 306–311 (2002).
[CrossRef]

Liu, Q.

Mende, J.

J. Mende, E. Schmid, J. Speiser, G. Spindler, and A. Giesen, “Thin-disk laser-power scaling to kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
[CrossRef]

Najafi, M.

M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009).
[CrossRef]

Patel, M.

B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
[CrossRef]

Sabbaghzadeh, J.

M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009).
[CrossRef]

Schmid, E.

J. Mende, E. Schmid, J. Speiser, G. Spindler, and A. Giesen, “Thin-disk laser-power scaling to kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
[CrossRef]

Sepehr, A.

M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009).
[CrossRef]

Shang, J.

Speiser, J.

J. Mende, E. Schmid, J. Speiser, G. Spindler, and A. Giesen, “Thin-disk laser-power scaling to kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
[CrossRef]

Spindler, G.

J. Mende, E. Schmid, J. Speiser, G. Spindler, and A. Giesen, “Thin-disk laser-power scaling to kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
[CrossRef]

Stewen, C.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1 kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Vitali, V. A.

D. C. Brown and V. A. Vitali, “Yb:YAG kinetics model including saturation and power conservation,” IEEE J. Quantum Electron. 47, 3–12 (2011).
[CrossRef]

Wang, H.

Yu, H.

H. Yu, G. Bourdet, and S. Ferre, “Comprehensive modeling of the temperature-related laser performances of the amplifiers of the LUCIA laser,” Appl. Opt. 44, 6413 (2005).
[CrossRef]

Zhu, C.

Zhu, G.

Zhu, X.

Appl. Opt.

IEEE J. Quantum Electron.

D. C. Brown and V. A. Vitali, “Yb:YAG kinetics model including saturation and power conservation,” IEEE J. Quantum Electron. 47, 3–12 (2011).
[CrossRef]

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hugel, “A 1 kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

C. Lim and Y. Izawa, “Modeling of end-pumped cw quasi-three-level lasers,” IEEE J. Quantum Electron. 38, 306–311 (2002).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009).
[CrossRef]

R. J. Beach, “CW theory of quasi-three level end-pumped laser oscillators,” Opt. Commun. 123, 385–393 (1995).
[CrossRef]

Proc. SPIE

D. A. Copeland, “Optical extraction model and optimal outcoupling for a CW quasi-three level thin disk laser,” Proc. SPIE 7912, 79120D (2011).
[CrossRef]

A. Giesen, “Thin-disk solid-state lasers,” Proc. SPIE 5620, 112–127 (2004).
[CrossRef]

A. Giesen, “Results and scaling laws of thin disk lasers,” Proc. SPIE 5332, 212–227 (2004).
[CrossRef]

D. Havrilla and M. Holzer, “High power disk lasers-advances and applications,” Proc. SPIE 7912, 79120F (2011).
[CrossRef]

B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
[CrossRef]

J. Mende, E. Schmid, J. Speiser, G. Spindler, and A. Giesen, “Thin-disk laser-power scaling to kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
[CrossRef]

Other

K. Contag, S. Erhard, and A. Giesen, “Calculations of optimum design parameters for Yb:YAG thin disk lasers,” in Advanced Solid State Lasers, Technical Digest (CD) (Optical Society of America, 2000), pp. 124–130.

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Figures (9)

Fig. 1.
Fig. 1.

Schematic drawing of the thin-disk crystal divided along thickness with equal space.

Fig. 2.
Fig. 2.

Laser radiation propagating in W-shaped resonator for a thin-disk laser with multiple-disk configuration.

Fig. 3.
Fig. 3.

Schematic of the pump optical rays bouncing inside the thin-disk crystal.

Fig. 4.
Fig. 4.

Corresponding orientation of axes and thermal boundary conditions of a thin-disk module.

Fig. 5.
Fig. 5.

Stable temperature distribution inside the thin-disk module.

Fig. 6.
Fig. 6.

Laser output intensity and laser intensity in the resonator as a function of output coupler reflectivity for a different number of thin disks in one resonator: (Rr,P=98%, Rf,P=99.5%, RCT,L=99.8%, Rd,L=99%, RRM,L=99.5%).

Fig. 7.
Fig. 7.

Dependence of laser output intensity and output coupler reflectivity for different pump intensities and a different number of thin disks in one resonator.

Fig. 8.
Fig. 8.

Laser output intensity, threshold intensity, and optical-to-optical efficiency as a function of pump intensity for a different number of thin disks at the optimal value of coupler reflectivity.

Fig. 9.
Fig. 9.

Laser output intensity as a function of output coupler reflectivity for various reflectivity of connection mirrors.

Tables (1)

Tables Icon

Table 1. Basic Parameters of Each Thin-Disk Module Used for This Model

Equations (41)

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

gj1=gj2=gj3=gjk=gjNuD=gj,
{IL+(OC)==Rd,LNuD·Rr,LNuD1·IL+(RM)·exp[NuD×2·(j=1MgjΔl)]IL(RM)==Rd,LNuD·Rr,LNuD1·IL(OC)·exp[NuD×2·(j=1MgjΔl)],
{IL(OC)=IL+(OC)·ROC,LIL+(RM)=IL(RM)·RRM,L,
j=1M(gjΔl)=12·NuDln(1Rd,LNuD·Rr,LNuD1·ROC,LRRM,L).
{Ip,i(L)=IP,0·Rr,Pi1·Rf,Pi1·exp[2(i1)j=1MδjΔl]Ip,i+(0)=IP,0·Rf,Pexp(j=1MδjΔl){Rr,Pi1·Rf,Pi1·exp[2(i1)j=1MδjΔl]}(IP,1(L)=IP,0),
{IP,i+(0)=Rf,PIP,i(0)IP,i+1(L)=Rr,PIP,i+(L),
{2TYAGr2+1rTYAGr+2TYAGz2=QkYAGH(rPr)TYAG(r,z)r|r=R=0TYAG(r,z)z|z=z0=0kYAGTYAG(r,z)z|z=0=kCuWTCuW(r,z)z|z=0(for disk crystal),
{2TCuWr2+1rTCuWr+2TCuWz2=0TCuW(r,z)r|r=R=0TCuW(r,z)z|z=z1=hkCuW[TCuW(r,0)Tf](forCuWheat sink),
H(rPr)={10<r<rP0rP<r<R.
TYAG(r,0)=TCuW(r,0).
TYAG(r,z)=A0+B0z+n=1[Anexp(xn(0)Rz)+Bnexp(xn(0)Rz)]·J0(xn(0)Rr)12z2[G0+n=1Gn·J0(xn(0)Rr)],
TCuW(r,z)=Tf+A0+B0z+n=1[Anexp(xn(0)Rz)+Bnexp(xn(0)Rz)]·J0(xn(0)Rr),
A0=Tf+A0=Tf+(1+hz1kCuW)z0QrP2hR2;B0=z0Qk2R20rPrdr=z0QrP2kYAGR2A0=kCuWh(1+hz1kCuW)B0=(1+hz1kCuW)z0QrP2hR2;B0=kYAGkCuWB0=z0QrP2kCuWR2An=2fn(hHnbkCuWEnb)(kCuWEnahHna){[(1+kCuWkYAG)(hHnbkCuWEnb)(kCuWEnahHna)+(1kCuWkYAG)]Ena+[(1kCuWkYAG)(hHnbkCuWEnb)(kCuWEnahHna)+(1+kCuWkYAG)]Enb}Bn=2fn{[(1+kCuWkYAG)(hHnbkCuWEnb)(kCuWEnahHna)+(1kCuWkYAG)]Ena+[(1kCuWkYAG)(hHnbkCuWEnb)(kCuWEnahHna)+(1+kCuWkYAG)]Enb}An=12[(1+kCuWkYAG)An+(1kCuWkYAG)Bn]=Bn2[(1+kCuWkYAG)(hHnbkCuWEnb)(kCuWEnahHna)+(1kCuWkYAG)]Bn=12[(1kCuWkYAG)An+(1+kCuWkYAG)Bn]=Bn2[(1kCuWkYAG)(hHnbkCuWEnb)(kCuWEnahHna)+(1+kCuWkYAG)]G0=QrP2kYAGR2Gn=2QrPkYAG[J0(xn(0))]2[xn(0)]RJ1(xn(0)RrP).
fAij=exp(EAiKTj)q=13exp(EAqKTj),
fZij=exp(EZiKTj)q=03exp(EZqKTj),
σL(1030,T)=[0.95334+33.608exp(T92.82465)]×1020cm2,
σP(941,T)=[0.207+0.637exp(T273288)]×1020cm2,
N1j(t,z)t=N0j(t,z)t=σPj(fojPN0jf1jPN1j)Pj+σLj(fojLN0jf1jLN1j)LjN1jτj,
ILj,i(z)z=±σLj(fojLN0j(z)f1jLN1j(z))ILj,i(z),
IPj,i(z)z=±σPj(fojPN0j(z)f1jPN1j(z))IPj,i(z),
Pj=i=1N(IPj,i++IPj,i)hvPLj2·(ILj++ILj)hvL,
N0j(z)=σPjf1jPPj+σLjf1jLLj+1τjσPj(f0jP+f1jP)Pj+σLj(f0jL+f1jL)Lj+1τjNt,
N1j(z)=σPjf0jPPj+σLjf0jLLjσPj(f0jP+f1jP)Pj+σLj(f0jL+f1jL)Lj+1τjNt,
σLj(fojL+f1jL)1IPj,i±(z)dIPj,i±(z)dzσPj(fojP+f1jP)1ILj±(z)dILj±(z)dz=±NtσLjσPj(f0jLf1jPf1jLf0jP),
σLj(f0jLN0jf1jL×N1j)dIPj,i±(z)IPj,i±(z)=σPj(f0jPN0jf1jP×N1j)dILj±(z)ILj±(z).
Δj=(f0jLf1jPf1jLf0jP).
0L(δjΔl)=j=1M(δjΔl)=j=1M[σPj(fojP+f1jP)σLj(fojL+f1jL)gjΔl]+j=1M[σPjΔj(fojL+f1jL)NtΔl]=σPFPσLFLj=1M(gjΔl)+NtFLj=1M(σPjΔjΔl)=σPFPσLFL12·NuDln(1Rd,LNuD·Rr,LNuD1·ROC,LRRM,L)+NtFLj=1M(σPjΔjΔl),
1ΔjfojLσPjτjdIPj,i±(z)IPj,i±(z)+i=1N(IPj,i++IPj,i)hvPdIPj,i±(z)IPj,i±(z)=1ΔjfojPσLjτjdILj±(z)ILj±(z)2(ILj++ILj)hvLdILj±(z)ILj±(z).
Aj,k+=i=1N(IPj,i++IPj,i)hvPdIPj,i+(z)IPj,i+(z)=i=1N1hvP(1+IPj,iIPj,i+)dIPj,i+(z),
Bj,k+=2(ILj++ILj)hvLdILj±(z)ILj±(z)=2hvL(1+ILjILj,i+)dILj+(z),
Cj,k+=1Δj(fojLσPjτjdIPj,i+(z)IPj,i+(z)fojPσLjτjdILj+(z)ILj+(z)).
Aj,k++Cj,k+=Bj,k+.
NuD×Ak++NuD×Ck+=NuD×Bk+=k=1NuD(Bk+).
Ak+=2·j=1M(IP+(lj1)IP+(lj)Aj,k+)=2hvPi=1N[IP,i+(L)IP,i+(0)+IP,i(0)IP,i(L)]=2hvPIP,0i=1N{Rr,Pi1·Rf,Pi1·exp[2(i1)j=1MδjΔl]}·(exp(j=1MδjΔl)Rf,P+1)·(exp(j=1MδjΔl)1),
k=1NuD(Bk+)=k=1NuD[L+L(IL+(lj1)IL+(lj)Bj,k+)]=2hvL{(1ROC,LRRM,L)(1+ROC,LRRM,L)+[(1Rr)(NuD1)+(1Rd)·NuD](1+ROC,L)}·IL+(OC),
Ck+=LL(IL+(lj)IL+(lj+1)Cj,k+)=LL(N1j(z)τjΔl)=Δlτ{1σLAvg·FL[LLσLjf0jLNt+12·NuD1Δlln(1Rd,LNuD·Rr,LNuD1·ROC,LRRM,L)]}=Δlτ{1σLAvg·FL[2·j=1MσLjf0jLNt+12·NuD1Δlln(1Rd,LNuD·Rr,LNuD1·ROC,LRRM,L)]}.
IL+(OC)=AB(NuD·IP,0+CA),
Iout=(1ROC,L)IL+(OC)=ηslope(NuD·IP,0Ith),
ηslope=AB(1ROC,L)=vLvP(1ROC,L)·i=1N{Rr,Pi1·Rf,Pi1·exp[2(i1)j=1MδjΔl]}·(exp(j=1MδjΔl)Rf,P+1)·(exp(j=1MδjΔl)1){(1ROC,LRRM,L)(1+ROC,LRRM,L)+[(1Rr)(NuD1)+(1Rd)·NuD](1+ROC,L)},
Ith=CA=hvPΔlτ·{1σLAvg·FL[2·NuD·j=1MσLjf0jLNt+12Δlln(1Rd,LNuD·Rr,LNuD1·ROC,LRRM,L)]}2×i=1N{Rr,Pi1·Rf,Pi1·exp[2(i1)j=1MδjΔl]}·(exp(j=1MδjΔl)Rf,P+1)·(exp(j=1MδjΔl)1).
IintI+(z)+I(z)=(1+ROC,L)·IL+(OC)=(1+ROC,L)(1ROC,L)·Iout.

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