Abstract

Here a transport model is used to simulate amplitude-only imaging and intensity-based quantitative phase imaging in a turbid medium. We derive an optical transfer function for propagation through a scattering medium. We also show that, as expected, scattering leads to a degradation in the spatial resolution in both forms of imaging, while the magnitude of the phase retrieved using a solution of the transport-of-intensity equation decreases as the optical density of the scattering medium increases.

© 2008 Optical Society of America

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  1. S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
    [CrossRef] [PubMed]
  2. R. Chandrasekhar, Radiative Transfer, (Oxford, 1950).
  3. A. Ishimaru, Wave propagation and scattering in random media Volume 1: Single scattering and transport theory (New York: Academic, 1978).
  4. J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).
  5. H. W. Lewis, "Multiple scattering in an infinite medium," Phys. Rev. 78, 526-529 (1950).
    [CrossRef]
  6. C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
    [CrossRef]
  7. A. Wax and J. E. Thomas, "Measurement of smoothed Wigner phase-space distribution for small-angle scattering in a turbid medium," J. Opt. Soc. Am. A 15, 1896-1908 (1998).
    [CrossRef]
  8. F. Dubois, L. Joannes and J.-C. Legros, "Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence," Appl. Opt. 38, 7085-7094 (1999).
    [CrossRef]
  9. F. Dubois, M.-L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
    [CrossRef] [PubMed]
  10. C.-C. Cheng and M. G. Raymer, "Long range saturation of spatial decoherence in wave-field transport in random multiple scattering media," Phys. Rev. Lett. 82, 4807-4810 (1999).
    [CrossRef]
  11. F. Dubois, M.-L. N. Requena, and C. Minetti, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
    [CrossRef] [PubMed]
  12. N. A. Beaudry and T. D. Milster, "Effects of object roughness on partially coherent image formation," Opt. Lett. 25, 454-456 (2000).
    [CrossRef]
  13. D. M. Marks, R. A. Stack, and D. J. Brady, "Astigmatic coherence sensor for digital imaging," Opt. Lett. 25, 1726-1728 (2000).
    [CrossRef]
  14. A. Momose, "Phase-sensitive imaging and phase tomography using X-ray interferometers," Opt. Express 11, 2303-2314 (2003).
    [CrossRef] [PubMed]
  15. M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005).
    [CrossRef]
  16. E. D. Barone-Nugent, A. Barty and, K. A. Nugent, "Quantitative phase-amplitude microscopy I: optical microscopy," J. Microsc. 206, 194-203 (2002).
    [CrossRef] [PubMed]
  17. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett.  23, 817-819 (1998).
    [CrossRef]
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).
  19. M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," J. Opt. Soc. Am. A 3, 1227-1238 (1986).
    [CrossRef]
  20. K.-H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
    [CrossRef]
  21. M. J. Bastiaans and T. Alieva, "Wigner distribution moments in fractional fourier transform systems," J. Opt. Soc. Am. A 19, 1763-1773 (2002).
    [CrossRef]
  22. C. K. Aruldoss, N. M. Dragomir, and A. Roberts, "Non-interferometric characterization of partially coherent scalar wavefields and application to scattered light," J. Opt. Soc. Am. A 24, 3189-3197 (2007).
    [CrossRef]
  23. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
    [CrossRef]
  24. M. R. Teague, "Deterministic phase retrieval: a Green’s function solution," J. Opt. Soc. Am. 731434-1441 (1983).
    [CrossRef]
  25. T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
    [CrossRef]
  26. D. Paganin and K. A. Nugent, "Non-interferometric phase imaging using partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
    [CrossRef]
  27. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
    [CrossRef] [PubMed]

2007 (1)

2005 (1)

M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005).
[CrossRef]

2004 (3)

2003 (1)

2002 (2)

M. J. Bastiaans and T. Alieva, "Wigner distribution moments in fractional fourier transform systems," J. Opt. Soc. Am. A 19, 1763-1773 (2002).
[CrossRef]

E. D. Barone-Nugent, A. Barty and, K. A. Nugent, "Quantitative phase-amplitude microscopy I: optical microscopy," J. Microsc. 206, 194-203 (2002).
[CrossRef] [PubMed]

2000 (3)

1999 (2)

C.-C. Cheng and M. G. Raymer, "Long range saturation of spatial decoherence in wave-field transport in random multiple scattering media," Phys. Rev. Lett. 82, 4807-4810 (1999).
[CrossRef]

F. Dubois, L. Joannes and J.-C. Legros, "Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence," Appl. Opt. 38, 7085-7094 (1999).
[CrossRef]

1998 (3)

1997 (1)

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

1995 (1)

1986 (1)

1984 (1)

K.-H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
[CrossRef]

1983 (1)

1950 (1)

H. W. Lewis, "Multiple scattering in an infinite medium," Phys. Rev. 78, 526-529 (1950).
[CrossRef]

Alfano, R. R.

M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005).
[CrossRef]

Alieva, T.

Alrubaiee, M.

M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005).
[CrossRef]

Arridge, S. R.

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Aruldoss, C. K.

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty and, K. A. Nugent, "Quantitative phase-amplitude microscopy I: optical microscopy," J. Microsc. 206, 194-203 (2002).
[CrossRef] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett.  23, 817-819 (1998).
[CrossRef]

Bastiaans, M. J.

Beaudry, N. A.

Brady, D. J.

Brenner, K.-H.

K.-H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
[CrossRef]

Brito, M.

M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005).
[CrossRef]

Cheng, C.-C.

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
[CrossRef]

C.-C. Cheng and M. G. Raymer, "Long range saturation of spatial decoherence in wave-field transport in random multiple scattering media," Phys. Rev. Lett. 82, 4807-4810 (1999).
[CrossRef]

Dragomir, N. M.

Dubois, F.

Gayen, S. K.

M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005).
[CrossRef]

Gureyev, T. E.

Hebden, J. C.

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Istasse, E.

Joannes, L.

Legros, J.-C.

Lewis, H. W.

H. W. Lewis, "Multiple scattering in an infinite medium," Phys. Rev. 78, 526-529 (1950).
[CrossRef]

Marks, D. M.

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

Milster, T. D.

Minetti, C.

Momose, A.

Monnom, O.

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, "Non-interferometric phase imaging using partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett.  23, 817-819 (1998).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942-1946 (1995).
[CrossRef]

Ojeda-Castaneda, J.

K.-H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
[CrossRef]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, "Non-interferometric phase imaging using partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett.  23, 817-819 (1998).
[CrossRef]

Raymer, M. G.

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
[CrossRef]

C.-C. Cheng and M. G. Raymer, "Long range saturation of spatial decoherence in wave-field transport in random multiple scattering media," Phys. Rev. Lett. 82, 4807-4810 (1999).
[CrossRef]

Requena, M.-L. N.

Roberts, A.

Stack, R. A.

Teague, M. R.

Thomas, J. E.

Wax, A.

Xu, M.

M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005).
[CrossRef]

J. Microsc. (2)

E. D. Barone-Nugent, A. Barty and, K. A. Nugent, "Quantitative phase-amplitude microscopy I: optical microscopy," J. Microsc. 206, 194-203 (2002).
[CrossRef] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

K.-H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Med. Biol. (1)

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Phys. Rev. (1)

H. W. Lewis, "Multiple scattering in an infinite medium," Phys. Rev. 78, 526-529 (1950).
[CrossRef]

Phys. Rev. A (1)

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

C.-C. Cheng and M. G. Raymer, "Long range saturation of spatial decoherence in wave-field transport in random multiple scattering media," Phys. Rev. Lett. 82, 4807-4810 (1999).
[CrossRef]

D. Paganin and K. A. Nugent, "Non-interferometric phase imaging using partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

Other (5)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

R. Chandrasekhar, Radiative Transfer, (Oxford, 1950).

A. Ishimaru, Wave propagation and scattering in random media Volume 1: Single scattering and transport theory (New York: Academic, 1978).

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Schematic showing the system discussed in Section 3. A partially coherent field is incident on a thin object with a complex transmission coefficient O(x). The transmitted field then propagates a distance, , through a scattering medium. We determine the intensity, I(x,0) and the longitudinal intensity derivative ∂I(x,0)/∂z at z=0.

Fig. 2.
Fig. 2.

Scattering optical transfer-like function, S(q⃗,0) for light propagating through a medium of thickness 10 mm containing 20 µm diameter spheres of relative index 1.20 with density 1.6×1011/m3 so that the medium has an optical density of 1.

Fig. 3.
Fig. 3.

The optical transfer function of the scattering medium consisting microspheres of sizes (a) 5 µm and (b) 20 µm.

Fig. 4.
Fig. 4.

Intensity of light transmitted through a scattering medium of total thickness 10 mm. In the center of this medium there is an amplitude mask with a double-slit transmission function given by the blue curve corresponding to zero scattering. (a) Shows the intensity when the scattering medium consists of 5 µm spheres and (b) 20 µm spheres.

Fig. 5.
Fig. 5.

The phase profile of the output field obtained through the spatial coherence function transmitted through two turbid media of same thickness (l2 =l1 =5 mm) containing spheres of diameter (a) 5 µm and (b) 20 µm.

Equations (26)

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G ( r 1 , r 2 , z ) = E ( r 1 , z , t ) E * ( r 2 , z , t )
J ( x , s , z ) = G ( x + s 2 , x s 2 , z ) .
B ( x , u , z ) = J ( x , s , z ) exp ( i u s ) d s
X ( q , s , z ) = 1 ( 2 π ) 4 d x B ( x , u , z ) exp [ i ( u s q x ) ] = 1 ( 2 π ) 2 d x J ( x , s , z ) exp ( i q x ) .
X ( q , s , z ) = X ( q , s z k q , 0 ) ,
J S ( x , s , 0 ) = d q e i q x S ( q , s ) I 0 ( q , s ) ,
I 0 ( q , s ) = 1 ( 2 π ) 2 d x e i q x J 0 ( x , s q k ) ,
S ( q , s ) = e μ T exp ( I 1 ( q , s ) ) ,
I 1 ( q , s ) = N σ N π θ 0 q exp [ k med 2 θ 0 2 4 ( s 2 q s 2 q 2 ) ]
× { erf ( k med θ 0 2 q q s ) erf [ k med θ 0 2 ( q s q q k med ) ] }
X S ( q , s , 0 ) = S ( q , s ) X 0 ( q , s k med q )
J obj ( x , s ) = J 0 ( x , s ) O ( x + s 2 ) O * ( x s 2 ) ,
X S ( q , s , 0 ) = 1 ( 2 π ) 2 S ( q , s ) d x A 2 ( x ) exp ( i s φ ( x ) ) exp ( i q x ) ,
X S ( q , s , z ) = 1 ( 2 π ) 2 S ( q , s z k q )
× dx A 2 ( x ) exp ( i ( s z k q ) φ ( x ) ) exp ( i q x ) .
I ( x , z ) = J ( x , 0 , z ) = 1 ( 2 π ) 2 d q d x ' A 2 ( x ' ) S ( q , z k q )
× exp ( i z k q φ ( x ' ) ) exp ( i q ( x x ' ) )
I ( x , 0 ) = 1 ( 2 π ) 2 d q d x ' A 2 ( x ' ) S ( q , 0 ) exp ( i q ( x x ' ) )
= 1 ( 2 π ) 2 d q S ( q , 0 ) I ˜ 0 ( q ) exp ( i q x ) ,
I ˜ 0 ( q ) = d x ' A 2 ( x ' ) exp ( i q x ' )
I ( x , 0 ) z = 1 ( 2 π ) 2 k d q exp ( i q x ) S ( q , 0 ) G ˜ ( q )
+ 1 ( 2 π ) 2 k d q ( i q x ' ) q s S ( q , 0 ) I ˜ 0 ( q )
I ( x , 0 ) z = 1 ( 2 π ) 2 k d q exp ( i q x ) σ ( q , 0 ) G ˜ ( q ) .
I ( x , 0 ) z = 1 k ( A 2 ( x ' ) φ ( x ' ) ) ,
S ( q , 0 ) = exp ( ( μ T + N σ N π θ 0 q erf ( q θ 0 2 ) ) ) .
J ( x , s ; 0 ) = 1 π a 2 exp [ ( x 2 + 1 4 s 2 ) a 2 ] .

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