Abstract

Plane wave fronts in a collimated beam of light are distorted in their passage through a region of variable density, and as a result a variable phase distribution is produced in an exit plane which lies just beyond the disturbance and is perpendicular to the direction of propagation of the collimated beam (optical axis). This phase distribution, ordinarily invisible due to the tremendous rapidity of optical oscillations, can be converted to an intensity distribution in which the maxima and minima of intensity correspond to points in the exit plane where the optical path length differs from that in an undisturbed portion of this plane (free field) by an integral multiple of half a wavelength. This can be done in a conventional schlieren system by forming an image of the exit (object) plane with a convex lens (or parabolic mirror) and then inserting a small absorbing object or other appropriate modification in the focal plane in such a way as to block the central maximum of the Fraunhofer pattern due to the free field. Very little disturbance light is cut off in the process since this light is refracted and does not go through the focal point. The blocking of free-field light sets up a diffraction process which causes this light to spread into the bordering disturbance image. The resulting interference produces the intensity band system mentioned earlier.

If the disturbance is two-dimensional, with density-gradient vectors perpendicular to the optical axis, the measurement of the phase distribution is equivalent to a measurement of density, by virtue of the Dale-Gladstone law, μ−1=Kρ, where μ is the index of refraction, ρ is the density, K is a constant. Measurements which have heretofore required the use of a Mach-Zehnder interferometer can, therefore, be made in a conventional schlieren system. The possibility of duplicating interferometer fringe-field experiments has also been investigated.

© 1957 Optical Society of America

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References

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  1. E. L. Gayhart and R. Prescott, J. Opt. Soc. Am. 39, 546–550 (1949).
    [Crossref]
  2. R. B. Kennard, J. Research Natl. Bur. Standards 8, 787–805 (1932).
    [Crossref]
  3. E. B. Temple, “The physical optical analysis and application of the schlieren interferometer,” , November, 1955.
  4. Bennett, Osterberg, Jupnik, and Richards, Phase Microscopy, Principles and Applications (John Wiley and Sons, Inc., New York, 1951).
  5. H. Bremmer, Communs. Pure Appl. Math. 4, 61–74 (1951).
    [Crossref]
  6. H. Bremmer, Physica 17, 11 (1951).
    [Crossref]
  7. H. N. Shafer, J. Soc. Motion Picture Engrs. 53, 524–544 (1949).
  8. H. T. Van de Vooren, J. Appl. Sci. Research B3, 18–28 (1952).

1952 (1)

H. T. Van de Vooren, J. Appl. Sci. Research B3, 18–28 (1952).

1951 (2)

H. Bremmer, Communs. Pure Appl. Math. 4, 61–74 (1951).
[Crossref]

H. Bremmer, Physica 17, 11 (1951).
[Crossref]

1949 (2)

H. N. Shafer, J. Soc. Motion Picture Engrs. 53, 524–544 (1949).

E. L. Gayhart and R. Prescott, J. Opt. Soc. Am. 39, 546–550 (1949).
[Crossref]

1932 (1)

R. B. Kennard, J. Research Natl. Bur. Standards 8, 787–805 (1932).
[Crossref]

Bennett,

Bennett, Osterberg, Jupnik, and Richards, Phase Microscopy, Principles and Applications (John Wiley and Sons, Inc., New York, 1951).

Bremmer, H.

H. Bremmer, Communs. Pure Appl. Math. 4, 61–74 (1951).
[Crossref]

H. Bremmer, Physica 17, 11 (1951).
[Crossref]

Gayhart, E. L.

Jupnik,

Bennett, Osterberg, Jupnik, and Richards, Phase Microscopy, Principles and Applications (John Wiley and Sons, Inc., New York, 1951).

Kennard, R. B.

R. B. Kennard, J. Research Natl. Bur. Standards 8, 787–805 (1932).
[Crossref]

Osterberg,

Bennett, Osterberg, Jupnik, and Richards, Phase Microscopy, Principles and Applications (John Wiley and Sons, Inc., New York, 1951).

Prescott, R.

Richards,

Bennett, Osterberg, Jupnik, and Richards, Phase Microscopy, Principles and Applications (John Wiley and Sons, Inc., New York, 1951).

Shafer, H. N.

H. N. Shafer, J. Soc. Motion Picture Engrs. 53, 524–544 (1949).

Temple, E. B.

E. B. Temple, “The physical optical analysis and application of the schlieren interferometer,” , November, 1955.

Van de Vooren, H. T.

H. T. Van de Vooren, J. Appl. Sci. Research B3, 18–28 (1952).

Communs. Pure Appl. Math. (1)

H. Bremmer, Communs. Pure Appl. Math. 4, 61–74 (1951).
[Crossref]

J. Appl. Sci. Research (1)

H. T. Van de Vooren, J. Appl. Sci. Research B3, 18–28 (1952).

J. Opt. Soc. Am. (1)

J. Research Natl. Bur. Standards (1)

R. B. Kennard, J. Research Natl. Bur. Standards 8, 787–805 (1932).
[Crossref]

J. Soc. Motion Picture Engrs. (1)

H. N. Shafer, J. Soc. Motion Picture Engrs. 53, 524–544 (1949).

Physica (1)

H. Bremmer, Physica 17, 11 (1951).
[Crossref]

Other (2)

E. B. Temple, “The physical optical analysis and application of the schlieren interferometer,” , November, 1955.

Bennett, Osterberg, Jupnik, and Richards, Phase Microscopy, Principles and Applications (John Wiley and Sons, Inc., New York, 1951).

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Figures (9)

Fig. 1
Fig. 1

A schematic schlieren system.

Fig. 2
Fig. 2

(a) Orientation of one-dimensional disturbance and adjacent free field considered in this paper; (b) orientatation of heated plate.

Fig. 3
Fig. 3

Focal-plane Fraunhofer amplitude patterns due to free field (a<x<b) and to disturbance (δ<x<a) in which density is a quadratic function of x. An example of the relatively small amount of disturbance light diffracted into the wire region [−Y<y<Y].

Fig. 4
Fig. 4

Three examples of the free field alteration function C2(z) representing the spreading of light produced by focal-plane wires of varying sizes. The portions of these curves in δ<z<a represent the free field light available for interference. “Proper” band systems are produced by curves like C2(2)(z) and C2(3)(z) which are positive in δ<z<a.

Fig. 5
Fig. 5

(a) Typical heated-plate phase distribution; (b) fictitious band system produced by the interference Aeikϕ(x)A; (d) actual band system produced in the disturbance image by the interference of free field and disturbance light, eikϕ(z)C2(z), where A=1 and C2(z) is the alteration function of Fig. 5(c).

Fig. 6
Fig. 6

Intensity band system in the image of a heated soldering iron.

Fig. 7
Fig. 7

Unmodified image of various opaque objects.

Fig. 8
Fig. 8

Image of the same objects modified due to the presence of a 0.006-in. wire in the focal plane.

Fig. 9
Fig. 9

Optical wedges used to create uniform background fringes.

Equations (23)

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μ - 1 = K ρ ,
[ μ 0 - μ ( x , x ) ] L .
2 π λ ϕ ( x , x ) = 2 π λ K L [ ρ 0 - ρ ( x , x ) ] .
ρ ( x , x ) = ρ 0 - ϕ ( x , x ) K L .
g ( y , y ) = - - f ( x , x ) e i k ( y / f ) x e i k ( y / f ) x d x d x
G ( α , α ) = g ( f α , f α ) = f ( x , x ) e i k α x e i k α x d x d x ,
S = Re Im g ( y , y ) e i k v t ,
u p = 1 λ 2 f 2 - - g ( y , y ) e - i k ( y / f ) z e - i k ( y / f ) z d y d y
u p = f ( x , x ) e i 2 π v ( x - z ) e i 2 π y ( x - z ) d x d x d v d v = f ( z , z ) .
f ( z , z ) = - - g 1 ( v , v ) e - i 2 π v z e - i 2 π v z d v d v = 1 λ 2 - - G ( α , a ) e - i k α z e - i k α z d α d α ,
h ( z , z ) = f ( z , z ) - 1 λ 2 - Δ Δ - Δ Δ G ( α , α ) e - i k α z e - i k α z d α d α = f ( z , z ) - C ( z , z )
G ( α , α ) = 2 b sin k α b k α b δ a e i k ϕ ( x ) e i k α x d x + a b e i k α x d x = 2 b sin k α b k α b { G δ a ( α ) + G α b ( α ) }
h ( z , z ) = f ( z , z ) - [ 2 b λ - Δ Δ sin k α b k α b e - i k α z d α ] × [ 1 λ - Δ Δ { G δ a ( α ) + G a b ( α ) } e - i k α z d α ] .
g ( α ) = G δ a ( α ) + G a b ( α ) ; h ( z ) = f ( z ) - 1 λ - Δ Δ G δ a ( α ) e - i k α z d α - 1 λ - Δ Δ G a b ( α ) e - i k α z d α = f ( z ) - C 1 ( z ) - C 2 ( z )             in             z < b .
ϕ ( x ) = n λ ( x - a ) 2 ( δ - a ) 2 .
( b - a ) sin [ k α ( b - a ) / 2 ] k α ( b - a / 2 ) ,
h ( z ) = f ( z ) - C 2 ( z ) .
G a b ( α ) = ( b - a ) sin k a ( b - a ) / 2 k α ( b - a ) / 2 exp ( i k α b + a 2 ) , C 2 ( z ) = 1 λ - Δ Δ ( b - a ) sin k α ( b - a ) / 2 k α ( b - a ) / 2 × exp [ i k α ( b + a 2 - z ) ] d α = 1 λ - Δ Δ ( b - a ) sin k α ( b - a ) / 2 × cos k α [ ( b + a ) / 2 - z ] k α ( b - a ) / 2 d α = k ( b - z ) 2 π - Δ Δ sin k α ( b - z ) k α ( b - z ) d α + k ( z - a ) 2 π - Δ Δ sin k α ( z - a ) k α ( z - a ) d α = 1 π 0 k Δ ( b - z ) sin t t d t + 1 π 0 k Δ ( z - a ) sin t t d t = 1 π [ S i k Δ ( b - z ) + S i k Δ ( z - a ) ] = 1 π [ S i π D λ f ( b - z ) + S i π D λ f ( z - a ) ] ,
S i θ = 0 θ ( sin t / t ) d t
h 1 ( z ) = 0 - C 2 ( z ) z < δ , z > b , z < b h 2 ( z ) = e i k ϕ ( z ) - C 2 ( z ) δ < z < a , z < b h 3 ( z ) = 1 - C 2 ( z ) a < z < b , z < b .
S 2 ( z ) = sin k [ v t + ϕ ( z ) ] - C 2 ( z ) sin k v t = Im h 2 ( z ) e i k v t .
I 1 = C 2 2 ( z ) z < δ , z > b I 2 = [ e i k ϕ ( z ) - C 2 ( z ) ] × [ e - i k ϕ ( z ) - C 2 ( z ) ] = [ 1 + C 2 2 ( z ) - 2 C 2 ( z ) cos k ϕ ( z ) ] δ < z < a I 3 = [ 1 - C 2 ( z ) ] 2 a < z < b .
D 1.2 ( λ f / π d )