Abstract

Numerical and analytical techniques are presented that allow three-dimensional, asymmetric, refractive index fields to be reconstructed from optical pathlength measurements, which can be obtained using multidirectional holographic interferometry. Analytical reconstruction techniques that have been used in radioaptronomy and electron microscopy for a number of years, and recently in interferometry, are presented in the context of interferometric applications in the refractionless limit. These techniques require that optical pathlength data be collected over a 180° angle of view. The required pathlength sampling rate is discussed. An efficient numerical procedure is developed for direct inversion of the data. Several numerical techniques are developed that do not require that data be collected over a full 180° angle of view. All such techniques require redundant data to achieve accurate reconstructions. The required degree of redundancy increases as the angle of view decreases. Numerical simulations using six different reconstruction techniques indicate that with a 180° angle of view, all are capable of providing accurate reconstructions. Four of the techniques were used to analyze simulated interferometric data recorded over an angle of view of less than 180°. Examples of reasonably accurate reconstructions using data with angles of view as low as 45° are presented.

© 1973 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  31. P. R. Smith, T. M. Peters, R. H. T. Bates, J. Phys. A 6, 361 (1973).
    [CrossRef]

1973

P. R. Smith, T. M. Peters, R. H. T. Bates, J. Phys. A 6, 361 (1973).
[CrossRef]

1972

B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
[CrossRef] [PubMed]

H. G. Junginger, W. van Haeringen, Opt. Commun. 5, 1 (1972).
[CrossRef]

1971

R. D. Matulka, D. J. Collins, J. Appl. Phys. 42, 1109 (1971).
[CrossRef]

A. Klug, Philos. Trans. Roy. Soc. London B261, 173 (1971).

G. Ramachandran, Proc. Indian Acad. Sci. 73, 14 (1971).

G. Ramachandran, A. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
[CrossRef] [PubMed]

1970

K. Iwata, R. Nagata, J. Opt. Soc. Am. 60, 133 (1970).
[CrossRef]

R. Gordon et al., J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

R. A. Crowther et al., Proc. Roy. Soc. London A317, 319 (1970).

L. H. Tanner, J. Sci. Instrum. 3, 987 (1970).
[CrossRef]

M. V. Berry, D. F. Gibbs, Proc. Roy. Soc. London A314, 143 (1970).

1969

1968

D. J. DeRosier, A. Klug, Nature 217, 130 (1968).
[CrossRef]

1967

W. T. Cochran et al., Proc. IEEE 55, 1664 (1967).
[CrossRef]

R. N. Bracewell, A. C. Riddle, Astrophys. J. 150, 427 (1967).
[CrossRef]

1966

1965

C. D. Maldonado, J. Math. Phys. 6, 1935 (1965).
[CrossRef]

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

1964

1962

D. P. Petersen, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

1957

S. F. Smerd, J. P. Wild, Philos. Mag. 8(2), 119 (1957).
[CrossRef]

1956

R. N. Bracewell, Aust. J. Phys. 9, 198 (1956).
[CrossRef]

Alwang, W.

W. Alwang et al., Item 1, Final Report, Pratt and Whitney, PWA-3942 [NAVAIR (Air-602) Contract], 1970.

Bates, R. H. T.

P. R. Smith, T. M. Peters, R. H. T. Bates, J. Phys. A 6, 361 (1973).
[CrossRef]

Berry, M. V.

M. V. Berry, D. F. Gibbs, Proc. Roy. Soc. London A314, 143 (1970).

Bracewell, R. N.

R. N. Bracewell, A. C. Riddle, Astrophys. J. 150, 427 (1967).
[CrossRef]

R. N. Bracewell, Aust. J. Phys. 9, 198 (1956).
[CrossRef]

R. N. Bracewell, The Fourier Transfer and Its Applications (McGraw-Hill, New York, 1965).

Cochran, W. T.

W. T. Cochran et al., Proc. IEEE 55, 1664 (1967).
[CrossRef]

Collins, D. J.

R. D. Matulka, D. J. Collins, J. Appl. Phys. 42, 1109 (1971).
[CrossRef]

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

Crowther, R. A.

R. A. Crowther et al., Proc. Roy. Soc. London A317, 319 (1970).

DeRosier, D. J.

D. J. DeRosier, A. Klug, Nature 217, 130 (1968).
[CrossRef]

Frieden, B. R.

Gibbs, D. F.

M. V. Berry, D. F. Gibbs, Proc. Roy. Soc. London A314, 143 (1970).

Golub, G.

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

Gordon, R.

R. Gordon et al., J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Grigull, U.

W. Hauf, U. Grigull, in Advances in Heat Transfer, J. P. Hartnett, T. F. Irvine, eds. (Academic Press, New York, 1970), Vol. 6.
[CrossRef]

Harris, J. L.

Hauf, W.

W. Hauf, U. Grigull, in Advances in Heat Transfer, J. P. Hartnett, T. F. Irvine, eds. (Academic Press, New York, 1970), Vol. 6.
[CrossRef]

Herman, G. T.

G. T. Herman et al., J. Theor. Biol. (to be published).

Hoppe, W.

W. Hoppe, Optik 29, 617 (1969).

Iwata, K.

Junginger, H. G.

H. G. Junginger, W. van Haeringen, Opt. Commun. 5, 1 (1972).
[CrossRef]

Klug, A.

A. Klug, Philos. Trans. Roy. Soc. London B261, 173 (1971).

D. J. DeRosier, A. Klug, Nature 217, 130 (1968).
[CrossRef]

Lakshminarayanan, A.

G. Ramachandran, A. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
[CrossRef] [PubMed]

Maldonado, C. D.

Matulka, R. D.

R. D. Matulka, D. J. Collins, J. Appl. Phys. 42, 1109 (1971).
[CrossRef]

Middleton, D.

D. P. Petersen, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Nagata, R.

Olsen, H. N.

Peters, T. M.

P. R. Smith, T. M. Peters, R. H. T. Bates, J. Phys. A 6, 361 (1973).
[CrossRef]

Petersen, D. P.

D. P. Petersen, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Ramachandran, G.

G. Ramachandran, Proc. Indian Acad. Sci. 73, 14 (1971).

G. Ramachandran, A. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
[CrossRef] [PubMed]

Reinsch, C.

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation-Linear Algebra (Springer-Verlag, Berlin, 1971), Vol. 2.
[CrossRef]

Riddle, A. C.

R. N. Bracewell, A. C. Riddle, Astrophys. J. 150, 427 (1967).
[CrossRef]

Rowley, P. D.

Smerd, S. F.

S. F. Smerd, J. P. Wild, Philos. Mag. 8(2), 119 (1957).
[CrossRef]

Smith, P. R.

P. R. Smith, T. M. Peters, R. H. T. Bates, J. Phys. A 6, 361 (1973).
[CrossRef]

Sweeney, D. W.

D. W. Sweeney, Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan (1972).

Tanner, L. H.

L. H. Tanner, J. Sci. Instrum. 3, 987 (1970).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

van Haeringen, W.

H. G. Junginger, W. van Haeringen, Opt. Commun. 5, 1 (1972).
[CrossRef]

Wild, J. P.

S. F. Smerd, J. P. Wild, Philos. Mag. 8(2), 119 (1957).
[CrossRef]

Wilkinson, J. H.

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation-Linear Algebra (Springer-Verlag, Berlin, 1971), Vol. 2.
[CrossRef]

Astrophys. J.

R. N. Bracewell, A. C. Riddle, Astrophys. J. 150, 427 (1967).
[CrossRef]

Aust. J. Phys.

R. N. Bracewell, Aust. J. Phys. 9, 198 (1956).
[CrossRef]

Inf. Control

D. P. Petersen, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

J. Appl. Phys.

R. D. Matulka, D. J. Collins, J. Appl. Phys. 42, 1109 (1971).
[CrossRef]

J. Math. Phys.

C. D. Maldonado, J. Math. Phys. 6, 1935 (1965).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. A

P. R. Smith, T. M. Peters, R. H. T. Bates, J. Phys. A 6, 361 (1973).
[CrossRef]

J. Sci. Instrum.

L. H. Tanner, J. Sci. Instrum. 3, 987 (1970).
[CrossRef]

J. Theor. Biol.

R. Gordon et al., J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Math. Comput.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

Nature

D. J. DeRosier, A. Klug, Nature 217, 130 (1968).
[CrossRef]

Numer. Math.

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

Opt. Commun.

H. G. Junginger, W. van Haeringen, Opt. Commun. 5, 1 (1972).
[CrossRef]

Optik

W. Hoppe, Optik 29, 617 (1969).

Philos. Mag.

S. F. Smerd, J. P. Wild, Philos. Mag. 8(2), 119 (1957).
[CrossRef]

Philos. Trans. Roy. Soc. London

A. Klug, Philos. Trans. Roy. Soc. London B261, 173 (1971).

Proc. IEEE

W. T. Cochran et al., Proc. IEEE 55, 1664 (1967).
[CrossRef]

Proc. Indian Acad. Sci.

G. Ramachandran, Proc. Indian Acad. Sci. 73, 14 (1971).

Proc. Natl. Acad. Sci. USA

G. Ramachandran, A. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
[CrossRef] [PubMed]

Proc. Roy. Soc. London

M. V. Berry, D. F. Gibbs, Proc. Roy. Soc. London A314, 143 (1970).

R. A. Crowther et al., Proc. Roy. Soc. London A317, 319 (1970).

Other

D. W. Sweeney, Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan (1972).

J. H. Wilkinson, C. Reinsch, Handbook for Automatic Computation-Linear Algebra (Springer-Verlag, Berlin, 1971), Vol. 2.
[CrossRef]

R. N. Bracewell, The Fourier Transfer and Its Applications (McGraw-Hill, New York, 1965).

G. T. Herman et al., J. Theor. Biol. (to be published).

W. Alwang et al., Item 1, Final Report, Pratt and Whitney, PWA-3942 [NAVAIR (Air-602) Contract], 1970.

W. Hauf, U. Grigull, in Advances in Heat Transfer, J. P. Hartnett, T. F. Irvine, eds. (Academic Press, New York, 1970), Vol. 6.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Notation for Fourier analysis of the pathlength data. (a) In the spatial plane a ray is specified by the parameters ρ and θ. (b) In the frequency plane the pathlength data of all rays with viewing angle θ determine the values of the transform along a radial line through the origin with angle θ + π/2.

Fig. 2
Fig. 2

Notation for the discrete Fourier transform. (a) In the spatial plane the refractive index field lies within a rectangular region of extent Lx by Ly. (b) In the frequency plane the Fourier-transform is effectively within a rectangular region of extent 2Bx by 2By.

Fig. 3
Fig. 3

Notation for the Sinc series expansion of the refractive index field.

Fig. 4
Fig. 4

Notation for the frequency plane restoration technique. (a) Spatial plane, (b) frequency plane.

Fig. 5
Fig. 5

Isometric plot of double-Gaussian function. (a) Input function, (b) reconstruction by frequency plane restoration technique with a 45° angle of view (see Table I for other parameters).

Fig. 6
Fig. 6

Isometric plot of cosine function. (a) Input function, (b) reconstruction by Sinc method with 90° angle of view (see Table II for other parameters).

Fig. 7
Fig. 7

Isometric plot of Rayleigh distribution. (a) Input function, (b) reconstruction by grid method with 180° angle of view (see Table III for other parameters).

Tables (3)

Tables Icon

Table I Comparison of Reconstruction Techniques for Double Gaussian Function: T(x,y) = T0 + 5. exp[−8.(x − 0.5)2 − 25.(y − 0.28)2] + 10. exp[−8.(x − 0.5)2 − 25.(y − 0.72)2]

Tables Icon

Table II Comparison of Reconstruction Techniques for Cosine Function: T(x,y) = T0 + 2.5{1 − cos[4π(x − 0.5)]} × {1 − cos[4π(y − 0.5)]}

Tables Icon

Table III Comparison of Reconstruction Techniques for Rayleigh Function: T(x,y) = T0 + 90 e xy exp[−4.5(x2 + y2)]

Equations (55)

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s i f ( x , y , z ) d s i = ϕ i ,
s f ( x , y ) d s = ϕ ( ρ , θ ) ,
ϕ ( ρ , 0 ) = ϕ ( y , 0 ) = - + f ( x , y ) d x ,
F y [ ϕ ( y , 0 ) ] = - + exp ( - 2 π f y y ) d y - + f ( x , y ) d x = - + - + f ( x , y ) exp [ - i 2 π ( f x x + f y y ) ] d x d y f x = 0 .
F ρ [ ϕ ( ρ , θ ) ] = F ( R , β ) = F ( R , θ + π / 2 ) = - + ϕ ( ρ , θ ) exp ( - i 2 π R ρ ) d ρ .
f ( x , y ) = - + - + F ( f x , f y ) exp [ i 2 π ( f x x + f y y ) ] d f x d f y .
x = r cos α , y = r sin α , f x = R cos β = - R sin θ , f y = R sin β = R cos θ .
f p ( r , α ) = - π / 2 + π / 2 d θ - + R F p ( R , θ ) × exp [ + i 2 π R r sin ( α - θ ) ] d R ,
R R Sgn ( R ) ( - i 2 π R ) [ i π Sgn ( R ) ] / 2 π 2 ,
Sgn ( R ) { + 1 if R > 0 , 0 if R = 0 , - 1 if R < 0.
f p ( r , α ) = 1 2 π 2 - π / 2 + π / 2 d θ - + d R × ( - i 2 π R ) [ i π Sgn ( R ) ] F p ( R , θ ) exp ( i 2 π R z ) .
f p ( r , α ) = 1 2 π 2 - π / 2 + π / 2 d θ [ ϕ ( z , θ ) z ] * 1 z ,
f p ( r , α ) = 1 2 π 2 - π / 2 + π / 2 d θ - + ϕ ( ρ , θ ) / ρ r sin ( α - θ ) - ρ d ρ .
F ( m L x , n L y ) = 1 4 B x B y l = - + k = - + f ( l 2 B x , k 2 B y ) × exp [ - i 2 π ( m l 2 B x L x + n k 2 B y L y ) ]
f ( l 2 B x , k 2 B y ) = 1 L x L y m = - n = - F ( m L x , n L y ) × exp [ - i 2 π ( m l 2 B x L x + n k 2 B y L y ) ] ,
f ( x , y ) = l = - + k = - + f ( l 2 B x , k 2 B y ) sinc [ 2 B x ( x - l 2 B x ) ] × sinc [ 2 B y ( y - k 2 B y ) ] ,
- + ( ϕ ( ρ , θ ) / ρ ) d ρ r sin ( α - θ ) - ρ = ϕ ( z , θ ) z * 1 z | z = r sin ( α - θ ) = 2 π 2 F f z - 1 { f z F z [ ϕ ( z , θ ) ] } z = r sin ( α - θ ) .
Δ θ = Δ β tan - 1 [ 1 / ( B y L x ) ] .
f ¯ ( x , y ) = m = 0 M - 1 n = 0 N - 1 a m n H m n ( x , y )
m = 0 M - 1 n = 0 N - 1 a m n s H m n ( x , y ) d x = ϕ ( ρ , θ ) .
s f ( x , y ) d s = s f ¯ ( x , y ) d s .
f ¯ ( x , y ) = m = 0 M - 1 n = 0 N - 1 f ( l x m , l y n ) × sinc [ 1 l x ( x - l x m ) ] · sinc [ 1 l y ( y - l y n ) ] .
m = 0 M - 1 n = 0 N - 1 f ¯ ( l x m , l y n ) s sinc [ 1 l x ( x - l x m ) ] , × sinc [ 1 l y ( y - l y n ) ] d s = ϕ ( ρ i , θ j ) .
m = 0 M - 1 n = 0 N - 1 f ¯ ( l x m , l y n ) - + sinc ( x - l x m l x ) sinc ( y - l y n l y ) × ( 1 + a j 2 ) d x = ϕ ( ρ i , θ j ) ,
y = a j x + b i j
m = 0 M - 1 n = 0 N - 1 f ¯ ( l x m , l y n ) - + sinc ( x l x ) sinc ( a j x + b i j l y ) × ( 1 + a j 2 ) 1 / 2 d x ϕ ( ρ i , θ j ) .
( 1 + a j 2 ) 1 / 2 m = 0 M - 1 n = 0 N - 1 f ¯ ( l x m , l y n ) - + sinc ( x l x ) × sinc [ a j l y ( q i j - x ) ] d x = ϕ ( ρ i , θ j ) ,
- + sinc ( x l x ) sinc [ a j l y ( q i j - x ) ] d x = F - 1 [ l x l y a j rect ( l x f q ) Rect ( l y f q a j ) ] .
rect ( x ) { 1 if x 1 / 2 , 0 if x > 1 / 2 ,
b i j = ρ i sec ( θ j ) + m l x tan ( θ j ) - n l y , c i = ρ i + m l x ,
x = c i
m = 0 M - 1 n = 0 N - 1 W m n ( ρ i , θ j ) f ¯ ( l x m , l y n ) = ϕ ( ρ i , θ j ) ,
W m n ( ρ i , θ j ) = { ( 1 + tan 2 θ j ) 1 / 2 l x sinc [ ( ρ i sec θ j + l x m tan θ j - l y n ) / l y ] for 0 tan θ j l y / l x , ( 1 + tan 2 θ j ) 1 / 2 ( l y / tan θ j ) sinc [ ( ρ i sec θ j + l x m tan θ j - l y n ) / l x tan θ j ] for l y / l x < tan θ j < , l y sinc [ ( ρ i + m l x ) / l x ]             for tan θ j = .
θ eff = tan - 1 [ a j ( l x / l y ) ] .
f ¯ ( x , y ) = m = 0 m - 1 n = 0 n - 1 f ¯ ( l x m , l y n ) × rect [ 1 l x ( x - l x m ) ] rect [ l l y ( y - l y n ) ] ,
m = 0 M - 1 n = 0 N - 1 W m n ( ρ i , θ j ( ρ i , θ j ) f ¯ ( l x m , l y n ) = ϕ ( ρ i , θ j ) ,
W ( ρ i , θ j ) = { ( 1 + tan 2 θ j ) 1 / 2 l x for b i j ( l y - l x tan θ j ) / 2 and tan θ j l y / l x , ( 1 + tan 2 θ j ) 1 / 2 l y / tan θ j for b i j ( l x tan θ j - l y ) / 2 and tan θ j > l y / l x , [ ( 1 + tan 2 θ j ) 1 / 2 / tan θ j ] { ( l x tan θ j + l y ) / 2 - b i j ) } for l y - l x tan θ j / 2 < b i j l y / l x tan θ j / 2 , l y             for c i < l x / 2 and tan θ j = , 0             for b i j > ( l x tan θ j + l y ) / 2 ,
b i j and c i
f ( x , y ) = m = - + n = - + G m n exp [ i 2 π ( m x / L x + n y / L y ) ] ,
G m n = 1 L x L y - L y 2 + L y 2 - L x 2 + L x 2 f ( x , y ) × exp [ - i 2 π ( m x L x + n y L y ) ] d x d y .
F ( f x , f y ) = L x L y m = - + n = - + G m n sinc [ L x ( m L x - f x ) ] × sinc [ L y ( n L y - f y ) ] .
[ W ] i j [ G ] j = [ F ] i ,
[ W ] i j [ f ] j = [ ϕ ] i ,
[ W ] i j [ f ] j - [ ϕ ] i = min ,
[ f ] j = min ,
x i = ( x i t x i ) 1 / 2 = [ i = 1 l ( x i ) 2 ] 1 / 2
[ f ] j = ( [ W ] j i [ W ] i j ) - 1 [ W ] j i [ ϕ ] i .
- + - + f ( x , y ) - f ¯ ( x , y ) 2 d x d y = min ,
f ¯ ( x , y ) = f ( x , y ) * 4 B x B y [ sinc ( 2 B x x ) sinc ( 2 B y y ) ] ,
i { - [ f ( x , y ) - f ¯ ( x , y ) ] d s i } 2 = min ,
Δ ϕ i ( k ) = ϕ i - W i j f j ( k )
f j ( k + 1 ) = f j ( k ) + ( c Δ ϕ i / α i ) [ W i j ]
α i = j = 1 J ( W i j ) 2
R ( k ) = [ W ] i j [ f ] j k - [ ϕ ] i .
e av = 1 N i = 1 N e i ,

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