Abstract

Equations are presented for determining the optical properties of an absorbing uniaxial crystal by measuring the reflectance from a basal plane as a function of the angle of incidence and the angle of polarization. The technique is applied to a pyrolytic graphite crystal to determine the optical properties at 632.8 nm. The results are calculated numerically by use of a combination of the Taylor-series and gradient methods. The best-fit values found for the optical constants of pyrolytic graphite at 632.8 nm are n0 = 2.55, ne = 1.78, κ0 = 0.66, and κe = 0.00.

© 1970 Optical Society of America

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References

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  1. J. T. McCartney and S. Ergun, Fuel 37, 272 (1958).
  2. E. A. Taft and H. R. Philipp, Phys. Rev. 138, A197 (1965).
    [Crossref]
  3. S. Ergun, Nature 213, 136 (1967).
    [Crossref]
  4. D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, Inc., New York, 1964), p. 611.
  6. L. P. Mosteller and F. Wooten, J. Opt. Soc. Am. 58, 511 (1968).
    [Crossref]
  7. H. O. Hartley, Technometrics 3, 269 (1961).
    [Crossref]
  8. D. W. Marquardt, Chem. Eng. Progr. 55,(6), 65 (1959).

1968 (1)

1967 (1)

S. Ergun, Nature 213, 136 (1967).
[Crossref]

1965 (1)

E. A. Taft and H. R. Philipp, Phys. Rev. 138, A197 (1965).
[Crossref]

1963 (1)

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963).
[Crossref]

1961 (1)

H. O. Hartley, Technometrics 3, 269 (1961).
[Crossref]

1959 (1)

D. W. Marquardt, Chem. Eng. Progr. 55,(6), 65 (1959).

1958 (1)

J. T. McCartney and S. Ergun, Fuel 37, 272 (1958).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, Inc., New York, 1964), p. 611.

Ergun, S.

S. Ergun, Nature 213, 136 (1967).
[Crossref]

J. T. McCartney and S. Ergun, Fuel 37, 272 (1958).

Hartley, H. O.

H. O. Hartley, Technometrics 3, 269 (1961).
[Crossref]

Marquardt, D. W.

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963).
[Crossref]

D. W. Marquardt, Chem. Eng. Progr. 55,(6), 65 (1959).

McCartney, J. T.

J. T. McCartney and S. Ergun, Fuel 37, 272 (1958).

Mosteller, L. P.

Philipp, H. R.

E. A. Taft and H. R. Philipp, Phys. Rev. 138, A197 (1965).
[Crossref]

Taft, E. A.

E. A. Taft and H. R. Philipp, Phys. Rev. 138, A197 (1965).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, Inc., New York, 1964), p. 611.

Wooten, F.

Chem. Eng. Progr. (1)

D. W. Marquardt, Chem. Eng. Progr. 55,(6), 65 (1959).

Fuel (1)

J. T. McCartney and S. Ergun, Fuel 37, 272 (1958).

J. Opt. Soc. Am. (1)

J. Soc. Indust. Appl. Math. (1)

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963).
[Crossref]

Nature (1)

S. Ergun, Nature 213, 136 (1967).
[Crossref]

Phys. Rev. (1)

E. A. Taft and H. R. Philipp, Phys. Rev. 138, A197 (1965).
[Crossref]

Technometrics (1)

H. O. Hartley, Technometrics 3, 269 (1961).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, Inc., New York, 1964), p. 611.

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Figures (2)

Fig. 1
Fig. 1

Experimental arrangement used for determining reflectance at oblique incidence. (a) He–Ne gas laser, (b) polarizer, (c) partially reflecting mirror, (d) diaphragm, (e) sample, (f) frosted glass, (g) movable arm, (h) first detector, (i) vernier, (j) chopper, and (k) second detector.

Fig. 2
Fig. 2

Reflectance of the basal plane of pyrolytic graphite vs angle of incidence for TE and TM waves. Solid line is the theoretical curve using Eq. (3). Dotted line is the theoretical curve using Eq. (3), assuming a 5° error in the angle of the plane of polarization. Circles are the data points.

Tables (1)

Tables Icon

Table I Summary of values of optical constants of pyrolytic graphite at 632.8 nm.

Equations (16)

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ˆ e = e + i ( 4 π / ω ) σ e ˆ 0 = 0 + i ( 4 π / ω ) σ 0 ,
n ˆ = ˆ 1 2 = n ( 1 + i k ) = n 2 - k 2             and             σ / v = n 2 k ,
R = ρ TM 2 cos 2 Ψ + ρ TE 2 sin 2 Ψ ,
ρ TM = n ˆ e n ˆ 0 cos θ - n a ( n ˆ e 2 - n a 2     sin 2 θ ) 1 2 n ˆ e n ˆ 0 cos θ + n a ( n ˆ e 2 - n a 2     sin 2 θ ) 1 2
ρ TE = n a cos θ - ( n ˆ 0 2 - n a 2 sin 2 θ ) 1 2 n a cos θ + ( n ˆ 0 2 - n a 2 sin 2 θ ) 1 2 ;
Φ ( n 0 , n e , κ 0 , κ e ) = i = 1 N [ R i - R 0 i ] 2
R T i = R ( θ i , Ψ i , n 0 + δ n 0 , n e + δ n e , κ 0 + δ κ 0 , κ e + δ κ e ) = R 0 i + R 0 i n 0 δ n 0 + R 0 i n e δ n e + R 0 i κ 0 δ κ 0 + R 0 i k e δ κ e ,
Φ = i = 1 N ( R i - R T i ) 2 .
A δ = G ,
A = P T P
P = ( R 01 n 0 R 02 n 0 R 0 N n 0 R 01 n e R 02 n e R 0 N n e R 01 κ 0 R 02 κ 0 R 0 N κ 0 R 01 κ e R 02 κ e R 0 N κ e )
G = ( i = 1 N ( R i - R 0 i ) R 0 i n 0 i = 1 N ( R i - R 0 i ) R 0 i n e i = 1 N ( R i - R 0 i ) R 0 i κ 0 i = 1 N ( R i - R 0 i ) R 0 i κ e )
δ = ( δ n 0 , δ n e , δ κ 0 , δ κ e ) T ,
δ = - ( Φ n 0 , Φ n e , Φ κ 0 , Φ κ e ) T .
δ = - 2 G .
( A + λ I ) δ = G ,