Abstract

The 3-D scattering potential of microscopic objects is reconstructed as well as the 2-D equivalent object from amplitude and phase data of the scattered field. The experimental setup and results are presented and discussed alongside corresponding computer simulations. In the computational simulation, several small coated spheres were assumed, and the scatter field data were determined by the Mie diffraction theory. In the experiment, a small sphere of 40-μm diam was used, and the scattered field was measured by interferometric methods. Two alternative methods were used to record the scattered field data; their sensitivity to phase quantization is discussed. Since in the experimental situation the scattered field data can only be determined in a very restricted range, a distinct smear of the PSF of this imaging process results. We describe the consequences of this drawback in terms of this PSF. On the other hand, we show that this allows a reduction of computation time in the digital inversion of the scattered field data.

© 1979 Optical Society of America

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References

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  1. E. Wolf, Opt. Commun. 1, 153 (1969).
    [Crossref]
  2. R. M. Lewis, IEEE Trans. Antennas Propag. AP-17, 308 (1969).
    [Crossref]
  3. L. D. Faddeyev, B. Seckler, J. Math. Phys. 4, 72 (1963).
    [Crossref]
  4. J. W. Goodman, R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967).
    [Crossref]
  5. W. H. Carter, J. Opt. Soc. Am. 60, 306 (1970).
    [Crossref]
  6. W. H. Carter, P.-C. Ho, Appl. Opt. 13, 162 (1974).
    [Crossref] [PubMed]
  7. D. K. Lam et al., Can. J. Phys. 54, 1925 (1976); H. G. Schmidt-Weinmar, J. Opt. Soc. Am. 65, 1059 (1975).
    [Crossref]
  8. R. Dändliker, K. Weiss, Opt. Commun. 1, 323 (1970).
    [Crossref]
  9. G. Schulz, in Recent Advances in Optical Physics, B. Havelka, J. Blabla, Eds. (Society of Czechoslovakia Mathematics and Physics, Prague, 1976), p. 559.
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973).
  12. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Washington, D.C., 1964).
  13. J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 13, 150 (1969).
    [Crossref]
  14. W. J. Dallas, Appl. Opt. 10, 673 (1971).
    [Crossref] [PubMed]
  15. A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
    [Crossref]
  16. E. Evans, Opt. Commun. 2, 317 (1970).
    [Crossref]

1976 (1)

D. K. Lam et al., Can. J. Phys. 54, 1925 (1976); H. G. Schmidt-Weinmar, J. Opt. Soc. Am. 65, 1059 (1975).
[Crossref]

1974 (1)

1971 (1)

1970 (3)

R. Dändliker, K. Weiss, Opt. Commun. 1, 323 (1970).
[Crossref]

W. H. Carter, J. Opt. Soc. Am. 60, 306 (1970).
[Crossref]

E. Evans, Opt. Commun. 2, 317 (1970).
[Crossref]

1969 (3)

E. Wolf, Opt. Commun. 1, 153 (1969).
[Crossref]

R. M. Lewis, IEEE Trans. Antennas Propag. AP-17, 308 (1969).
[Crossref]

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 13, 150 (1969).
[Crossref]

1967 (1)

J. W. Goodman, R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967).
[Crossref]

1963 (1)

L. D. Faddeyev, B. Seckler, J. Math. Phys. 4, 72 (1963).
[Crossref]

1951 (1)

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Washington, D.C., 1964).

Aden, A. L.

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Carter, W. H.

Champeney, D. C.

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973).

Dallas, W. J.

Dändliker, R.

R. Dändliker, K. Weiss, Opt. Commun. 1, 323 (1970).
[Crossref]

Evans, E.

E. Evans, Opt. Commun. 2, 317 (1970).
[Crossref]

Faddeyev, L. D.

L. D. Faddeyev, B. Seckler, J. Math. Phys. 4, 72 (1963).
[Crossref]

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 13, 150 (1969).
[Crossref]

J. W. Goodman, R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Ho, P.-C.

Kerker, M.

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Lam, D. K.

D. K. Lam et al., Can. J. Phys. 54, 1925 (1976); H. G. Schmidt-Weinmar, J. Opt. Soc. Am. 65, 1059 (1975).
[Crossref]

Lawrence, R. W.

J. W. Goodman, R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967).
[Crossref]

Lewis, R. M.

R. M. Lewis, IEEE Trans. Antennas Propag. AP-17, 308 (1969).
[Crossref]

Schulz, G.

G. Schulz, in Recent Advances in Optical Physics, B. Havelka, J. Blabla, Eds. (Society of Czechoslovakia Mathematics and Physics, Prague, 1976), p. 559.

Seckler, B.

L. D. Faddeyev, B. Seckler, J. Math. Phys. 4, 72 (1963).
[Crossref]

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 13, 150 (1969).
[Crossref]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Washington, D.C., 1964).

Weiss, K.

R. Dändliker, K. Weiss, Opt. Commun. 1, 323 (1970).
[Crossref]

Wolf, E.

E. Wolf, Opt. Commun. 1, 153 (1969).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. W. Goodman, R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967).
[Crossref]

Can. J. Phys. (1)

D. K. Lam et al., Can. J. Phys. 54, 1925 (1976); H. G. Schmidt-Weinmar, J. Opt. Soc. Am. 65, 1059 (1975).
[Crossref]

IBM J. Res. Dev. (1)

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 13, 150 (1969).
[Crossref]

IEEE Trans. Antennas Propag. (1)

R. M. Lewis, IEEE Trans. Antennas Propag. AP-17, 308 (1969).
[Crossref]

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

J. Math. Phys. (1)

L. D. Faddeyev, B. Seckler, J. Math. Phys. 4, 72 (1963).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (3)

R. Dändliker, K. Weiss, Opt. Commun. 1, 323 (1970).
[Crossref]

E. Evans, Opt. Commun. 2, 317 (1970).
[Crossref]

E. Wolf, Opt. Commun. 1, 153 (1969).
[Crossref]

Other (4)

G. Schulz, in Recent Advances in Optical Physics, B. Havelka, J. Blabla, Eds. (Society of Czechoslovakia Mathematics and Physics, Prague, 1976), p. 559.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Washington, D.C., 1964).

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Figures (23)

Fig. 1
Fig. 1

Accessible Fourier space data points K, in a scattering experiment with a constant incident beam direction, are on the surface of the Ewald sphere.

Fig. 2
Fig. 2

(a) A sphere of radius r contains 50% of the total energy flux of the PSF; (b) locus of the first node in the 3-D PSF.

Fig. 3
Fig. 3

Amplitude P and intensity |P|2 of a point image if the transfer function is a sphere of radius K0. P is normalized to ( 4 π / 3 ) K 0 3, hence P = 3[j1(rK0)]/(rK0); j1(rK0) is the spherical Bessel function.12

Fig. 4
Fig. 4

Pointspread function if the transfer function is a spherical shell of outer radius K0 and inner radius Ki: (a) Ki/K0 = 0; (b) Ki/K0 = 0.8; and (c) Ki/K0 = 0.95. The graphs are normalized to ( 4 π / 3 ) K 0 3.

Fig. 5
Fig. 5

Multidirectional illumination k01, k02, and k03, e.g., gives access to scattered field data on discrete pieces e1, e2, and e3 of Ewald spheres.

Fig. 6
Fig. 6

Far-field scattering approximation geometry. k(s) is approximately parallel to rr′.

Fig. 7
Fig. 7

Mach-Zehnder- interferometer applied to measure the scattered field data: L is the laser; BS is the beam splitter; C is the glass plate used as compensator; O is the object; and R is the reference beam.

Fig. 8
Fig. 8

Phase measurement method 1. Eight-level quantized phase measurement from four interferograms obtained with four different reference waves. Top row indicates the relative phases of the reference waves belonging to the interferograms used to find the object wave phase as indicated in the row below. The result is π arg [ A l , m ( s ) ] ( 3 π ) / 4.

Fig. 9
Fig. 9

Phase measurement method 2: O is the object; R is the reference wave focus; and P is the arbitrary point in the interferogram plane.

Fig. 10
Fig. 10

Simulation of false images in the case of eight-level phase quantization: (a) original object; (b) third-order false image; and (c) seventh-order false image.

Fig. 11
Fig. 11

Reconstructed images in the case of (a) eight-level, (b) four-level, and (c) two-level phase quantization.

Fig. 12
Fig. 12

Locus of the reconstructed scattering potential of nonabsorbing objects plotted in the complex domain. The scattering potential is normalized to F/[ik(n − 1)] according to Eq. (28): (a) × represents the homogeneous sphere: 20-μm diam, n = 1.02; (b) + represents the coated sphere: core of 10-μm diam, n = 1.04; coat of 20-μm diam, n = 1.02; (c) ○ represents the coated sphere: core of 10-μm diam, n = 1.00; coat of 20-μm diam, n = 1.02; (d) △ represents the coated sphere according to (b) but with sixteen-level phase quantization; (e) □ represents the coated sphere according to (b) but with eight-level phase quantization; (f) ● represents the coated sphere according to (b) but with four-level phase quantization; and g) | represents the coated sphere according to (b) but with two-level phase quantization.

Fig. 13
Fig. 13

Planar section in Fourier space is accessible, if the incident angle is matched to the angle of observation of the scattered wave.

Fig. 14
Fig. 14

Real part (a) and imaginary part (b) of the reconstructed 2-D equivalent object of the coated sphere of Fig. 11(b).

Fig. 15
Fig. 15

(a) Bright field, (b) phase contrast, and (c) interference contrast images of the coated sphere of Fig. 12(b).

Fig. 16
Fig. 16

Threefold diffraction pattern of a spherical object (top). Below are printouts of the amplitudes and the phases in the scattered field determined by method 1.

Fig. 17
Fig. 17

Amplitude and phase in a section (plane nine of thirty-two) through the scattered field data in Fourier space. Three Ewald spheres according to threefold illumination are easily recognizable.

Fig. 18
Fig. 18

Three sections through 3-D reconstructions of a homogeneous sphere of 40-μm diam and n = 1.02. Shown are the sections along the planes with indices 16, 17, and 32.

Fig. 19
Fig. 19

Computational reconstruction of a 2-D object 2. The phase of the scattered wave was determined by two different methods: (a) method 1; (b) method 2.

Fig. 20
Fig. 20

Microscopic interferogram of the 3-D object.

Fig. 21
Fig. 21

Phase ϕ(x,y) of the reconstructed 2-D equivalent object of the 3-D object.

Fig. 22
Fig. 22

Threefold object illumination via a grating.

Fig. 23
Fig. 23

Three of sixty-four sectional planes through the reconstructed object. Plane thirty-three is the center of the reconstruction. Left column displays amplitude; right column displays the phase of the scattering potential.

Equations (37)

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F ( r ) = F T - 1 [ U ( K ) ] .
F r , s , t = l = 0 L - 1 m = 0 M - 1 n = 0 N - 1 U l , m , n × exp ( - 2 π i r l L ) exp ( - 2 π i s m M ) exp ( - 2 π i n t N ) , r = 0 , 1 , , L - 1 , s = 0 , 1 , , M - 1 , t = 0 , 1 , , N - 1 ,
F ^ l , m , t = n = 0 N - 1 U l , m , n · exp ( - 2 π i n t N ) , F ^ l , s , t = m = 0 M - 1 F ^ l , m , t · exp ( - 2 π i m s M ) , F r , s , t = l = 0 L - 1 F ^ ^ l , s , t · exp ( - 2 π i r l L ) .
F r , s , t = l = 0 L - 1 F ^ ^ l , s , t · exp ( - 2 π i r l M )
F ¯ s , t = r = 0 N - 1 F r , s , t ,
F ^ ^ l , s , t = r = 0 N - 1 F r , s , t · exp 2 π i r l N = F ¯ s , t if l = 0.
U ¯ l , m = all n U l , m , n ,
F r , s , t = 0 = F T ( all n U l , m , n ) ,
F 2 m = l = 0 L / 2 - 1 [ G l + H l ] exp ( - 2 π i l m L / 2 ) ,
F 2 m + 1 = l = 0 L / 2 - 1 [ G l - H l ] × exp [ - 2 π i l ( m + ½ ) L / 2 ]             m = 0 , , L / 2 - 1.
2 - D FFT : 4096 as 64 2 : 42.7 sec , 2 - D FFT : 4096 as 64 2 featuring skip - if - zeros : 26.8 sec , 3 - D FFT : 4096 as 16 3 , featuring skip - if - zeros : 22.1 sec .
U ( K ) = 1 K K 0 = 0 otherwise .
P ( r K 0 ) = 4 π K 0 3 j 1 ( r K 0 ) r K 0 ,
U ( K ) 2 d 3 K = 1 ( 2 π ) 3 P ( r ) 2 d 3 r = 4 π 3 K 0 3 .
U ( K ) = 1 K i K K 0 = 0 otherwise .
U ( K ) = U 0 ( K ) - U i ( K ) with U 0 ( K ) = 1 K K 0 = 0 otherwise , and U i ( K ) = 1 K K i = 0 otherwise ,
P shell ( r ) = 4 π K 0 3 j 1 ( r K 0 ) r K 0 - 4 π K i 3 j 1 ( r K i ) r K i .
rect x a = 1 - a 2 x a 2 = 0 otherwise .
F T - 1 ( rect K z - K 0 / 2 K 0 ) = sinc ( K 0 z ) · exp ( i π K 0 z ) .
P ( r ) = P shell ( r ) * sinc ( K 0 z ) * sinc ( a x ) * sinc ( b y ) · exp ( - i π K 0 z ) .
P ( r ) = P shell ( r ) * S ( r ) · exp ( - i π K 0 z ) , with S ( r ) = sinc ( a x ) · sinc ( b y ) · sinc ( K 0 z ) .
U ( s ) ( r ) = - 1 4 π F ( r ) U ( i ) ( r ) G ( r - r ) d 3 r .
G ( r - r ) = exp ( i k 0 r - r ) r - r .
k 0 r - r = k ( s ) ( r - r ) k ( s ) ( r - r ) k ( s ) = k 0 ,
U ( s ) ( r ) = - 1 4 π F ( r ) U ( i ) ( r ) exp [ i k ( s ) ( r - r ) ] r d 3 r ,
U ( s ) ( r ) = - 1 4 π A ( i ) exp [ i k ( s ) r ] r F ( r ) exp ( i Kr ) ] d 3 r ,
I l , m ; j = | A l , m ( s ) exp ( i φ l , m ) + exp ( i j π 2 ) | 2             j = 0 , 1 , 2 , 3.
φ l , m = arctan ( I l , m ; 2 - I l , m ; 4 I l , m ; 1 - I l , m ; 3 ) .
O P - R P = [ ( x + d 2 ) 2 + f 2 ] 1 / 2 - [ ( x - d 2 ) 2 + f 2 ] 1 / 2 = f + 1 2 ( x + d 2 ) 2 f - [ f + 1 2 ( x - d 2 ) 2 f ] = d x f i f ( x ± d 2 ) 2 f 2 .
A l , m ( s ) = 1 2 [ ( I l , m ; max ) 1 / 2 - ( I l , m ; min ) 1 / 2 ] i f A l , m ( s ) A l , m ( s ) ,
F ˜ ( x ) = sinc ( 1 J ) j b j F j ( x ) ,
b j = ( - 1 ) j / ( j J + 1 ) , F m ( x ) = - U ( K ) exp [ i ( j J + 1 ) ϕ ( K ) ] exp ( 2 π i K x ) d K .
E φ = - i exp ( - i k 2 r ) k 2 r sin φ S 1 ( ϑ ) E ϑ = i exp ( - i k 2 r ) k 2 r cos φ S 2 ( ϑ ) } .
U = E = i k 2 r exp ( - i k 2 r ) [ sin 2 φ S 1 ( ϑ ) + cos 2 φ S 2 ( ϑ ) ] .
ϕ ( x , y ) = k - 0 [ n ( x , y , z ) - 1 ] d z ,
F ( r ) = - k 2 [ n 2 ( r ) - 1 ] exp { ( i k ) - z [ n ( x , y , z ) - 1 ] d z } .
O ( x , y ) = - 0 F ( r ) d z = F i k ( n - 1 ) { 1 - cos [ k ( n - 1 ) d ( x , y ) ] + i sin [ k ( n - 1 ) d ( x , y ) ] } ,

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