Abstract

We report on the design and performance of a ZnSe tetra-prism for homogeneous substrate heating using a continuous wave CO2 laser beam in pulsed laser deposition experiments. We discuss here three potential designs for homogenizing prisms and use ray-tracing modeling to compare their operation to an alternative square-tapered beam-pipe design. A square-pyramidal tetra-prism design was found to be optimal and was subjected to modeling and experimental testing to determine the influence of interference and diffraction effects on the homogeneity of the resultant intensity profile produced at the substrate surface. A heat diffusion model has been used to compare the temperature distributions produced when using various different source intensity profiles. The modeling work has revealed the importance of substrate thickness as a thermal diffuser in producing a resultant homogeneous substrate temperature distribution.

© 2008 Optical Society of America

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  1. K. H. Wu, C. L. Lee, J. Y. Juang, T. M. Uen, and Y. S. Gou, “In situ growth of Y1Ba2Cu3O7-x superconducting thin films using a pulsed neodymium:yttrium aluminium garnet laser with CO2 laser heated substrates,” Appl. Phys. Lett. 58, 1089-1091 (1991).
    [CrossRef]
  2. P. E. Dyer, A. Issa, P. H. Key, and P. Monk, “A cw CO2 laser substrate heater for superconducting thin-film deposition,” Supercond. Sci. Technol. 3, 472-475 (1990).
    [CrossRef]
  3. T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).
  4. J. A. Hopkins, F. A. Schwartz, M. H. McCay, T. D. McCay, N. B. Dahotre, and J. B. Bible, “Apparatus and method for producing an improved laser beam,” U.S. patent 6,016,227 (18 January 2000).
  5. A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
    [CrossRef]
  6. A. Delmas and J. C. Li, “Study of an optical device used to homogenize a laser beam. Application to emissivity measurements on semitransparent materials at high temperature,” Int. J. Thermophys. 24, 1427-1439 (2003).
    [CrossRef]
  7. S. J. Barrington and R. W. Eason, “Homogeneous substrate heating using a CO2 laser with feedback, rastering, and temperature monitoring,” Rev. Sci. Instrum. 71, 4223-4225(2000).
    [CrossRef]
  8. D. K. Fork, “Single binary optical element beam homogenizer,” U.S. patent 5,986,807 (16 November 1999).
  9. M. A. Dakhil, Y. M. Hassan, and A. M. Samour, “Folding of Gaussian beam and a proposal for an optical device,” Opt. Commun. 238, 163-168 (2004).
    [CrossRef]
  10. J. P. Sercel and M. von Dadelszen, “Practical UV excimer laser image system illuminators,” in Laser Beam Shaping Applications, F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds. (CRC Press, 2006), pp. 113-156.
  11. Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44-46(1983).
    [CrossRef]
  12. N. P. Padture and P. G. Klemens, “Low thermal conductivity in garnets,” J. Am. Cerm. Soc 80, 1018-1020 (1997).
  13. G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10 to 55 000 cm−1 wave numbers,” Phys. Rev. 177, 1308-1314 (1969).
    [CrossRef]

2004 (1)

M. A. Dakhil, Y. M. Hassan, and A. M. Samour, “Folding of Gaussian beam and a proposal for an optical device,” Opt. Commun. 238, 163-168 (2004).
[CrossRef]

2003 (2)

A. Delmas and J. C. Li, “Study of an optical device used to homogenize a laser beam. Application to emissivity measurements on semitransparent materials at high temperature,” Int. J. Thermophys. 24, 1427-1439 (2003).
[CrossRef]

T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).

2000 (1)

S. J. Barrington and R. W. Eason, “Homogeneous substrate heating using a CO2 laser with feedback, rastering, and temperature monitoring,” Rev. Sci. Instrum. 71, 4223-4225(2000).
[CrossRef]

1997 (2)

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

N. P. Padture and P. G. Klemens, “Low thermal conductivity in garnets,” J. Am. Cerm. Soc 80, 1018-1020 (1997).

1991 (1)

K. H. Wu, C. L. Lee, J. Y. Juang, T. M. Uen, and Y. S. Gou, “In situ growth of Y1Ba2Cu3O7-x superconducting thin films using a pulsed neodymium:yttrium aluminium garnet laser with CO2 laser heated substrates,” Appl. Phys. Lett. 58, 1089-1091 (1991).
[CrossRef]

1990 (1)

P. E. Dyer, A. Issa, P. H. Key, and P. Monk, “A cw CO2 laser substrate heater for superconducting thin-film deposition,” Supercond. Sci. Technol. 3, 472-475 (1990).
[CrossRef]

1983 (1)

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44-46(1983).
[CrossRef]

1969 (1)

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10 to 55 000 cm−1 wave numbers,” Phys. Rev. 177, 1308-1314 (1969).
[CrossRef]

Anderson, A. A.

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Barrington, S. J.

S. J. Barrington and R. W. Eason, “Homogeneous substrate heating using a CO2 laser with feedback, rastering, and temperature monitoring,” Rev. Sci. Instrum. 71, 4223-4225(2000).
[CrossRef]

Bible, J. B.

J. A. Hopkins, F. A. Schwartz, M. H. McCay, T. D. McCay, N. B. Dahotre, and J. B. Bible, “Apparatus and method for producing an improved laser beam,” U.S. patent 6,016,227 (18 January 2000).

Chrenko, R. M.

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10 to 55 000 cm−1 wave numbers,” Phys. Rev. 177, 1308-1314 (1969).
[CrossRef]

Dahotre, N. B.

J. A. Hopkins, F. A. Schwartz, M. H. McCay, T. D. McCay, N. B. Dahotre, and J. B. Bible, “Apparatus and method for producing an improved laser beam,” U.S. patent 6,016,227 (18 January 2000).

Dakhil, M. A.

M. A. Dakhil, Y. M. Hassan, and A. M. Samour, “Folding of Gaussian beam and a proposal for an optical device,” Opt. Commun. 238, 163-168 (2004).
[CrossRef]

Delmas, A.

A. Delmas and J. C. Li, “Study of an optical device used to homogenize a laser beam. Application to emissivity measurements on semitransparent materials at high temperature,” Int. J. Thermophys. 24, 1427-1439 (2003).
[CrossRef]

Dyer, P. E.

P. E. Dyer, A. Issa, P. H. Key, and P. Monk, “A cw CO2 laser substrate heater for superconducting thin-film deposition,” Supercond. Sci. Technol. 3, 472-475 (1990).
[CrossRef]

Eason, R. W.

S. J. Barrington and R. W. Eason, “Homogeneous substrate heating using a CO2 laser with feedback, rastering, and temperature monitoring,” Rev. Sci. Instrum. 71, 4223-4225(2000).
[CrossRef]

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Ebata, K.

T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).

Fork, D. K.

D. K. Fork, “Single binary optical element beam homogenizer,” U.S. patent 5,986,807 (16 November 1999).

Fotakis, C.

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Fuse, K.

T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).

Gou, Y. S.

K. H. Wu, C. L. Lee, J. Y. Juang, T. M. Uen, and Y. S. Gou, “In situ growth of Y1Ba2Cu3O7-x superconducting thin films using a pulsed neodymium:yttrium aluminium garnet laser with CO2 laser heated substrates,” Appl. Phys. Lett. 58, 1089-1091 (1991).
[CrossRef]

Grivas, C.

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Hassan, Y. M.

M. A. Dakhil, Y. M. Hassan, and A. M. Samour, “Folding of Gaussian beam and a proposal for an optical device,” Opt. Commun. 238, 163-168 (2004).
[CrossRef]

Hickey, L. M. B.

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Hirai, T.

T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).

Hopkins, J. A.

J. A. Hopkins, F. A. Schwartz, M. H. McCay, T. D. McCay, N. B. Dahotre, and J. B. Bible, “Apparatus and method for producing an improved laser beam,” U.S. patent 6,016,227 (18 January 2000).

Issa, A.

P. E. Dyer, A. Issa, P. H. Key, and P. Monk, “A cw CO2 laser substrate heater for superconducting thin-film deposition,” Supercond. Sci. Technol. 3, 472-475 (1990).
[CrossRef]

Itagaki, Y.

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44-46(1983).
[CrossRef]

Jelinek, M.

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Juang, J. Y.

K. H. Wu, C. L. Lee, J. Y. Juang, T. M. Uen, and Y. S. Gou, “In situ growth of Y1Ba2Cu3O7-x superconducting thin films using a pulsed neodymium:yttrium aluminium garnet laser with CO2 laser heated substrates,” Appl. Phys. Lett. 58, 1089-1091 (1991).
[CrossRef]

Kawamura, Y.

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44-46(1983).
[CrossRef]

Key, P. H.

P. E. Dyer, A. Issa, P. H. Key, and P. Monk, “A cw CO2 laser substrate heater for superconducting thin-film deposition,” Supercond. Sci. Technol. 3, 472-475 (1990).
[CrossRef]

Klemens, P. G.

N. P. Padture and P. G. Klemens, “Low thermal conductivity in garnets,” J. Am. Cerm. Soc 80, 1018-1020 (1997).

Lane, D.

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Lee, C. L.

K. H. Wu, C. L. Lee, J. Y. Juang, T. M. Uen, and Y. S. Gou, “In situ growth of Y1Ba2Cu3O7-x superconducting thin films using a pulsed neodymium:yttrium aluminium garnet laser with CO2 laser heated substrates,” Appl. Phys. Lett. 58, 1089-1091 (1991).
[CrossRef]

Li, J. C.

A. Delmas and J. C. Li, “Study of an optical device used to homogenize a laser beam. Application to emissivity measurements on semitransparent materials at high temperature,” Int. J. Thermophys. 24, 1427-1439 (2003).
[CrossRef]

McCay, M. H.

J. A. Hopkins, F. A. Schwartz, M. H. McCay, T. D. McCay, N. B. Dahotre, and J. B. Bible, “Apparatus and method for producing an improved laser beam,” U.S. patent 6,016,227 (18 January 2000).

McCay, T. D.

J. A. Hopkins, F. A. Schwartz, M. H. McCay, T. D. McCay, N. B. Dahotre, and J. B. Bible, “Apparatus and method for producing an improved laser beam,” U.S. patent 6,016,227 (18 January 2000).

Monk, P.

P. E. Dyer, A. Issa, P. H. Key, and P. Monk, “A cw CO2 laser substrate heater for superconducting thin-film deposition,” Supercond. Sci. Technol. 3, 472-475 (1990).
[CrossRef]

Namba, S.

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44-46(1983).
[CrossRef]

Nanba, H.

T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).

Okada, T.

T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).

Oliver, D. W.

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10 to 55 000 cm−1 wave numbers,” Phys. Rev. 177, 1308-1314 (1969).
[CrossRef]

Padture, N. P.

N. P. Padture and P. G. Klemens, “Low thermal conductivity in garnets,” J. Am. Cerm. Soc 80, 1018-1020 (1997).

Roberts, S.

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10 to 55 000 cm−1 wave numbers,” Phys. Rev. 177, 1308-1314 (1969).
[CrossRef]

Rogers, K.

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Samour, A. M.

M. A. Dakhil, Y. M. Hassan, and A. M. Samour, “Folding of Gaussian beam and a proposal for an optical device,” Opt. Commun. 238, 163-168 (2004).
[CrossRef]

Schwartz, F. A.

J. A. Hopkins, F. A. Schwartz, M. H. McCay, T. D. McCay, N. B. Dahotre, and J. B. Bible, “Apparatus and method for producing an improved laser beam,” U.S. patent 6,016,227 (18 January 2000).

Sercel, J. P.

J. P. Sercel and M. von Dadelszen, “Practical UV excimer laser image system illuminators,” in Laser Beam Shaping Applications, F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds. (CRC Press, 2006), pp. 113-156.

Shiozaki, M.

T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).

Slack, G. A.

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10 to 55 000 cm−1 wave numbers,” Phys. Rev. 177, 1308-1314 (1969).
[CrossRef]

Toyoda, K.

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44-46(1983).
[CrossRef]

Uen, T. M.

K. H. Wu, C. L. Lee, J. Y. Juang, T. M. Uen, and Y. S. Gou, “In situ growth of Y1Ba2Cu3O7-x superconducting thin films using a pulsed neodymium:yttrium aluminium garnet laser with CO2 laser heated substrates,” Appl. Phys. Lett. 58, 1089-1091 (1991).
[CrossRef]

von Dadelszen, M.

J. P. Sercel and M. von Dadelszen, “Practical UV excimer laser image system illuminators,” in Laser Beam Shaping Applications, F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds. (CRC Press, 2006), pp. 113-156.

Wu, K. H.

K. H. Wu, C. L. Lee, J. Y. Juang, T. M. Uen, and Y. S. Gou, “In situ growth of Y1Ba2Cu3O7-x superconducting thin films using a pulsed neodymium:yttrium aluminium garnet laser with CO2 laser heated substrates,” Appl. Phys. Lett. 58, 1089-1091 (1991).
[CrossRef]

Appl. Phys. Lett. (1)

K. H. Wu, C. L. Lee, J. Y. Juang, T. M. Uen, and Y. S. Gou, “In situ growth of Y1Ba2Cu3O7-x superconducting thin films using a pulsed neodymium:yttrium aluminium garnet laser with CO2 laser heated substrates,” Appl. Phys. Lett. 58, 1089-1091 (1991).
[CrossRef]

Int. J. Thermophys. (1)

A. Delmas and J. C. Li, “Study of an optical device used to homogenize a laser beam. Application to emissivity measurements on semitransparent materials at high temperature,” Int. J. Thermophys. 24, 1427-1439 (2003).
[CrossRef]

Opt. Commun. (2)

M. A. Dakhil, Y. M. Hassan, and A. M. Samour, “Folding of Gaussian beam and a proposal for an optical device,” Opt. Commun. 238, 163-168 (2004).
[CrossRef]

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a Gaussian laser beam,” Opt. Commun. 48, 44-46(1983).
[CrossRef]

Phys. Rev. (1)

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10 to 55 000 cm−1 wave numbers,” Phys. Rev. 177, 1308-1314 (1969).
[CrossRef]

Rev. Sci. Instrum. (1)

S. J. Barrington and R. W. Eason, “Homogeneous substrate heating using a CO2 laser with feedback, rastering, and temperature monitoring,” Rev. Sci. Instrum. 71, 4223-4225(2000).
[CrossRef]

SEI Tech. Rev. (1)

T. Hirai, K. Fuse, M. Shiozaki, T. Okada, K. Ebata, and H. Nanba, “Characteristics of ZnSe aspheric beam homogenizer for CO2 laser,” SEI Tech. Rev. 55, 71-77 (2003).

Supercond. Sci. Technol. (1)

P. E. Dyer, A. Issa, P. H. Key, and P. Monk, “A cw CO2 laser substrate heater for superconducting thin-film deposition,” Supercond. Sci. Technol. 3, 472-475 (1990).
[CrossRef]

Thin Solid Films (1)

A. A. Anderson, R. W. Eason, M. Jelinek, C. Grivas, D. Lane, K. Rogers, L. M. B. Hickey, and C. Fotakis, “Growth of Ti:sapphire single crystal thin films by pulsed laser deposition,” Thin Solid Films 300, 68-71 (1997)
[CrossRef]

Other (4)

N. P. Padture and P. G. Klemens, “Low thermal conductivity in garnets,” J. Am. Cerm. Soc 80, 1018-1020 (1997).

J. A. Hopkins, F. A. Schwartz, M. H. McCay, T. D. McCay, N. B. Dahotre, and J. B. Bible, “Apparatus and method for producing an improved laser beam,” U.S. patent 6,016,227 (18 January 2000).

D. K. Fork, “Single binary optical element beam homogenizer,” U.S. patent 5,986,807 (16 November 1999).

J. P. Sercel and M. von Dadelszen, “Practical UV excimer laser image system illuminators,” in Laser Beam Shaping Applications, F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds. (CRC Press, 2006), pp. 113-156.

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Figures (16)

Fig. 1
Fig. 1

Diagram of a vertical substrate holder design based on machined alumina ceramic tubes.

Fig. 2
Fig. 2

(a) Cross section of a square-tapered beam-pipe with some rays drawn to indicate its basic operation. (b) Map showing how the beam-pipe device reflects different parts of the beam onto the substrate.

Fig. 3
Fig. 3

Modeled averaged intensity beam profiles at (a)  1 mm and (b)  40 mm from the end of the square-tapered beam-pipe.

Fig. 4
Fig. 4

(a) Line drawing of the tetra-prism. (b) Map showing how the tetra-prism translates different parts of the beam.

Fig. 5
Fig. 5

(a) Line drawing of the square-top tetra-prism. (b) Map showing how the square-top tetra-prism translates different parts of the beam.

Fig. 6
Fig. 6

(a) Line drawing of the square-top octa-prism. (b) Map showing how the square-top octa-prism translates different parts of the beam.

Fig. 7
Fig. 7

Averaged modeled intensity profiles for each prism design under consideration at 50%, 100%, and 150% of the working distance in each case: (a) tetra-prism, (b) square-top tetra-prism, and (c) square-top octa-prism.

Fig. 8
Fig. 8

Illustration of parameters used in Eq. (1).

Fig. 9
Fig. 9

Averaged modeled intensity profiles for three different initial beam waists to illustrate the effect of changing beam size: (a)  3.0 mm , (b)  9.0 mm , and (c)  27.0 mm .

Fig. 10
Fig. 10

(a) Untransformed Gaussian beam source function with w = 5.0 mm and P = 5.6 W . (b) Resultant modeled temperature distribution of the substrate deposition face, standard deviation = 43 ° K , mean = 978 ° K , and range = 173 ° K .

Fig. 11
Fig. 11

(a) Flat intensity heat source function with P = 7.2 W . (b) Resultant modeled temperature distribution of the substrate deposition face, standard deviation = 7.1 ° K , mean = 1063 ° K , and range = 34 ° K .

Fig. 12
Fig. 12

(a) Raster-scanned beam source function with w = 1.0 mm , d = 1.6 mm , and P = 6.9 W . (b) Resultant modeled temperature distribution of the substrate deposition face; standard deviation = 14 ° K , mean = 1052 ° K , and range = 63 ° K .

Fig. 13
Fig. 13

(a) Recombined incoherent Gaussian beam source function with w = 7.7 mm , d = 5.0 mm , and P = 11.0 W . (b) Resultant modeled temperature distribution of the substrate deposition face, standard deviation = 2.7 ° K , mean = 1076 ° K , and range = 16 ° K .

Fig. 14
Fig. 14

(a) Recombined Gaussian with Fresnel diffraction with w = 9.5 mm , d = 5.6 mm , and P = 9.5 W . (b) Resultant modeled temperature distribution of the substrate deposition face, standard deviation = 2.2 ° K , mean = 1076 ° K , and range = 14 ° K .

Fig. 15
Fig. 15

Picture of the intensity profile generated by a square-pyramidal tetra-prism using temperature sensitive liquid crystal film. Note that some saturation has occurred in the center of the most intense spots shown.

Fig. 16
Fig. 16

Variation of the temperature inhomogeneity due to source modulation for substrate thicknesses of 0.1, 0.25, 0.5, and 1.0 mm .

Tables (1)

Tables Icon

Table 1 Summary of Modeling Results for the Different Homogenization Techniques Under Consideration

Equations (8)

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sin 1 ( n 1 θ n 2 ) θ = tan 1 ( 2 h 2 D )
k 2 T = Q ,
n · ( k T ) = ε σ ( T amb 4 T 4 ) ,
Q = 2 P π w 2 exp [ 2 ( x 2 + y 2 ) w 2 ] δ ( z ) ,
Q = 2 P 36 π w 2 i , j = 1 3 [ exp ( 2 w 2 { [ x + ( i 0.5 ) d ] 2 + [ y + ( j 0.5 ) d ] 2 } ) + exp ( 2 w 2 { [ x + ( i 0.5 ) d ] 2 + [ y ( j 0.5 ) d ] 2 } ) + exp ( 2 w 2 { [ x ( i 0.5 ) d ] 2 + [ y + ( j 0.5 ) d ] 2 } ) + exp ( 2 w 2 { [ x ( i 0.5 ) d ] 2 + [ y ( j 0.5 ) d ] 2 } ) ] ,
Q = 2 P π w 2 ( exp { 2 w 2 [ ( x + a ) 2 + ( y + a ) 2 ] } ( x > a , y > a ) + exp { 2 w 2 [ ( x + a ) 2 + ( y a ) 2 ] } ( x > a , y < a ) + exp { 2 w 2 [ ( x a ) 2 + ( y + a ) 2 ] } ( x < a , y > a ) + exp { 2 w 2 [ ( x a ) 2 + ( y a ) 2 ] } ( x < a , y < a ) ) .
Q = A k f k f h + cos ( k f h ) sin ( k f h ) cos 2 ( k f y ) δ ( z ) ,
Δ = [ T mod ( y ) T flat ( y ) ] max   values [ T mod ( y ) T flat ( y ) ] min   values

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