Abstract

An efficient beam splitter for a Michelson interferometer can be made from a pair of prisms, using the process of frustrated total internal reflection. Although the phases and irradiances of the beams reflected and transmitted by the beam splitter depend on the polarization, the phase difference between the two interferometer beams vanishes for all polarizations and the transmittance can be made polarization insensitive by suitable design. The interferometer transmits radiant power within a wavelength band approximately two octaves wide, and rejects all other radiant power. The design, laboratory tests, and astronomical applications of an interferometer for the millimeter and submillimeter regions are discussed. An observation of the day-sky spectrum is presented.

© 1975 Optical Society of America

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References

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  1. I. Newton, Opticks (London, 1730), Book II, Obs. 1, 2, 4, 8 and Book III, Qu. 29.
  2. E. E. Hall, Phys. Rev. 15, 73 (1902).
  3. C. Schaefer and G. Gross, Ann. Phys. 32, 648 (1910).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), Ch. I.
  5. H. Wolters, in Handbuch der Physik, Vol. 24, edited by S. Flügge (Springer, Berlin, 1956), p. 461.
    [Crossref]
  6. N. S. Kapany and J. J. Burke, Optical Waveguides, (Academic, New York, 1972), Ch. 3.
  7. N. J. Harrick, Phys. Rev. Lett. 4, 224 (1960); Phys. Rev. 125, 1165 (1962).
    [Crossref]
  8. W. Leeb, Appl. Opt. 13, 17 (1974).
    [Crossref] [PubMed]
  9. N. J. Harrick, Appl. Opt. 2, 1203 (1963).
    [Crossref]
  10. H. D. Raker and G. R. Valenzuela, IRE Trans. MTT-10, 392 (1962).
    [Crossref]
  11. W. Culshaw and D. S. Jones, Proc. Phys. Soc. Lond. 66B859 (1953).
  12. V. A. Kuznetsov, V. N. Listvin, V. T. Potapov, and V. A. Strakhov, Opt. Spectrosk. 31, 151 (1971) [Opt. Spectrosc. 31, 77 (1971)].
  13. W. H. Steel, Interferometry (Cambridge U. P., London, 1967), p. 84.
  14. J. Chamberlain, Infrared Phys. 11, 25 (1971).
    [Crossref]
  15. J. Chamberlain and H. A. Gebbie, Infrared Phys. 11, 57 (1971).
    [Crossref]
  16. J. E. Harries and P. A. R. Ade, Infrared Phys. 12, 81 (1972).
    [Crossref]

1974 (1)

1972 (1)

J. E. Harries and P. A. R. Ade, Infrared Phys. 12, 81 (1972).
[Crossref]

1971 (3)

V. A. Kuznetsov, V. N. Listvin, V. T. Potapov, and V. A. Strakhov, Opt. Spectrosk. 31, 151 (1971) [Opt. Spectrosc. 31, 77 (1971)].

J. Chamberlain, Infrared Phys. 11, 25 (1971).
[Crossref]

J. Chamberlain and H. A. Gebbie, Infrared Phys. 11, 57 (1971).
[Crossref]

1963 (1)

1962 (1)

H. D. Raker and G. R. Valenzuela, IRE Trans. MTT-10, 392 (1962).
[Crossref]

1960 (1)

N. J. Harrick, Phys. Rev. Lett. 4, 224 (1960); Phys. Rev. 125, 1165 (1962).
[Crossref]

1953 (1)

W. Culshaw and D. S. Jones, Proc. Phys. Soc. Lond. 66B859 (1953).

1910 (1)

C. Schaefer and G. Gross, Ann. Phys. 32, 648 (1910).
[Crossref]

1902 (1)

E. E. Hall, Phys. Rev. 15, 73 (1902).

Ade, P. A. R.

J. E. Harries and P. A. R. Ade, Infrared Phys. 12, 81 (1972).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), Ch. I.

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides, (Academic, New York, 1972), Ch. 3.

Chamberlain, J.

J. Chamberlain, Infrared Phys. 11, 25 (1971).
[Crossref]

J. Chamberlain and H. A. Gebbie, Infrared Phys. 11, 57 (1971).
[Crossref]

Culshaw, W.

W. Culshaw and D. S. Jones, Proc. Phys. Soc. Lond. 66B859 (1953).

Gebbie, H. A.

J. Chamberlain and H. A. Gebbie, Infrared Phys. 11, 57 (1971).
[Crossref]

Gross, G.

C. Schaefer and G. Gross, Ann. Phys. 32, 648 (1910).
[Crossref]

Hall, E. E.

E. E. Hall, Phys. Rev. 15, 73 (1902).

Harrick, N. J.

N. J. Harrick, Appl. Opt. 2, 1203 (1963).
[Crossref]

N. J. Harrick, Phys. Rev. Lett. 4, 224 (1960); Phys. Rev. 125, 1165 (1962).
[Crossref]

Harries, J. E.

J. E. Harries and P. A. R. Ade, Infrared Phys. 12, 81 (1972).
[Crossref]

Jones, D. S.

W. Culshaw and D. S. Jones, Proc. Phys. Soc. Lond. 66B859 (1953).

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides, (Academic, New York, 1972), Ch. 3.

Kuznetsov, V. A.

V. A. Kuznetsov, V. N. Listvin, V. T. Potapov, and V. A. Strakhov, Opt. Spectrosk. 31, 151 (1971) [Opt. Spectrosc. 31, 77 (1971)].

Leeb, W.

Listvin, V. N.

V. A. Kuznetsov, V. N. Listvin, V. T. Potapov, and V. A. Strakhov, Opt. Spectrosk. 31, 151 (1971) [Opt. Spectrosc. 31, 77 (1971)].

Newton, I.

I. Newton, Opticks (London, 1730), Book II, Obs. 1, 2, 4, 8 and Book III, Qu. 29.

Potapov, V. T.

V. A. Kuznetsov, V. N. Listvin, V. T. Potapov, and V. A. Strakhov, Opt. Spectrosk. 31, 151 (1971) [Opt. Spectrosc. 31, 77 (1971)].

Raker, H. D.

H. D. Raker and G. R. Valenzuela, IRE Trans. MTT-10, 392 (1962).
[Crossref]

Schaefer, C.

C. Schaefer and G. Gross, Ann. Phys. 32, 648 (1910).
[Crossref]

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U. P., London, 1967), p. 84.

Strakhov, V. A.

V. A. Kuznetsov, V. N. Listvin, V. T. Potapov, and V. A. Strakhov, Opt. Spectrosk. 31, 151 (1971) [Opt. Spectrosc. 31, 77 (1971)].

Valenzuela, G. R.

H. D. Raker and G. R. Valenzuela, IRE Trans. MTT-10, 392 (1962).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), Ch. I.

Wolters, H.

H. Wolters, in Handbuch der Physik, Vol. 24, edited by S. Flügge (Springer, Berlin, 1956), p. 461.
[Crossref]

Ann. Phys. (1)

C. Schaefer and G. Gross, Ann. Phys. 32, 648 (1910).
[Crossref]

Appl. Opt. (2)

Infrared Phys. (3)

J. Chamberlain, Infrared Phys. 11, 25 (1971).
[Crossref]

J. Chamberlain and H. A. Gebbie, Infrared Phys. 11, 57 (1971).
[Crossref]

J. E. Harries and P. A. R. Ade, Infrared Phys. 12, 81 (1972).
[Crossref]

IRE Trans. (1)

H. D. Raker and G. R. Valenzuela, IRE Trans. MTT-10, 392 (1962).
[Crossref]

Opt. Spectrosk. (1)

V. A. Kuznetsov, V. N. Listvin, V. T. Potapov, and V. A. Strakhov, Opt. Spectrosk. 31, 151 (1971) [Opt. Spectrosc. 31, 77 (1971)].

Phys. Rev. (1)

E. E. Hall, Phys. Rev. 15, 73 (1902).

Phys. Rev. Lett. (1)

N. J. Harrick, Phys. Rev. Lett. 4, 224 (1960); Phys. Rev. 125, 1165 (1962).
[Crossref]

Proc. Phys. Soc. Lond. (1)

W. Culshaw and D. S. Jones, Proc. Phys. Soc. Lond. 66B859 (1953).

Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), Ch. I.

H. Wolters, in Handbuch der Physik, Vol. 24, edited by S. Flügge (Springer, Berlin, 1956), p. 461.
[Crossref]

N. S. Kapany and J. J. Burke, Optical Waveguides, (Academic, New York, 1972), Ch. 3.

I. Newton, Opticks (London, 1730), Book II, Obs. 1, 2, 4, 8 and Book III, Qu. 29.

W. H. Steel, Interferometry (Cambridge U. P., London, 1967), p. 84.

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Figures (11)

FIG. 1
FIG. 1

Optical geometry for the FTIR beam splitter.

FIG. 2
FIG. 2

Reflectance R of the FTIR beam splitter as a function of λ/d = (wavelength)/(prism spacing) for the two polarization states (⊥ and ∥) and for various angles of incidence. Curves are for prisms of refractive index n = 1.5; for higher n, the curves have a similar appearance, but are shifted to the right.

FIG. 3
FIG. 3

Phase shift δ r of the reflected wave with respect to the incident wave, for perpendicular polarization, as a function of λ/d = (wavelength)/(prism spacing). Curves are given for various angles of incidence i, assuming a prism material n = 1.5.

FIG. 4
FIG. 4

Optical paths within a Michelson interferometer using an FTIR beam splitter.

FIG. 5
FIG. 5

Dependence of interferometer efficiency on polarization, interferometer angle of incidence θ, and prism refractive index n. The efficiency is plotted against λ/d, the wavelength in units of the prism-gap spacing.

FIG. 6
FIG. 6

Beam-splitter efficiency for two important cases. (A) Efficiency τ = 1 2 ( τ + τ ) for unpolarized radiant flux with the 90° prisms of Fig. 5. Efficiency is calculated for the laboratory instrument with polyethylene prisms (n = 1.5) and d = 0.325 mm. Curves for angles of incidence between 0° and 3° cannot be distinguished because of the curve’s width. (B) Efficiency for the rhomboidal beam splitter (Fig. 7), with ψ chosen so that τ = τ. As in curve (A), n = 1.5 and d = 0.324 mm, and curves for angles of incidence between 0° and 3° cannot be distinguished because of the curve’s width.

FIG. 7
FIG. 7

FTIE beam splitter designed to eliminate differences of efficiency for the two states of polarization. A refractive index n = 1.5 requires a prism angle ψ = 51.7°.

FIG. 8
FIG. 8

Optics for the first frustrated-total-internal-reflection Fourier-transform-infrared, or (FTIR)2, interferometer.

FIG. 9
FIG. 9

Spectra of a mercury-arc lamp, showing atmospheric absorption lines, taken under identical conditions, except that the upper curve was obtained with a beam splitter consisting of a conventional Mylar film (100 µm thick), whereas the lower curve was obtained with an FTIR cube.

FIG. 10
FIG. 10

Optical system for the interferometer used on the Queen Mary College telescope.

FIG. 11
FIG. 11

Interferogram and transform spectrum of the day sky.

Equations (21)

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E 1 = E 1 exp { i [ ω t 2 π n σ ( x sini + y cosi ) ] } , E 2 = E 2 exp { i [ ω t 2 π n σ ( x sini y cosi ) ] } , E 5 = E 5 exp { i [ ω t 2 π n σ ( x sini + ( y d ) cosi ) ] } ,
ω = 2 π f = angular frequency , σ = 1 / λ = ω / 2 π c = wave number , c = velocity of light .
E 2 = E 1 ( n 2 1 ) sinh α d ( 1 + n 2 2 n 2 sin 2 i ) sinh α d 2 i n cosi ( n 2 sin 2 i 1 ) 1 / 2 cosh α d , E 5 = E 1 2 i n cosi ( n 2 sin 2 i 1 ) 1 / 2 ( 1 + n 2 2 n 2 sin 2 i ) sinh α d 2 i n cosi ( n 2 sin 2 i 1 ) 1 / 2 cosh α d ,
E 2 = E 1 ( n 2 1 ) ( n 2 sin 2 i cos 2 i ) sinh α d [ n 2 + 1 ( n 4 + 1 ) sin 2 i ] sinh α d 2 i n cosi ( n 2 sin 2 i 1 ) 1 / 2 cosh α d , E 5 = E 1 2 i n cosi ( n 2 sin 2 i 1 ) 1 / 2 [ n 2 + 1 ( n 4 + 1 ) sin 2 i ] sinh α d 2 i n cosi ( n 2 sin 2 i 1 ) 1 / 2 cosh α d .
R = ( n 2 1 ) 2 sinh 2 α d ( n 2 1 ) 2 sinh 2 α d + 4 n 2 cos 2 i ( n 2 sin 2 i 1 ) , R = ( n 2 1 ) 2 ( n 2 sin 2 i cos 2 i ) 2 sinh 2 α d ( n 2 1 ) 2 ( n 2 sin 2 i cos 2 i ) 2 sinh 2 α d + 4 n 2 cos 2 i ( n 2 sin 2 i 1 ) , T = 4 n 2 cos 2 i ( n 2 sin 2 i 1 ) ( n 2 1 ) 2 sinh 2 α d + 4 n 2 cos 2 i ( n 2 sin 2 i 1 ) , T = 4 n 2 cos 2 i ( n 2 sin 2 i 1 ) ( n 2 1 ) 2 ( n 2 sin 2 i cos 2 i ) 2 sinh 2 α d + 4 n 2 cos 2 i ( n 2 sin 2 i 1 ) .
tan δ r = Im ( E 2 / E 1 ) Re ( E 2 / E 1 ) ,
tan δ t = Im ( E 5 / E 1 ) Re ( E 5 / E 1 ) .
tan δ r = 2 n cos i ( n 2 sin 2 i 1 ) 1 / 2 coth α d 1 + n 2 2 n 2 sin 2 i , tan δ t = ( 1 + n 2 2 n 2 sin 2 i ) tanh α d 2 n cos i ( n 2 sin 2 i 1 ) 1 / 2 , tan δ r = 2 n cos i ( n 2 sin 2 i 1 ) 1 / 2 coth α d 1 + n 2 ( n 4 + 1 ) sin 2 i , tan δ t = [ 1 + n 2 ( n 4 + 1 ) sin 2 i ] tanh α d 2 n cos i ( n 2 sin 2 i 1 ) 1 / 2 .
δ t δ r = ± π / 2 , δ t δ r = ± π / 2 .
I = [ R ( i 1 ) T ( i 2 ) ] 1 / 2 exp { i 2 π σ ( 3 2 n L sec θ + s 1 ) + [ δ r ( i 1 ) + δ t ( i 2 ) ] } + [ T ( i 1 ) R ( i 2 ) ] 1 / 2 exp { i 2 π σ ( 3 2 n L sec θ + s 2 ) + [ δ t ( i 1 ) + δ r ( i 2 ) ] } ,
I = I I * = R ( i 1 ) T ( i 2 ) + T ( i 1 ) R ( i 2 ) + 2 [ R ( i 1 ) T ( i 2 ) T ( i 1 ) R ( i 2 ) ] 1 / 2 × cos [ 4 π σ ( l 2 l 1 ) cos θ δ t ( i 1 ) δ r ( i 2 ) + δ r ( i 1 ) + δ t ( i 2 ) ] .
I = [ R ( i 1 ) T ( i 2 ) + T ( i 1 ) R ( i 2 ) ] + 2 [ R ( i 1 ) T ( i 2 ) T ( i 1 ) R ( i 2 ) ] 1 / 2 cos [ 4 π σ ( l 2 l 1 ) ] ,
τ = 4 [ R ( i 1 ) T ( i 2 ) T ( i 1 ) R ( i 2 ) ] 1 / 2
cosi 1 = cos ψ cos θ + sin ψ sin θ cos φ , cosi 2 = cos ψ cos θ sin ψ sin θ cos φ .
E 3 = E 3 exp { i [ ω t 2 π σ ( x sini + y cosi ) ] } , E 4 = E 4 exp { i [ ω t 2 π σ ( x sini ( y d ) cosi ) ] } ,
E 1 + E 2 = E 3 + E 4 , H 1 cosi H 2 cosi = H 3 cosi H 4 cosi ,
E 3 + E 4 = E 5 , H 3 cosi H 4 cosi = H 5 cosi .
E 1 + E 2 = E 3 + e i β d E 4 , n cosi E 1 n cosi E 2 = cosi E 3 cosi E 3 cosi e i β d E 4 , e i β d E 3 + E 4 = E 5 , cosi e i β d E 3 cosi E 4 = n cosi E 5 ,
E 2 = E 1 ( n 2 1 ) ( e i β d e i β d ) e i β d ( n cosi + cosi ) 2 e i β d ( n cosi cosi ) 2 ,
E 5 = E 1 4 n cosi cosi e i β d ( n cosi + cosi ) 2 e i β d ( n cosi cosi ) 2 .
α = i β = ± 2 π σ ( n 2 sin 2 i 1 ) 1 / 2 ,