Abstract

A multichannel interferometer that uses a rotating grating to both disperse and Doppler shift an incident far-infrared laser beam is described. The laser beam is diffracted into a fan array of many (~ 10) beams that can be directed to probe the plasma along distinct chords. The phase-shift information is multiplexed in the frequency domain so that a single detector suffices to sense the recombined probing beams.

© 1992 Optical Society of America

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References

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  1. D. Veron, ‘High sensitivity HCN laser interferometer for plasma electron density measurements,” Opt. Commun. 10, 95–98 (1974).
    [CrossRef]
  2. P. E. Young, D. P. Neikirk, P. P. Tong, D. B. Rutledge, N. C. Luhmann, “Multichannel far-infrared phase imaging for fusion plasmas,” Rev. Sci. Instrum. 56, 81–89 (1984).
    [CrossRef]
  3. W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
    [CrossRef]
  4. D. Veron, “Submillimeter interferometry of high density plasmas,” in Infrared and Millimeter Waves, K. J. Button, ed. (Academic, New York, 1979), Vol. 2, pp. 69–135.
  5. J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
    [CrossRef]
  6. J. Howard, “Novel scanning interferometer for two-dimensional plasma density measurements,” Rev. Sci. Instrum. 61, 1086–1094 (1990).
    [CrossRef]
  7. H. Ikuno, K. Yasuura, “Improved point matching method with application to scattering from a periodic structure,” IEEE Trans. Antennas Propagat. AP-21, 657–662 (1973).
    [CrossRef]
  8. T. Matsuda, Y. Okuno, “Computer-aided algorithm based on the Yasuura method for analysis of diffraction by a grating,” J. Opt. Soc. Am. A 7, 1693–1700 (1990).
    [CrossRef]
  9. J. P. Hugonin, R. Petit, M. Cadilhac, “Plane wave expansions used to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–598 (1981).
    [CrossRef]
  10. N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propagat. AP-17, 747–751 (1969).
    [CrossRef]
  11. L. B. Whitbourn, “A 337 μm density interferometer for the LT4 Tokamak,” Int. J. Infrared Millimeter Waves 5, 625–635 (1984).
    [CrossRef]
  12. H. Dammann, K. Görtler “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  13. D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
    [CrossRef]
  14. J. Howard, W. A. Peebles, N. C. Luhmann, “The use of polarization transforming reflectors for far-infrared and millimeter waves,” Int. J. Infrared Millimeter Waves 7, 1591–1603 (1986).
    [CrossRef]

1990

J. Howard, “Novel scanning interferometer for two-dimensional plasma density measurements,” Rev. Sci. Instrum. 61, 1086–1094 (1990).
[CrossRef]

T. Matsuda, Y. Okuno, “Computer-aided algorithm based on the Yasuura method for analysis of diffraction by a grating,” J. Opt. Soc. Am. A 7, 1693–1700 (1990).
[CrossRef]

1988

J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
[CrossRef]

1987

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

1986

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

J. Howard, W. A. Peebles, N. C. Luhmann, “The use of polarization transforming reflectors for far-infrared and millimeter waves,” Int. J. Infrared Millimeter Waves 7, 1591–1603 (1986).
[CrossRef]

1984

P. E. Young, D. P. Neikirk, P. P. Tong, D. B. Rutledge, N. C. Luhmann, “Multichannel far-infrared phase imaging for fusion plasmas,” Rev. Sci. Instrum. 56, 81–89 (1984).
[CrossRef]

L. B. Whitbourn, “A 337 μm density interferometer for the LT4 Tokamak,” Int. J. Infrared Millimeter Waves 5, 625–635 (1984).
[CrossRef]

1981

1974

D. Veron, ‘High sensitivity HCN laser interferometer for plasma electron density measurements,” Opt. Commun. 10, 95–98 (1974).
[CrossRef]

1973

H. Ikuno, K. Yasuura, “Improved point matching method with application to scattering from a periodic structure,” IEEE Trans. Antennas Propagat. AP-21, 657–662 (1973).
[CrossRef]

1971

H. Dammann, K. Görtler “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

1969

N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propagat. AP-17, 747–751 (1969).
[CrossRef]

Amitay, N.

N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propagat. AP-17, 747–751 (1969).
[CrossRef]

Bengston, R. D.

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

Brower, D. L.

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

Cadilhac, M.

Choi, D. W.

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

Dammann, H.

H. Dammann, K. Görtler “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Doyle, E. J.

J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
[CrossRef]

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

Galindo, V.

N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propagat. AP-17, 747–751 (1969).
[CrossRef]

Görtler, K.

H. Dammann, K. Görtler “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Howard, J.

J. Howard, “Novel scanning interferometer for two-dimensional plasma density measurements,” Rev. Sci. Instrum. 61, 1086–1094 (1990).
[CrossRef]

J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
[CrossRef]

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

J. Howard, W. A. Peebles, N. C. Luhmann, “The use of polarization transforming reflectors for far-infrared and millimeter waves,” Int. J. Infrared Millimeter Waves 7, 1591–1603 (1986).
[CrossRef]

Hugonin, J. P.

Ikuno, H.

H. Ikuno, K. Yasuura, “Improved point matching method with application to scattering from a periodic structure,” IEEE Trans. Antennas Propagat. AP-21, 657–662 (1973).
[CrossRef]

Joyce, G.

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

Kim, S. K.

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

Lehecka, T.

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

Luhmann, N. C.

J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
[CrossRef]

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

J. Howard, W. A. Peebles, N. C. Luhmann, “The use of polarization transforming reflectors for far-infrared and millimeter waves,” Int. J. Infrared Millimeter Waves 7, 1591–1603 (1986).
[CrossRef]

P. E. Young, D. P. Neikirk, P. P. Tong, D. B. Rutledge, N. C. Luhmann, “Multichannel far-infrared phase imaging for fusion plasmas,” Rev. Sci. Instrum. 56, 81–89 (1984).
[CrossRef]

Matsuda, T.

Neikirk, D. P.

P. E. Young, D. P. Neikirk, P. P. Tong, D. B. Rutledge, N. C. Luhmann, “Multichannel far-infrared phase imaging for fusion plasmas,” Rev. Sci. Instrum. 56, 81–89 (1984).
[CrossRef]

Okuno, Y.

Peebles, W. A.

J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
[CrossRef]

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

J. Howard, W. A. Peebles, N. C. Luhmann, “The use of polarization transforming reflectors for far-infrared and millimeter waves,” Int. J. Infrared Millimeter Waves 7, 1591–1603 (1986).
[CrossRef]

Petit, R.

Powers, E. J.

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

Reibeiz, G.

J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
[CrossRef]

Rutledge, D. B.

P. E. Young, D. P. Neikirk, P. P. Tong, D. B. Rutledge, N. C. Luhmann, “Multichannel far-infrared phase imaging for fusion plasmas,” Rev. Sci. Instrum. 56, 81–89 (1984).
[CrossRef]

Savage, R. L.

J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
[CrossRef]

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

Tong, P. P.

P. E. Young, D. P. Neikirk, P. P. Tong, D. B. Rutledge, N. C. Luhmann, “Multichannel far-infrared phase imaging for fusion plasmas,” Rev. Sci. Instrum. 56, 81–89 (1984).
[CrossRef]

Veron, D.

D. Veron, ‘High sensitivity HCN laser interferometer for plasma electron density measurements,” Opt. Commun. 10, 95–98 (1974).
[CrossRef]

D. Veron, “Submillimeter interferometry of high density plasmas,” in Infrared and Millimeter Waves, K. J. Button, ed. (Academic, New York, 1979), Vol. 2, pp. 69–135.

Whitbourn, L. B.

L. B. Whitbourn, “A 337 μm density interferometer for the LT4 Tokamak,” Int. J. Infrared Millimeter Waves 5, 625–635 (1984).
[CrossRef]

Yasuura, K.

H. Ikuno, K. Yasuura, “Improved point matching method with application to scattering from a periodic structure,” IEEE Trans. Antennas Propagat. AP-21, 657–662 (1973).
[CrossRef]

Young, P. E.

P. E. Young, D. P. Neikirk, P. P. Tong, D. B. Rutledge, N. C. Luhmann, “Multichannel far-infrared phase imaging for fusion plasmas,” Rev. Sci. Instrum. 56, 81–89 (1984).
[CrossRef]

IEEE Trans. Antennas Propagat.

H. Ikuno, K. Yasuura, “Improved point matching method with application to scattering from a periodic structure,” IEEE Trans. Antennas Propagat. AP-21, 657–662 (1973).
[CrossRef]

N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propagat. AP-17, 747–751 (1969).
[CrossRef]

Int. J. Infrared Millimeter Waves

L. B. Whitbourn, “A 337 μm density interferometer for the LT4 Tokamak,” Int. J. Infrared Millimeter Waves 5, 625–635 (1984).
[CrossRef]

W. A. Peebles, R. L. Savage, D. L. Brower, S. K. Kim, T. Lehecka, J. Howard, E. J. Doyle, N. C. Luhmann, “Plasma diagnostic applications on the TEXT Tokamak using a high-power twin-frequency optically pumped far-infrared laser,” Int. J. Infrared Millimeter Waves 8, 1355–1363 (1987).
[CrossRef]

J. Howard, W. A. Peebles, N. C. Luhmann, “The use of polarization transforming reflectors for far-infrared and millimeter waves,” Int. J. Infrared Millimeter Waves 7, 1591–1603 (1986).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

D. Veron, ‘High sensitivity HCN laser interferometer for plasma electron density measurements,” Opt. Commun. 10, 95–98 (1974).
[CrossRef]

H. Dammann, K. Görtler “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Rev. Sci. Instrum.

D. W. Choi, E. J. Powers, R. D. Bengston, G. Joyce, D. L. Brower, N. C. Luhmann, W. A. Peebles, “Digital complex demodulation applied to interferometry,” Rev. Sci. Instrum. 57, 1989–1991 (1986).
[CrossRef]

P. E. Young, D. P. Neikirk, P. P. Tong, D. B. Rutledge, N. C. Luhmann, “Multichannel far-infrared phase imaging for fusion plasmas,” Rev. Sci. Instrum. 56, 81–89 (1984).
[CrossRef]

J. Howard, E. J. Doyle, G. Reibeiz, R. L. Savage, W. A. Peebles, N. C. Luhmann, “Density profile reconstructions from 2-D interferometric data on microtor using novel tomographic analysis techniques,” Rev. Sci. Instrum. 59, 2135–2138 (1988).
[CrossRef]

J. Howard, “Novel scanning interferometer for two-dimensional plasma density measurements,” Rev. Sci. Instrum. 61, 1086–1094 (1990).
[CrossRef]

Other

D. Veron, “Submillimeter interferometry of high density plasmas,” in Infrared and Millimeter Waves, K. J. Button, ed. (Academic, New York, 1979), Vol. 2, pp. 69–135.

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Figures (8)

Fig. 1
Fig. 1

Diffraction geometry showing relationship between incident and reflected rays.

Fig. 2
Fig. 2

Groove profiles for (a) scanning and (b) multiorder (#3) gratings.

Fig. 3
Fig. 3

Measured reflected beam profiles for groove patterns #l–#3 at a distance 0.3 m from the grating. The superimposed gray bars show the computed relative intensities, and the positions of the minimum and maximum design orders m1 and mN are indicated on the plot.

Fig. 4
Fig. 4

Computed reflected beam intensities for modified groove pattern #1 for (a) θi = −45°, P polarization; (b) θi = −45°, S polarization; and (c) θi = −55°, S polarization.

Fig. 5
Fig. 5

Diagram showing reflection of diffracted orders from the successively inclined subfacets that constitute the modified piece-wise linear grating profile #1. The nonreflecting surface is parallel to the incident wave vector (heavy line). The lengths of the solid lines emanating from the center of each subfacet represent the relative reflected beam intensities (S polarization), while the dashed lines are the subfacet normals. Radiation in orders not corresponding to one of the subfacets is shown as emanating from the groove center.

Fig. 6
Fig. 6

Measured reflected beam profile for groove pattern # 1 at a distance 0.3 m from the grating and incidence angle −45°. The superimposed gray bars show the corresponding computed relative intensities.

Fig. 7
Fig. 7

Test grating interferometer configuration.

Fig. 8
Fig. 8

Fringe bursts from the ninth-order diffracted beam for d = 4-mm (#1) and d = 5-mm (#3) grooves. In each case, the intermediate frequency is nine times greater than the small amplitude modulation at the groove frequency.

Equations (19)

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

sin θ m = sin θ i + m K / k 0 .
Ω m = m K R ω cos γ ,
δθ m = δξ ( 1 cos θ c cos θ m )
y = η ( x ) = q = 1 Q F q ( α q x + β q ) ,
H 1 = k ˆ ψ i ( x , y ) ,
ψ i ( x , y ) = exp ( j k i x X + j k i y y ) ,
( k i x , k i y ) = k 0 ( sin θ i , cos θ i ) .
Ψ r = m = H m ψ m ( x , y ) ,
ψ m ( x , y ) = exp ( j k m x x + j k m y y ) ,
k m x = k i x + m K , k m y = ( k 0 2 k m x 2 ) 1 / 2 ,
Ψ r ( S ) = Ψ i ( S ) ,
Ψ r ( S ) v = Ψ i ( S ) v
Ψ r ( M ) = m = M m = M H m ( M ) ψ m ( x , y )
δ ( M ) = 1 M p m = M m = M | H m ( M ) H m ( M 1 ) | 2 | H m ( M ) | 2 ,
Pattern # 1 : d = 4 mm , ( m 1 , m N ) = ( 4 , 15 ) ; Pattern # 2 : d = 4 mm , ( m 1 , m N ) = ( 6 , 14 ) ; Pattern #3: d = 5 mm , ( m 1 , m N ) = ( 6 , 18 ) .
u LO = a LO exp ( j ω LO t ) ,
u 1 = a 1 exp [ j ω 1 t + j ϕ 1 ( t ) ] , u 2 = a 2 exp [ j ω 2 t + j ϕ 2 ( t ) ] ,
i ( t ) = | u 1 ( t ) + u 2 ( t ) + u LO ( t ) | 2 ,
i ( t ) = a LO a 1 cos [ ( Ω LO Ω 1 ) t ϕ 1 ( t ) ] + a LO a 2 cos [ ( Ω LO 2 Ω 1 ) t ϕ 2 ( t ) ] + a 1 a 2 cos [ Ω 1 t + ϕ 1 ( t ) ϕ 2 ( t ) ] .

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