Abstract

The beam splitter of the all-reflection Michelson interferometer consists of a combination of three parallel diffraction gratings. This paper extends the analysis of the instrument to include the effects of lateral errors in the grating adjustment (i.e., displacements parallel to the grating faces and perpendicular to the grooves). Such errors are shown to introduce a phase shift independent of wavenumber and proportional to grating order number. Tests of an instrument designed for Fourier transform spectroscopy in the 500–1000-μm spectral range are reported and shown to be in agreement with the analysis. For wavenumbers which pass through the instrument in 2 or more orders, cross-order interference effects are expected to occur which cause rapid variations in the efficiency vs wavenumber curve. This possibility should be eliminated in the design of a practical instrument. The resolution of the test instrument (1.6 cm−1) was insufficient to reveal this effect.

© 1978 Optical Society of America

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References

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  1. G. H. C. Freeman, Opt. Commun. 14, 274 (1975).
    [CrossRef]
  2. R. A. Kruger, L. W. Anderson, F. L. Roesler, J. Opt. Soc. Am. 62, 938 (1972).
    [CrossRef]
  3. R. A. Kruger, L. W. Anderson, F.L. Roesler, Appl. Opt. 12, 533 (1973).
    [CrossRef] [PubMed]
  4. F. L. Roesler, R. A. Kruger, L. W. Anderson, in Space Optics, ISMN 0-309-02144-8 (National Academy of Sciences, Washington, D.C., 1974).
  5. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972, Chap. 12).

1975 (1)

G. H. C. Freeman, Opt. Commun. 14, 274 (1975).
[CrossRef]

1973 (1)

1972 (1)

Anderson, L. W.

R. A. Kruger, L. W. Anderson, F.L. Roesler, Appl. Opt. 12, 533 (1973).
[CrossRef] [PubMed]

R. A. Kruger, L. W. Anderson, F. L. Roesler, J. Opt. Soc. Am. 62, 938 (1972).
[CrossRef]

F. L. Roesler, R. A. Kruger, L. W. Anderson, in Space Optics, ISMN 0-309-02144-8 (National Academy of Sciences, Washington, D.C., 1974).

Bell, R. J.

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972, Chap. 12).

Freeman, G. H. C.

G. H. C. Freeman, Opt. Commun. 14, 274 (1975).
[CrossRef]

Kruger, R. A.

R. A. Kruger, L. W. Anderson, F.L. Roesler, Appl. Opt. 12, 533 (1973).
[CrossRef] [PubMed]

R. A. Kruger, L. W. Anderson, F. L. Roesler, J. Opt. Soc. Am. 62, 938 (1972).
[CrossRef]

F. L. Roesler, R. A. Kruger, L. W. Anderson, in Space Optics, ISMN 0-309-02144-8 (National Academy of Sciences, Washington, D.C., 1974).

Roesler, F. L.

R. A. Kruger, L. W. Anderson, F. L. Roesler, J. Opt. Soc. Am. 62, 938 (1972).
[CrossRef]

F. L. Roesler, R. A. Kruger, L. W. Anderson, in Space Optics, ISMN 0-309-02144-8 (National Academy of Sciences, Washington, D.C., 1974).

Roesler, F.L.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

G. H. C. Freeman, Opt. Commun. 14, 274 (1975).
[CrossRef]

Other (2)

F. L. Roesler, R. A. Kruger, L. W. Anderson, in Space Optics, ISMN 0-309-02144-8 (National Academy of Sciences, Washington, D.C., 1974).

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972, Chap. 12).

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Figures (10)

Fig. 1
Fig. 1

All-reflection interferometer for use as a Fourier transform spectrometer. G1, G2, and G3 are diffraction gratings with identical groove spacings. M1 and M2 are plane front surface mirrors, and M is a concave mirror.

Fig. 2
Fig. 2

Lateral translational phase shift for normal incidence. When the grating is shifted left a distance , the diffracted wave pattern moves with it, resulting in a phase shift 2π∊σ sinθ as measured between fixed observation points P and P′.

Fig. 3
Fig. 3

Estimated efficiency as a function of wavenumber for the all-reflection interferometer, showing peaks at grating blaze wave-numbers (from Ref. 2).

Fig. 4
Fig. 4

Displacement of beam-splitter grating from a symmetry position by a distance .

Fig. 5
Fig. 5

Cross-order interference between two beams of light with the same wavenumber, but traveling separately through one arm of the interferometer in order of diffraction k andk + 1 simultaneously. In going from G1 to G2, the (k + 1) order beam travels a greater geometric distance than the kth order beam. In addition, a phase difference due to the LTPS arises between the two beams because they do not arrive at the same position on G2.

Fig. 6
Fig. 6

The lateral translational effect is evident in this plot of the interferometer output as a function of . The data were obtained using the 2.2-mm klystron source and the d = 3.81-mm grating set. A maximum occurs whenever the total phase difference is an integer multiple of 2π The symmetry positions of the grating are indicated by the numbered vertical dashes. Δ ≈ +0.3 mm.

Fig. 7
Fig. 7

Top: Interferogram obtained using 2.2-mm and 3.2-mm klystrons simultaneously and the d = 6.35-mm grating set. Bottom: Spectrum obtained by double-sided transformation of interferogram. Peaks appear at 3.16 cm−1 and 4.15 cm−1.

Fig. 8
Fig. 8

Interferograms of a mercury arc source obtained using gratings with a groove spacing d = 3.81 mm, and (a) ∊/d = 0, (b) ∊/d = 0.026, (c) ∊/d = 0.125. The reference Δ position is denoted by the arrow. Adjacent peaks are separated in path difference by 2.42 mm for all three cases. In part (d), the spectrum obtained by transformation of interferogram in (c) is shown. The numbers identify each peak with a diffraction order.

Fig. 9
Fig. 9

Above: Interferogram of a mercury arc source, using gratings with d = 6.35 mm. The peaks are separated by 3.97 mm in Δ. Below: Transform of the interferogram. The numbers identify each peak with a diffraction order.

Fig. 10
Fig. 10

Interferogram peak positions in Δ as a function of lateral grating displacement. The lines drawn have a slope given by −4 sinθB where θB = 40°.

Equations (27)

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ϕ = | 2 π k / d | ,
I ( Δ ) I ( ) = 0 B ( σ ) cos 2 π σ Δ d σ ,
B c ( σ ) = Δ max Δ max d Δ A ( Δ ) [ I ( Δ ) I ( ) ] cos ( 2 π σ Δ ) ,
ϕ ( k ) = ( 8 π k ) / d
/ d = m / 4 k ,
/ d = m / 4 ,
ϕ T = ϕ ( k ) + 2 π σ Δ ,
ϕ T k = 2 π k σ 1 ( Δ + 4 sin θ B ) .
σ 1 ( Δ n + 4 sin θ B ) = n
Δ n = [ n / ( σ 1 ) ] 4 sin θ B ,
D ( σ ) = k = 0 δ ( σ k σ 1 ) ,
B ( σ ) = [ D ( σ ) W ( σ ) ] E ( σ ) ,
Φ ( σ ) = ϕ ( k ) , k + 1 2 σ / σ 1 > k 1 2 .
I ( Δ ) I ( ) = 0 B ( σ ) cos [ 2 π σ Δ + Φ ( σ ) ] d σ ,
I ( Δ ) I ( σ ) = 1 2 B ( σ ) exp [ i Φ ( σ ) ] exp ( i 2 π σ Δ ) d σ .
B ( σ ) exp [ i Φ ( σ ) ] = E ( σ ) exp [ i Φ ( σ ) ] k [ δ ( σ k σ 1 ) W ( σ ) ] .
B ( σ ) exp ( i Φ ) = E ( σ ) ] k [ { exp ( i ϕ ( k ) ] δ ( σ k σ 1 ) } W ( σ ) ]
B exp ( i ϕ ) = [ D exp ( i ϕ ) W ] E .
I ( Δ ) I ( ) = 1 2 { W ¯ [ D exp ( i ϕ ) ] ¯ } Ē ,
[ D exp ( i ϕ ) ] ¯ = k = exp [ i 2 π k σ 1 ( Δ + 4 sin θ B ) ] ,
[ D exp ( i ϕ ) ] ¯ = 1 σ 1 n = δ ( Δ + 4 sin θ B n σ 1 ) .
I ( Δ ) I ( ) = 1 2 σ 1 [ W ¯ n δ ( Δ + 4 sin θ B n σ 1 ) ] Ē .
Δ n = n σ 1 4 sin θ B ,
Δ n + 1 Δ n = 1 / σ 1 = d sin θ B ,
Δ o = 4 sin θ B .
4 π k d τ = 4 π S d [ ( k + 1 ) tan θ k + 1 k tan θ k ] .
ϕ k + 1 ϕ k = 4 π σ S ( cos θ k cos θ k + 1 ) ,

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