Abstract

Applications of quantitative schlieren techniques to pulsed high current electric arcs are described. A new formalism for the Abel inversion is derived that does not start from the small angle approximation. The numerical treatment of experimental data is given special attention. A pulsed argon laser was constructed that gives enough background intensity to outshine a 2000-A arc. Several high speed schlieren techniques were tried to obtain quantitative information about the density and temperature in the neighborhood of the arc.

© 1972 Optical Society of America

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References

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  1. M. P. Freeman, S. Katz, J. Opt. Soc. Am. 50, 826 (1960).
    [CrossRef]
  2. K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
    [CrossRef]
  3. R. Gorenflo, Y. Kovets, Numer. Math. 8, 392 (1966).
    [CrossRef]
  4. M. Kock, J. Richter, Ann. Phys. Leipzig 24, 30 (1969).
    [CrossRef]
  5. H. Schardin, Ergeb. Exakt. Natur. 20, 303 (1942).
    [CrossRef]
  6. H. Wolter, Handbuch der Physik XXIV, Grundlagen der Optik (Springer, Berlin, 1956), p. 572.
  7. M. Neumann, DVL-91 (Deutsche Versuchsanst. f. Luftfahrt, 1961).
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 121.
  9. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D. C., 1968).
  10. D. J. H. Wort, CLM-R 27 (Culham, 1963).
  11. F. Keilmann, IPP IV/4 (Garching, 1970).
  12. H. Salzmann, LGI 68, 25 (Frascati, 1968).
  13. S. I. Herlitz, Ark. Fys. 23, 571 (1963).
  14. C. D. Maldonado, A. P. Caron, H. N. Olsen, J. Opt. Soc. Am. 55, 1247 (1965).
    [CrossRef]
  15. G. N. Minerbo, M. E. Levy, SIAM, J. Numer. Anal. 6, 598 (1969).
    [CrossRef]
  16. Internal report, available on request.
  17. C. van Trigt, in Proc. 10th. Conf. on Phenomena in Ionized Gases (Oxford U. P., London, 1971), p. 396.
  18. U. Kogelschatz, E. Schade, in Proc. 10th Int. Conf. on Phenomena in Ionized Gases (Oxford U. P., London, 1971), p. 198.
  19. W. Bötticher, U. Kogelschatz, E. Schade, Z. Naturforsch. (in press).
  20. G. Herziger, W. Seelig, Z. Phys. 215, 437 (1968).
    [CrossRef]
  21. G. Herziger, W. Seelig, Z. Phys. 219, 5 (1969).
    [CrossRef]
  22. V. Ronchi, Atti della Fondazzione Georgio Ronchi 13, 368 (1958).
  23. G. S. Speak, D. J. Walters, A.R.C.-Reports and Memoranda 2859 1950 (U. K. Aeronautical Research Council, 1950).
  24. F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), p. 182.
  25. H. Wolter, Ann. Phys. Leipzig 7, 341 (1950).
    [CrossRef]
  26. K. Behringer, W. Kollmar, J. Mentel, Z. Phys. 215, 127 (1968).
    [CrossRef]
  27. L. G. Longsworth, J. Am. Chem. Soc. 91, 929 (1929).

1969 (3)

M. Kock, J. Richter, Ann. Phys. Leipzig 24, 30 (1969).
[CrossRef]

G. N. Minerbo, M. E. Levy, SIAM, J. Numer. Anal. 6, 598 (1969).
[CrossRef]

G. Herziger, W. Seelig, Z. Phys. 219, 5 (1969).
[CrossRef]

1968 (2)

G. Herziger, W. Seelig, Z. Phys. 215, 437 (1968).
[CrossRef]

K. Behringer, W. Kollmar, J. Mentel, Z. Phys. 215, 127 (1968).
[CrossRef]

1966 (1)

R. Gorenflo, Y. Kovets, Numer. Math. 8, 392 (1966).
[CrossRef]

1965 (1)

1963 (1)

S. I. Herlitz, Ark. Fys. 23, 571 (1963).

1961 (1)

1960 (1)

1958 (1)

V. Ronchi, Atti della Fondazzione Georgio Ronchi 13, 368 (1958).

1950 (1)

H. Wolter, Ann. Phys. Leipzig 7, 341 (1950).
[CrossRef]

1942 (1)

H. Schardin, Ergeb. Exakt. Natur. 20, 303 (1942).
[CrossRef]

1929 (1)

L. G. Longsworth, J. Am. Chem. Soc. 91, 929 (1929).

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D. C., 1968).

Behringer, K.

K. Behringer, W. Kollmar, J. Mentel, Z. Phys. 215, 127 (1968).
[CrossRef]

Bockasten, K.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 121.

Bötticher, W.

W. Bötticher, U. Kogelschatz, E. Schade, Z. Naturforsch. (in press).

Caron, A. P.

Freeman, M. P.

Gorenflo, R.

R. Gorenflo, Y. Kovets, Numer. Math. 8, 392 (1966).
[CrossRef]

Herlitz, S. I.

S. I. Herlitz, Ark. Fys. 23, 571 (1963).

Herziger, G.

G. Herziger, W. Seelig, Z. Phys. 219, 5 (1969).
[CrossRef]

G. Herziger, W. Seelig, Z. Phys. 215, 437 (1968).
[CrossRef]

Katz, S.

Keilmann, F.

F. Keilmann, IPP IV/4 (Garching, 1970).

Kock, M.

M. Kock, J. Richter, Ann. Phys. Leipzig 24, 30 (1969).
[CrossRef]

Kogelschatz, U.

W. Bötticher, U. Kogelschatz, E. Schade, Z. Naturforsch. (in press).

U. Kogelschatz, E. Schade, in Proc. 10th Int. Conf. on Phenomena in Ionized Gases (Oxford U. P., London, 1971), p. 198.

Kollmar, W.

K. Behringer, W. Kollmar, J. Mentel, Z. Phys. 215, 127 (1968).
[CrossRef]

Kovets, Y.

R. Gorenflo, Y. Kovets, Numer. Math. 8, 392 (1966).
[CrossRef]

Levy, M. E.

G. N. Minerbo, M. E. Levy, SIAM, J. Numer. Anal. 6, 598 (1969).
[CrossRef]

Longsworth, L. G.

L. G. Longsworth, J. Am. Chem. Soc. 91, 929 (1929).

Maldonado, C. D.

Mentel, J.

K. Behringer, W. Kollmar, J. Mentel, Z. Phys. 215, 127 (1968).
[CrossRef]

Minerbo, G. N.

G. N. Minerbo, M. E. Levy, SIAM, J. Numer. Anal. 6, 598 (1969).
[CrossRef]

Neumann, M.

M. Neumann, DVL-91 (Deutsche Versuchsanst. f. Luftfahrt, 1961).

Olsen, H. N.

Richter, J.

M. Kock, J. Richter, Ann. Phys. Leipzig 24, 30 (1969).
[CrossRef]

Ronchi, V.

V. Ronchi, Atti della Fondazzione Georgio Ronchi 13, 368 (1958).

Salzmann, H.

H. Salzmann, LGI 68, 25 (Frascati, 1968).

Schade, E.

U. Kogelschatz, E. Schade, in Proc. 10th Int. Conf. on Phenomena in Ionized Gases (Oxford U. P., London, 1971), p. 198.

W. Bötticher, U. Kogelschatz, E. Schade, Z. Naturforsch. (in press).

Schardin, H.

H. Schardin, Ergeb. Exakt. Natur. 20, 303 (1942).
[CrossRef]

Seelig, W.

G. Herziger, W. Seelig, Z. Phys. 219, 5 (1969).
[CrossRef]

G. Herziger, W. Seelig, Z. Phys. 215, 437 (1968).
[CrossRef]

Speak, G. S.

G. S. Speak, D. J. Walters, A.R.C.-Reports and Memoranda 2859 1950 (U. K. Aeronautical Research Council, 1950).

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D. C., 1968).

van Trigt, C.

C. van Trigt, in Proc. 10th. Conf. on Phenomena in Ionized Gases (Oxford U. P., London, 1971), p. 396.

Walters, D. J.

G. S. Speak, D. J. Walters, A.R.C.-Reports and Memoranda 2859 1950 (U. K. Aeronautical Research Council, 1950).

Weinberg, F. J.

F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), p. 182.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 121.

Wolter, H.

H. Wolter, Ann. Phys. Leipzig 7, 341 (1950).
[CrossRef]

H. Wolter, Handbuch der Physik XXIV, Grundlagen der Optik (Springer, Berlin, 1956), p. 572.

Wort, D. J. H.

D. J. H. Wort, CLM-R 27 (Culham, 1963).

Ann. Phys. Leipzig (2)

M. Kock, J. Richter, Ann. Phys. Leipzig 24, 30 (1969).
[CrossRef]

H. Wolter, Ann. Phys. Leipzig 7, 341 (1950).
[CrossRef]

Ark. Fys. (1)

S. I. Herlitz, Ark. Fys. 23, 571 (1963).

Atti della Fondazzione Georgio Ronchi (1)

V. Ronchi, Atti della Fondazzione Georgio Ronchi 13, 368 (1958).

Ergeb. Exakt. Natur. (1)

H. Schardin, Ergeb. Exakt. Natur. 20, 303 (1942).
[CrossRef]

J. Am. Chem. Soc. (1)

L. G. Longsworth, J. Am. Chem. Soc. 91, 929 (1929).

J. Opt. Soc. Am. (3)

Numer. Math. (1)

R. Gorenflo, Y. Kovets, Numer. Math. 8, 392 (1966).
[CrossRef]

SIAM, J. Numer. Anal. (1)

G. N. Minerbo, M. E. Levy, SIAM, J. Numer. Anal. 6, 598 (1969).
[CrossRef]

Z. Phys. (3)

G. Herziger, W. Seelig, Z. Phys. 215, 437 (1968).
[CrossRef]

G. Herziger, W. Seelig, Z. Phys. 219, 5 (1969).
[CrossRef]

K. Behringer, W. Kollmar, J. Mentel, Z. Phys. 215, 127 (1968).
[CrossRef]

Other (13)

G. S. Speak, D. J. Walters, A.R.C.-Reports and Memoranda 2859 1950 (U. K. Aeronautical Research Council, 1950).

F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), p. 182.

Internal report, available on request.

C. van Trigt, in Proc. 10th. Conf. on Phenomena in Ionized Gases (Oxford U. P., London, 1971), p. 396.

U. Kogelschatz, E. Schade, in Proc. 10th Int. Conf. on Phenomena in Ionized Gases (Oxford U. P., London, 1971), p. 198.

W. Bötticher, U. Kogelschatz, E. Schade, Z. Naturforsch. (in press).

H. Wolter, Handbuch der Physik XXIV, Grundlagen der Optik (Springer, Berlin, 1956), p. 572.

M. Neumann, DVL-91 (Deutsche Versuchsanst. f. Luftfahrt, 1961).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), p. 121.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D. C., 1968).

D. J. H. Wort, CLM-R 27 (Culham, 1963).

F. Keilmann, IPP IV/4 (Garching, 1970).

H. Salzmann, LGI 68, 25 (Frascati, 1968).

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Figures (16)

Fig. 1
Fig. 1

Beam deflection caused by a cylindrical refractive index distribution.

Fig. 2
Fig. 2

Illustration of deflection parameters.

Fig. 3
Fig. 3

Deflection curve corresponding to parabolic refractive index distribution and transformed curves α*(λ).

Fig. 4
Fig. 4

Family of deflection curves αν(y) and corresponding refractive index distributions.

Fig. 5
Fig. 5

Comparison of numerical methods.

Fig. 6
Fig. 6

Schematic of electric arc and nozzle geometry.

Fig. 7
Fig. 7

Schematic of laser discharge.

Fig. 8
Fig. 8

Optical setup.

Fig. 9
Fig. 9

Snapshot of isophots taken with an image converter (1-μsec exposure time). The pictures were taken with different grids or masks in the focal plane of L2 indicated below each picture. One electrode of the arc is shown on the right and the nozzle on the left.

Fig. 10
Fig. 10

Streak recording of the isophots of a stable arc and a pulsating arc.

Fig. 11
Fig. 11

Connection between grid spacing and fringe pattern.

Fig. 12
Fig. 12

Schematic illustrating the moving slit method.

Fig. 13
Fig. 13

Repetitive scan of the deflection curve.

Fig. 14
Fig. 14

Deflection curve of the arc under investigation.

Fig. 15
Fig. 15

Refractive index distribution.

Fig. 16
Fig. 16

Density and temperature in the neighborhood of the arc.

Equations (45)

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d ψ / d r = y n 0 / [ r ( r 2 n 2 - y 2 n 0 2 ) 1 2 ] ,
r 0 n ( r 0 ) = y n 0 .
Δ ψ = 2 y n 0 r 0 d r r ( r 2 n 2 - y 2 n 0 2 ) 1 2 .
Δ ψ = π .
α ( y ) = π - 2 y n 0 r 0 d r r ( r 2 n 2 - y 2 n 0 2 ) 1 2 .
s = [ r n ( r ) ] / n 0 .
1 = ( 1 / n 0 ) { n [ r ( s ) ] + r ( s ) n [ r ( s ) ] } ( d r / d s ) ,
1 r ( s ) d r d s = 1 s - n [ r ( s ) ] n [ r ( s ) ] d r d s .
α ( y ) = π - 2 y y d s s ( s 2 - y 2 ) 1 2 + 2 y y R f ( s ) ( s 2 - y 2 ) 1 2 d s
f ( s ) = { n [ r ( s ) ] / n [ r ( s ) ] } ( d r / d s ) .
2 y y d s s ( s 2 - y 2 ) 1 2 = 2 y [ 1 y cos - 1 ( y s ) ] s = y s = = π .
α ( y ) = 2 y y R f ( s ) ( s 2 - y 2 ) 1 2 d s .
1 π s R α ( y ) ( y 2 - s 2 ) 1 2 d y = 1 π s R d y ( y 2 - s 2 ) 1 2 2 y y R f ( t ) ( t 2 - y 2 ) 1 2 d t = s R f ( t ) d t 1 π s t 2 y d y [ ( t 2 - y 2 ) ( y 2 - s 2 ) ] 1 2 = s R f ( t ) d t .
s t 2 y d y [ ( t 2 - y 2 ) ( y 2 - s 2 ) ] 1 2 = a b d z [ ( b - z ) ( z - z ) ] 1 2 = 2 [ tan - 1 ( z - a b - z ) 1 2 ] z = a z = b = π .
s R f ( t ) d t = - ln n [ r ( s ) ] n 0 .
n [ r ( s ) ] = n 0 exp [ - 1 π s R α ( y ) ( y 2 - s 2 ) 1 2 d y ] ,
r ( s ) = n 0 s n [ r ( s ) ] = s exp [ 1 π s R α ( y ) ( y 2 - s 2 ) 1 2 d y ] .
ϕ ( s ) = 1 π s R α ( y ) ( y 2 - s 2 ) 1 2 d y ,
n ( s ) = n 0 exp [ - ϕ ( s ) ] , r ( s ) = s exp [ ϕ ( s ) ] .
exp [ - ϕ ( s ) ] 1 - ϕ ( s )
r ( s ) s n ( s ) n ( r ) n 0 exp [ - ϕ ( r ) ] n 0 [ 1 - ϕ ( r ) ] ,
Δ r ( s ) r ( s ) = exp [ ϕ ( s ) ] - 1 exp [ ϕ ( s ) ] = 1 - exp [ - ϕ ( s ) ] ϕ ( s ) ,
Δ n ( s ) n ( s ) = n 0 exp [ - ϕ ( s ) ] - n 0 [ 1 - ϕ ( s ) ] n 0 exp [ - ϕ ( s ) ] = 1 - exp [ ϕ ( s ) ] [ 1 - ϕ ( s ) ] ϕ 2 ( s ) .
y = s cosh λ or λ = cosh - 1 ( y / s )             ( s 0 ) ,
ϕ ( s ) = 1 π λ = 0 Λ α * ( λ ) d λ
α ( y ) = 2 cos - 1 y - sin - 1 { 1 - - 2 y 2 [ ( 1 - ) 2 + 4 y 2 ] 1 2 } - π 2 .
n ( r ) = { [ 1 - ( 1 - r 2 ) ] 1 2 for 0 r 1 , 1 for r 1 ,
α ^ ν ( y ) = y ( 1 - y 2 ) ν - 1 2 .
ϕ ν ( s ) = 1 π s 1 y ( 1 - y 2 ) ν - 1 2 ( y 2 - s 2 ) 1 2 d y = 1 2 π F ν - 1 2 , - 1 2 ( s 2 , 1 ) ,
F μ , ν ( a , b ) = a b ( b - x ) μ ( x - a ) ν d x .
x = ( b - a ) z + a
F μ , ν ( a , b ) = ( b - a ) μ + ν + 1 F μ , ν ( 0 , 1 ) ,
F μ , ν ( 0 , 1 ) = 0 1 ( 1 - z ) μ z ν d z = Γ ( μ + 1 ) Γ ( ν + 1 ) Γ ( μ + ν + 2 )
ϕ ν ( s ) = 1 2 π Γ ( ν + 1 2 ) Γ ( 1 2 ) Γ ( ν + 1 ) ( 1 - s 2 ) ν = A ν ( 1 - s 2 ) ν .
A ν = ( 2 ν - 1 ) ! 2 2 ν ( ν - 1 ) ! ν ! .
n ν ( s ) = n 0 exp [ - ( 1 - s 2 ) ν ] , r ν ( s ) = s exp [ ( 1 - s 2 ) ν ] ,
n ν ( r ) n 0 exp [ - ( 1 - r 2 ) ν ] n 0 [ 1 - ( 1 - r 2 ) ν ] .
α ν ( y ) = - α ν ( - y )             and             α ν ( 0 ) = α ν ( 1 ) = 0.
α ( y ) = y K = 1 ( 4 K - 1 ) b K P 2 K - 1 [ ( 1 - y 2 ) 1 2 ] ,
b K = 0 1 β [ ( 1 - z 2 ) 1 2 ] P 2 K - 1 ( z ) d z
β ( y ) = α ( y ) / y .
ϕ ( s ) = 1 2 b 1 + K = 1 a K P 2 K ( s ) ,
a K = ( - 1 ) K ( 2 K ) ! 4 K K ! ( K - 1 ) ! ( b K + 2 K + 1 2 K b K + 1 ) .
n - 1 = K ρ ,             ( K :             constant ) ,
n - 1 = - 4.46 × 10 - 14 N e λ 2 .

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