Abstract

The theory of moire deflectometry is derived using the Fourier series approach. The advantages and disadvantages of the infinite fringe limit are demonstrated. The approach of fringe deviation is compared to the intensity measurement, which is selected as being more suitable for flames. An experimental technique based on pointwise scanning is developed to obviate the deficiencies of the video system used previously. Moire data of CH4/air flame are interpreted, and a full map of the temperature and gas density in the flame is produced. A comparison between moire deflectometry and related techniques is given.

© 1983 Optical Society of America

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References

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  1. Y. Nishijima, G. Oster, J. Opt. Soc. Am. 54, 1 (1964).
    [CrossRef]
  2. G. Oster, M. Wasserman, C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
    [CrossRef]
  3. O. Kafri, Opt. Lett. 5, 555 (1980).
    [CrossRef] [PubMed]
  4. E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, Appl. Opt. 20, 4263 (1981).
    [CrossRef] [PubMed]
  5. E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, “Measurements of Temperature Distributions in Methane–Air Flame by Moire Deflectometry,” to be published in the Nineteenth Symposium (International) on Combustion, 1983 volume.
  6. S. Yokozeki, Y. Kusaka, K. Patorski, Appl. Opt. 15, 2223 (1976); K. Patorski, S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 15, 443 (1976).
    [CrossRef] [PubMed]
  7. J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. London Sect. B 70, 486 (1957).
    [CrossRef]
  8. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  9. J. Stricker, O. Kafri, AIAA J. 20, 820 (1982).
    [CrossRef]
  10. O. Kafri, A. Livnat, E. Keren, Appl. Opt. 21, 3884 (1982).
    [CrossRef] [PubMed]
  11. C. H. Palmer, B. Z. Hollmann, Appl. Opt. 11, 780 (1972); S. Yokozeki, K. Patorski, Appl. Opt. 17, 2541 (1978).
    [CrossRef] [PubMed]
  12. O. Kafri, E. Margalit, Appl. Opt. 20, 2344 (1981).
    [CrossRef] [PubMed]
  13. F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), Chap. 6.
  14. See, for example, L. H. Tanner, J. Sci. Instrum. 42, 834 (1965); J. Sci. Instrum. 43, 878 (1966); J. E. Creeden, R. M. Fristrom, C. Grunfelder, F. J. Weinberg, J. Phys. D 5, 1063 (1972); R. D. Small, V. A. Sernas, R. H. Page, Appl. Opt. 11, 858 (1972); W. Merzkirch, W. Erdmann, Appl. Phys. 2, 119 (1973); R. D. Flack, J. Phys. E 14, 409 (1981).
    [CrossRef] [PubMed]
  15. O. Kafri, A. Livnat, E. Keren, “Optical Second Differentiation by Shearing Moire Deflectometry,” submitted for publication.
  16. S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971).
    [CrossRef] [PubMed]

1982 (2)

1981 (2)

1980 (1)

1976 (1)

1972 (1)

1971 (1)

1965 (1)

See, for example, L. H. Tanner, J. Sci. Instrum. 42, 834 (1965); J. Sci. Instrum. 43, 878 (1966); J. E. Creeden, R. M. Fristrom, C. Grunfelder, F. J. Weinberg, J. Phys. D 5, 1063 (1972); R. D. Small, V. A. Sernas, R. H. Page, Appl. Opt. 11, 858 (1972); W. Merzkirch, W. Erdmann, Appl. Phys. 2, 119 (1973); R. D. Flack, J. Phys. E 14, 409 (1981).
[CrossRef] [PubMed]

1964 (2)

1957 (1)

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. London Sect. B 70, 486 (1957).
[CrossRef]

Bar-Ziv, E.

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, Appl. Opt. 20, 4263 (1981).
[CrossRef] [PubMed]

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, “Measurements of Temperature Distributions in Methane–Air Flame by Moire Deflectometry,” to be published in the Nineteenth Symposium (International) on Combustion, 1983 volume.

Cowley, J. M.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. London Sect. B 70, 486 (1957).
[CrossRef]

Glatt, I.

Hollmann, B. Z.

Kafri, O.

J. Stricker, O. Kafri, AIAA J. 20, 820 (1982).
[CrossRef]

O. Kafri, A. Livnat, E. Keren, Appl. Opt. 21, 3884 (1982).
[CrossRef] [PubMed]

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, Appl. Opt. 20, 4263 (1981).
[CrossRef] [PubMed]

O. Kafri, E. Margalit, Appl. Opt. 20, 2344 (1981).
[CrossRef] [PubMed]

O. Kafri, Opt. Lett. 5, 555 (1980).
[CrossRef] [PubMed]

O. Kafri, A. Livnat, E. Keren, “Optical Second Differentiation by Shearing Moire Deflectometry,” submitted for publication.

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, “Measurements of Temperature Distributions in Methane–Air Flame by Moire Deflectometry,” to be published in the Nineteenth Symposium (International) on Combustion, 1983 volume.

Keren, E.

O. Kafri, A. Livnat, E. Keren, Appl. Opt. 21, 3884 (1982).
[CrossRef] [PubMed]

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, Appl. Opt. 20, 4263 (1981).
[CrossRef] [PubMed]

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, “Measurements of Temperature Distributions in Methane–Air Flame by Moire Deflectometry,” to be published in the Nineteenth Symposium (International) on Combustion, 1983 volume.

O. Kafri, A. Livnat, E. Keren, “Optical Second Differentiation by Shearing Moire Deflectometry,” submitted for publication.

Kusaka, Y.

Livnat, A.

O. Kafri, A. Livnat, E. Keren, Appl. Opt. 21, 3884 (1982).
[CrossRef] [PubMed]

O. Kafri, A. Livnat, E. Keren, “Optical Second Differentiation by Shearing Moire Deflectometry,” submitted for publication.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

Margalit, E.

Moodie, A. F.

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. London Sect. B 70, 486 (1957).
[CrossRef]

Nishijima, Y.

Oster, G.

Palmer, C. H.

Patorski, K.

Sgulim, S.

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, “Measurements of Temperature Distributions in Methane–Air Flame by Moire Deflectometry,” to be published in the Nineteenth Symposium (International) on Combustion, 1983 volume.

Stricker, J.

J. Stricker, O. Kafri, AIAA J. 20, 820 (1982).
[CrossRef]

Suzuki, T.

Tanner, L. H.

See, for example, L. H. Tanner, J. Sci. Instrum. 42, 834 (1965); J. Sci. Instrum. 43, 878 (1966); J. E. Creeden, R. M. Fristrom, C. Grunfelder, F. J. Weinberg, J. Phys. D 5, 1063 (1972); R. D. Small, V. A. Sernas, R. H. Page, Appl. Opt. 11, 858 (1972); W. Merzkirch, W. Erdmann, Appl. Phys. 2, 119 (1973); R. D. Flack, J. Phys. E 14, 409 (1981).
[CrossRef] [PubMed]

Wasserman, M.

Weinberg, F. J.

F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), Chap. 6.

Yokozeki, S.

Zwerling, C.

AIAA J. (1)

J. Stricker, O. Kafri, AIAA J. 20, 820 (1982).
[CrossRef]

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

J. Sci. Instrum. (1)

See, for example, L. H. Tanner, J. Sci. Instrum. 42, 834 (1965); J. Sci. Instrum. 43, 878 (1966); J. E. Creeden, R. M. Fristrom, C. Grunfelder, F. J. Weinberg, J. Phys. D 5, 1063 (1972); R. D. Small, V. A. Sernas, R. H. Page, Appl. Opt. 11, 858 (1972); W. Merzkirch, W. Erdmann, Appl. Phys. 2, 119 (1973); R. D. Flack, J. Phys. E 14, 409 (1981).
[CrossRef] [PubMed]

Opt. Lett. (1)

Proc. Phys. Soc. London Sect. B (1)

J. M. Cowley, A. F. Moodie, Proc. Phys. Soc. London Sect. B 70, 486 (1957).
[CrossRef]

Other (4)

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, “Measurements of Temperature Distributions in Methane–Air Flame by Moire Deflectometry,” to be published in the Nineteenth Symposium (International) on Combustion, 1983 volume.

F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), Chap. 6.

O. Kafri, A. Livnat, E. Keren, “Optical Second Differentiation by Shearing Moire Deflectometry,” submitted for publication.

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Figures (12)

Fig. 1
Fig. 1

Three possible positions of two Ronchi gratings: (1) and (2) refer to G1 and G2; (a) χ = p; (b) χ = 0; and (c) χ = ½P.

Fig. 2
Fig. 2

Light rays traversing a diverging lens L and passing through two Ronchi gratings G1 and G2.

Fig. 3
Fig. 3

Light ray passing through a phase object and deflected by an angle ϕ.

Fig. 4
Fig. 4

Schematic description of an optical system for determining deflection mapping by the moire effect using a video detector.

Fig. 5
Fig. 5

Schematic description of a setup for determining deflection mapping by the moire effect using a photomultiplier tube.

Fig. 6
Fig. 6

An oscilloscope trace of I(y) of a nonhorizontal moire pattern at a fixed z, case (3), using a video detector.

Fig. 7
Fig. 7

I vs y of a horizontal moire pattern at fixed z when the gratings were translated with respect to one another, case (2), and I measured by a PMT.

Fig. 8
Fig. 8

Moire patterns of a CH4/air flame with total gas flow rate of 300 liter/h and fuel-to-air ratio of 1:8: (a) θ = 0, the dark mode χ = P/2; (b) θ = 0, the bright mode χ = 0; (c) θ ≠ 0.

Fig. 9
Fig. 9

Intensity profiles I(y,z) at a fixed z value: (a) χ = P/2; (b) χ = 0; (c) 0 < χ < P/2; and (d) the intensity of the shadow at the same z with no gratings.

Fig. 10
Fig. 10

(a) Parallel light beam traversing a concentric double lens, L1, converging and L2, diverging; (b) the intensity profile of the light beam at a distance l from the double lens.

Fig. 11
Fig. 11

Normalized intensity profiles of the moire pattern of a CH4/air flame; χ has an arbitrary value in the region 0 < χ < P/2.

Fig. 12
Fig. 12

Radial temperature profiles of the Ch4/air flame at different z values.

Equations (34)

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G 1 ( y , z ) = 1 2 + 2 π n = 0 sin [ 2 π ( 2 n + 1 ) ( y p z 2 p ) ] / ( 2 n + 1 ) , G 2 ( y , z ) = 1 2 + 2 π n = 0 sin [ 2 π ( 2 n + 1 ) ( y χ p + z 2 p ) ] / ( 2 n + 1 ) ;
P = P 0 / cos ( θ / 2 ) ; P = P / 2 sin ( θ / 2 ) ,
M ( y , z ) = 1 P y y + P G 1 ( y , z ) G 2 ( y , z ) d y ,
M ( y , z ) = 1 4 + 2 π 2 n = 0 cos [ 2 π ( 2 n + 1 ) ( z P χ P ) ] / ( 2 n + 1 ) 2 .
Δ = m P 2 / λ m = 1,2,3 , ,
G 1 ( y , z ) = ( 1 Δ l ) 2 { 1 2 + 2 π n = 0 sin [ 2 π ( 1 Δ l ) ( 2 n + 1 ) × ( y P z 2 P ) ] / ( 2 n + 1 ) } .
M ( y , z ) = ( 1 Δ l ) 2 ( 1 4 + 2 π 2 n = 0 cos { 2 π ( 2 n + 1 ) × [ Δ l · y P + ( 1 Δ 2 l ) z P χ P ] } / ( 2 n + 1 ) 2 ) .
1 2 ( 1 Δ l ) 2 .
z = z 0 2 Δ 2 l Δ · P P · y ,
f = Δ ( 1 α P P 1 2 ) ,
f = Δ α · P P .
ϕ ( y , z ) = y l ( y , z ) .
ϕ ( y , z ) = 1 Δ P P [ z 0 z ( y ) ] .
I ( y , z ) = 1 2 + 2 π 2 n = 0 cos { 2 π ( 2 n + 1 ) ( Δ P ϕ ( y , z ) + z P χ P ] } / ( 2 n + 1 ) 2 ;
d d s ( n d r d s ) = n ,
ϕ = 1 n s s o s e n y d s ,
ϕ = 1 n s x o x e n y d x ;
ϕ ( y , z ) = 2 y n s y r o n ( r , z ) r d r ( r 2 y 2 ) 1 / 2 ,
n ( r , z ) = n s n s π r r o ϕ ( y , z ) ( y 2 r 2 ) 1 / 2 d y .
I ( y , z ) = 1 2 + 4 π 2 n = 0 cos { 2 π ( 2 n + 1 ) [ Δ P ϕ ( y , z ) χ P ] } / ( 2 n + 1 ) 2 .
I ( y , z ) = | [ 1 2 | η ( y , z ) 1 2 | ] | ; for | η 1 2 | 1 ,
I ( y , z ) = 2 | η ( y , z ) | ; for | η | 1 2 .
I ( y , z ) = | ( 1 2 | η ( y , z ) | ) | ; for | η | 1.
I ( y , z ) = | ( 1 2 | η ( y , z ) χ P | ) | ; for | η χ P | 1 .
I ( y , z ) = 1 2 + 2 η ( y , z ) , for | η | 1 4 .
I n ( r , z ) r r 0 ϕ ( y , z ) ( y 2 r 2 ) 1 / 2 d y
I n ( r ) = r q ϕ ( y ) ϕ ( r ) ( y 2 r 2 ) 1 / 2 d y + ϕ ( r ) r q d y ( y 2 r 2 ) 1 / 2 + q r 0 ϕ ( y ) ( y 2 r 2 ) 1 / 2 d y ,
I n 3 ( r , q ) = h 2 ϕ ( q ) ( q 2 r 2 ) 1 / 2 + h m ϕ ( q + m h ) [ ( q + m h ) 2 r 2 ] 1 / 2 ,
I n 2 ( r , q ) = ϕ ( r ) arccos h ( q / r ) .
lim y r ϕ ( y ) ϕ ( r ) ( y 2 r 2 ) 1 / 2 = ϕ ( y ) y | y = r · y r ( y r ) 1 / 2 ( y + r ) 1 / 2 = 0.
I n ( r ) = ϕ ( r ) [ arccos h ( 1 + h r ) h 2 ( h 2 + 2 r h ) 1 / 2 ] + h m ( r + m h ) [ ( r + m h ) 2 r 2 ] 1 / 2 .
I ( y , z ) = I ( y , z ) / I 0 ( y , z ) .
ϕ ( y , z ) = 1 n s n ( y , z ) n ( y + Δ y , z ) Δ y d x ,
ϕ ( y , z ) = 1 n s n ( y , z ) y d x ,

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