Abstract

For many years channel spectra, caused by multiple reflections of light between the faces of flat samples of optical material, have been used to determine refractive indices. Interferometers are excellent for this measurement, particularly in the far ir spectral region where their superior sensitivity and spectral resolution are required. The theory of the method is developed and the limitations are discussed. Experimentally determined refractive indices of silicon, germanium, and fused quartz are presented. These indices have been determined by these methods from data obtained with the Aerospace lamellar grating interferometer.

© 1967 Optical Society of America

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References

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  1. S. Roberts, D. D. Coon, J. Opt. Soc. Am. 52, 1023 (1962).
    [CrossRef]
  2. E. V. Loewenstein, J. Opt. Soc. Am. 51, 108 (1961).
    [CrossRef]
  3. E. E. Bell, Infrared Phys. 6, 57 (1966).
    [CrossRef]
  4. E. E. Russell, E. E. Bell, Infrared Phys. 6, 75 (1966).
    [CrossRef]
  5. E. E. Russell, E. E. Bell, J. Opt. Soc. Am. 57, 341 (1967).
    [CrossRef]
  6. R. T. Hall, D. Vrabec, J. M. Dowling, Appl. Opt. 5, 1147 (1966).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
  8. A. Sommerfield, Optics (Academic Press Inc., New York, 1964), p. 47.
  9. D. Malz, Exptl. Tech. Physik 4, 257 (1965).
  10. R. T. Hall, J. M. Dowling, J. Chem. Phys. 45, 1899 (1966).
    [CrossRef]
  11. J. E. Chamberlain, Appl. Opt. 6, 980 (1967), gives a discussion of the corrections required if a converging sample beam is used. These corrections arise from different effective path lengths in the sample, which we have corrected for, and different path lengths in the interferometer itself. Since our interferometer operates in essentially parallel light, this second correction is not applicable to our measurements. Thus our corrections differ slightly from those of Ref. 5.
    [CrossRef] [PubMed]
  12. J. H. Aronson, H. G. McLinden, P. J. Gielisse, Phys. Rev. 135, A785 (1964).
    [CrossRef]

1967 (2)

1966 (4)

R. T. Hall, D. Vrabec, J. M. Dowling, Appl. Opt. 5, 1147 (1966).
[CrossRef] [PubMed]

E. E. Bell, Infrared Phys. 6, 57 (1966).
[CrossRef]

E. E. Russell, E. E. Bell, Infrared Phys. 6, 75 (1966).
[CrossRef]

R. T. Hall, J. M. Dowling, J. Chem. Phys. 45, 1899 (1966).
[CrossRef]

1965 (1)

D. Malz, Exptl. Tech. Physik 4, 257 (1965).

1964 (1)

J. H. Aronson, H. G. McLinden, P. J. Gielisse, Phys. Rev. 135, A785 (1964).
[CrossRef]

1962 (1)

1961 (1)

Aronson, J. H.

J. H. Aronson, H. G. McLinden, P. J. Gielisse, Phys. Rev. 135, A785 (1964).
[CrossRef]

Bell, E. E.

E. E. Russell, E. E. Bell, J. Opt. Soc. Am. 57, 341 (1967).
[CrossRef]

E. E. Bell, Infrared Phys. 6, 57 (1966).
[CrossRef]

E. E. Russell, E. E. Bell, Infrared Phys. 6, 75 (1966).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Chamberlain, J. E.

Coon, D. D.

Dowling, J. M.

Gielisse, P. J.

J. H. Aronson, H. G. McLinden, P. J. Gielisse, Phys. Rev. 135, A785 (1964).
[CrossRef]

Hall, R. T.

Loewenstein, E. V.

Malz, D.

D. Malz, Exptl. Tech. Physik 4, 257 (1965).

McLinden, H. G.

J. H. Aronson, H. G. McLinden, P. J. Gielisse, Phys. Rev. 135, A785 (1964).
[CrossRef]

Roberts, S.

Russell, E. E.

E. E. Russell, E. E. Bell, J. Opt. Soc. Am. 57, 341 (1967).
[CrossRef]

E. E. Russell, E. E. Bell, Infrared Phys. 6, 75 (1966).
[CrossRef]

Sommerfield, A.

A. Sommerfield, Optics (Academic Press Inc., New York, 1964), p. 47.

Vrabec, D.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Appl. Opt. (2)

Exptl. Tech. Physik (1)

D. Malz, Exptl. Tech. Physik 4, 257 (1965).

Infrared Phys. (2)

E. E. Bell, Infrared Phys. 6, 57 (1966).
[CrossRef]

E. E. Russell, E. E. Bell, Infrared Phys. 6, 75 (1966).
[CrossRef]

J. Chem. Phys. (1)

R. T. Hall, J. M. Dowling, J. Chem. Phys. 45, 1899 (1966).
[CrossRef]

J. Opt. Soc. Am. (3)

Phys. Rev. (1)

J. H. Aronson, H. G. McLinden, P. J. Gielisse, Phys. Rev. 135, A785 (1964).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

A. Sommerfield, Optics (Academic Press Inc., New York, 1964), p. 47.

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Figures (9)

Fig. 1
Fig. 1

The function F(n) = ln{(n + 1)3/[8n(n − 1)]} for determining the maximum products of absorption α and thickness h for which the channel spectrum method is a feasible means of obtaining refractive index.

Fig. 2
Fig. 2

Transmittance T(ν) of fused quartz 2.14 mm thick showing both channels and transmittance change owing to changing absorption.

Fig. 3
Fig. 3

Upper: Fused double beam quartz difference interferogram (left) before editing and spectrum transformed from it. Lower: Same data after removing noise not located near the signature.

Fig. 4
Fig. 4

Absorption constant α = 4πkν and real refractive index for fused quartz.

Fig. 5
Fig. 5

Interferogram and spectrum portions for silicon sample. Because of the high refractive index and low absorption, four signatures are visible indicating light undergoing up to eight internal reflections is contributing to the spectrum.

Fig. 6
Fig. 6

Absorption constant α = 4πkν and real refractive index for silicon. △ = 0.194067-cm thick sample. ○ = 0.641495-cm thick sample.

Fig. 7
Fig. 7

Mean transmittance τ(ν) 0.2-cm thick sample of silicon: ⊙ data obtained with PE 301 grating spectrophotometer; ▲ taken with lamellar grating interferometer.

Fig. 8
Fig. 8

Absorption coefficient α = 4πkν and real refractive index n for germanium. △ = 0.193839-cm thick sample. ○ = 0.622931-cm thick sample.

Fig. 9
Fig. 9

Mean transmittance τ(ν) of 0.2-cm thick sample of germanium: ⊙ data obtained with PE 301 grating spectrometer. ▲ data obtained with lamellar grating interferometer.

Equations (12)

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A ( ν ) = t ^ 2 { 1 + l = 1 r ^ 2 l exp [ i ( 2 π l × 2 n ^ ν h cos β ) ] } exp ( 2 i π ν n ^ h cos β )
T ( ν ) = A × A * ,
T ( ν ) = τ ( ν ) ( 1 + 2 l = 1 ρ l cos l θ ) ,
τ ( ν ) = 4 n ^ / ( n ^ + 1 ) 2 2 exp ( - α h cos β ) / ( 1 - ρ 2 ) , = 16 ( n 2 + k 2 ) exp ( - α h cos β ) / { [ ( n + 1 ) 2 + k 2 ] 2 ( 1 - ρ 2 ) } ,
ρ = [ ( n - 1 ) 2 + k 2 ] exp ( - α h cos β ) / [ ( n + 1 ) 2 + k 2 ] ,
θ = 4 π n h ν cos β + δ = 4 π n h ν cos β + tan - 1 [ 2 k / ( n 2 + k 2 - 1 ) ] ,
T ( ν ) = l = 0 A l cos l θ ,
F ( x ) = 0 τ ( ν ) ( 1 + 2 l = 1 ρ l cos l θ ) cos 2 π ν x d ν ,
2 n ν max h cos β + δ / 2 π = m .
e - 2 α h > ( 1 + n ) 6 64 n 2 ( n - 1 ) 2 > ( 1 + n ) 6 64 n 2 ( n - 1 ) 2 [ 1 + ( 1 - n 2 ) 2 / 64 n 2 ] .
T ( ν ) = τ ( ν ) [ 1 + 2 l = 1 ρ l ( 1 - cos β m ) β sin β cos l θ ( β ) d β ] .
β sin β cos l θ ( β ) d β = [ [ cos l 4 π n h ( cos β m + 1 2 ) ] sin y , y ,

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