Abstract

In this communication, we present an analytic expression of the thermal load in a cylindrical laser rod. We consider a pump beam with Gaussian temporal and spatial profile, which permits, using superposition of the single pulse solution, an explicit calculation of the optical path length difference across the radial direction of the rod and of the transient thermal focal length changes for a variable pump repetition rate and pulse width. We have chosen to model Ti:Al2O3 as a specific example, however our solution is completely general and can be applied to any materials with cylindrical geometry employing a stable laser cavity design.

© 2015 Optical Society of America

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References

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  1. V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
    [Crossref]
  2. F. Salin, C. L. Blanc, J. Squier, and C. Barty, “Thermal eigenmode amplifiers for diffraction-limited amplification of ultrashort pulses,” Opt. Lett. 23(9), 718–720 (1998).
    [Crossref] [PubMed]
  3. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, 1988), Chap. 7.
  4. R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34(6), 1046–1053 (1998).
    [Crossref]
  5. R. Lausten and P. Balling, “Thermal lensing in pulsed laser amplifiers: an analytical model,” J. Opt. Soc. Am. B 20(7), 1479–1485 (2003).
    [Crossref]
  6. U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state saser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
    [Crossref]
  7. H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford Univ., 1948), pp. 191.
  8. P. F. Moulton, “Spectroscopic and laser characteristics of Ti:Al2O3,” J. Opt. Soc. Am. B 3(1), 125–133 (1986).
    [Crossref]
  9. M. L. Boas, Mathematical Methods in the Physical Sciences (John Wiley & Sons, 1993).
  10. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1998), Chap. 4.
  11. G. N. Watson, Theory of Bessel Functions 2nd edition (Cambridge University, 1922), pp. 393.
  12. P. A. Schulz and S. R. Henion, “Liquid-nitrogen-cooled Ti:Al2O3 laser,” IEEE J. Quantum Electron. 27(4), 1039–1047 (1991).
    [Crossref]
  13. E. Wyss, M. Roth, T. Graf, and H. P. Weber, “Thermooptical compensation methods for high-power lasers, “IEEE,” Quantum Electron. 38(12), 1620–1628 (2002).
    [Crossref]

2006 (1)

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

2003 (1)

2002 (1)

E. Wyss, M. Roth, T. Graf, and H. P. Weber, “Thermooptical compensation methods for high-power lasers, “IEEE,” Quantum Electron. 38(12), 1620–1628 (2002).
[Crossref]

1998 (2)

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34(6), 1046–1053 (1998).
[Crossref]

F. Salin, C. L. Blanc, J. Squier, and C. Barty, “Thermal eigenmode amplifiers for diffraction-limited amplification of ultrashort pulses,” Opt. Lett. 23(9), 718–720 (1998).
[Crossref] [PubMed]

1991 (1)

P. A. Schulz and S. R. Henion, “Liquid-nitrogen-cooled Ti:Al2O3 laser,” IEEE J. Quantum Electron. 27(4), 1039–1047 (1991).
[Crossref]

1988 (1)

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state saser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

1986 (1)

Balling, P.

Barty, C.

Blanc, C. L.

Buoncristiani, A. M.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state saser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

Byvik, C. E.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state saser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

Farrukh, U. O.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state saser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

Graf, T.

E. Wyss, M. Roth, T. Graf, and H. P. Weber, “Thermooptical compensation methods for high-power lasers, “IEEE,” Quantum Electron. 38(12), 1620–1628 (2002).
[Crossref]

Henion, S. R.

P. A. Schulz and S. R. Henion, “Liquid-nitrogen-cooled Ti:Al2O3 laser,” IEEE J. Quantum Electron. 27(4), 1039–1047 (1991).
[Crossref]

Lausten, R.

Lee, J.

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

Mac Donald, M.

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34(6), 1046–1053 (1998).
[Crossref]

Malphurs, W.

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

Moulton, P. F.

Neuenschwander, B.

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34(6), 1046–1053 (1998).
[Crossref]

Ramanathan, V.

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

Reitze, D. H.

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

Roos, M. B.

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34(6), 1046–1053 (1998).
[Crossref]

Roth, M.

E. Wyss, M. Roth, T. Graf, and H. P. Weber, “Thermooptical compensation methods for high-power lasers, “IEEE,” Quantum Electron. 38(12), 1620–1628 (2002).
[Crossref]

Salin, F.

Schulz, P. A.

P. A. Schulz and S. R. Henion, “Liquid-nitrogen-cooled Ti:Al2O3 laser,” IEEE J. Quantum Electron. 27(4), 1039–1047 (1991).
[Crossref]

Squier, J.

Wang, X.

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

Weber, H. P.

E. Wyss, M. Roth, T. Graf, and H. P. Weber, “Thermooptical compensation methods for high-power lasers, “IEEE,” Quantum Electron. 38(12), 1620–1628 (2002).
[Crossref]

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34(6), 1046–1053 (1998).
[Crossref]

Weber, R.

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34(6), 1046–1053 (1998).
[Crossref]

Williams, L.

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

Wyss, E.

E. Wyss, M. Roth, T. Graf, and H. P. Weber, “Thermooptical compensation methods for high-power lasers, “IEEE,” Quantum Electron. 38(12), 1620–1628 (2002).
[Crossref]

Xu, S.

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

IEEE J. Quantum Electron. (3)

R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinally diode laser-pumped Nd:YAG rods,” IEEE J. Quantum Electron. 34(6), 1046–1053 (1998).
[Crossref]

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state saser rods,” IEEE J. Quantum Electron. 24(11), 2253–2263 (1988).
[Crossref]

P. A. Schulz and S. R. Henion, “Liquid-nitrogen-cooled Ti:Al2O3 laser,” IEEE J. Quantum Electron. 27(4), 1039–1047 (1991).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Quantum Electron. (1)

E. Wyss, M. Roth, T. Graf, and H. P. Weber, “Thermooptical compensation methods for high-power lasers, “IEEE,” Quantum Electron. 38(12), 1620–1628 (2002).
[Crossref]

Rev. Sci. Instrum. (1)

V. Ramanathan, J. Lee, S. Xu, X. Wang, L. Williams, W. Malphurs, and D. H. Reitze, “Analysis of thermal aberrations in a high average power single-stage Ti:sapphire regenerative chirped pulse amplifier: Simulation and experiment,” Rev. Sci. Instrum. 77(10), 103103 (2006).
[Crossref]

Other (5)

H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford Univ., 1948), pp. 191.

W. Koechner, Solid-State Laser Engineering (Springer-Verlag, 1988), Chap. 7.

M. L. Boas, Mathematical Methods in the Physical Sciences (John Wiley & Sons, 1993).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1998), Chap. 4.

G. N. Watson, Theory of Bessel Functions 2nd edition (Cambridge University, 1922), pp. 393.

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Figures (3)

Fig. 1
Fig. 1 (a) Thermal relaxation of the laser rod of a single input pulse in a cylindrical geometry (0.7-cm long, 0.25-cm radius). For this calculation, we used a time duration, τ=160ns and a pulse energy of 10mJ. (b) Longitudinal temperature distribution shows the initial exponential (rd curve) and temperature distribution at various times at r = 0.
Fig. 2
Fig. 2 Thermal buildup when the pulse interval time is less than the thermal relaxation time. For the calculation: (a) three respectively pulsed lasers are introduced to the laser rod with 20μs pulse separation; (b) five thousand pulsed lasers are introduced to the laser with 5-kHz repetition rate. All parameters for the calculation are the same as those given for Fig. 1.
Fig. 3
Fig. 3 (a) Transient focal length change of one single pulse for 1 ms (solid red), 1 μs (solid green) and 1 ns (solid blue) time duration. Focal length changes of (b) 100 Hz and (c) 5 kHz repetitively pulsed pump laser input for a 1ms, 10mJ pulse. All of the parameters used in this calculation except for the pulse duration and the repetition rate are equal for all three cases and identical to those used in Fig. 1.

Equations (38)

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C T t κ 2 T=S(r,t)
S(r,t)=N e 2 r 2 ω 2 e αz e 2 (tΔt) 2 τ h 2
0 l N e 2 r 2 ω 2 e αz e 2 (tΔt) 2 τ h 2 dxdydzdt =N ω 2 π 2 1 e αl α π 2 τ h =A
N= 2A π ω 2 τ h α 1 e αl 2 π
S(r,t)= 2A π ω 2 τ h α 1 e αl 2 π e 2 r 2 ω 2 e αz [ e 2 (tΔt) 2 τ h 2 + e 2 (t2Δt) 2 τ h 2 + e 2 (t3Δt) 2 τ h 2 + e 2 (t4Δt) 2 τ h 2 ] = 2A π ω 2 τ h α 1 e αl 2 π e 2 r 2 ω 2 e αz q=1 e 2 (tqΔt) 2 τ h 2
T(r=a,z,t)=Constant
dT dz | z=0,l =0
C T(r,z,t) t κ( 2 r 2 + 1 r r + 2 z 2 )T(r,z,t)= 2A π ω 2 τ h α 1 e αl 2 π e 2 r 2 ω 2 e αz e 2 (tΔt) 2 τ h 2
1 r d dr [r d J 0 ( k n r a ) dr ]+ k n 2 a 2 J 0 ( k n r a )=0
0 a r J 0 ( k n r a ) J 0 ( k m r a )dr= a 2 2 J 1 ( k n ) 2 δ nm = a 2 2 J 0 ( k n ) 2 δ nm
T(r,z,t)= m=0 n=1 A n J 0 ( k n r a ) [ B m cos( l m z)+ C m sin( l m z)] φ nm (t)
C T(r,z,t) t κ 2 T(r,z,t) = n=1 A n J 0 ( k n r a ) m=0 B m cos( mπ l z)(C d φ nm (t) dt +κ k n 2 a 2 φ nm (t)+κ ( mπ l ) 2 φ nm (t)) = 2A π ω 2 τ h α 1 e αl 2 π e 2 r 2 ω 2 e αz e 2 (tΔt) 2 τ h 2
e 2 r 2 ω 2 = n=1 C n J 0 ( k n r a ) C n = 2 a 2 J 1 ( k n ) 2 0 a e 2 r 2 ω 2 r J 0 ( k n r a )dr
C n ω 2 2 a 2 J 1 (k ) n 2 e ω 2 k n 2 8 a 2
0 l cos( mπ l z)cos( nπ l z)dz= l 2 δ mn m=n0, =l δ 0n m=n=0
e αz = m=0 D m cos( mπ l z)
D 0 = e αl 1 αl D m = 2(cos(mπ) e αl 1) αl(1+ ( mπ αl ) 2 )
A n = 2A π ω 2 τ h α 1 e αl C n
B n = D n
(C d φ mn (t) dt +κ k n 2 a 2 φ mn (t)+κ ( mπ l ) 2 φ mn (t))= 2 π e 2 (tΔt) 2 τ h 2
C d φ nm (t) dt + κ nm φ nm (t)= 2 π e 2 (tΔt) 2 τ h 2
d e κ nm C t φ nm (t) dt = e κ nm C t C 2 π e 2 (tΔt) 2 τ h 2
φ nm (t)= e κ nm C t 1 C 2 π t= t=t [ e κ nm C t e 2 (tΔt) 2 τ h 2 ]dt
φ nm (t)= e κ nm C t C 2 π e τ h 2 8 ( κ nm C + 4Δt τ h 2 ) 2 2Δ t 2 τ h 2 t= t=t [ e 2 τ h 2 (t τ h 2 4 ( κ nm C + 4Δt τ h 2 )) 2 ]dt
φ nm (t)= τ h C π e κ nm C t e τ h 2 8 ( κ nm C + 4Δt τ h 2 ) 2 2Δ t 2 τ h 2 t = t = 2 τ h (t τ h 2 4 ( κ nm C + 4Δt τ h 2 )) e t 2 dt .
erf(t)= 2 π 0 t e t 2 dt
φ nm (t)= τ h 2C e κ nm C t e τ h 2 8 ( κ nm C + 4Δt τ h 2 ) 2 2Δ t 2 τ h 2 [(1θ( t ))(1erf( t ))+θ( t )(1+erf( t ))]
T(r,z,t)= m=0 n=1 A n J 0 ( k n r a ) B m cos( mπ l z) φ nm (t)
A n = 2A π ω 2 τ h α 1 e αl C n
B n = D n
φ nm (t)= τ h 2C e κ nm C t e τ h 2 8 ( κ nm C + 4Δt τ h 2 ) 2 2Δ t 2 τ h 2 [(1θ( t ))(1erf( t ))+θ( t )(1+erf( t ))]
φ mn (t) | general = φ mn (t)+ φ mn (tΔt)+ φ mn (t2Δt)+= q=0 φ mn (tqΔt)
l(r,t)= z=0 l n(T(r,z,t))dz
n(T)n( T C )+ n T | T C (T T C )
Δl(r,t)= n T | T C z=0 l [T(r,z,t)T(r=0,z,t)]dz
z=0 l T(r,z,t)dz = n=1 A n J 0 ( k n r a ) 1 e αl α φ n0 (t)
Δl(r,t) n T | T C n=1 A n ((1 ( k n r 2a ) 2 J 0 (0)) 1 e αl α φ n0 (t) ) = r 2 n T | T C n=1 αA (1 e αl ) a 2 π τ h J 1 (k ) n 2 e ω 2 k n 2 8 a 2 ( k n 2a ) 2 1 e αl α φ n0 (t) = r 2 n T | T C n=1 A k n 2 e ω 2 k n 2 8 a 2 φ n0 (t) 4 a 4 π τ h J 1 (k ) n 2
f= 1 n T | T C n=1 A k n 2 e ω 2 k n 2 8 a 2 φ n0 (t)(n1) 2 a 4 π τ h J 1 (k ) n 2

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