Abstract

A quantitative analysis of the effect on image reconstruction of discarding the amplitude information contained in a wavefront reflected by a diffusely reflecting, coherently illuminated surface is given. The image reconstruction from a phase record alone is analyzed for the perfect and imperfect phase-matching cases.

© 1970 Optical Society of America

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References

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  1. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
    [CrossRef]
  2. This restriction is introduced mostly to simplify the mathematical formulation and does not have any particular physical significance.
  3. J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
    [CrossRef]
  4. For a discussion of the central-limit theorem see, for example, D. Middleton, Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Sec. 7.7-3.
  5. It can, for example, be derived from Ref. 4, Sec. 9.1, Eq. (9.33).
  6. See, for example, Ref. 4, problem 9.1(d). [The term 14π2 before the summation sign in Eq. (8) is apparently wrong and should not be there.]

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[CrossRef]

1968 (1)

Goodman, J. W.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[CrossRef]

Knight, G. R.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[CrossRef]

Middleton, D.

For a discussion of the central-limit theorem see, for example, D. Middleton, Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Sec. 7.7-3.

IBM J. Res. Develop. (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (4)

This restriction is introduced mostly to simplify the mathematical formulation and does not have any particular physical significance.

For a discussion of the central-limit theorem see, for example, D. Middleton, Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Sec. 7.7-3.

It can, for example, be derived from Ref. 4, Sec. 9.1, Eq. (9.33).

See, for example, Ref. 4, problem 9.1(d). [The term 14π2 before the summation sign in Eq. (8) is apparently wrong and should not be there.]

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Equations (21)

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exp [ i φ ( ξ 1 , η 1 ) - i φ ( ξ 2 , η 2 ) ] = δ ( ξ 1 - ξ 2 ) δ ( η 1 - η 2 ) ,
U ( u , v ) = 1 λ d I 0 1 2 ( ξ , η ) × exp { i [ π λ d [ ( u - ξ ) 2 + ( v - η ) 2 ] + φ ( ξ , η ) ] } d ξ d η ,
U r ( u , v ) U r ( u + μ , v + ν ) = U i ( u , ν ) U i ( u + μ , ν + ν ) = R 11 ( u , v ; μ , ν )
U r ( u , v ) U i ( u + μ , v + ν ) = - U r ( u + μ , v + ν ) U i ( u , v ) = R 12 ( u , v ; μ , ν ) ,
R 11 ( u , v ; μ , ν ) = 1 2 λ 2 d 2 I 0 ( ξ , η ) × cos { π λ d [ μ 2 + ν 2 + 2 ( u - ξ ) μ + 2 ( v - η ) ν ] } d ξ d η
R 12 ( u , v ; μ , ν ) = 1 2 λ 2 d 2 I 0 ( ξ , η ) × sin { π λ d [ μ 2 + ν 2 + 2 ( u - ξ ) μ + 2 ( v - η ) ν ] } d ξ d η .
Ī ( x , y ) = A λ 2 d 2 exp [ i θ ( u + μ , v + ν ) - i θ ( u , v ) ] × exp { - i π λ d [ μ 2 + ν 2 + 2 ( u - x ) μ + 2 ( v - y ) ν ] } × d u d v d μ d ν .
exp [ i θ ( u + μ , v + ν ) - i θ ( u , v ) ] = ( α + i β ) 1 4 π 2 F 1 ( 1 2 ; 1 2 ; 2 ; α 2 + β 2 ) ,
α = R 11 ( u , v ; μ , ν ) / R 11 ( u , v ; 0 , 0 ) , β = R 12 ( u , v ; μ , ν ) / R 11 ( u , v ; 0 , 0 ) ;
F 2 1 ( 1 2 ; 1 2 ; 2 ; α 2 + β 2 ) = 1 + 1 8 ( α 2 + β 2 ) + 3 / 64 ( α 2 + β 2 ) 2 +
α + i β = exp { i ( π / λ d ) [ μ 2 + ν 2 + 2 μ ν + 2 ν v ] } 2 λ 2 d 2 R 11 ( 0 , 0 ; 0 , 0 ) × I o ( ξ , η ) exp [ - i 2 π λ d ( μ ξ + ν η ) ] d ξ d η .
Ī ( x , y ) = A S 4 π 2 J ( ω 1 , ω 2 ) 1 4 π F 2 1 ( 1 2 ; 1 2 ; 2 ; ) J ( ω 1 ω 2 ) 2 × exp [ i ( ω , x + ω 2 y ) ] d ω 1 d ω 2 ,
J ( ω 1 , ω 2 ) = I 0 ( x , y ) exp [ - i ( ω 1 x + ω 2 y ) ] d x d y / I 0 ( x , y ) d x d y
I 0 ( x , y ) = I 0 ( x , y ) / I 0 ( x , y ) d x d y ,
Ī ( x , y ) = A S 1 4 π [ I 0 + 1 8 I 0 * ( I 0 * * I 0 ) + ( 3 / 64 ) I 0 * ( I 0 * * I 0 ) * ( I 0 * * I 0 ) + ] .
F ( 0 , 0 ) = f ( x , y ) d x d y .
exp [ i c θ ( u + μ , v + ν ) - i c θ ( u , v ) ] = m = 0 m ( α 2 + β 2 ) m / 2 m ! G m ( c ) Γ 2 ( m 2 + 1 ) F 2 1 × ( m 2 ; m 2 ; m + 1 ; α 2 + β 2 ) ,
G m ( c ) = 1 - cos [ 2 π ( c + m ) ] 4 π 2 ( c + m ) 2 exp ( - i m ϕ 0 ) + 1 - cos [ 2 π ( c - m ) ] 4 π 2 ( c - m ) 2 exp ( i m ϕ 0 ) , 0 = 1 ,             m = 2             for             m 0 ,             and             ϕ o = tan - 1 ( β / α ) .
Ī m ( x , y ) = A S T m ( c ) 4 π 2 × J m ( ω 1 , ω 2 ) F 2 1 ( m 2 ; m 2 ; m + 1 ; J ( ω 1 , ω 2 ) 2 ) × exp [ i ( ω 1 x + ω 2 y ) ] d ω 1 d ω 2 ,
T m ( c ) = Γ 2 ( 1 2 m + 1 ) m ! 1 - cos [ 2 π ( c - m ) ] 2 π 2 ( c - m ) 2 .
Ī m ( x , y ) = [ A S T m ( c ) / 4 π 2 ] [ B m + a 1 B m * ( I 0 * * I 0 ) + a 2 B m * ( I 0 * * I 0 ) * ( I 0 * * I 0 ) + ] ,