Abstract

This paper concerns tables of Fresnel intensity reflectances which resulted from studies based on classical electrodynamics. The tables contain approximately 2500 indices of refraction N=nki; i.e., n=0.1(0.1)4.0, k=0(0.1)6.0 for angles of incidence θ0=0°(5°)85°. Series of graphs illustrate the solutions of the Fresnel equations. The supplementary discussion describes in detail the occurrence of reflection characteristics for which (a) the amplitude of the wave oscillating parallel to the plane of incidence is a minimum, (b) the degree of polarization is a maximum, and (c) the two amplitudes of the reflected wave have a 90° difference of phase. The numerical solutions are also presented in graphs that provide means, in addition to those already in general use, of determining the indices of refraction of solid materials.

© 1967 Optical Society of America

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References

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  1. Herbert B. Holl, “The Effect of Radiation Force on Satellites of Convex Shape,” NASA TN D-604 (May1961).
  2. Herbert B. Holl, “The Reflection of Electromagnetic Radiation (Based on Classical Electrodynamics),” Vol. I (AD422882), Vol. II (AD600720), “Appendix-Tables of Radiation Reflection Functions” U. S. Army Missile Command, Redstone Arsenal, Alabama, (15March1963).
  3. W. König, “Elektromagnetische Lichttheorie,” in Handbuch der Physik (Springer, Berlin, 1928), Vol. XX.
  4. R. Minkowski, “Theorie der Reflexion, Brechung und Dispersion,” in Müller-Pouillet’s Lehrbuch der Physik (Friedr. Vieweg und Sohn Akt. Ges., Braunschweig, 1929).
  5. Julius A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941).
  6. Max Born and Emil Wolf, Principles of Optics (Pergamon Press, New York, 1959).
  7. Richard Tousey, J. Opt. Soc. Am. 29, 235 (1939).
    [Crossref]
  8. Parry Moon, J. Math. Phys. 19, 1 (1940).
  9. Ivan Šimon, J. Opt. Soc. Am. 41, 336 (1951).
    [Crossref]
  10. D. G. Avery, Proc. Phys. Soc. (London) 65, 425 (1952).
    [Crossref]
  11. Gerhard Heilmann, Physik 152, 368 (1958).
    [Crossref]
  12. Kozo Ishiguro, Taizo Sasaki, and Sadao Nomura, Sci. Papers of the College of General Education, University of Tokyo10, 207 (1960).
  13. S. P. F. Humphreys-Owen, Proc. Phys. Soc. (London) 77, 949 (1961).
    [Crossref]
  14. H. G. Häfele, Physik 168, 530 (1962).
    [Crossref]
  15. A. P. Prishivalko, Reflection of Light from Absorbing Media (Academy of Sciences of White Russian SSR, Minsk, 1963).
  16. Antonín. Vašíček, Tables of Determination of Optical Constants from the Intensities of Reflected Light (Nakladatelství Ceskoslovenské Akademie VĚD, Praha, 1964).
  17. The information about Dr. A. P. Prishivalko’s work was furnished by Professor Dr. Antonín Vašíček, University Brno, Czeckoslavakia, who also supplied a copy of the former’s book which contained the tables. The author is deeply indebted to Dr. Vašíček for these kindnesses.
  18. Background details and complete sequence of equations on which this discussion is based can be found in Ref. 2.
  19. Expressions for the first and second Brewster angles can be obtained by differentiation of expressions based on Eqs. (2) and (3). To do this, however, requires tedious analytical labor, and it has previously been considered impossible to solve this problem explicitly (Refs. 20, 21) so that only approximations were sought. The problem found new consideration recently, and was explicitly solved for the first Brewster angle by Humphreys-Owen13 and for the second Brewster angle by Potter.22
  20. C. Boeckner, J. Opt. Soc. Am. and Rev. Sci. Instr. 19, 7 (1929).
    [Crossref]
  21. P. H. Miller and J. R. Johnson, Physica 20, 11, 1026 (1954).
    [Crossref]
  22. Roy F. Potter, J. Opt. Soc. Am. 54, 904 (1964).
    [Crossref]
  23. These tables were for internal use only and were not published in Holl.2
  24. Paul Drude, The Theory of Optics (Dover Publications, Inc., New York, 1959).
  25. A similar equation given by König3 [Eq. (169b), p. 246] is found to be incorrect.
  26. Herbert B. Holl, Numerical Solutions of the Fresnel Equations in the Optical Region, Symposium on Thermal Radiation of Solids, NASA SP-55, AF ML-TDR-64-159 (National Aeronautics & Space Administration, Washington, D. C., 1965), p. 45.
  27. R. W. Wood, Phys. Rev. 44, 353 (1933).
    [Crossref]
  28. R. de L. Kronig [Nature, Feb.10, 211 (1934)] obtained, for potassium, N=0.294−0.0058i for λ=2727 Å.
    [Crossref]
  29. Felix Jentzsch-Gräfe [Verhandl. Deut. Phys. Ges. 21, 361, (1919)] thoroughly investigated the minimum reflectance of natural light for real index of refraction. His paper presented an exact mathematical formulation for this case and tabulated the values for θ0 and R=min from n=3.7321 to n=9.507.

1964 (1)

1962 (1)

H. G. Häfele, Physik 168, 530 (1962).
[Crossref]

1961 (1)

S. P. F. Humphreys-Owen, Proc. Phys. Soc. (London) 77, 949 (1961).
[Crossref]

1958 (1)

Gerhard Heilmann, Physik 152, 368 (1958).
[Crossref]

1954 (1)

P. H. Miller and J. R. Johnson, Physica 20, 11, 1026 (1954).
[Crossref]

1952 (1)

D. G. Avery, Proc. Phys. Soc. (London) 65, 425 (1952).
[Crossref]

1951 (1)

1940 (1)

Parry Moon, J. Math. Phys. 19, 1 (1940).

1939 (1)

1934 (1)

R. de L. Kronig [Nature, Feb.10, 211 (1934)] obtained, for potassium, N=0.294−0.0058i for λ=2727 Å.
[Crossref]

1933 (1)

R. W. Wood, Phys. Rev. 44, 353 (1933).
[Crossref]

1929 (1)

C. Boeckner, J. Opt. Soc. Am. and Rev. Sci. Instr. 19, 7 (1929).
[Crossref]

1919 (1)

Felix Jentzsch-Gräfe [Verhandl. Deut. Phys. Ges. 21, 361, (1919)] thoroughly investigated the minimum reflectance of natural light for real index of refraction. His paper presented an exact mathematical formulation for this case and tabulated the values for θ0 and R=min from n=3.7321 to n=9.507.

Avery, D. G.

D. G. Avery, Proc. Phys. Soc. (London) 65, 425 (1952).
[Crossref]

Boeckner, C.

C. Boeckner, J. Opt. Soc. Am. and Rev. Sci. Instr. 19, 7 (1929).
[Crossref]

Born, Max

Max Born and Emil Wolf, Principles of Optics (Pergamon Press, New York, 1959).

Drude, Paul

Paul Drude, The Theory of Optics (Dover Publications, Inc., New York, 1959).

Häfele, H. G.

H. G. Häfele, Physik 168, 530 (1962).
[Crossref]

Heilmann, Gerhard

Gerhard Heilmann, Physik 152, 368 (1958).
[Crossref]

Holl, Herbert B.

Herbert B. Holl, “The Effect of Radiation Force on Satellites of Convex Shape,” NASA TN D-604 (May1961).

Herbert B. Holl, “The Reflection of Electromagnetic Radiation (Based on Classical Electrodynamics),” Vol. I (AD422882), Vol. II (AD600720), “Appendix-Tables of Radiation Reflection Functions” U. S. Army Missile Command, Redstone Arsenal, Alabama, (15March1963).

Herbert B. Holl, Numerical Solutions of the Fresnel Equations in the Optical Region, Symposium on Thermal Radiation of Solids, NASA SP-55, AF ML-TDR-64-159 (National Aeronautics & Space Administration, Washington, D. C., 1965), p. 45.

Humphreys-Owen, S. P. F.

S. P. F. Humphreys-Owen, Proc. Phys. Soc. (London) 77, 949 (1961).
[Crossref]

Ishiguro, Kozo

Kozo Ishiguro, Taizo Sasaki, and Sadao Nomura, Sci. Papers of the College of General Education, University of Tokyo10, 207 (1960).

Jentzsch-Gräfe, Felix

Felix Jentzsch-Gräfe [Verhandl. Deut. Phys. Ges. 21, 361, (1919)] thoroughly investigated the minimum reflectance of natural light for real index of refraction. His paper presented an exact mathematical formulation for this case and tabulated the values for θ0 and R=min from n=3.7321 to n=9.507.

Johnson, J. R.

P. H. Miller and J. R. Johnson, Physica 20, 11, 1026 (1954).
[Crossref]

König, W.

W. König, “Elektromagnetische Lichttheorie,” in Handbuch der Physik (Springer, Berlin, 1928), Vol. XX.

Kronig, R. de L.

R. de L. Kronig [Nature, Feb.10, 211 (1934)] obtained, for potassium, N=0.294−0.0058i for λ=2727 Å.
[Crossref]

Miller, P. H.

P. H. Miller and J. R. Johnson, Physica 20, 11, 1026 (1954).
[Crossref]

Minkowski, R.

R. Minkowski, “Theorie der Reflexion, Brechung und Dispersion,” in Müller-Pouillet’s Lehrbuch der Physik (Friedr. Vieweg und Sohn Akt. Ges., Braunschweig, 1929).

Moon, Parry

Parry Moon, J. Math. Phys. 19, 1 (1940).

Nomura, Sadao

Kozo Ishiguro, Taizo Sasaki, and Sadao Nomura, Sci. Papers of the College of General Education, University of Tokyo10, 207 (1960).

Potter, Roy F.

Prishivalko, A. P.

A. P. Prishivalko, Reflection of Light from Absorbing Media (Academy of Sciences of White Russian SSR, Minsk, 1963).

Sasaki, Taizo

Kozo Ishiguro, Taizo Sasaki, and Sadao Nomura, Sci. Papers of the College of General Education, University of Tokyo10, 207 (1960).

Šimon, Ivan

Stratton, Julius A.

Julius A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941).

Tousey, Richard

Vašícek, Antonín.

Antonín. Vašíček, Tables of Determination of Optical Constants from the Intensities of Reflected Light (Nakladatelství Ceskoslovenské Akademie VĚD, Praha, 1964).

Wolf, Emil

Max Born and Emil Wolf, Principles of Optics (Pergamon Press, New York, 1959).

Wood, R. W.

R. W. Wood, Phys. Rev. 44, 353 (1933).
[Crossref]

J. Math. Phys. (1)

Parry Moon, J. Math. Phys. 19, 1 (1940).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. and Rev. Sci. Instr. (1)

C. Boeckner, J. Opt. Soc. Am. and Rev. Sci. Instr. 19, 7 (1929).
[Crossref]

Nature (1)

R. de L. Kronig [Nature, Feb.10, 211 (1934)] obtained, for potassium, N=0.294−0.0058i for λ=2727 Å.
[Crossref]

Phys. Rev. (1)

R. W. Wood, Phys. Rev. 44, 353 (1933).
[Crossref]

Physica (1)

P. H. Miller and J. R. Johnson, Physica 20, 11, 1026 (1954).
[Crossref]

Physik (2)

H. G. Häfele, Physik 168, 530 (1962).
[Crossref]

Gerhard Heilmann, Physik 152, 368 (1958).
[Crossref]

Proc. Phys. Soc. (London) (2)

D. G. Avery, Proc. Phys. Soc. (London) 65, 425 (1952).
[Crossref]

S. P. F. Humphreys-Owen, Proc. Phys. Soc. (London) 77, 949 (1961).
[Crossref]

Verhandl. Deut. Phys. Ges. (1)

Felix Jentzsch-Gräfe [Verhandl. Deut. Phys. Ges. 21, 361, (1919)] thoroughly investigated the minimum reflectance of natural light for real index of refraction. His paper presented an exact mathematical formulation for this case and tabulated the values for θ0 and R=min from n=3.7321 to n=9.507.

Other (16)

These tables were for internal use only and were not published in Holl.2

Paul Drude, The Theory of Optics (Dover Publications, Inc., New York, 1959).

A similar equation given by König3 [Eq. (169b), p. 246] is found to be incorrect.

Herbert B. Holl, Numerical Solutions of the Fresnel Equations in the Optical Region, Symposium on Thermal Radiation of Solids, NASA SP-55, AF ML-TDR-64-159 (National Aeronautics & Space Administration, Washington, D. C., 1965), p. 45.

A. P. Prishivalko, Reflection of Light from Absorbing Media (Academy of Sciences of White Russian SSR, Minsk, 1963).

Antonín. Vašíček, Tables of Determination of Optical Constants from the Intensities of Reflected Light (Nakladatelství Ceskoslovenské Akademie VĚD, Praha, 1964).

The information about Dr. A. P. Prishivalko’s work was furnished by Professor Dr. Antonín Vašíček, University Brno, Czeckoslavakia, who also supplied a copy of the former’s book which contained the tables. The author is deeply indebted to Dr. Vašíček for these kindnesses.

Background details and complete sequence of equations on which this discussion is based can be found in Ref. 2.

Expressions for the first and second Brewster angles can be obtained by differentiation of expressions based on Eqs. (2) and (3). To do this, however, requires tedious analytical labor, and it has previously been considered impossible to solve this problem explicitly (Refs. 20, 21) so that only approximations were sought. The problem found new consideration recently, and was explicitly solved for the first Brewster angle by Humphreys-Owen13 and for the second Brewster angle by Potter.22

Kozo Ishiguro, Taizo Sasaki, and Sadao Nomura, Sci. Papers of the College of General Education, University of Tokyo10, 207 (1960).

Herbert B. Holl, “The Effect of Radiation Force on Satellites of Convex Shape,” NASA TN D-604 (May1961).

Herbert B. Holl, “The Reflection of Electromagnetic Radiation (Based on Classical Electrodynamics),” Vol. I (AD422882), Vol. II (AD600720), “Appendix-Tables of Radiation Reflection Functions” U. S. Army Missile Command, Redstone Arsenal, Alabama, (15March1963).

W. König, “Elektromagnetische Lichttheorie,” in Handbuch der Physik (Springer, Berlin, 1928), Vol. XX.

R. Minkowski, “Theorie der Reflexion, Brechung und Dispersion,” in Müller-Pouillet’s Lehrbuch der Physik (Friedr. Vieweg und Sohn Akt. Ges., Braunschweig, 1929).

Julius A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941).

Max Born and Emil Wolf, Principles of Optics (Pergamon Press, New York, 1959).

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Figures (13)

Fig. 1
Fig. 1

Reflectance curves R1 and R2 for k=0(1.0)6.0, for values of n as follows: (a) 0.5; (b) 1.0; (c) 1.5; (d) 2.0; (e) 2.5; (f) 3.0; (g) 3.5; (h) 4.0 [(a) additionally shows k values of 0.2, 0.4, 0.5, 0.8)].

Fig. 2
Fig. 2

Reflectance curves R = 1 2 ( R 1 + R 2 ) for k=0(1.0)6.0), for values of n as follows: (a) 0.5; (b) 1.0; (c) 1.5; (d) 2.0; (e) 2.5; (f) 3.0; (g) 3.5; (h) 4.0 [(a) additionally shows k values of 0.2, 0.4, 0.5, 0.8)].

Fig. 3
Fig. 3

The function R=[(n−1)2+k2]/[(n+1)2+k2] in the first and fourth quadrants. R is physically interpreted as the reflectance at normal incidence.

Fig. 4
Fig. 4

Relation between the reflectance R1 and the complex index of refraction N=nki, for angles of incidence θ0 as follows: (a) 10°; (b) 20°; (c) 30°; (d) 40°; (e) 50°; (f) 60°; (g) 70°; (h) 75°; (i) 80°; (j) 85°.

Fig. 5
Fig. 5

Relation between the reflectance R2 and the complex index of refraction N=nki, for angles of incidence θ0 as follows: (a) 10°; (b) 20°; (c) 30°; (d) 40°; (e) 50°; (f) 60°; (g) 70°; (h) 75°; (i) 80°; (j) 85°.

Fig. 6
Fig. 6

Relation between the reflectance R = 1 2 ( R 1 + R 2 ) and the complex index of refraction N=nki for angles of incidence θ0 as follows: (a) 10°; (b) 20°; (c) 30°; (d) 40°; (e) 50°; (f) 60°; (g) 70°; (h) 75°; (i) 80°; (j) 85°.

Fig. 7
Fig. 7

Relation between the degree of polarization P=(R1R2)/(R1+R2) and the complex index of refraction N=nki for angles of incidence as follows: (a) 10°; (b) 20°; (c) 30°; (d) 40°; (e) 50°; (f) 60°; (g) 70°; (h) 75°; (i) 80°; (j) 85°.

Fig. 8
Fig. 8

Illustration of the definition of the first, second, and third Brewster angles of incidence, for N=0.6−0.6i.

Fig. 9
Fig. 9

Relation between Brewster angles and complex index of refraction N=nki, as follows: (a) first Brewster angle (where R2 is a minimum); (b) second Brewster angle (where P is a maximum); (c) third Brewster angle (where the phase difference δ is 90°).

Fig. 10
Fig. 10

The third Brewster angle of incidence equals the principal angle of incidence, for small values of the index of refraction N=nki. χ denotes region of multiple principal angles of incidence.

Fig. 11
Fig. 11

Plots of the phase change δ for small values of k in and close to the region of multiple principal angles of incidence. (a) k=0.01; n=(1) 0.26, (2) 0.30, (3) 0.34, (4) 0.38, (5) 0.42. (b) k=0.02; n=(1) 0.30, (2) 0.34, (3) 0.38, (4) 0.42. (c) k=0.03; n=(1) 0.37, (2) 0.40, (3) 0.43. (d) k=0.04; n=(1) 0.38, (2) 0.41, (3) 0.44. (e) k=0.05; n=(1) 0.40, (2) 0.42, (3) 0.44. (f) k=0.06, n=(1) 0.40, (2) 0.42, (3) 0.44.

Fig. 12
Fig. 12

Relation among characteristic angles of incidence, reflectance or degree of polarization, and the complex index of refraction N=nki, as follows: (a) first Brewster angle of incidence (where R2 is minimum) and reflectance (R2)min; (b) second Brewster angle of incidence (where P is a maximum) and the degree of polarization Pmax; (c) third Brewster angle of incidence (where the phase difference=90°) and the reflectance R2; (d) angle of incidence for minimum reflectance of natural light [where R = 1 2 ( R 1 + R 2 ) is a minimum] and the reflectance Rmin.

Fig. 13
Fig. 13

Particular example of the determination of N from experimental data (—θ0=80°; – – – –θ0=70°). N1 at 80°, R1=0.945. R2=0.408, P=0.400. N1 at 70°: R1=0.894, R2=0.449, P=0.333. N2: R1=0.805, R2=0.404, P=0.333. N3: R1=0.850, R2=0.3675, P=0.400.

Tables (2)

Tables Icon

Table I Published tables and graphs of reflectance.

Tables Icon

Table II Reflectance characteristics at special angles of incidence.

Equations (13)

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sin θ 0 sin θ 1 = sin θ 0 sin ( φ ± χ i ) = N = n k i = n ( 1 κ i ) .
R 1 = ( a - cos θ 0 ) 2 + b 2 ( a + cos θ 0 ) 2 + b 2
R 2 = ( a - cos θ 0 ) 2 + b 2 ( a + cos θ 0 ) 2 + b 2 · ( a - sin θ 0 tan θ 0 ) 2 + b 2 ( a + sin θ 0 tan θ 0 ) 2 + b 2 .
a 2 = 1 2 { n 2 - k 2 - sin 2 θ 0 + [ 4 n 2 k 2 + ( n 2 - k 2 - sin 2 θ 0 ) 2 ] 1 2 } ,
b 2 = 1 2 { - n 2 + k 2 + sin 2 θ 0 + [ 4 n 2 k 2 + ( n 2 - k 2 - sin 2 θ 0 ) 2 ] 1 2 } .
R = 1 2 ( R 1 + R 2 ) ,
P = R 1 - R 2 R 1 + R 2 = 1 - R 2 / R 1 1 + R 2 / R 1 ,
( n - 1 + R 1 - R ) 2 + k 2 = 4 R ( 1 - R ) 2 ,
tan δ = 2 b sin θ 0 tan θ 0 sin 2 θ 0 tan 2 θ 0 - ( a 2 + b 2 ) .
( n 2 + k 2 ) 2 - 2 sin 2 θ 0 ( n 2 - k 2 ) = sin 4 θ 0 tan 4 θ 0 - sin 4 θ 0 .
tan 8 θ 0 - ( A - 2 B + 1 ) tan 4 θ 0 - 2 ( A - B ) tan 2 θ 0 - A = 0 ,
e 2 = [ ( R 1 + R 2 ) 2 - 4 R 1 R 2 sin 2 δ ] 1 2 ,
θ 1 * < θ 2 * < θ 3 * .