Abstract

We propose a novel optical sectioning method for optical scanning holography, which is performed in phase space by using Wigner distribution functions together with the fractional Fourier transform. The principle of phase-space optical sectioning for one-dimensional signals, such as slit objects, and two-dimensional signals, such as rectangular objects, is first discussed. Computer simulation results are then presented to substantiate the proposed idea.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462-463 (1985).
  2. C. S. Vikram and M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149-153 (1984).
    [CrossRef]
  3. D. A. Agard, “Optical sectioning microscopy: celluar architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191-219 (1984).
    [CrossRef] [PubMed]
  4. K. R. Castleman, Digital Image Processing (Prentice-Hall, 1979).
  5. E. N. Leith, W.-C. Chein, K. D. Mils, B. D. Athey, and D. S. Dilworth, “Optical sectioning by holographic coherence imaging: a generalized analysis,” J. Opt. Soc. Am. A 20, 380-387(2003).
    [CrossRef]
  6. T. Kim, “Optical sectioning by optical scanning holography and a Wiener filter,” Appl. Opt. 45, 872-879 (2006).
    [CrossRef] [PubMed]
  7. T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 521-527 (1985).
    [CrossRef]
  8. T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344 (1995).
    [CrossRef]
  9. T.-C. Poon and T. Kim, “Optical image recognition of three-dimensional objects,” Appl. Opt. 38, 370-381 (1999).
    [CrossRef]
  10. T. -C. Poon, Optical Scanning Holography with Matlab (Springer, 2007).
    [PubMed]
  11. T. -C. Poon and T. Kim, Engineering Optics with Matlab (World Scientific, 2006).
    [PubMed]
  12. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.
  13. H. M. Ozaktas, Z Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  14. R. F. O'Connell, “The Wigner distribution function--50th birthday,” Found. Phys. 13, 83-92 (1983).
    [CrossRef]
  15. J. Hahn, H. Kim, and B. Lee, “Optical implementation of iterative fractional Fourier transform algorithm,” Opt. Express 14, 11103-11104 (2006).
    [CrossRef] [PubMed]
  16. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).
  17. H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am. A 21, 2353-2365(2004).
    [CrossRef]

2006 (2)

2004 (1)

2003 (1)

1999 (1)

1995 (1)

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344 (1995).
[CrossRef]

1985 (2)

1984 (2)

C. S. Vikram and M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149-153 (1984).
[CrossRef]

D. A. Agard, “Optical sectioning microscopy: celluar architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191-219 (1984).
[CrossRef] [PubMed]

Agard, D. A.

D. A. Agard, “Optical sectioning microscopy: celluar architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191-219 (1984).
[CrossRef] [PubMed]

Athey, B. D.

Billet, M. L.

C. S. Vikram and M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149-153 (1984).
[CrossRef]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, 1979).

Caulfield, H. J.

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462-463 (1985).

Chein, W.-C.

Dilworth, D. S.

Doh, K.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344 (1995).
[CrossRef]

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).

Hahn, J.

Kim, H.

Kim, T.

Kutay, M. A.

H. M. Ozaktas, Z Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Lee, B.

Leith, E. N.

Mils, K. D.

O'Connell, R. F.

R. F. O'Connell, “The Wigner distribution function--50th birthday,” Found. Phys. 13, 83-92 (1983).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, Z Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Poon, T. -C.

T. -C. Poon, Optical Scanning Holography with Matlab (Springer, 2007).
[PubMed]

T. -C. Poon and T. Kim, Engineering Optics with Matlab (World Scientific, 2006).
[PubMed]

Poon, T.-C.

Schilling, B.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344 (1995).
[CrossRef]

Shinoda, K.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344 (1995).
[CrossRef]

Suzuki, Y.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344 (1995).
[CrossRef]

Vikram, C. S.

C. S. Vikram and M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149-153 (1984).
[CrossRef]

Wu, M.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344 (1995).
[CrossRef]

Yang, B.

Zalevsky, Z

H. M. Ozaktas, Z Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Annu. Rev. Biophys. Bioeng. (1)

D. A. Agard, “Optical sectioning microscopy: celluar architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191-219 (1984).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. B (1)

C. S. Vikram and M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149-153 (1984).
[CrossRef]

Found. Phys. (1)

R. F. O'Connell, “The Wigner distribution function--50th birthday,” Found. Phys. 13, 83-92 (1983).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Eng. (2)

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344 (1995).
[CrossRef]

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462-463 (1985).

Opt. Express (1)

Other (6)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).

K. R. Castleman, Digital Image Processing (Prentice-Hall, 1979).

T. -C. Poon, Optical Scanning Holography with Matlab (Springer, 2007).
[PubMed]

T. -C. Poon and T. Kim, Engineering Optics with Matlab (World Scientific, 2006).
[PubMed]

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.

H. M. Ozaktas, Z Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Single-lens optical system representing holographic reconstruction of real images by a complex hologram obtained by OSH. Note that the defocused planes are modeled by the FFRT, whereas the in-focus plane is represented by a pure Fourier transform.

Fig. 2
Fig. 2

Evolutionary profiles of FRFTs of example sectional signals (a)  f 1 ( x ) and (b)  f 2 ( x ) .

Fig. 3
Fig. 3

(a) Two slits, (b) amplitude profile of the complex hologram φ ( x ) given by Eq. (23), (c)  s z 1 ( x ) given by Eq. (24a), (d)  s z 2 ( x ) given by Eq. (24b).

Fig. 4
Fig. 4

(a) WDF of s z 1 ( x ) : W f 1 + f 2 , a ( x , μ ) , (b)  W f 1 ( x , μ ) , (c)  W f 2 , a ( x , μ ) , (d)  C f 1 + f 2 , a ( x , μ ) .

Fig. 5
Fig. 5

(a) WDF of s z 2 ( x ) : W f 1 , a + f 2 ( x , μ ) , (b)  W f 1 , a ( x , μ ) , (c)  W f 2 ( x , μ ) , (d)  C f 1 , a + f 2 ( x , μ ) .

Fig. 6
Fig. 6

WDF filtering and signal reconstruction: (a) filtered WDF W 1 of W f 1 + f 2 , a ( x , μ ) , (b)  s z 1 ( x ) and its line trace along x, (c) filtered signal f ¯ 1 and its line trace along x.

Fig. 7
Fig. 7

WDF filtering and signal reconstruction: (a) filtered WDF W 2 of W f 1 , a + f 2 ( x , μ ) , (b)  s z 2 ( x ) and its line trace along x, (c) filtered signal f ¯ 2 and its line trace along x

Fig. 8
Fig. 8

(a) Two slits, (b) amplitude profile of the complex hologram φ ( x ) given by Eq. (23), (c)  s z 1 ( x ) given by Eq. (24a), (d)  s z 2 ( x ) given by Eq. (24b).

Fig. 9
Fig. 9

(a) WDF of s z 1 ( x ) : W f 1 + f 2 , a ( x , μ ) , (b)  W f 1 ( x , μ ) , (c)  W f 2 , a ( x , μ ) , (d)  C f 1 + f 2 , a ( x , μ ) .

Fig. 10
Fig. 10

(a) WDF of s z 2 ( x ) : W f 1 , a + f 2 ( x , μ ) , (b)  W f 1 , - a ( x , μ ) , (c)  W f 2 ( x , μ ) , (d)  C f 1 , - a + f 2 ( x , μ ) .

Fig. 11
Fig. 11

Concept of WDF filtering: filtering of (a)  W f 1 + f 2 , a ( x , μ ) and (b)  W f 1 , a + f 2 ( x , μ ) .

Fig. 12
Fig. 12

WDF filtering and signal reconstruc tion: (a) filter structure, (b) filtered WDF W 1 of W f 1 + f 2 , a ( x , μ ) , (c)  s z 1 ( x ) and its line trace along x, (d) filtered signal f ¯ 1 and its line trace along x.

Fig. 13
Fig. 13

WDF filtering and signal reconstruction: (a) filter structure, (b) filtered WDF W 2 of W f 1 , a + f 2 ( x , μ ) , (c)  s z 2 ( x ) and its line trace along x (d) filtered signal f ¯ 2 and its line trace along x.

Fig. 14
Fig. 14

Example 2-D signals: (a)  f ( x 1 , x 2 ) , (b)  f 0.5 ( x 1 , x 2 ) , (c)  g ( x 1 , x 2 ) , (d)  g 0.5 ( x 1 , x 2 ) , (e)  f ( x 1 , x 2 ) + g 0.5 ( x 1 , x 2 ) , (f)  f 0.5 ( x 1 , x 2 ) + g ( x 1 , x 2 ) .

Fig. 15
Fig. 15

WDFs of f ( x 1 , x 2 ) + g 0.5 ( x 1 , x 2 ) at specific phase-space points (a)  ( x 2 , μ 2 ) = ( 0 , 0 ) and (b)  ( x 1 , μ 1 ) = ( 0 , 0 ) .

Fig. 16
Fig. 16

(a) Unfiltered sectional signal f ( x 1 , x 2 ) + g 0.5 ( x 1 , x 2 ) and (b) filtered sectional signal f ˜ ( x 1 , x 2 ) . (c) Unfiltered sectional signal f 0.5 ( x 1 , x 2 ) + g ( x 1 , x 2 ) and (d) filtered sectional signal g ˜ ( x 1 , x 2 ) .

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

φ ( x , y ) = | O ( x , y ; z ) | 2 k o 2 π z exp [ j k 0 2 z ( x 2 + y 2 ) ] d z ,
φ ( x , y ) = I ( x , y ) k o 2 π z 0 exp [ j k 0 2 z 0 ( x 2 + y 2 ) ] .
φ ( x , y ) h ( x , y , z ) = I ( x , y ) k o 2 π z 0 exp [ j k 0 2 z 0 ( x 2 + y 2 ) ] h ( x , y ; z ) ,
h ( x , y , z ) = j k o 2 π z exp [ j k 0 2 z ( x 2 + y 2 ) ]
φ ( x , y ) h ( x , y , z 0 ) = I ( x , y ) k o 2 π z 0 exp [ j k 0 2 z 0 ( x 2 + y 2 ) ] h ( x , y ; z 0 ) I ( x , y ) δ ( x , y ) = I ( x , y ) .
s z R = φ ( x , y ) h ( x , y ; z R ) = | O ( x , y ; z R ) | 2 + z z R | O ( x , y ; z ) | 2 h ( x , y ; z z R ) d z = f ( x , y ; z R ) + z z R f ( x , y ; z ) h ( x , y ; z z R ) d z ,
W f ( x 1 , x 2 , μ 1 , μ 2 ) = f ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) f * ( x 1 x 1 / 2 , x 2 x 2 / 2 ) e j 2 π ( μ 1 x 1 + μ 2 x 2 ) d x 1 d x 2 .
f ( x 1 , x 2 ) = 1 f * ( x m , 1 , x m , 2 ) W f ( x 1 + x m , 1 2 , x 2 + x m , 2 2 , μ 1 , μ 2 ) e j 2 π ( μ 1 ( x 1 x m , 1 ) + μ 2 ( x 2 x m , 2 ) ) d μ 1 d μ 2 .
F r t ( a ) { f ( x 1 , x 2 ) } = f a ( x 1 , x 2 ) = K a ( x 1 , x 1 ) K a ( x 2 , x 2 ) f ( x 1 , x 2 ) d x 1 d x 2 ,
K a ( x , x ) = { 1 j cot χ exp ( j π [ cot χ x 2 2 csc χ x x + cot χ x 2 ] ) for     χ π m δ ( x x ) for     χ = 2 π m δ ( x + x ) for     χ = 2 π m ± π ,
W f a ( x 1 , μ 1 , x 2 , μ 2 ) = W f a ( x 1 cos χ μ 1 sin χ , x 1 sin χ + μ 1 cos χ , x 2 cos χ μ 2 sin χ , x 2 sin χ + μ 2 cos χ ) .
f a ( x 1 , x 2 ) = - - h ( a ) ( x 1 , x 2 , x 1 , x 2 ) f ( x 1 , x 2 ) d x 1 d x 2 ,
h ( a ) ( x 1 , x 2 , x 1 , x 2 ) = csc ( χ ) s 2 e j π / 2 exp ( j π s 2 [ cot ( χ ) ( x 2 + y 2 ) 2 csc ( χ ) ( x x + y y ) + cot ( χ ) ( x 2 + y 2 ) ] ) .
| O ( x , y ; z ) | 2 = f 1 ( x 1 , x 2 , z 1 ) δ ( z z 1 ) + f 2 ( x 1 , x 2 , z 2 ) δ ( z z 2 ) ,
φ ( x 1 , x 2 ) = f 1 ( x 1 , x 2 , z 1 ) h ( a 1 ) ( x 1 , x 2 , z 1 ) + f 2 ( x 1 , x 2 , z 2 ) h ( a 2 ) ( x 1 , x 2 , z 2 ) ,
s z 1 ( x 1 , x 2 ) = φ ( x 1 , x 2 ) h ( a 1 ) * ( x 1 , x 2 ) = φ ( x 1 , x 2 ) h ( a 1 ) ( x 1 , x 2 ) = f 1 ( x 1 , x 2 , z 1 ) h ( a 1 ) ( x 1 , x 2 ) h ( a 1 ) ( x 1 , x 2 ) + f 2 ( x 1 , x 2 , z 2 ) h ( a 2 ) ( x 1 , x 2 ) h ( a 1 ) ( x 1 , x 2 ) = f 1 ( x 1 , x 2 , z 1 ) + f 2 ( x 1 , x 2 , z 2 ) h ( a 2 a 1 ) ( x 1 , x 2 ) ,
s z 2 ( x , y ) = φ ( x , y ) h ( a 2 ) * ( x , y ) = φ ( x , y ) h ( a 2 ) ( x , y ) = f 1 ( x , y , z 1 ) h ( a 1 ) ( x , y ) h ( a 2 ) ( x , y ) + f 2 ( x , y , z 2 ) h ( a 2 ) ( x , y ) h ( a 2 ) ( x , y ) = f 1 ( x , y , z 1 ) h ( a 1 a 2 ) ( x , y ) + f 2 ( x , y , z 2 ) .
h ( x 1 , x 2 , x 1 , x 2 ) = j λ f exp ( j π λ f [ 2 ( x 1 x 1 + x 2 x 2 ) + ( 1 d f ) ( x 1 2 + x 2 2 ) ] ) ,
z 1 = f Δ z ,
z 2 = f + Δ z .
a 1 = 2 π cos 1 ( Δ z f ) ,
a 2 = 2 π cos 1 ( Δ z f ) = 2 a 1 ,
s z 1 ( x 1 , x 2 ) = f 1 ( x 1 , x 2 , z 1 ) + f 2 ( x 1 , x 2 , z 2 ) h ( 2 2 a 1 ) ( x 1 , x 2 ) ,
s z 2 ( x 1 , x 2 ) = f 1 ( x 1 , x 2 , z 1 ) h ( 2 a 1 2 ) ( x 1 , x 2 ) + f 2 ( x 1 , x 2 , z 2 ) .
W f ( x , μ ) = f ( x + x / 2 ) f * ( x x / 2 ) e j 2 π μ x d x ,
f ( x ) = 1 f * ( x m ) W ( x + x m 2 , μ ) e i 2 π μ ( x + x m ) d μ .
W f a ( x , μ ) = W f a ( x cos χ μ sin χ , x sin χ + μ cos χ ) .
φ ( x ) = f 1 ( x , z 1 ) h ( 3 / 4 ) ( x ) + f 2 ( x , z 2 ) h ( 5 / 4 ) ( x ) ,
s z 1 ( x ) = f 1 ( x , z 1 ) + f 2 ( x , z 2 ) h ( 0.5 ) ( x ) = f 1 ( x , z 1 ) + f 2 , 0.5 ( x , z 2 ) ,
s z 2 ( x ) = f 1 ( x , z 1 ) h ( 0.5 ) ( x ) + f 2 ( x , z 2 ) = f 1 , 0.5 ( x , z 1 ) + f 2 ( x , z 2 ) .
W f 1 + f 2 , a ( x , μ ) = [ f 1 ( x + x / 2 ) + f 2 , a ( x + x / 2 ) ] [ f 1 * ( x x / 2 ) + f 2 , a * ( x x / 2 ) ] e j 2 π μ x d x = W f 1 ( x , μ ) + W f 2 , a ( x , μ ) + 2 Re { e j 4 π μ x f 2 , a ( x 2 ) f 1 * ( 2 x x 2 ) e j 2 π μ x d x } = W f 1 ( x , μ ) + W f 2 , a ( x , μ ) + C f 1 + f 2 , a ( x , μ ) ,
W f 1 , a + f 2 ( x , μ ) = [ f 1 , a ( x + x / 2 ) + f 2 ( x + x / 2 ) ] [ f 1 , a * ( x x / 2 ) + f 2 * ( x x / 2 ) ] e j 2 π μ x d x = W f 1 , a ( x , μ ) + W f 2 ( x , μ ) + 2 Re { e j 4 π μ x f 1 , a ( x 2 ) f 2 * ( 2 x - x 2 ) e j 2 π μ x d x } = W f 1 , a ( x , μ ) + W f 2 ( x , μ ) + C f 1 , a + f 2 ( x , μ ) .
f ( x 1 , x 2 ) = f 1 ( x 1 ) f 2 ( x 2 ) ,
g ( x 1 , x 2 ) = g 1 ( x 1 ) g 2 ( x 2 ) ,
W f + g a ( x 1 , x 2 , μ 1 , μ 2 ) = [ f ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) + g a ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) ] [ f * ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) + g a * ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) ] d x 1 d x 2 = W f 1 ( x 1 , μ 1 ) W f 2 ( x 2 , μ 2 ) + C f 1 , g 1 , a * ( x 1 , μ 1 ) C f 2 , g 2 , a * ( x 2 , μ 2 ) + C g 1 , a , f 1 * ( x 1 , μ 1 ) C g 2 , a , f 2 * ( x 2 , μ 2 ) + W g 1 , a ( x 1 , μ 1 ) W g 2 , a ( x 2 , μ 2 ) ,
W f a + g ( x 1 , x 2 , μ 1 , μ 2 ) = [ f a ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) + g ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) ] [ f a * ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) + g * ( x 1 + x 1 / 2 , x 2 + x 2 / 2 ) ] d x 1 d x 2 = W f 1 , a ( x 1 , μ 1 ) W f 2 , a ( x 2 , μ 2 ) + C f 1 , a , g 1 * ( x 1 , μ 1 ) C f 2 , a , g 2 * ( x 2 , μ 2 ) + C g 1 , f 1 , a * ( x 1 , μ 1 ) C g 2 , f 2 , a * ( x 2 , μ 2 ) + W g 1 ( x 1 , μ 1 ) W g 2 ( x 2 , μ 2 ) .
W 1 = W f + g a ( x 1 , x 2 , μ 1 , μ 2 ) = a 1 W f 1 ( x 1 , μ 1 ) + b 1 W g 1 , a ( x 1 , μ 1 ) + 2 Re [ c 1 C f 1 , g 1 , a * ( x 1 , μ 1 ) ] .

Metrics