Abstract

In the study of a flexible fluid prism, it was noticed that the coherence of a collimated beam arising from a filtered Hg lamp (Δν≤100 Å) was greatly reduced, even for prism angles of only a few degrees (7° or less). A mathematical description for the effect of thin prisms on the coherence was accordingly developed and used as a guide to further study of the effect. This description, as well as the confirming experimental results, is presented in this paper.

© 1965 Optical Society of America

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References

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  1. A. Smith and R. Whitney, presented at the Spring Meeting of the Optical Society of America, Washington, D. C. (1964).[J. Opt. Soc. Am. 54, 571A (1964)].
  2. B. J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957).
    [Crossref]
  3. M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1964), Chap. 10.
  4. R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).
    [Crossref]
  5. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

1957 (1)

1954 (1)

R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).
[Crossref]

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1964), Chap. 10.

Bracewell, R. N.

R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).
[Crossref]

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

Roberts, J. A.

R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).
[Crossref]

Smith, A.

A. Smith and R. Whitney, presented at the Spring Meeting of the Optical Society of America, Washington, D. C. (1964).[J. Opt. Soc. Am. 54, 571A (1964)].

Thompson, B. J.

Whitney, R.

A. Smith and R. Whitney, presented at the Spring Meeting of the Optical Society of America, Washington, D. C. (1964).[J. Opt. Soc. Am. 54, 571A (1964)].

Wolf, E.

B. J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1964), Chap. 10.

Australian J. Phys. (1)

R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).
[Crossref]

J. Opt. Soc. Am. (1)

Other (3)

M. Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1964), Chap. 10.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

A. Smith and R. Whitney, presented at the Spring Meeting of the Optical Society of America, Washington, D. C. (1964).[J. Opt. Soc. Am. 54, 571A (1964)].

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Figures (8)

F. 1
F. 1

Fraunhofer patterns produced by a beam passing through the flexible coupling.

F. 2
F. 2

Densitometer traces of Fraunhofer patterns shown in Fig. 1 made along the direction of smear.

F. 3
F. 3

Fraunhofer patterns produced by a circular aperture illuminated with radiation of varying degrees of coherence. (a) coherence 0.99, pinhole size 25μ; (b) coherence 0.70, pinhole size 106μ; (c) coherence 0.30, pinhole size 190μ; (d) coherence 0, pinhole size 430μ.

F. 4
F. 4

Densitometer traces of Fraunhofer patterns shown in Fig. 3.

F. 5
F. 5

Interference patterns produced by two pinholes placed in the beam emerging from a prism.

F. 6
F. 6

Densitometer traces of interference patterns shown in Fig. 5.

F. 7
F. 7

Coherence of radiation emerging from a prism is compared to a theoretical curve for various beam deflection angles.

F. 8
F. 8

Computed intensity variation in an interference pattern produced by two pinholes placed in a beam emerging from a prism.

Equations (40)

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Γ 12 ( τ ) = V 1 ( t + τ ) V 2 * ( t )
Γ ̂ 12 ( ν ) = Γ 12 ( τ ) e 2 π i ν τ d τ .
Γ ̂ 12 ( ν ) = { f ( ν ) e i k ( z 1 z 2 ) } ,
Γ ̂ 11 ( ν ) f ( ν ) ;
Γ ̂ l ( x 1 , x 2 ) = f ( ν ) e i k · ( r 1 r 2 ) .
k = ( 2 π / λ ) [ β y + δ z ] ,
β = α ( n 1 ) and δ = [ 1 α 2 ( n 1 ) 2 ] 1 2 .
x = x , y = ξ cos θ + η sin θ , z = η cos θ ξ sin θ .
sin θ = α ( n ¯ 1 ) .
Γ ̂ ( x 1 , x 2 , ν ) = f ( ν ) exp { ( 2 π i / λ ) × [ ( ξ 2 ξ 1 ) ( β β ¯ ) + ( η 1 η 2 ) ( β β ¯ + 1 ) ] } .
Γ ̂ ( x 1 , x 2 , ν ) = f ( ν ) exp { ( 2 π i / λ ) [ ( ξ 2 ξ 1 ) ( β β ¯ ) ] } .
Γ ( x 1 , x 2 , τ ) = 0 f ( ν ) × exp { ( 2 π i / λ ) [ ( ξ 2 ξ 1 ) ( β β ¯ ) ] } e 2 π i ν τ d ν .
n ( λ ) = a + b λ ,
β = α ( n 1 ) = α [ ( a 1 ) + b λ ]
β β ¯ = α b ( λ λ ¯ ) .
Γ ( ξ 1 , ξ 2 , τ ) = exp [ 2 π i α b ( ξ 2 ξ 1 ) ] 0 f ( ν ) × exp { 2 π i [ ( ξ 2 ξ 1 ) ( α b λ / c ) + τ ] ν } d ν .
Γ ( ξ 1 , ξ 2 ξ τ ) = exp [ 2 π i α b ( ξ 2 ξ 1 ) ] × f ̂ [ ( α b λ ¯ / c ) ( ξ 2 ξ 1 ) + τ ] .
Γ ( ξ 1 , ξ 2 , 0 ) = exp [ 2 π i α b ( ξ 2 ξ 1 ) ] × f ̂ [ ( α b λ ¯ / c ) ( ξ 2 ξ 1 ) ] .
Γ ( ξ 1 , ξ 2 , ν ) = f ( ν ) exp { 2 π i ( ξ 2 ξ 1 ) [ α b ( α b λ ¯ / λ ) ] } = f ( ν ) exp [ ( 2 π i / λ ) ( ξ 2 ξ 1 ) A ] ,
A = α b λ α b λ ¯ .
Î ( x , ν ) = aperture f ( ν ) exp [ ( 2 π i / λ ) A ( ξ 2 ξ 1 ) ] × exp [ 2 π i λ ( ξ 1 2 ξ 2 2 2 f ) ] exp [ 2 π i λ ( r 1 r 2 ) ] d ξ 1 d ξ 2
Î ( x , ν ) = aperture f ( ν ) × exp { + 2 π i λ [ ( ξ 1 ξ 2 ) ( A x f ) ] } d ξ 1 d ξ 2 .
I ( x ) = 0 aperture f ( ν ) × exp { + 2 π i λ [ ( ξ 1 ξ 2 ) ( A x f ) ] } d ξ 1 d ξ 2 d ν .
I ( x ) = 0 f ( ν ) | U [ ( A x f ) ν c ] | 2 d ν .
U ( x ) = aperture e 2 π i ξ x d ξ ,
x = A f .
Δ λ / λ = 1 / α b d .
T ( ξ 1 , ξ 2 ) = [ δ ( ξ 1 ± e ) ] [ δ ( ξ 2 ± e ) ] ;
δ ( ξ ± e ) = δ ( ξ + e ) + δ ( ξ e ) .
I ( x ) = 0 aperture f ( ν ) exp { + 2 π i λ [ ( ξ 1 ξ 2 ) ( A x f ) ] } × δ ( ξ 1 ± e ) δ ( ξ 2 ± e ) d ξ 1 d ξ 2 d ν .
I ( x ) = f ( ν ) ( 2 + exp { + 2 π i λ [ 2 e ( A x f ) ] } + exp { 2 π i λ [ 2 e ( A x f ] } ) d ν .
I ( x ) = 2 f ̂ ( 0 ) ( 1 + | f { ( 2 e / c ) [ α b λ ¯ ( x / f ) ] } | f ̂ ( 0 ) × cos 2 π { 2 e b α + ϕ { ( 2 e / c ) [ α b λ ¯ ( x / f ) ] } ) .
f ( ν ) = | f ( ν ) | e 2 π i ϕ ( ν ) .
V = ( I max I min ) / ( I max + I min )
f ( ν ) = δ ( ν ν 1 ) + δ ( ν ν 2 ) ,
f ( τ ) = e 2 π i ν 1 τ + e 2 π i ν 2 τ .
V | x = 0 = cos 2 π Δ ν ( 2 e / c ) [ α b λ ¯ ( x / f ) ] | x = 0 ,
ν 1 = ν ¯ + Δ ν and ν 2 = ν ¯ Δ ν .
Δ ν / ν = 1 / 2 α b d = Δ λ / λ .
Γ ̂ 11 ( ν ) = f ( ν ) .