Abstract

The synthesis of a multiple-peak spatial degree of coherence is demonstrated. This degree of coherence enables us to scan different sample points on different altitudes simultaneously and thus decreases the acquisition time. The multipeak degree of coherence is also used for imaging through an absorbing layer with different thicknesses or different indices of refraction along the layer. All our experiments are performed with a quasi-monochromatic light source. Therefore problems of dispersion and inhomogeneous absorption are avoided. Our experimental results are presented.

© 2005 Optical Society of America

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References

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  1. Y. Teramura, K. Suzuki, M. Suzuki, F. Kannari, “Low coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974–5980 (1999).
    [CrossRef]
  2. J. Rosen, A. Yariv, “General theorem of spatial coherence: application to three-dimensional imaging,” J. Opt. Soc. Am. A 13, 2091–2095 (1996).
    [CrossRef]
  3. J. Rosen, M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
    [CrossRef]
  4. W. Wang, H. Kozaki, J. Rosen, M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt. 41, 1962–1971 (2002).
    [CrossRef] [PubMed]
  5. M. Gokhler, Z. Duan, J. Rosen, M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830–836 (2003).
    [CrossRef]
  6. C. Dunsby, P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D 36, R207–R227 (2003).
    [CrossRef]
  7. E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
    [CrossRef] [PubMed]
  8. C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
    [CrossRef]
  9. V. P. Ryabukho, D. V. Lyakin, M. I. Lobachev, “The effects of temporal and longitudinal spatial coherence in a disbalanced-arm interferometer,” Tech. Phys. Lett. 30, 64–67 (2004).
    [CrossRef]
  10. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]

2004 (1)

V. P. Ryabukho, D. V. Lyakin, M. I. Lobachev, “The effects of temporal and longitudinal spatial coherence in a disbalanced-arm interferometer,” Tech. Phys. Lett. 30, 64–67 (2004).
[CrossRef]

2003 (3)

M. Gokhler, Z. Duan, J. Rosen, M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830–836 (2003).
[CrossRef]

C. Dunsby, P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D 36, R207–R227 (2003).
[CrossRef]

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

2002 (1)

2000 (1)

J. Rosen, M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
[CrossRef]

1999 (1)

Y. Teramura, K. Suzuki, M. Suzuki, F. Kannari, “Low coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974–5980 (1999).
[CrossRef]

1996 (1)

1991 (1)

E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
[CrossRef] [PubMed]

1964 (1)

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

Ansari, Z.

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

Chen, C.

E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
[CrossRef] [PubMed]

Chen, Y.

E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
[CrossRef] [PubMed]

Dilworth, D.

E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
[CrossRef] [PubMed]

Duan, Z.

M. Gokhler, Z. Duan, J. Rosen, M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830–836 (2003).
[CrossRef]

Dunsby, C.

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

C. Dunsby, P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D 36, R207–R227 (2003).
[CrossRef]

French, P. M. W.

C. Dunsby, P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D 36, R207–R227 (2003).
[CrossRef]

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

Gokhler, M.

M. Gokhler, Z. Duan, J. Rosen, M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830–836 (2003).
[CrossRef]

Gu, Y.

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

Kannari, F.

Y. Teramura, K. Suzuki, M. Suzuki, F. Kannari, “Low coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974–5980 (1999).
[CrossRef]

Kozaki, H.

Leith, E. N.

E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
[CrossRef] [PubMed]

Lobachev, M. I.

V. P. Ryabukho, D. V. Lyakin, M. I. Lobachev, “The effects of temporal and longitudinal spatial coherence in a disbalanced-arm interferometer,” Tech. Phys. Lett. 30, 64–67 (2004).
[CrossRef]

Lopez, J.

E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
[CrossRef] [PubMed]

Lyakin, D. V.

V. P. Ryabukho, D. V. Lyakin, M. I. Lobachev, “The effects of temporal and longitudinal spatial coherence in a disbalanced-arm interferometer,” Tech. Phys. Lett. 30, 64–67 (2004).
[CrossRef]

McCutchen, C. W.

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

Meloch, M. R.

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

Nolte, D. D.

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

Peng, L.

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

Rosen, J.

Ryabukho, V. P.

V. P. Ryabukho, D. V. Lyakin, M. I. Lobachev, “The effects of temporal and longitudinal spatial coherence in a disbalanced-arm interferometer,” Tech. Phys. Lett. 30, 64–67 (2004).
[CrossRef]

Sun, P.-C.

E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
[CrossRef] [PubMed]

Suzuki, K.

Y. Teramura, K. Suzuki, M. Suzuki, F. Kannari, “Low coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974–5980 (1999).
[CrossRef]

Suzuki, M.

Y. Teramura, K. Suzuki, M. Suzuki, F. Kannari, “Low coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974–5980 (1999).
[CrossRef]

Takeda, M.

M. Gokhler, Z. Duan, J. Rosen, M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830–836 (2003).
[CrossRef]

W. Wang, H. Kozaki, J. Rosen, M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt. 41, 1962–1971 (2002).
[CrossRef] [PubMed]

J. Rosen, M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
[CrossRef]

Teramura, Y.

Y. Teramura, K. Suzuki, M. Suzuki, F. Kannari, “Low coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974–5980 (1999).
[CrossRef]

Wang, W.

Yariv, A.

Yu, P.

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

Appl. Opt. (2)

Y. Teramura, K. Suzuki, M. Suzuki, F. Kannari, “Low coherence interferometry with synthesis of coherence function,” Appl. Opt. 38, 5974–5980 (1999).
[CrossRef]

J. Rosen, M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

C. Dunsby, P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D 36, R207–R227 (2003).
[CrossRef]

Opt. Eng. (1)

M. Gokhler, Z. Duan, J. Rosen, M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830–836 (2003).
[CrossRef]

Opt. Lett. (1)

E. N. Leith, C. Chen, Y. Chen, J. Lopez, P.-C. Sun, D. Dilworth, “Imaging through scattering media using spatial incoherence techniques,” Opt. Lett. 16, 1820–1822 (1991).
[CrossRef] [PubMed]

Opt. Commun. (1)

C. Dunsby, Y. Gu, Z. Ansari, P. M. W. French, L. Peng, P. Yu, M. R. Meloch, D. D. Nolte, “High-speed depth-sectioned wide-field imaging using low-coherence photorefractive holographic microscopy,” Opt. Commun. 219, 87–99 (2003).
[CrossRef]

Tech. Phys. Lett. (1)

V. P. Ryabukho, D. V. Lyakin, M. I. Lobachev, “The effects of temporal and longitudinal spatial coherence in a disbalanced-arm interferometer,” Tech. Phys. Lett. 30, 64–67 (2004).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic of the interferometric system used for the optical spatial coherence profilometry. (b) Structure of the sample arm in the system shown in (a) for the experiment of imaging through the absorbing media. NDF, neutral-density filter.

Fig. 2
Fig. 2

Shift of the focal point of lens L1 owing to parallel dielectric media.

Fig. 3
Fig. 3

(a) Set of FZPs with different values of γn for different angular segments. (b) Schematics of the complex degree of coherence in relation to two mirror positions. (c) Absolute values of the subtraction of two fringe images taken with βm equal 0 and π.

Fig. 4
Fig. 4

(a) Set of FZPs with different values of γn for different angular segments. (b) Schematics of the complex degree of coherence in relation to three mirror positions. (c) Absolute value of the subtraction between two fringe images taken with βm equal 0 and π.

Fig. 5
Fig. 5

(a) Image hologram recorded by the CCD when the path difference between reference and sample mirrors was Δz = 2.3 mm. (b) Digitally reconstructed image from the hologram shown in (a). (c) Image hologram recorded by the CCD when the path difference between reference and sample mirrors was Δz = 4 mm. (d) Digitally reconstructed image from the hologram shown in (c).

Fig. 6
Fig. 6

(a) Set of FZP’s with different values of γn for different angular segments. (b) Schematics of the complex degree of coherence in relation to the structure of the sample mirror covered by absorbing media. (c) Images reconstructed digitally from the holograms recorded by the CCD when the interferometer was illuminated by the FZPs shown in (a).

Tables (2)

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Table 1 Fringe Visibilities Measured in the Experiment with Two Sample Mirrors

Tables Icon

Table 2 Fringe Visibilities Measured in the Experiment with Three Sample Mirrors

Equations (11)

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μ ( Δ z ) = I s ( x s , y s ) exp [ j 2 π Δ z λ f 2 ( x s 2 + y s 2 ) ] d x s d y s I s ( x s , y s ) d x s d y s ,
I s ( x s , y s ) = 1 + n = 0 N 1 [ rect ( θ n π / N π / N ) + rect ( θ n π / N π π / N ) ] cos { π γ n ( x s 2 + y s 2 ) + β n } , x s 2 + y x 2 R ,
μ ( Δ z ) sinc ( Δ z R 2 2 λ f 2 ) * [ 2 δ ( Δ z ) + 1 N n = 1 N exp ( j β n ) × δ ( Δ z + γ n λ f 2 2 ) + exp ( j β n ) δ ( Δ z γ n λ f 2 2 ) ] ,
Δ z n = γ n λ f 2 / 2 .
Δ z min = 2 λ f 2 / R 2 .
z ¯ = h / tan α = d ( tan α tan β ) / tan α = d ( n 1 ) n ,
u ( x , y , z ) = u s ( x s , y s ) j λ f exp [ j 2 π ( z + 2 f ) λ j 2 π λ f × ( x s x + y s y ) + j π z λ f 2 ( x s 2 + y s 2 ) ] ,
u S ( x , y , L + Δ z ) = u s ( x x , y s ) j λ f × exp [ j 2 π L + 2 Δ z + 2 d ( n 1 ) + 2 f λ j 2 λ λ f ( x s x + y s y ) + j π ( L + 2 Δ z 2 z ¯ ) λ f 2 ( x s 2 + y s 2 ) ] .
I ( x , y , L ) = | u s ( x , y , L + 2 Δ z ) + u ( x , y , L ) | 2 d x s d y s = A ( 1 + | μ ( Δ x , Δ y , Δ z ) | cos { 2 π λ f ( x Δ x + y Δ y ) + 4 π [ Δ z + d ( n 1 ) ] λ + ϕ ( Δ x , Δ y , 2 Δ z ) + π ( L + 2 Δ z 2 z ¯ ) λ f 2 × ( Δ x 2 + Δ y 2 ) } ) ,
μ ( Δ z , Δ x , Δ y ) = 1 I s ( x s , y s ) d x s d y s × I s ( x s , y s ) exp [ j 2 π ( Δ z z ¯ ) λ f 2 × ( x s 2 + y s 2 ) j 2 π ( L + 2 Δ z 2 z ¯ ) λ f 2 × ( x s Δ x + y s Δ y ) ] d x s d y s .
μ ( Δ z ) = 1 I s ( x s , y s ) d x s d y s × I s ( x s , y s ) exp [ j 2 π ( Δ z z ¯ ) λ f 2 × ( x s 2 + y s 2 ) ] d x s d y s .

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