Abstract

A new setup for plasma diagnostics is presented that is based on real-time holographic interferometry. The hologram is used as a holographic optical element (HOE) that combines the properties of a hologram, of a lens, and of a grating simultaneously. The HOE is responsible for the formation of the interference pattern, and, in addition, acts as an imaging element and prevents most of the plasma radiation from reaching the interferogram detection system. The spectral and imaging properties of this HOE are calculated numerically, and this numeric procedure is tested experimentally. We applied the HOE–interferometry technique to the measurement of the electron density in a brightly radiating high-pressure xenon lamp. The principle of this experiment, two-wavelength interferometry, is described, and the results of the measurement are presented and discussed.

© 1996 Optical Society of America

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References

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  1. W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
    [CrossRef]
  2. O. Bryngdahl, W.-H. Lee, “Shearing interferometry in polar coordinates,” J. Opt. Soc. Am. 64, 1606–1615 (1974).
    [CrossRef]
  3. G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 93–167.
    [CrossRef]
  4. L. Hesselink, “Optical tomography,” in Handbook of Flow Visualization, W.-J. Yang, ed. (Hemisphere, New York, 1989), pp. 307–329.
  5. J. N. Latta, “Computer-based analysis of hologram imagery and aberrations. I. Hologram types and their nonchromatic aberrations,” Appl. Opt. 10, 599–608 (1971).
    [CrossRef] [PubMed]
  6. J. N. Latta, “Computer-based analysis of hologram imagery and aberrations. II. Aberrations induced by a wavelength shift,” Appl. Opt. 10, 609–618 (1971).
    [CrossRef] [PubMed]
  7. J. N. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2698–2710 (1971).
    [CrossRef] [PubMed]
  8. D. Close, “Optically recorded holographic optical elements,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), pp. 573–585.
  9. R. Alpher, D. White, “Optical refractivity of high-temperature gases,” Phys. Fluids 2, 153–169 (1959).
    [CrossRef]
  10. M. Mitchener, C. Kruger, Partially Ionized Gases (Wiley, New York, 1975).
  11. G. Pretzler, H. Jäger, T. Neger, H. Philipp, J. Woisetschläger, “Comparison of different methods of Abel inversion using computer simulated and experimental side-on data,” Z. Naturforsch. Teil A 47, 955–970 (1992).
  12. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  13. A. Džubur, D. Vukicević, “Ultrahigh resolution sandwich holography,” Appl. Opt. 23, 1474 (1984).
    [CrossRef]
  14. H. Philipp, T. Neger, H. Jäger, J. Woisetschläger, “Optical tomography of phase objects by holographic interferometry,” Measurement 10 (4), 170–181 (1992).
    [CrossRef]
  15. G. Pretzler, “A new method for numerical Abel-inversion,” Z. Naturforsch. Teil A 46, 639–641 (1991).
  16. V. E. Gavrilov, “The continuous absorption (radiation) spectrum of a pulsed-discharge plasma in a closed quartz tube with a xenon filling,” Opt. Spectros. 59, 5 (1985).
  17. J. Woisetschläger, H. Jäger, T. Neger, K. Widmann, “Investigation of the population inversion in a He–Ne laser discharge by heterodyne holographic interferometry,” Appl. Phys. B 54, 132–135 (1992).
    [CrossRef]
  18. K. Widmann, G. Pretzler, J. Woisetschläger, T. Neger, H. Jäger, “Application of holographic optical elements to plasma diagnostics,” in Holographics International ‘92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 712–718 (1992).

1992 (3)

G. Pretzler, H. Jäger, T. Neger, H. Philipp, J. Woisetschläger, “Comparison of different methods of Abel inversion using computer simulated and experimental side-on data,” Z. Naturforsch. Teil A 47, 955–970 (1992).

H. Philipp, T. Neger, H. Jäger, J. Woisetschläger, “Optical tomography of phase objects by holographic interferometry,” Measurement 10 (4), 170–181 (1992).
[CrossRef]

J. Woisetschläger, H. Jäger, T. Neger, K. Widmann, “Investigation of the population inversion in a He–Ne laser discharge by heterodyne holographic interferometry,” Appl. Phys. B 54, 132–135 (1992).
[CrossRef]

1991 (1)

G. Pretzler, “A new method for numerical Abel-inversion,” Z. Naturforsch. Teil A 46, 639–641 (1991).

1985 (1)

V. E. Gavrilov, “The continuous absorption (radiation) spectrum of a pulsed-discharge plasma in a closed quartz tube with a xenon filling,” Opt. Spectros. 59, 5 (1985).

1984 (1)

1982 (1)

1974 (1)

1971 (3)

1959 (1)

R. Alpher, D. White, “Optical refractivity of high-temperature gases,” Phys. Fluids 2, 153–169 (1959).
[CrossRef]

Alpher, R.

R. Alpher, D. White, “Optical refractivity of high-temperature gases,” Phys. Fluids 2, 153–169 (1959).
[CrossRef]

Bryngdahl, O.

Close, D.

D. Close, “Optically recorded holographic optical elements,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), pp. 573–585.

Džubur, A.

Gavrilov, V. E.

V. E. Gavrilov, “The continuous absorption (radiation) spectrum of a pulsed-discharge plasma in a closed quartz tube with a xenon filling,” Opt. Spectros. 59, 5 (1985).

Hesselink, L.

L. Hesselink, “Optical tomography,” in Handbook of Flow Visualization, W.-J. Yang, ed. (Hemisphere, New York, 1989), pp. 307–329.

Ina, H.

Jäger, H.

H. Philipp, T. Neger, H. Jäger, J. Woisetschläger, “Optical tomography of phase objects by holographic interferometry,” Measurement 10 (4), 170–181 (1992).
[CrossRef]

G. Pretzler, H. Jäger, T. Neger, H. Philipp, J. Woisetschläger, “Comparison of different methods of Abel inversion using computer simulated and experimental side-on data,” Z. Naturforsch. Teil A 47, 955–970 (1992).

J. Woisetschläger, H. Jäger, T. Neger, K. Widmann, “Investigation of the population inversion in a He–Ne laser discharge by heterodyne holographic interferometry,” Appl. Phys. B 54, 132–135 (1992).
[CrossRef]

K. Widmann, G. Pretzler, J. Woisetschläger, T. Neger, H. Jäger, “Application of holographic optical elements to plasma diagnostics,” in Holographics International ‘92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 712–718 (1992).

Kobayashi, S.

Kruger, C.

M. Mitchener, C. Kruger, Partially Ionized Gases (Wiley, New York, 1975).

Latta, J. N.

Lee, W.-H.

O. Bryngdahl, W.-H. Lee, “Shearing interferometry in polar coordinates,” J. Opt. Soc. Am. 64, 1606–1615 (1974).
[CrossRef]

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

Mitchener, M.

M. Mitchener, C. Kruger, Partially Ionized Gases (Wiley, New York, 1975).

Neger, T.

G. Pretzler, H. Jäger, T. Neger, H. Philipp, J. Woisetschläger, “Comparison of different methods of Abel inversion using computer simulated and experimental side-on data,” Z. Naturforsch. Teil A 47, 955–970 (1992).

J. Woisetschläger, H. Jäger, T. Neger, K. Widmann, “Investigation of the population inversion in a He–Ne laser discharge by heterodyne holographic interferometry,” Appl. Phys. B 54, 132–135 (1992).
[CrossRef]

H. Philipp, T. Neger, H. Jäger, J. Woisetschläger, “Optical tomography of phase objects by holographic interferometry,” Measurement 10 (4), 170–181 (1992).
[CrossRef]

K. Widmann, G. Pretzler, J. Woisetschläger, T. Neger, H. Jäger, “Application of holographic optical elements to plasma diagnostics,” in Holographics International ‘92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 712–718 (1992).

Philipp, H.

G. Pretzler, H. Jäger, T. Neger, H. Philipp, J. Woisetschläger, “Comparison of different methods of Abel inversion using computer simulated and experimental side-on data,” Z. Naturforsch. Teil A 47, 955–970 (1992).

H. Philipp, T. Neger, H. Jäger, J. Woisetschläger, “Optical tomography of phase objects by holographic interferometry,” Measurement 10 (4), 170–181 (1992).
[CrossRef]

Pretzler, G.

G. Pretzler, H. Jäger, T. Neger, H. Philipp, J. Woisetschläger, “Comparison of different methods of Abel inversion using computer simulated and experimental side-on data,” Z. Naturforsch. Teil A 47, 955–970 (1992).

G. Pretzler, “A new method for numerical Abel-inversion,” Z. Naturforsch. Teil A 46, 639–641 (1991).

K. Widmann, G. Pretzler, J. Woisetschläger, T. Neger, H. Jäger, “Application of holographic optical elements to plasma diagnostics,” in Holographics International ‘92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 712–718 (1992).

Schulz, G.

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 93–167.
[CrossRef]

Schwider, J.

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 93–167.
[CrossRef]

Takeda, M.

Vukicevic, D.

White, D.

R. Alpher, D. White, “Optical refractivity of high-temperature gases,” Phys. Fluids 2, 153–169 (1959).
[CrossRef]

Widmann, K.

J. Woisetschläger, H. Jäger, T. Neger, K. Widmann, “Investigation of the population inversion in a He–Ne laser discharge by heterodyne holographic interferometry,” Appl. Phys. B 54, 132–135 (1992).
[CrossRef]

K. Widmann, G. Pretzler, J. Woisetschläger, T. Neger, H. Jäger, “Application of holographic optical elements to plasma diagnostics,” in Holographics International ‘92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 712–718 (1992).

Woisetschläger, J.

J. Woisetschläger, H. Jäger, T. Neger, K. Widmann, “Investigation of the population inversion in a He–Ne laser discharge by heterodyne holographic interferometry,” Appl. Phys. B 54, 132–135 (1992).
[CrossRef]

G. Pretzler, H. Jäger, T. Neger, H. Philipp, J. Woisetschläger, “Comparison of different methods of Abel inversion using computer simulated and experimental side-on data,” Z. Naturforsch. Teil A 47, 955–970 (1992).

H. Philipp, T. Neger, H. Jäger, J. Woisetschläger, “Optical tomography of phase objects by holographic interferometry,” Measurement 10 (4), 170–181 (1992).
[CrossRef]

K. Widmann, G. Pretzler, J. Woisetschläger, T. Neger, H. Jäger, “Application of holographic optical elements to plasma diagnostics,” in Holographics International ‘92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 712–718 (1992).

Appl. Opt. (4)

Appl. Phys. B (1)

J. Woisetschläger, H. Jäger, T. Neger, K. Widmann, “Investigation of the population inversion in a He–Ne laser discharge by heterodyne holographic interferometry,” Appl. Phys. B 54, 132–135 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

Measurement (1)

H. Philipp, T. Neger, H. Jäger, J. Woisetschläger, “Optical tomography of phase objects by holographic interferometry,” Measurement 10 (4), 170–181 (1992).
[CrossRef]

Opt. Spectros. (1)

V. E. Gavrilov, “The continuous absorption (radiation) spectrum of a pulsed-discharge plasma in a closed quartz tube with a xenon filling,” Opt. Spectros. 59, 5 (1985).

Phys. Fluids (1)

R. Alpher, D. White, “Optical refractivity of high-temperature gases,” Phys. Fluids 2, 153–169 (1959).
[CrossRef]

Z. Naturforsch. Teil A (2)

G. Pretzler, H. Jäger, T. Neger, H. Philipp, J. Woisetschläger, “Comparison of different methods of Abel inversion using computer simulated and experimental side-on data,” Z. Naturforsch. Teil A 47, 955–970 (1992).

G. Pretzler, “A new method for numerical Abel-inversion,” Z. Naturforsch. Teil A 46, 639–641 (1991).

Other (6)

M. Mitchener, C. Kruger, Partially Ionized Gases (Wiley, New York, 1975).

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 93–167.
[CrossRef]

L. Hesselink, “Optical tomography,” in Handbook of Flow Visualization, W.-J. Yang, ed. (Hemisphere, New York, 1989), pp. 307–329.

D. Close, “Optically recorded holographic optical elements,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), pp. 573–585.

K. Widmann, G. Pretzler, J. Woisetschläger, T. Neger, H. Jäger, “Application of holographic optical elements to plasma diagnostics,” in Holographics International ‘92, Y. N. Denisyuk, F. Wyrowski, eds., Proc. SPIE1732, 712–718 (1992).

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Production and operation of a HOE to reduce the intensity of the radiation from the investigated object at the detector position. a: Production of the hologram. The object O is removed from the object beam OB. The reference beam RB, that goes through the hologram H, is focused into the point F by the lens L. b: Reconstruction of the reference beam. If the reference beam is reconstructed by the object beam, any wave-front deformation in OB will pass over to RB and lead to an interferogram if the original RB is still present. c: Reduction of the effect of the object radiation. The white object radiation is spectrally decomposed by the hologram into the spectrum SP (from red R to violet V). Only that part that has the wavelength and the direction of the laser light passes the diaphragm D. The shielding S prevents direct illumination of D.

Fig. 2
Fig. 2

Geometry for the numerical and experimental investigation of the imaging properties of a HOE. The object beam OB is parallel and has an angle of incidence α0 on the HOE. The reference beam RB falls in perpendicular to the HOE and is focused into the point F by the lens L (f is the distance HOE − F). An object point OP (distance o to the HOE), in the object O is imaged into IP (distance i from the HOE).

Fig. 3
Fig. 3

Experimental results concerning imaging properties of the HOE. Relation of measured object point distances o and image line distances i produced by a HOE with α0 = 60.2° and f = 60.0 cm (see Fig. 2). For both the distance of the vertical image line v and the distance of the horizontal image line h, the agreement of the measured points (circles and squares, respectively) with the numerically achieved results (presented as lines) is very good.

Fig. 4
Fig. 4

Spectral behavior of the HOE: p, plasma width; o, plasma distance; α0, angle of incidence of the object beam; P2 and P3, edges of the radiating region of the object; P1 and P4, edges of the field of view; and Pi′, images of these points on the camera C. The point PC is only hit by light from the point PH on the hologram, which is illuminated by the whole object (shaded area). For each point of the object, only light with a certain wavelength can reach PC by way of PH and the diaphragm D. Therefore a certain wavelength range λmin <λ <λmax of the plasma radiation can be found on this point of the camera.

Fig. 5
Fig. 5

Spectral distribution of the object radiation that hits a horizontal line on the camera. 1 maximal and 2 minimal wavelength of the incoming object radiation; λ0, laser wavelength; Δλ, total bandwidth of the object radiation reaching the camera; Pi′, images of certain points in the object plane (see Fig. 4). The curves 1 and 2 were calculated for p = 2 cm, o = 1.00 m, α0 = 60°, and λ0 = 632.8 nm (cf. Fig. 4).

Fig. 6
Fig. 6

Achromatic astigmatism for a divergent light beam falling on the HOE. The positions of the vertical focal line, v (curve, calculated values; circles, measured values), and of the horizontal focal line, h (curve, calculated values; squares, measured values) in the x, y-plane of the setup (see Fig. 2), if divergent light with different wavelengths falls on the hologram. The hologram is at the position (0, 0); the setup is shown in Fig. 20 = 60°, o = 1.18 m). The calculated directions of the central beams of the diffracted light are also indicated for different wavelengths.

Fig. 7
Fig. 7

Experimental setup for the determination of the electron-density distribution in a high-pressure Xe lamp. 1, Ar+ laser (λb = 457.9 nm); 2, He–Ne laser (λr = 632.8 nm); 3, monomode fiber; 4, high-pressure xenon lamp (450 W); 5, hologram; 6, diaphragm; 7, CCD camera; 8, beam splitter; 9, mirror; 10, beam expander; 11, rotatable mirror; 12, lens; 13, optical table.

Fig. 8
Fig. 8

Real-time holographic two-wavelength interferogram of the burning Xe lamp. Upper half of figure: interferogram using blue light (λ = 457.9 nm); lower half of figure: interferogram using red light (λ = 632.8 nm). Both interferograms display the same part of the object and were recorded simultaneously with one camera.

Fig. 9
Fig. 9

Spatial distribution of the integral electron density in two horizontal planes of a high-pressure Xe-lamp discharge. The curves show the difference of the products φi × λi (φ-measured phase shifts, λ-laser wavelengths) that are proportional to the electron density integrated along the path l of the light rays [cf. Eq. (8)]. Solid line: height 0.2 mm above the cathode. Dashed line: height 1.0 mm above the cathode.

Fig. 10
Fig. 10

Radial distribution of the electron density in two horizontal planes of a high-pressure Xe-lamp discharge. Electron density (Ne) in the Xe plasma achieved by Abel inversion of the functions in Fig. 9. Solid line: height 0.2 mm above the cathode. Dashed line: height 1.0 mm above the cathode.

Equations (8)

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G V ( G V / 2 + S ) = 0
T 1 = [ S 1 2 - G 3 ( 2 S 3 + G 3 ) - G 2 ( 2 S 2 + G 2 ) ] 1 / 2 T 2 = S 2 + G 2 T 3 = S 3 + G 3 .
1 / i h = 1 / f - 1 / o 1 / i v = 1 / f - cos 2 α 0 ( 1 / o ) .
Δ λ 2 λ 0 ( p / o × tan α 0 ) .
n - 1 = j ( n - 1 ) j .
( n - 1 ) e = - e 2 λ 2 8 π 2 0 m e c 2 N e ,
φ = 2 π λ l [ n ( l ) - n 0 ] d l
φ 1 λ 1 - φ 2 λ 2 = - e 2 4 π 0 m e c 2 ( λ 1 2 - λ 2 2 ) l N e d l ,

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