Abstract

Powerful, long-pulse lasers have a variety of applications. In many applications, optical elements are employed to direct, focus, or collimate the beam. Typically the optic is suspended in a gaseous environment (e.g., air) and can cool by convection. The variation of the optic temperature with time is obtained by combining the effects of laser heating, thermal conduction, and convective loss. Characteristics of the solutions in terms of the properties of the optic material, laser beam parameters, and the environment are discussed and compared with measurements at the Naval Research Laboratory, employing kW-class, 1 µm wavelength, continuous wave lasers and optical elements made of fused silica or BK7 glass. The calculated results are in good agreement with the measurements, given the approximations in the analysis and the expected variation in the absorption coefficients of the glasses used in the experiments.

© 2012 Optical Society of America

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References

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  1. J. Peñano, P. Sprangle, A. Ting, R. Fischer, B. Hafizi, and P. Serafim, “Optical quality of high-power laser beams in lenses,” J. Opt. Soc. Am. B 26, 503–510 (2009).
    [CrossRef]
  2. L. D. Landau and E. M. Lifshitz, “Thermal conduction in an incompressible fluid,” in Fluid Mechanics (Pergamon, 1975).
  3. L. D. Landau and E. M. Lifshitz, “The general equation of heat transfer,” in Fluid Mechanics (Pergamon, 1975).
  4. L. D. Landau and E. M. Lifshitz, “Free convection,” in Fluid Mechanics (Pergamon, 1975).
  5. A. Nakayama and H. Koyama, “An integral method for free convection from a vertical heated surface in a thermally stratified porous medium,” Wärme-Stoffübertrag 21, 297–300 (1987).
    [CrossRef]
  6. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Use of the equations of change to solve steady-state problems” in Transport Phenomena (Wiley, 2002).
  7. Y. Çengel and R. H. Turner, “Equation of motion and the Grashof number,” Fundamentals of Thermal-Fluid Sciences (McGraw-Hill, 2005).
  8. S. Ostrach, “An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force,” National Advisory Committee for Aeronautics Report Number 1111 (1953).
  9. P. Sprangle, A. Ting, J. Peñano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
    [CrossRef]
  10. V. Loriette and C. Boccara, “Absorption of low-loss optical materials measured at 1064 nm by a position-modulated ollinear photothermal detection technique,” Appl. Opt. 42, 649–656 (2003).
    [CrossRef]
  11. S. R. Nersisyan, N. V. Tabiryan, and C. M. Stickley, “Characterization of glass and high-power near-infrared CW laser beams using nonlinear optical techniques,” Opt. Eng. 45, 104301 (2006).
    [CrossRef]

2009 (2)

P. Sprangle, A. Ting, J. Peñano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

J. Peñano, P. Sprangle, A. Ting, R. Fischer, B. Hafizi, and P. Serafim, “Optical quality of high-power laser beams in lenses,” J. Opt. Soc. Am. B 26, 503–510 (2009).
[CrossRef]

2006 (1)

S. R. Nersisyan, N. V. Tabiryan, and C. M. Stickley, “Characterization of glass and high-power near-infrared CW laser beams using nonlinear optical techniques,” Opt. Eng. 45, 104301 (2006).
[CrossRef]

2003 (1)

1987 (1)

A. Nakayama and H. Koyama, “An integral method for free convection from a vertical heated surface in a thermally stratified porous medium,” Wärme-Stoffübertrag 21, 297–300 (1987).
[CrossRef]

Bird, R. B.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Use of the equations of change to solve steady-state problems” in Transport Phenomena (Wiley, 2002).

Boccara, C.

Çengel, Y.

Y. Çengel and R. H. Turner, “Equation of motion and the Grashof number,” Fundamentals of Thermal-Fluid Sciences (McGraw-Hill, 2005).

Fischer, R.

P. Sprangle, A. Ting, J. Peñano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

J. Peñano, P. Sprangle, A. Ting, R. Fischer, B. Hafizi, and P. Serafim, “Optical quality of high-power laser beams in lenses,” J. Opt. Soc. Am. B 26, 503–510 (2009).
[CrossRef]

Hafizi, B.

J. Peñano, P. Sprangle, A. Ting, R. Fischer, B. Hafizi, and P. Serafim, “Optical quality of high-power laser beams in lenses,” J. Opt. Soc. Am. B 26, 503–510 (2009).
[CrossRef]

P. Sprangle, A. Ting, J. Peñano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Koyama, H.

A. Nakayama and H. Koyama, “An integral method for free convection from a vertical heated surface in a thermally stratified porous medium,” Wärme-Stoffübertrag 21, 297–300 (1987).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, “The general equation of heat transfer,” in Fluid Mechanics (Pergamon, 1975).

L. D. Landau and E. M. Lifshitz, “Free convection,” in Fluid Mechanics (Pergamon, 1975).

L. D. Landau and E. M. Lifshitz, “Thermal conduction in an incompressible fluid,” in Fluid Mechanics (Pergamon, 1975).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, “Thermal conduction in an incompressible fluid,” in Fluid Mechanics (Pergamon, 1975).

L. D. Landau and E. M. Lifshitz, “Free convection,” in Fluid Mechanics (Pergamon, 1975).

L. D. Landau and E. M. Lifshitz, “The general equation of heat transfer,” in Fluid Mechanics (Pergamon, 1975).

Lightfoot, E. N.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Use of the equations of change to solve steady-state problems” in Transport Phenomena (Wiley, 2002).

Loriette, V.

Nakayama, A.

A. Nakayama and H. Koyama, “An integral method for free convection from a vertical heated surface in a thermally stratified porous medium,” Wärme-Stoffübertrag 21, 297–300 (1987).
[CrossRef]

Nersisyan, S. R.

S. R. Nersisyan, N. V. Tabiryan, and C. M. Stickley, “Characterization of glass and high-power near-infrared CW laser beams using nonlinear optical techniques,” Opt. Eng. 45, 104301 (2006).
[CrossRef]

Ostrach, S.

S. Ostrach, “An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force,” National Advisory Committee for Aeronautics Report Number 1111 (1953).

Peñano, J.

J. Peñano, P. Sprangle, A. Ting, R. Fischer, B. Hafizi, and P. Serafim, “Optical quality of high-power laser beams in lenses,” J. Opt. Soc. Am. B 26, 503–510 (2009).
[CrossRef]

P. Sprangle, A. Ting, J. Peñano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Serafim, P.

Sprangle, P.

J. Peñano, P. Sprangle, A. Ting, R. Fischer, B. Hafizi, and P. Serafim, “Optical quality of high-power laser beams in lenses,” J. Opt. Soc. Am. B 26, 503–510 (2009).
[CrossRef]

P. Sprangle, A. Ting, J. Peñano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Stewart, W. E.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Use of the equations of change to solve steady-state problems” in Transport Phenomena (Wiley, 2002).

Stickley, C. M.

S. R. Nersisyan, N. V. Tabiryan, and C. M. Stickley, “Characterization of glass and high-power near-infrared CW laser beams using nonlinear optical techniques,” Opt. Eng. 45, 104301 (2006).
[CrossRef]

Tabiryan, N. V.

S. R. Nersisyan, N. V. Tabiryan, and C. M. Stickley, “Characterization of glass and high-power near-infrared CW laser beams using nonlinear optical techniques,” Opt. Eng. 45, 104301 (2006).
[CrossRef]

Ting, A.

J. Peñano, P. Sprangle, A. Ting, R. Fischer, B. Hafizi, and P. Serafim, “Optical quality of high-power laser beams in lenses,” J. Opt. Soc. Am. B 26, 503–510 (2009).
[CrossRef]

P. Sprangle, A. Ting, J. Peñano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

Turner, R. H.

Y. Çengel and R. H. Turner, “Equation of motion and the Grashof number,” Fundamentals of Thermal-Fluid Sciences (McGraw-Hill, 2005).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

P. Sprangle, A. Ting, J. Peñano, R. Fischer, and B. Hafizi, “Incoherent combining and atmospheric propagation of high-power fiber lasers for directed-energy applications,” IEEE J. Quantum Electron. 45, 138–148 (2009).
[CrossRef]

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

Opt. Eng. (1)

S. R. Nersisyan, N. V. Tabiryan, and C. M. Stickley, “Characterization of glass and high-power near-infrared CW laser beams using nonlinear optical techniques,” Opt. Eng. 45, 104301 (2006).
[CrossRef]

Wärme-Stoffübertrag (1)

A. Nakayama and H. Koyama, “An integral method for free convection from a vertical heated surface in a thermally stratified porous medium,” Wärme-Stoffübertrag 21, 297–300 (1987).
[CrossRef]

Other (6)

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Use of the equations of change to solve steady-state problems” in Transport Phenomena (Wiley, 2002).

Y. Çengel and R. H. Turner, “Equation of motion and the Grashof number,” Fundamentals of Thermal-Fluid Sciences (McGraw-Hill, 2005).

S. Ostrach, “An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force,” National Advisory Committee for Aeronautics Report Number 1111 (1953).

L. D. Landau and E. M. Lifshitz, “Thermal conduction in an incompressible fluid,” in Fluid Mechanics (Pergamon, 1975).

L. D. Landau and E. M. Lifshitz, “The general equation of heat transfer,” in Fluid Mechanics (Pergamon, 1975).

L. D. Landau and E. M. Lifshitz, “Free convection,” in Fluid Mechanics (Pergamon, 1975).

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Figures (8)

Fig. 1.
Fig. 1.

Collimated laser beam propagating along the z -axis is incident on focusing optic. The y -axis is along the vertical and g = g e y is the gravitational field. Laser energy is absorbed by the optic, heating the surrounding air and resulting in convective upwelling of air.

Fig. 2.
Fig. 2.

Plot of temperature rise [K] versus time [s] for fused silica optic irradiated with a P laser = 1.5 kW laser beam, assuming the absorption coefficient α F SiO 2 = 4 × 10 3 m 1 .

Fig. 3.
Fig. 3.

(a) False color plot of the difference Δ T [K] between the local air temperature and the ambient air temperature in the z - y plane. (b) Lineout of Δ T versus z at y = 0.2 cm (blue curve) and y = 2 cm (green curve).

Fig. 4.
Fig. 4.

Plot of stream function ψ ( y , z ) . Vertical and horizontal airflow velocities are given by v y ψ / z and v Z ψ / y , respectively.

Fig. 5.
Fig. 5.

(a) Contour plot of vertical velocity in the z - y plane. The velocity vanishes on the optic surface (no-slip boundary condition) and at large distances from the optic, with a maximum in between. (b) Contour plot of horizontal velocity in the z - y plane. Air flows in towards the optic ( v Z < 0 ) and acquires a vertical component as it approaches the optic. (c) Lineouts of velocity components at y = 1 cm .

Fig. 6.
Fig. 6.

Infrared camera image of BK-7 optic irradiated with 540 W laser. This image is used to measure the surface temperature of the optic.

Fig. 7.
Fig. 7.

Comparison between measured (points) and theoretical (solid curve) temperature rise [K] versus time [s] for UV-grade fused silica optic irradiated with a P laser = 1.156 kW laser beam for absorption coefficient α F SiO 2 = 1.0 × 10 3 m 1 .

Fig. 8.
Fig. 8.

Comparison between measured (points) and theoretical (solid curve) temperature rise [K] versus time [s] for BK7 optic for absorption coefficient α BK 7 = 3.0 × 10 2 m 1 . (a)  P laser = 230 W , (b)  P laser = 540 W .

Equations (24)

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ρ optic C optic T optic t = α optic I · q ,
ρ t + · ( ρ v ) = 0 ,
v t + ( v · ) v = ( 1 / ρ ) p + v 2 v g e y ,
T t + ( v · ) T = χ 2 T .
ρ ˜ t + ( v · ) ρ ˜ + ρ 0 · v = 0 ,
v t + ( v · ) v = ( 1 / ρ 0 ) p ˜ + v 2 v + β g T ˜ e y ,
T ˜ t + ( v · ) T = χ 2 T ˜ ,
· v = 0 ,
( v · ) v = ( 1 / ρ 0 ) p ˜ + v 2 v + β g T ˜ e y ,
( v · ) T ˜ = χ 2 T ˜ .
v y y + v z z = 0 ,
v y v y y + v z v y z = v 2 v y z 2 + β g ( T T 0 ) ,
v y T y + v z T z = χ 2 T z 2 ,
C [ β g ( T optic T 0 ) / 4 v 2 ] 1 / 4 .
ϕ + 3 ϕ ϕ 2 ϕ 2 + θ = 0 ,
θ + 3 P ϕ θ = 0 ,
q conv ( z = 0 ) = κ θ ( 0 ) C T optic T 0 y 1 / 4 e z ,
Δ T t α optic P laser ( 1 e 2 R 2 / R laser 2 ) π R 2 ρ optic C optic F 5 / 4 N κ π R ρ optic C optic l optic Δ T 2 F χ optic R R optic Δ T .
Δ T t α optic I laser ρ optic C optic .
θ ( ξ ) 1 + θ 0 0 ξ d ξ 1 exp [ 1 2 P ( ξ 1 4 / 4 ϕ 0 ξ 1 3 ) ] ,
ϕ ( ξ ) ξ 3 / 6 + ϕ 0 ξ 2 / 2 .
θ ( ξ ) A 3 P ϕ exp ( 3 P ϕ ξ ) ,
ϕ ( ξ ) ϕ + B ( 3 ϕ ) 2 exp ( 3 ϕ ξ ) + A P 3 ( 1 P ) ( 3 ϕ ) 4 exp ( 3 P ϕ ξ ) ,
ϕ ( ξ ) ϕ + 1 ( 3 ϕ ) 2 [ B ˜ + A ˜ ( 3 ϕ ) 2 ( 2 + 3 ϕ ξ ) ] exp ( 3 ϕ ξ ) ,

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