Abstract

The measurement of wavelengths is subject to systematic errors caused by phase shifts on reflection. This paper gives methods for calculating these phase shifts for metallic and dielectric mirrors from knowledge of the easily obtained reflectivity or transmission spectrum. The Kramers–Kronig relations, as originated by H. W. Bode [ Network Analysis and Feedback Amplifier Design ( Van Nostrand, Princeton, N.J., 1945)], apply to dielectric and metallic-mirrors in many cases. The success of metallic surfaces for interferometry is a direct consequence of the Kramers–Kronig relations. A specific example of a phase-shift calculation for a multilayer-dielectric-coated mirror is given.

© 1985 Optical Society of America

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References

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  1. P. Woods, K. Shotton, W. Rowley, “Frequency determination of visible laser light by interferometric comparison with upconverted CO2 laser radiation,” Appl. Opt. 17, 1048–1054 (1978);W. R. C. Rowley, “Laser wavelength measurements,” Radio Sci. 14, 585–591 (1979).
    [CrossRef] [PubMed]
  2. C. R. Pollock, D. A. Jennings, F. R. Petersen, J. S. Wells, R. E. Drullinger, E. C. Beaty, K. M. Evenson, “Direct frequency measurements of transitions at 520 THz (576 nm) in iodine and 260 THz (1.15 μm) in neon,” Opt. Lett. 8, 133–135 (1984).
    [CrossRef]
  3. D. A. Jennings, C. R. Pollock, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells, J. L. Hall, H. P. Layer, “Direct frequency measurement of the I2-stabilized He–Ne 473-THz (633 nm) laser,” Opt. Lett. 8, 136–138 (1984).
    [CrossRef]
  4. J. M. Bennett, “Precise method for measuring the absolute phase change on reflection,” J. Opt. Soc. Am. 54, 612–624 (1964).This paper contains an excellent summary of the theory of reflection from metal surfaces.
    [CrossRef]
  5. H. Buisson, Ch. Fabry, “Mesures de longueurs d’onde pour l’establissement d’un système de repères spectroscopiques,” J. Phys. (Paris) 7, 169–195 (1908).
  6. D. H. Rank, E. R. Shull, J. M. Bennett, T. A. Wiggins, “Interferometric wavelength measurement in the infra red by the method of exact orders. Precision measurement of the index of refraction of air at 1.65 μ,” J. Opt. Soc. Am. 43, 952–956 (1953).
    [CrossRef]
  7. R. L. Barger, J. L. Hall, “Wavelength of the 3.39-μm laser-saturated absorption line of methane,” Appl. Phys. Lett. 22, 196–199 (1973).
    [CrossRef]
  8. R. W. Stanley, K. L. Andrew, “Use of dielectric coatings in absolute wavelength measurements with a Fabry–Perot interferometer,” J. Opt. Soc. Am. 54, 625–627 (1964).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  10. H. A. Kramers, Atti Congr. Fis. Como, 545–557 (1927);R. de L. Kronig, “On the theory of dispersion of x-rays,” J. Opt. Soc. Am. 12, 547 (1926).
    [CrossRef]
  11. H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, Princeton, N.J., 1945).
  12. G. Hass, L. Hadley, “Optical properties of metals,” in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1972).
  13. S. Seely, Electron-Tube Circuits, 2nd ed. (McGraw-Hill, New York, 1958), pp. 133–137.
  14. D. H. Rank, H. E. Bennett, “Problem of phase variation with wavelength in dielectric films. Extension of interferometric standards in the infra-red,” J. Opt. Soc. Am. 45, 69–73 (1955).
    [CrossRef]
  15. S. Penselin, A. Steudel, “Fabry–Perot interferometer verspiegelungen aus dielektrischen vielfachschichten,” Z. Phys. 142, 21–41 (1955). (Intensity only.)
    [CrossRef]
  16. C. DuFour, “Épaisseur optique d’un étalon ou d’un filtre inter-férentiel de Fabry–Perot à miroirs semitransparents complexes,” Rev. Opt. 31, 1–16 (1952). (Intensity and phase.)
  17. J. S. Seeley, “Resolving power of multilayer filters,” J. Opt. Soc. Am. 54, 342–346 (1964). (Phase.)
    [CrossRef]
  18. C. J. Koester, “Phase shift effects of Fabry–Perot interferometry,” J. Res. Nat. Bur. Stand. 64A, 191–200 (1960).
    [CrossRef]
  19. C. F. Bruce, P. E. Ciddor, “Phase dispersion in multilayer films,” J. Opt. Soc. Am. 50, 295–299 (1960).
    [CrossRef]
  20. For a general discussion and literature citations, see Refs. 9, 21, 22, and 23.
  21. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955;reprinted by Dover, New York, 1965).
  22. H. A. Macleond, Thin-Film Optical Filters (Elsevier, New York, 1969).
  23. H. M. Liddell, “Computer-aided techniques for the design of multilayer filters” (Hilger, Bristol, 1981).(Design techniques with digital computers and electrical filters.)
  24. P. W. Baumeister, F. A. Jenkins, “Dispersion of the phase change for dielectric multilayers. Application to the interference filter,” J. Opt. Soc. Am. 47, 57–61 (1957).
    [CrossRef]
  25. R. P. Netterfield, R. C. Schaeffer, W. G. Sainty, “Coating Fabry-Perot interferometer plates with broadband multilayer dielectric mirrors,” Appl. Opt. 19, 3010–3017 (1980).
    [CrossRef] [PubMed]
  26. P. W. Baumeister, J. M. Stone, “Broad-band multilayer film for Fabry–Perot interferometers,” J. Opt. Soc. Am. 46, 228–229 (1956).
    [CrossRef]
  27. O. S. Heavens, H. M. Liddell, “Staggered broad-band reflecting multilayers,” Appl. Opt. 5, 373–376 (1966).
    [CrossRef] [PubMed]
  28. S. R. Amin, C. D. Caldwell, W. Lichten, “Crossed beam spectroscopy of hydrogen: a new value for the Rydberg consant,” Phys. Rev. Lett. 47, 1234–1238 (1981).
    [CrossRef]
  29. S. R. Amin, “Crossed beam spectroscopy of neon: precision measurement of three 20Ne wavelengths,” J. Opt. Soc. Am. 74, 862–863 (1983).
    [CrossRef]
  30. J. C. Bergquist, H.-U. Daniel, “A wideband frequency-offset-locked dye laser spectrometer using a Schottky barrier mixer,” Opt. Commun. 48, 327–333 (1984).
    [CrossRef]
  31. K. W. Meissner, “Interference spectroscopy,” J. Opt. Soc. Am. 31, 405–427 (1941).
    [CrossRef]
  32. R. Badoual, E. Pelletier, “Miroirs insensibles aux defauts superficiels,” C. R. Acad. Sci. Ser. B 262, 937 (1966).

1984 (3)

1983 (1)

S. R. Amin, “Crossed beam spectroscopy of neon: precision measurement of three 20Ne wavelengths,” J. Opt. Soc. Am. 74, 862–863 (1983).
[CrossRef]

1981 (1)

S. R. Amin, C. D. Caldwell, W. Lichten, “Crossed beam spectroscopy of hydrogen: a new value for the Rydberg consant,” Phys. Rev. Lett. 47, 1234–1238 (1981).
[CrossRef]

1980 (1)

1978 (1)

1973 (1)

R. L. Barger, J. L. Hall, “Wavelength of the 3.39-μm laser-saturated absorption line of methane,” Appl. Phys. Lett. 22, 196–199 (1973).
[CrossRef]

1966 (2)

O. S. Heavens, H. M. Liddell, “Staggered broad-band reflecting multilayers,” Appl. Opt. 5, 373–376 (1966).
[CrossRef] [PubMed]

R. Badoual, E. Pelletier, “Miroirs insensibles aux defauts superficiels,” C. R. Acad. Sci. Ser. B 262, 937 (1966).

1964 (3)

1960 (2)

C. J. Koester, “Phase shift effects of Fabry–Perot interferometry,” J. Res. Nat. Bur. Stand. 64A, 191–200 (1960).
[CrossRef]

C. F. Bruce, P. E. Ciddor, “Phase dispersion in multilayer films,” J. Opt. Soc. Am. 50, 295–299 (1960).
[CrossRef]

1957 (1)

1956 (1)

1955 (2)

D. H. Rank, H. E. Bennett, “Problem of phase variation with wavelength in dielectric films. Extension of interferometric standards in the infra-red,” J. Opt. Soc. Am. 45, 69–73 (1955).
[CrossRef]

S. Penselin, A. Steudel, “Fabry–Perot interferometer verspiegelungen aus dielektrischen vielfachschichten,” Z. Phys. 142, 21–41 (1955). (Intensity only.)
[CrossRef]

1953 (1)

1952 (1)

C. DuFour, “Épaisseur optique d’un étalon ou d’un filtre inter-férentiel de Fabry–Perot à miroirs semitransparents complexes,” Rev. Opt. 31, 1–16 (1952). (Intensity and phase.)

1941 (1)

1927 (1)

H. A. Kramers, Atti Congr. Fis. Como, 545–557 (1927);R. de L. Kronig, “On the theory of dispersion of x-rays,” J. Opt. Soc. Am. 12, 547 (1926).
[CrossRef]

1908 (1)

H. Buisson, Ch. Fabry, “Mesures de longueurs d’onde pour l’establissement d’un système de repères spectroscopiques,” J. Phys. (Paris) 7, 169–195 (1908).

Amin, S. R.

S. R. Amin, “Crossed beam spectroscopy of neon: precision measurement of three 20Ne wavelengths,” J. Opt. Soc. Am. 74, 862–863 (1983).
[CrossRef]

S. R. Amin, C. D. Caldwell, W. Lichten, “Crossed beam spectroscopy of hydrogen: a new value for the Rydberg consant,” Phys. Rev. Lett. 47, 1234–1238 (1981).
[CrossRef]

Andrew, K. L.

Badoual, R.

R. Badoual, E. Pelletier, “Miroirs insensibles aux defauts superficiels,” C. R. Acad. Sci. Ser. B 262, 937 (1966).

Barger, R. L.

R. L. Barger, J. L. Hall, “Wavelength of the 3.39-μm laser-saturated absorption line of methane,” Appl. Phys. Lett. 22, 196–199 (1973).
[CrossRef]

Baumeister, P. W.

Beaty, E. C.

Bennett, H. E.

Bennett, J. M.

Bergquist, J. C.

J. C. Bergquist, H.-U. Daniel, “A wideband frequency-offset-locked dye laser spectrometer using a Schottky barrier mixer,” Opt. Commun. 48, 327–333 (1984).
[CrossRef]

Bode, H. W.

H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, Princeton, N.J., 1945).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Bruce, C. F.

Buisson, H.

H. Buisson, Ch. Fabry, “Mesures de longueurs d’onde pour l’establissement d’un système de repères spectroscopiques,” J. Phys. (Paris) 7, 169–195 (1908).

Caldwell, C. D.

S. R. Amin, C. D. Caldwell, W. Lichten, “Crossed beam spectroscopy of hydrogen: a new value for the Rydberg consant,” Phys. Rev. Lett. 47, 1234–1238 (1981).
[CrossRef]

Ciddor, P. E.

Daniel, H.-U.

J. C. Bergquist, H.-U. Daniel, “A wideband frequency-offset-locked dye laser spectrometer using a Schottky barrier mixer,” Opt. Commun. 48, 327–333 (1984).
[CrossRef]

Drullinger, R. E.

DuFour, C.

C. DuFour, “Épaisseur optique d’un étalon ou d’un filtre inter-férentiel de Fabry–Perot à miroirs semitransparents complexes,” Rev. Opt. 31, 1–16 (1952). (Intensity and phase.)

Evenson, K. M.

Fabry, Ch.

H. Buisson, Ch. Fabry, “Mesures de longueurs d’onde pour l’establissement d’un système de repères spectroscopiques,” J. Phys. (Paris) 7, 169–195 (1908).

Hadley, L.

G. Hass, L. Hadley, “Optical properties of metals,” in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1972).

Hall, J. L.

Hass, G.

G. Hass, L. Hadley, “Optical properties of metals,” in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1972).

Heavens, O. S.

O. S. Heavens, H. M. Liddell, “Staggered broad-band reflecting multilayers,” Appl. Opt. 5, 373–376 (1966).
[CrossRef] [PubMed]

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955;reprinted by Dover, New York, 1965).

Jenkins, F. A.

Jennings, D. A.

Koester, C. J.

C. J. Koester, “Phase shift effects of Fabry–Perot interferometry,” J. Res. Nat. Bur. Stand. 64A, 191–200 (1960).
[CrossRef]

Kramers, H. A.

H. A. Kramers, Atti Congr. Fis. Como, 545–557 (1927);R. de L. Kronig, “On the theory of dispersion of x-rays,” J. Opt. Soc. Am. 12, 547 (1926).
[CrossRef]

Layer, H. P.

Lichten, W.

S. R. Amin, C. D. Caldwell, W. Lichten, “Crossed beam spectroscopy of hydrogen: a new value for the Rydberg consant,” Phys. Rev. Lett. 47, 1234–1238 (1981).
[CrossRef]

Liddell, H. M.

O. S. Heavens, H. M. Liddell, “Staggered broad-band reflecting multilayers,” Appl. Opt. 5, 373–376 (1966).
[CrossRef] [PubMed]

H. M. Liddell, “Computer-aided techniques for the design of multilayer filters” (Hilger, Bristol, 1981).(Design techniques with digital computers and electrical filters.)

Macleond, H. A.

H. A. Macleond, Thin-Film Optical Filters (Elsevier, New York, 1969).

Meissner, K. W.

Netterfield, R. P.

Pelletier, E.

R. Badoual, E. Pelletier, “Miroirs insensibles aux defauts superficiels,” C. R. Acad. Sci. Ser. B 262, 937 (1966).

Penselin, S.

S. Penselin, A. Steudel, “Fabry–Perot interferometer verspiegelungen aus dielektrischen vielfachschichten,” Z. Phys. 142, 21–41 (1955). (Intensity only.)
[CrossRef]

Petersen, F. R.

Peterson, F. R.

Pollock, C. R.

Rank, D. H.

Rowley, W.

Sainty, W. G.

Schaeffer, R. C.

Seeley, J. S.

Seely, S.

S. Seely, Electron-Tube Circuits, 2nd ed. (McGraw-Hill, New York, 1958), pp. 133–137.

Shotton, K.

Shull, E. R.

Stanley, R. W.

Steudel, A.

S. Penselin, A. Steudel, “Fabry–Perot interferometer verspiegelungen aus dielektrischen vielfachschichten,” Z. Phys. 142, 21–41 (1955). (Intensity only.)
[CrossRef]

Stone, J. M.

Wells, J. S.

Wiggins, T. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Woods, P.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

R. L. Barger, J. L. Hall, “Wavelength of the 3.39-μm laser-saturated absorption line of methane,” Appl. Phys. Lett. 22, 196–199 (1973).
[CrossRef]

Atti Congr. Fis. Como (1)

H. A. Kramers, Atti Congr. Fis. Como, 545–557 (1927);R. de L. Kronig, “On the theory of dispersion of x-rays,” J. Opt. Soc. Am. 12, 547 (1926).
[CrossRef]

C. R. Acad. Sci. Ser. B (1)

R. Badoual, E. Pelletier, “Miroirs insensibles aux defauts superficiels,” C. R. Acad. Sci. Ser. B 262, 937 (1966).

J. Opt. Soc. Am. (10)

S. R. Amin, “Crossed beam spectroscopy of neon: precision measurement of three 20Ne wavelengths,” J. Opt. Soc. Am. 74, 862–863 (1983).
[CrossRef]

P. W. Baumeister, F. A. Jenkins, “Dispersion of the phase change for dielectric multilayers. Application to the interference filter,” J. Opt. Soc. Am. 47, 57–61 (1957).
[CrossRef]

K. W. Meissner, “Interference spectroscopy,” J. Opt. Soc. Am. 31, 405–427 (1941).
[CrossRef]

P. W. Baumeister, J. M. Stone, “Broad-band multilayer film for Fabry–Perot interferometers,” J. Opt. Soc. Am. 46, 228–229 (1956).
[CrossRef]

C. F. Bruce, P. E. Ciddor, “Phase dispersion in multilayer films,” J. Opt. Soc. Am. 50, 295–299 (1960).
[CrossRef]

J. M. Bennett, “Precise method for measuring the absolute phase change on reflection,” J. Opt. Soc. Am. 54, 612–624 (1964).This paper contains an excellent summary of the theory of reflection from metal surfaces.
[CrossRef]

R. W. Stanley, K. L. Andrew, “Use of dielectric coatings in absolute wavelength measurements with a Fabry–Perot interferometer,” J. Opt. Soc. Am. 54, 625–627 (1964).
[CrossRef]

D. H. Rank, H. E. Bennett, “Problem of phase variation with wavelength in dielectric films. Extension of interferometric standards in the infra-red,” J. Opt. Soc. Am. 45, 69–73 (1955).
[CrossRef]

D. H. Rank, E. R. Shull, J. M. Bennett, T. A. Wiggins, “Interferometric wavelength measurement in the infra red by the method of exact orders. Precision measurement of the index of refraction of air at 1.65 μ,” J. Opt. Soc. Am. 43, 952–956 (1953).
[CrossRef]

J. S. Seeley, “Resolving power of multilayer filters,” J. Opt. Soc. Am. 54, 342–346 (1964). (Phase.)
[CrossRef]

J. Phys. (Paris) (1)

H. Buisson, Ch. Fabry, “Mesures de longueurs d’onde pour l’establissement d’un système de repères spectroscopiques,” J. Phys. (Paris) 7, 169–195 (1908).

J. Res. Nat. Bur. Stand. (1)

C. J. Koester, “Phase shift effects of Fabry–Perot interferometry,” J. Res. Nat. Bur. Stand. 64A, 191–200 (1960).
[CrossRef]

Opt. Commun. (1)

J. C. Bergquist, H.-U. Daniel, “A wideband frequency-offset-locked dye laser spectrometer using a Schottky barrier mixer,” Opt. Commun. 48, 327–333 (1984).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

S. R. Amin, C. D. Caldwell, W. Lichten, “Crossed beam spectroscopy of hydrogen: a new value for the Rydberg consant,” Phys. Rev. Lett. 47, 1234–1238 (1981).
[CrossRef]

Rev. Opt. (1)

C. DuFour, “Épaisseur optique d’un étalon ou d’un filtre inter-férentiel de Fabry–Perot à miroirs semitransparents complexes,” Rev. Opt. 31, 1–16 (1952). (Intensity and phase.)

Z. Phys. (1)

S. Penselin, A. Steudel, “Fabry–Perot interferometer verspiegelungen aus dielektrischen vielfachschichten,” Z. Phys. 142, 21–41 (1955). (Intensity only.)
[CrossRef]

Other (8)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, Princeton, N.J., 1945).

G. Hass, L. Hadley, “Optical properties of metals,” in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1972).

S. Seely, Electron-Tube Circuits, 2nd ed. (McGraw-Hill, New York, 1958), pp. 133–137.

For a general discussion and literature citations, see Refs. 9, 21, 22, and 23.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955;reprinted by Dover, New York, 1965).

H. A. Macleond, Thin-Film Optical Filters (Elsevier, New York, 1969).

H. M. Liddell, “Computer-aided techniques for the design of multilayer filters” (Hilger, Bristol, 1981).(Design techniques with digital computers and electrical filters.)

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Figures (5)

Fig. 1
Fig. 1

Location of optically reflecting surface of various metals. Sources of data, theoretical curves calculated from data in Ref. 12 and from Ref. 4, Eq. (5); experimental data for 128-Å film, Ref. 4. All thin films are on quartz substrate.

Fig. 2
Fig. 2

Plot of reflection-coefficient amplitude for Al. Sources, fresh Al (calculated from data in Ref. 12); one-year old Al (experimental data in Ref. 12). RLC circuit is shown in Fig. 3.

Fig. 3
Fig. 3

One of four identical sections of a model circuit to mimic reflection coefficient of a fresh-Al films. See also Fig. 2.

Fig. 4
Fig. 4

Measured and calculated reflection amplitudes for a quarter-wave stack. The reflection phase shifts are calculated from the best fit with the following parameters: λ0 = 5800 Å, nH/nL = 1.74, 11 layers.

Fig. 5
Fig. 5

Calculated reflection amplitudes and phase shifts for a staggered 15-layer dielectric stack. Parameters are in Table 1. Source, Refs. 24 and 26.

Tables (4)

Tables Icon

Table 1 Characteristic Wavelengths of the 15 Layers of a Dielectric Stacka

Tables Icon

Table 2 Measured and Calculated Phase Shifts for a Quarter-Wave Stacka

Tables Icon

Table 3 Precision in Locating the Position of a Surface

Tables Icon

Table 4 Frequency and Wavelength of some He–Ne Laser Lines (c = 299,792,458 m/sec)

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

2 n t cos θ = m λ ,
Δ d = λ δ 4 π ,
r = n 1 n 0 n 1 + n 0 .
r ( ω ) = 2 ω π 0 r ( ω ) r ( ) ( ω ) ω 2 d ω ,
V OUT ( ω ) V IN ( ω ) = exp [ A ( ω ) + i B ( ω ) ] ,
B ( ω c ) = 2 ω c π 0 A ( ω ) A ( ω c ) ω 2 ω c 2 d ω .
r ( ω ) = E r ( ω ) / E i ( ω ) = [ n 1 ( ω ) n 0 ( ω ) ] / [ n 1 ( ω ) + n 0 ( ω ) ] .
R = | r | 2 ,
T = 1 R .
B ( ω c ) 2 ω c π ω h A ( ω ) A ( ω c ) ω 2 ω c 2 d ω , B ( ω c ) ω c π ω h ln R ( ω ) ω 2 d ω ,
A = { constant , ω < ω 0 k ω , ω > ω 0 .
B = k π ln | ω + ω 0 ω ω 0 | , ω < ω 0 .
B 2 π k ω ω 0 , ω ω 0 .
tan ϕ = { n 0 / [ n L ( n 1 ) ] } sin θ ,
θ = π λ 0 / λ .
B ( ω c ) = 2 ω c π ( 0 ω 1 A ω 2 ω c 2 d ω + ω 2 A d ω ω 2 ω c 2 ) .
d B d θ = 4 A π 2 ( ω 2 ω 1 ) .
r = r 1 + r 2 exp ( i Δ ) 1 + r 1 r 2 exp ( i Δ ) ,
Δ = 2 π λ 2 n 1 t ,
r 1 = n 0 n 1 n 0 + n 1 , r 2 = n 1 n 2 n 1 + n 2 ,
r = r 1 + exp [ i ( ϕ Δ ) ] 1 + r 1 exp [ ( i ϕ Δ ) ] = exp ( i θ ) .
r = r 1 exp ( i Δ ) 1 r 1 exp ( i Δ ) .
r r 1 1 + i Δ 1 r 1 + r 1 i Δ 1 i Δ ( r 1 + 1 1 r 1 ) .
r ( 1 + i 4 π n 0 t λ ) exp ( i 4 π n 0 t λ ) .
n 0 = 1 , n 1 = 1.52 , n 2 = 0.82 5.99 i ,
r 1 = 0.20635 , r 2 = 0.9378 exp ( i 152 ° ) = 0.9378 exp ( i 28 ° ) ,
Δ = 20.04 ° .
n 0 = 1 n 1 = 1 , n 2 = 0.82 5.99 i ,
r 1 = 0 , r 2 = 0.96 exp ( i 161 ° ) .
147.34 = 161.38 720 x 5460 Å ,
x = 106 Å .

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