Abstract

We develop models of laser interactions with composite materials consisting of fibers embedded within a matrix. A ray-trace model is shown to determine the absorptivity, absorption depth, and optical power enhancement within the material, as well as the angular distribution of the reflected light. We also develop a macroscopic model, which provides physical insight and overall results. We show that the parameters in this model can be determined from the ray trace model.

© 2013 Optical Society of America

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References

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  1. R. K. Freeman, F. A. Rigby, and N. Morley, “Temperature-dependent reflectance of plated metals and composite materials under laser irradiation,” J. Thermophys. Heat Transfer 14, 305–312 (2000).
    [CrossRef]
  2. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. f. Techn. Physik 12, 593–601 (1931).
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  4. Ya. B. Zeldovich and Yu. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (Academic, 1967).
  5. P. Mudgett and L. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [CrossRef]
  6. FRED is distributed by Photon Engineering, LLC, Tucson, AZ.
  7. A. Borghesi and G. Guizzetti, “Graphite (C),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1981), Vol. II, pp. 449–468.
  8. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).
  9. W. L. Wolfe and G. J. Zissis, The Infrared Handbook (Infrared Information Analysis Center, 1985).

2000

R. K. Freeman, F. A. Rigby, and N. Morley, “Temperature-dependent reflectance of plated metals and composite materials under laser irradiation,” J. Thermophys. Heat Transfer 14, 305–312 (2000).
[CrossRef]

1971

1931

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. f. Techn. Physik 12, 593–601 (1931).

Borghesi, A.

A. Borghesi and G. Guizzetti, “Graphite (C),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1981), Vol. II, pp. 449–468.

Freeman, R. K.

R. K. Freeman, F. A. Rigby, and N. Morley, “Temperature-dependent reflectance of plated metals and composite materials under laser irradiation,” J. Thermophys. Heat Transfer 14, 305–312 (2000).
[CrossRef]

Guizzetti, G.

A. Borghesi and G. Guizzetti, “Graphite (C),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1981), Vol. II, pp. 449–468.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Kubelka, P.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. f. Techn. Physik 12, 593–601 (1931).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Morley, N.

R. K. Freeman, F. A. Rigby, and N. Morley, “Temperature-dependent reflectance of plated metals and composite materials under laser irradiation,” J. Thermophys. Heat Transfer 14, 305–312 (2000).
[CrossRef]

Mudgett, P.

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. f. Techn. Physik 12, 593–601 (1931).

Raizer, Yu. P.

Ya. B. Zeldovich and Yu. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (Academic, 1967).

Richards, L.

Rigby, F. A.

R. K. Freeman, F. A. Rigby, and N. Morley, “Temperature-dependent reflectance of plated metals and composite materials under laser irradiation,” J. Thermophys. Heat Transfer 14, 305–312 (2000).
[CrossRef]

Wolfe, W. L.

W. L. Wolfe and G. J. Zissis, The Infrared Handbook (Infrared Information Analysis Center, 1985).

Zeldovich, Ya. B.

Ya. B. Zeldovich and Yu. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (Academic, 1967).

Zissis, G. J.

W. L. Wolfe and G. J. Zissis, The Infrared Handbook (Infrared Information Analysis Center, 1985).

Appl. Opt.

J. Thermophys. Heat Transfer

R. K. Freeman, F. A. Rigby, and N. Morley, “Temperature-dependent reflectance of plated metals and composite materials under laser irradiation,” J. Thermophys. Heat Transfer 14, 305–312 (2000).
[CrossRef]

Zeits. f. Techn. Physik

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeits. f. Techn. Physik 12, 593–601 (1931).

Other

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Ya. B. Zeldovich and Yu. P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (Academic, 1967).

FRED is distributed by Photon Engineering, LLC, Tucson, AZ.

A. Borghesi and G. Guizzetti, “Graphite (C),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1981), Vol. II, pp. 449–468.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

W. L. Wolfe and G. J. Zissis, The Infrared Handbook (Infrared Information Analysis Center, 1985).

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Figures (14)

Fig. 1.
Fig. 1.

Schematic geometry of the model for the case of carbon fibers. The fibers, seen head-on, are first arranged in uniform rows and then slightly perturbed. The two mirrors account for the limited extent of the beam. The fibers or portions of them, outside the mirrors are not included in the ray tracing.

Fig. 2.
Fig. 2.

Index of refraction of carbon (graphite, ordinary mode) versus wavelength, from data given in [7]. Upper line, real part; lower line, imaginary part.

Fig. 3.
Fig. 3.

Fresnel absorptivity of 0.8 μm light incident on carbon from epoxy. Top, angular dependence at 0.8 μm. Bottom, wavelength dependence at 45°.

Fig. 4.
Fig. 4.

Sample rays within the epoxy, with carbon fibers, after several steps. (In a step, each ray advances from one surface to the next.) The horizontal lines represent rays scattered from the epoxy surface, while the dotted lines represent rays that enter the composite and are scattered out.

Fig. 5.
Fig. 5.

Calculated absorptivity of the carbon–fiber composite, at a wavelength of 0.8 μm, for 40 fiber configurations. Upper points, p polarization; lower points, s polarization. The dotted lines give the averages (about 0.88 and 0.87, respectively). Each run involved 106 laser rays. Note that the vertical scale is truncated.

Fig. 6.
Fig. 6.

Calculated absorptivity of the carbon–fiber composite versus wavelength. Each point represents an average over 10 configurations, with 106 laser rays each. The jump near 1 μm is caused by the apparent irregularity in the graphite index of refraction in Fig. 2.

Fig. 7.
Fig. 7.

Calculated angular distributions of light reflected from the carbon–fiber composite, away from the specular (backward) direction. The jagged lines show the computed results, while the dotted lines give the best fit to a function proportional to cosθ. The curves are normalized so that the integral gives the total diffuse reflectivity. These results were calculated for one configuration and several million rays.

Fig. 8.
Fig. 8.

Local enhancement for a particular carbon–fiber configuration. Top, s polarization; bottom, p polarization. Note the shadows produced by the first row of fibers.

Fig. 9.
Fig. 9.

Average enhancement versus depth for the distributions of the previous figure.

Fig. 10.
Fig. 10.

Left: Fresnel transmission of 0.8 μm light incident from epoxy onto fused silica. Right: transmission for incidence from fused silica onto epoxy.

Fig. 11.
Fig. 11.

Left: Dominant scattering of a sample ray from fused silica. The incident ray arrives at θ=30° normal to the surface, and the outgoing ray is diverted from the path traveled by the incoming normal ray by a net angle of δθ=6.8°. Right: Example of scattering in a system with a smaller ratio of fiber/background indices of refraction. Note the multiple reflections and the sequence of interior isosceles triangles.

Fig. 12.
Fig. 12.

Typical geometry of the fused-silica composite model.

Fig. 13.
Fig. 13.

Local enhancement for fused silica fibers. The maximum enhancement is 2.1. The null areas (blue) along the sides indicate areas in which the enhancement was not calculated, due to complications caused by reflections.

Fig. 14.
Fig. 14.

Average enhancement versus depth for the distribution of the previous figure. The dashed line shows the result from the two-flux model.

Tables (1)

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Table 1. Overall Absorptivity of Carbon–Fiber Composite Versus Wavelength

Equations (10)

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R0=[(1ne,r)/(1+ne,r)]20.053.
uir=niPirlir/(cVi),
u0=(1R0)neP0/(cA0).
fi=niL0(1R0)neP0AirPirlir,
dF+dz=(K+S)F++SF,dFdz=(K+S)FSF+.
F+(0)=F0,F(h)=0.
F+(h)=F0(a21)ba2b21,F(0)=F0a(b21)a2b21,
a=(K+2S+p)/(K+2Sp),b=exp(ph).
K=(a1)lnb(a+1)h,S=2alnb(a21)h,
a=(1+R2T2+[(1+R2T2)24R2]1/2)/2R,b=(1R2+T2+[(1+R2T2)24R2]1/2)/2T.

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