Abstract

A novel quantitative method, based on the moiré effect, for mapping ray deflections of a collimated light beam is described and. demonstrated. This method, which does not require coherent light, can replace interferometric techniques in many cases. The proposed setup is simple, and the interpretation is straightforward. We demonstrate deflection mapping of a candle’s flame with a resolution of 5 × 10−5 rad and lens mapping with a resolution of 10−2 rad. An analysis of the ray deflection for index-of-refraction mapping is provided.

© 1980 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 256–370.
  2. F. Zernike, Z. Tech. Phys. 16, 454 (1935); Physica 9, 686 (1942).
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 425.
  4. O. Kafri, A. Livnat, Opt. Lett. 4, 314 (1979).
    [CrossRef] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 122.
  6. L. O. Heflinger, R. F. Warker, R. E. Brooks, J. Appl. Phys. 37, 642 (1966).
    [CrossRef]
  7. K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
    [CrossRef]
  8. E. Keren, Nuclear Research Centre-Negev, P.O. Box 9001, Beer-Sheva, Israel, personal communication.
  9. J. Shamir, Opt. Laser Technol. 2, 78 (1973).
    [CrossRef]
  10. Y. Nishijima, G. Oster, J. Opt. Soc. Am. 54, 1 (1964).
    [CrossRef]
  11. G. Oster, M. Wasserman, C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
    [CrossRef]

1979

1973

J. Shamir, Opt. Laser Technol. 2, 78 (1973).
[CrossRef]

1966

L. O. Heflinger, R. F. Warker, R. E. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

1964

1961

1935

F. Zernike, Z. Tech. Phys. 16, 454 (1935); Physica 9, 686 (1942).

Bockasten, K.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 256–370.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 425.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 122.

Brooks, R. E.

L. O. Heflinger, R. F. Warker, R. E. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

Heflinger, L. O.

L. O. Heflinger, R. F. Warker, R. E. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

Kafri, O.

Keren, E.

E. Keren, Nuclear Research Centre-Negev, P.O. Box 9001, Beer-Sheva, Israel, personal communication.

Livnat, A.

Nishijima, Y.

Oster, G.

Shamir, J.

J. Shamir, Opt. Laser Technol. 2, 78 (1973).
[CrossRef]

Warker, R. F.

L. O. Heflinger, R. F. Warker, R. E. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

Wasserman, M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 122.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 425.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 256–370.

Zernike, F.

F. Zernike, Z. Tech. Phys. 16, 454 (1935); Physica 9, 686 (1942).

Zwerling, C.

J. Appl. Phys.

L. O. Heflinger, R. F. Warker, R. E. Brooks, J. Appl. Phys. 37, 642 (1966).
[CrossRef]

J. Opt. Soc. Am.

Opt. Laser Technol.

J. Shamir, Opt. Laser Technol. 2, 78 (1973).
[CrossRef]

Opt. Lett.

Z. Tech. Phys.

F. Zernike, Z. Tech. Phys. 16, 454 (1935); Physica 9, 686 (1942).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 425.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 122.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 256–370.

E. Keren, Nuclear Research Centre-Negev, P.O. Box 9001, Beer-Sheva, Israel, personal communication.

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Figures (5)

Fig. 1
Fig. 1

Schematic description of the setup; the telescope (L1 and L2) produces a collimated light beam with passes through a phase object and then to a Ronchi ruling G1. The distorted shadow of G1 is projected on a second Ronchi ruling G2 separated from G1 by a distance Δ. The moiré pattern may be viewed on a matte screen adjusted to G2 by an observer on the right-hand side.

Fig. 2
Fig. 2

The moiré pattern observed when two Ronchi rulings G1 and G2 are tilted with an angle θ; if the grating G1 is moved a distance p/[cos(θ/2)] in the y direction, the moiré pattern will move a distance p′ = p/2 sin(θ/2) in the Z direction.

Fig. 3
Fig. 3

A distorted line of the moiré pattern; the distance hj,k is simply related by Eq. (3) to the deflection of a light ray at the location (yj,zk + hj,k) in the Y direction.

Fig. 4
Fig. 4

a, A deflection mapping of a candle’s flame; the flame radiation is not visible so that the hot gases above the flame are merging with the radiative hot gas (the flame). The hot gas creates a negative lens that has a negative focal length of around 20 m according to calculations from the deflectogram. b, As in a, with a different sensitivity and axis.

Fig. 5
Fig. 5

Low-resolution deflection mapping of a 13-cm positive lens.

Equations (6)

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p = p / 2 sin ( θ / 2 ) .
A = p / { p / [ cos ( θ / 2 ) ] } = 1 2 cot ( θ / 2 ) .
ϕ j , k = h j , k A Δ = h j , k 2 tan ( θ / 2 ) Δ ,
d d s [ n ( x , y , z ) d r d s ] = n ( x , y , z ) ,
d r d s = ϕ j , k ( x ) = 1 n ( x , y ) 0 x n ( u , y ) y d u ,
ϕ j , k 1 n f 0 xf ( n y ) z k , y j d x .

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