Abstract

A mathematical model has been developed which describes the silicon composition gradient produced in germanium–silicon alloy GRIN crystals formed via Czochralski crystal growing. This model is based on the naturally occurring segregation effect of silicon in germanium. The refractive index of the alloy is described in terms of its relation to the band gap energy, which is itself dependent on the silicon composition. A relationship between refractive index and silicon composition of the alloy is derived.

© 1988 Optical Society of America

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References

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  1. J. Czochralski, “Measuring the Velocity of Crystallization of Metals,” Z. Phys. Chem. 92, 219 (1918).
  2. J. J. Miceli, “Infrared Gradient Index Optics: Materials, Fabrication and Testing,” Ph.D. Thesis, U. Rochester (1982).
  3. D. P. Naughton, “Fabrication and Testing of Infrared Gradient Index Materials,” M. S. Thesis, U. Rochester (1986).
  4. E. R. Johnson, S. M. Christian, “Some Properties of Germanium-Silicon Alloys,” Phys. Rev. 95, 560 (1954).
    [CrossRef]
  5. J. J. Miceli, Ref. 2, p. 34.

1954 (1)

E. R. Johnson, S. M. Christian, “Some Properties of Germanium-Silicon Alloys,” Phys. Rev. 95, 560 (1954).
[CrossRef]

1918 (1)

J. Czochralski, “Measuring the Velocity of Crystallization of Metals,” Z. Phys. Chem. 92, 219 (1918).

Christian, S. M.

E. R. Johnson, S. M. Christian, “Some Properties of Germanium-Silicon Alloys,” Phys. Rev. 95, 560 (1954).
[CrossRef]

Czochralski, J.

J. Czochralski, “Measuring the Velocity of Crystallization of Metals,” Z. Phys. Chem. 92, 219 (1918).

Johnson, E. R.

E. R. Johnson, S. M. Christian, “Some Properties of Germanium-Silicon Alloys,” Phys. Rev. 95, 560 (1954).
[CrossRef]

Miceli, J. J.

J. J. Miceli, “Infrared Gradient Index Optics: Materials, Fabrication and Testing,” Ph.D. Thesis, U. Rochester (1982).

J. J. Miceli, Ref. 2, p. 34.

Naughton, D. P.

D. P. Naughton, “Fabrication and Testing of Infrared Gradient Index Materials,” M. S. Thesis, U. Rochester (1986).

Phys. Rev. (1)

E. R. Johnson, S. M. Christian, “Some Properties of Germanium-Silicon Alloys,” Phys. Rev. 95, 560 (1954).
[CrossRef]

Z. Phys. Chem. (1)

J. Czochralski, “Measuring the Velocity of Crystallization of Metals,” Z. Phys. Chem. 92, 219 (1918).

Other (3)

J. J. Miceli, “Infrared Gradient Index Optics: Materials, Fabrication and Testing,” Ph.D. Thesis, U. Rochester (1982).

D. P. Naughton, “Fabrication and Testing of Infrared Gradient Index Materials,” M. S. Thesis, U. Rochester (1986).

J. J. Miceli, Ref. 2, p. 34.

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Figures (8)

Fig. 1
Fig. 1

Czochralski crystal growing apparatus.

Fig. 2
Fig. 2

Phase diagram Ge/Si binary system.

Fig. 3
Fig. 3

Alloy band gap vs mole % Si.

Fig. 4
Fig. 4

Refractive index vs silicon concentration.

Fig. 5
Fig. 5

Germanium–silicon alloy gradient-index crystal.

Fig. 6
Fig. 6

Mole fraction Si vs length.

Fig. 7
Fig. 7

Band gap energy vs length.

Fig. 8
Fig. 8

Refractive index vs length.

Equations (16)

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C Si,S = K C 0 ( 1 g ) K 1 = 5 . 5 C 0 ( 1 g ) 4 . 5 ,
E alloy = 0 . 72 + 1 . 467 C Si,S eV .
N = n + i κ ,
n ( ω 0 ) = 1 + 2 π 0 ω κ ( ω ) ω 2 ω 0 2 d ω ,
κ ( ω 0 ) = 2 ω π 0 0 n ( ω ) ω 2 ω 0 2 d ω ,
κ ( ω ) = c / 2 ω α ( ω ) .
n ( ω 0 ) = 1 + C π ω gap α ( ω ) ω 2 ω 0 2 d ω .
n ( ω 0 ) = 1 + C π ω gap α ( ω ) ω 2 d ω .
α ( ω ) = [ 1 for ω > ω gap , 0 for ω < ω gap .
n ( ω 0 ) 1 + ω gap alloy ω gap Ge 1 ω 2 d ω .
n = 2 . 54 + 1 . 053 E alloy .
V ( l ) = π R 2 ( l ) Δ l ,
g ( l ) = V ( l ) / V 0 .
C Si,S ( l ) = 4 . 5 C 0 [ 1 g ( l ) ] 4 . 5 .
E alloy ( l ) = 0 . 72 + 1 . 467 C Si,S ( l ) .
n ( l ) = 2 . 54 + 1 . 053 / E alloy ( l ) .

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