Abstract

In this paper, we present the theory of three-dimensional (3D) imaging using partially coherent light under the nonparaxial condition. Using the linear system approach, we derive the image intensity in terms of the 3D nonparaxial transmission cross coefficient (TCC) and the transmission function defined in this paper. We present that the 3D TCC can be calculated by multiple applications of the 3D fast Fourier transform instead of the six-dimensional integral in the original formula. Using the simplified formula, we simulate phase contrast and Nomarski differential interference contrast (DIC) imaging of a transparent 3D object. Within our knowledge, the 3D model for the DIC based on the 3D nonparaxial TCC is the most rigorous approach that has been suggested. It demonstrates clearly the optical sectioning effect of DIC.

© 2011 Optical Society of America

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References

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2009 (2)

2008 (1)

2007 (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Meth. 4, 717–719 (2007).
[CrossRef]

1999 (1)

1994 (1)

1992 (1)

C. J. Cogswell and C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

1989 (1)

1985 (1)

1984 (2)

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

N. Streibl, “Depth transfer by an imaging system,” J. Mod. Opt. 31, 1233–1241 (1984).
[CrossRef]

1975 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1967 (2)

E. Brigham and R. Morrow, “The fast Fourier transform,” IEEE Spectrum 4, 63–70 (1967).
[CrossRef]

B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–65 (1967).
[CrossRef]

1966 (1)

1964 (1)

1955 (1)

G. Nomarski, “Differential microinterferometer with polarized waves,” J. Phys. Radium 16, 9–13 (1955).

Badizadegan, K.

Brigham, E.

E. Brigham and R. Morrow, “The fast Fourier transform,” IEEE Spectrum 4, 63–70 (1967).
[CrossRef]

Choi, W.

Cogswell, C. J.

C. J. Cogswell and C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

Conchello, J.

Dasari, R.

Fang-Yen, C.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. Dasari, and M. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Meth. 4, 717–719 (2007).
[CrossRef]

Feld, M.

Frieden, B. R.

Goodman, J.

J. Goodman, Statistical Optics (Wiley, 1984).

J. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

Gu, M.

Kalashnikov, M.

Kawata, S.

Kawata, Y.

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Meth. 4, 717–719 (2007).
[CrossRef]

Mao, X.

Marathay, A.

McCutchen, C.

Mehta, S. B.

Morrow, R.

E. Brigham and R. Morrow, “The fast Fourier transform,” IEEE Spectrum 4, 63–70 (1967).
[CrossRef]

Nomarski, G.

G. Nomarski, “Differential microinterferometer with polarized waves,” J. Phys. Radium 16, 9–13 (1955).

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Meth. 4, 717–719 (2007).
[CrossRef]

Preza, C.

Sheppard, C. J. R.

Snyder, D.

Streibl, N.

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[CrossRef]

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

N. Streibl, “Depth transfer by an imaging system,” J. Mod. Opt. 31, 1233–1241 (1984).
[CrossRef]

Sung, Y.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Yu, C.

IEEE Spectrum (1)

E. Brigham and R. Morrow, “The fast Fourier transform,” IEEE Spectrum 4, 63–70 (1967).
[CrossRef]

J. Microsc. (1)

C. J. Cogswell and C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81–101 (1992).
[CrossRef]

J. Mod. Opt. (1)

N. Streibl, “Depth transfer by an imaging system,” J. Mod. Opt. 31, 1233–1241 (1984).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

J. Phys. Radium (1)

G. Nomarski, “Differential microinterferometer with polarized waves,” J. Phys. Radium 16, 9–13 (1955).

Nat. Meth. (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Meth. 4, 717–719 (2007).
[CrossRef]

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Express (3)

Optik (1)

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

Other (2)

J. Goodman, Statistical Optics (Wiley, 1984).

J. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

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Figures (7)

Fig. 1
Fig. 1

Optical transfer of a point source.

Fig. 2
Fig. 2

(a) Modified 3D Shepp–Logan phantom, the magnitude of which is adjusted to the refractive index of usual biological samples. (b), (c) Horizontal and vertical cross sections, respectively. The scale bar is 5 μm .

Fig. 3
Fig. 3

C point ( p 1 ; p 2 ) in p 2 space for (a)  ( p 1 x , p 1 y , p 1 z ) = ( 0.46 , 0 , 0 ) , (b)  ( p 1 x , p 1 y , p 1 z ) = ( 0.68 , 0 , 0 ) , (c)  ( p 1 x , p 1 y , p 1 z ) = ( 1.84 , 1.84 , 1.84 ) . Unit of frequency is μm 1 .

Fig. 4
Fig. 4

Horizontal cross sections of bright field image reconstructed using (a) line and (b) point intersection terms (after subtracting background). x is horizontal and y is vertical.

Fig. 5
Fig. 5

2D pupil functions used for the calculation of the phase contrast in Figs. 6b, 6e: (a) condenser, (b) objective pupil function. The 3D pupil function is calculated by projecting the 2D pupil function onto the Ewald sphere. Dark gray region in (a) is opaque, and the colored ring in (b) induces phase retardation with some absorption of transmitted light.

Fig. 6
Fig. 6

Comparison of cross sections imaged by (a), (d) bright field, (b), (e) phase contrast, and (c), (f) Nomarski DIC. (a)–(c) Horizontal cross sections, (d)–(f) vertical cross sections. For phase contrast, the condenser and the objective pupil functions in Fig. 2 are used. For Nomarski DIC, shearing is given in the X direction with 2 Δ = λ / 4 NA , 2 ϕ = π / 2 . The optical axis is in the Z direction and the scale bar is 5 μm .

Fig. 7
Fig. 7

Cross sections of the effective transfer function for Nomarski DIC ( 2 Δ = λ / 4 NA , 2 ϕ = π / 2 ): (a)  U V cross section, (b)  U W cross section. ( U , V , W ) is the spatial frequency corresponding to the spatial coordinate ( X , Y , Z ) , in which X is the shearing direction and Z is the optical axis.

Equations (25)

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( 2 + k 0 2 ) G 0 ( x ) = δ ( x ) ,
U ( r ) = 1 i 4 π λ Ω A ( q ) e i 2 π q · r d Ω ,
U ˜ ( p ) = 1 i 4 π λ P obj ( u , v ) δ ( | p | 1 ) ,
u scat ( x ) = u in ( x ) f ( x ) U ( x x ) d 3 x ,
f ( x ) = k 0 2 ( n 0 2 n ( x ) 2 ) ,
u out ( r ) = u in ( r ) T ( r ) H ( r r ) d 3 r ,
J out ( r 1 ; r 2 ) = u out ( r 1 , t ) u out * ( r 2 , t ) .
J out ( r 1 ; r 2 ) = T ( r 1 ) T * ( r 2 ) J in ( r 1 ; r 2 ) H ( r 1 r 1 ) H * ( r 2 r 2 ) d 3 r d 3 r 2 ,
J in ( r 1 ; r 2 ) J in ( Δ r ) = J ˜ in ( p ) exp [ i 2 π ( p · Δ r ) ] d 3 p ,
J ˜ out ( p 1 ; p 2 ) J out ( r 1 ; r 2 ) exp [ i 2 π ( p 1 · r 1 + p 2 · r 2 ) ] d 3 r 1 d 3 r 2 .
J ˜ out ( p 1 ; p 2 ) = H ˜ ( p 1 ) H ˜ * ( p 2 ) J ˜ in ( p ) T ˜ ( p 1 p ) T ˜ * ( p 2 p ) d 3 p ,
I img ( r ) = J out ( r ; r ) = J ˜ out ( p 1 ; p 2 ) exp [ i 2 π ( p 1 + p 2 ) · r ] d 3 p 1 d 3 p 2 .
I img ( r ) = C ( p 1 ; p 2 ) T ˜ ( p 1 ) T ˜ * ( p 2 ) exp [ i 2 π ( p 1 p 2 ) · r ] d 3 p 1 d 3 p 2 ,
C ( p 1 ; 0 ) = ( P cond 3 D P obj 3 D * P obj 3 D ) ( p 1 ) C 1 ( p 1 ) ,
C ( 0 ; p 2 ) = C ( p 2 ; 0 ) * = C 1 ( p 2 ) * ,
( f * g ) ( p ) f * ( q ) g ( p + q ) d 3 q .
C 1 ( p ) = 4 π K arccos [ 1 M ( 2 cos α | w | + 1 ) ] ,
I img ( r ) I 0 + 2 Re [ α C 1 ( p ) T ˜ ( p ) exp [ i 2 π ( p · r ) ] d 3 p ] ,
I img ( r ) = I 0 + 2 Re [ α F 1 { C 1 ( p ) T ˜ ( p ) } ] ,
| p 1 | | p 2 | | p 1 p 2 | 2 r | p 1 × p 2 | ,
I img ( r ) k res 3 p 1 T ˜ ( p 1 ) { C point ( p 1 ; p 1 + p ) T ˜ * ( p 1 + p ) exp [ i 2 π ( p · r ) ] } d 3 p ,
T N ( r ) = T ( r + Δ e Δ ) exp ( i ϕ ) + T ( r Δ e Δ ) exp ( i ϕ ) ,
T ˜ N ( p ) = 2 T ˜ ( p ) cos ( 2 π m Δ ϕ ) .
I N ( r ) = I 0 + 2 Re [ α F 1 { C N ( p ) T ˜ ( p ) } ] ,
C K ( p ) = ( P cond 3 D P K 3 D * P K 3 D ) ( p ) ,

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