Abstract

Introducing a microscope objective in an interferometric setup induces a phase curvature on the resulting wavefront. In digital holography, the compensation of this curvature is often done by introducing an identical curvature in the reference arm and the hologram is then processed using a plane wave in the reconstruction. This physical compensation can be avoided, and several numerical methods exist to retrieve phase contrast images in which the microscope curvature is compensated. Usually, a digital array of complex numbers is introduced in the reconstruction process to perform this curvature correction. Different corrections are discussed in terms of their influence on the reconstructed image size and location in space. The results are presented according to two different expressions of the Fresnel transform, the single Fourier transform and convolution approaches, used to propagate the reconstructed wavefront from the hologram plane to the final image plane.

© 2006 Optical Society of America

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References

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  1. U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
    [CrossRef]
  2. E. Cuche, P. Marquet, and C. Depeursinge, "Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms," Appl. Opt. 38, 6994-7001 (1999).
    [CrossRef]
  3. P. Ferraro, S. D. Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
    [CrossRef] [PubMed]
  4. T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, "Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation," Appl. Opt. 45, 851-863 (2006).
    [CrossRef] [PubMed]
  5. T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, "Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram," Opt. Express 14, 4300-4306 (2006).
    [CrossRef] [PubMed]
  6. T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, "Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy," J. Opt. Soc. Am. A 23 (2006) (to be published).
  7. D.Malacara, ed., Optical Shop Testing, Wiley Series in Pure and Applied Optics (Wiley, 1992).
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  9. E. Cuche, P. Marquet, and C. Depeursinge, "Aperture apodization using cubic spline interpolation: application in digital holographic microscopy," Opt. Commun. 182, 59-69 (2000).
    [CrossRef]
  10. E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
    [CrossRef]
  11. J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
    [CrossRef]
  12. T. H. Demetrakopoulos and R. Mittra, "Digital and optical reconstruction of images from suboptical diffraction patterns," Appl. Opt. 13, 665-670 (1974).
    [CrossRef] [PubMed]

2006 (2)

2003 (1)

2002 (1)

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

2000 (2)

E. Cuche, P. Marquet, and C. Depeursinge, "Aperture apodization using cubic spline interpolation: application in digital holographic microscopy," Opt. Commun. 182, 59-69 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

1999 (1)

1974 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Aspert, N.

Bourquin, S.

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, "Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy," J. Opt. Soc. Am. A 23 (2006) (to be published).

Charrière, F.

Colomb, T.

Coppola, G.

Cuche, E.

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, "Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation," Appl. Opt. 45, 851-863 (2006).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, "Aperture apodization using cubic spline interpolation: application in digital holographic microscopy," Opt. Commun. 182, 59-69 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, "Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms," Appl. Opt. 38, 6994-7001 (1999).
[CrossRef]

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, "Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy," J. Opt. Soc. Am. A 23 (2006) (to be published).

Demetrakopoulos, T. H.

Depeursinge, C.

Ferraro, P.

Finizio, A.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

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

Grilli, S.

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Kühn, J.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Magro, C.

Marian, A.

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, "Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy," J. Opt. Soc. Am. A 23 (2006) (to be published).

Marquet, P.

Mittra, R.

Montfort, F.

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, "Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation," Appl. Opt. 45, 851-863 (2006).
[CrossRef] [PubMed]

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, "Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy," J. Opt. Soc. Am. A 23 (2006) (to be published).

Nicola, S. D.

Pierattini, G.

Schnars, U.

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Meas. Sci. Technol. (1)

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Opt. Commun. (1)

E. Cuche, P. Marquet, and C. Depeursinge, "Aperture apodization using cubic spline interpolation: application in digital holographic microscopy," Opt. Commun. 182, 59-69 (2000).
[CrossRef]

Opt. Express (1)

Other (3)

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, "Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy," J. Opt. Soc. Am. A 23 (2006) (to be published).

D.Malacara, ed., Optical Shop Testing, Wiley Series in Pure and Applied Optics (Wiley, 1992).

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

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Figures (6)

Fig. 1
Fig. 1

Standard configuration in holographic microscopy: The CCD defining the hologram plane is placed in front of the image obtained though the microscope objective (MO). d is the reconstruction distance.

Fig. 2
Fig. 2

Schema of the used notations.

Fig. 3
Fig. 3

(a) Phase reconstruction of the microlens recorded in a transmission DHM setup (diameter 240 μ m , height 21.15 μ m ), (b) two-dimensional unwrap of (a), (c) perspective representation of (b).

Fig. 4
Fig. 4

(a) Reconstruction in the image plane approach: The illumination beam is a plane wave propagating along the optical axis. The phase curvature is compensated in the image plane. The reconstructed image is not the image of the object through the MO (shown by a dashed line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the image plane in convolution and FT formulations, respectively.

Fig. 5
Fig. 5

(a) Reconstruction in the mixed approach: The illumination beam is a plane wave propagating along the reference wave axis. The first-order DPM is applied in the hologram plane and a second of higher orders in the image plane. The reconstructed image is not the image of the object through the MO (shown by a dashed line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the image plane in convolution and FT formulations, respectively. The white lines define the diameter of the microlens diameter to be compared with Fig. 6

Fig. 6
Fig. 6

(a) Reconstruction in the hologram plane approach: The illumination beam is a replica of the reference beam. The phase curvature is compensated in the hologram plane. The reconstructed image is not the image of the object through the MO (shown by a dashed line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the image plane in convolution and FT formulations, respectively. The white lines define the diameter of the microlens reconstructed with the mixed approach (Fig. 5).

Tables (1)

Tables Icon

Table 1 Summary of the Different Reconstructed Image Properties

Equations (64)

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I H ( x , y ) = O + R 2 = O 2 + R 2 + R O * + R * O ,
Ψ I ( x , y ) = exp ( i k d ) i λ d Ψ H ( ξ , η ) exp { i π λ d [ ( ξ x ) 2 + ( η y ) 2 ] } d ξ d η ,
F τ [ f ( x , y ) ] = 1 τ 2 f ( ξ , η ) exp { i π λ d [ ( ξ x ) 2 + ( η y ) 2 ] } d ξ d η .
Ψ I ( x , y ) = i exp ( i k d ) F λ d [ Ψ H ( x , y ) ] .
Ψ I ( x , y ) = exp ( i k d ) i λ d exp [ i π λ d ( x 2 + y 2 ) ] F T { Ψ H ( ξ , η ) exp [ i π λ d ( ξ 2 + η 2 ) ] } .
T x = λ d N pts T ξ , T y = λ d N pts T η .
Ψ I ( x , y ) = exp ( i k d ) i λ d [ Ψ H ( x , y ) ] exp [ i π λ d ( x 2 + y 2 ) ] .
Υ I ( x , y ) = Γ I i exp ( i k d ) F λ d [ Γ H Ψ H ( x , y ) ] .
Ψ id H = Γ id H R * O = R R * O = O ,
Ψ id I = i exp ( i k d id ) F λ d id [ Ψ id H ] ,
= i exp ( i k d id ) F λ d id [ O ] ,
S R = [ S R x , S R y , ( h r 2 S R x 2 S R y 2 ) 1 2 ] ,
S O = [ 0 , 0 , h o ] ,
R ( x , y ) = exp { i π λ h r [ ( x S R x ) 2 + ( y S R y ) 2 ] } .
O 0 ( x , y ) = exp [ i π λ h o ( x 2 + y 2 ) ] .
Γ id I = { F λ d id [ Ψ id , 0 H ] } * ,
= { F λ d id [ O 0 ] } * .
Υ id I = Γ id I Ψ id I ,
= { F λ d id [ O 0 ] } * F λ d id [ O ] .
Γ id I ( x , y ) = exp [ i π λ ( h o + d id ) ( x 2 + y 2 ) ] ,
Υ id I ( x , y ) = i exp ( i k d id ) exp [ i π λ ( h o + d id ) ( x 2 + y 2 ) ] F λ d id [ Ψ id H ] .
Ψ H = Γ H R * O .
Ψ I = i exp ( i k d ) F λ d [ Ψ H ] ,
= i exp ( i k d ) F λ d [ Γ H R * O ] .
Υ I = Γ I Ψ I ,
= i exp ( i k d ) Γ I F λ d [ Γ H R * O ] .
Γ H = exp { i π λ h d [ ( x S D x ) 2 + ( y S D y ) 2 ] } ,
S D = [ S D x , S D y , ( h d 2 S D x 2 S D y 2 ) 1 2 ] .
Γ I ( x , y ) = exp ( i π [ ( S D x S R x ) 2 + ( S D y S R y ) 2 ] λ ( h d h r ) ) × exp ( i π λ ( h + d M ) { 1 M 2 [ x h ( S R x h r S D x h d ) ] 2 + 1 M 2 [ y h ( S R y h r S D y h d ) ] 2 } ) × exp ( i π λ ( h 0 + d M ) { 1 M 2 [ x d ( S R x h r S D x h d ) ] 2 + 1 M 2 [ y d ( S R y h r S D y h d ) ] 2 } ) ,
h = h d h r h d h r , M = h d h .
Υ I ( x , y ) = Γ I ( x , y ) Ψ I ( x , y ) = i exp ( i k d ) 1 M F λ d M ( x , y ) exp ( i π λ ( h + d M ) { 1 M 2 [ x h ( S R x h r S D x h d ) ] 2 + 1 M 2 [ y h ( S R y h r S D y h d ) ] 2 } ) = exp [ i k ( d d M ) ] 1 M Υ id H ( x , y ) ,
sin θ = S R x 2 + S R y 2 h r S D x 2 + S D y 2 h d .
Γ H = R S D = S R , h d = h r lim h d h = h r , M = 1 ,
lim Γ H R Ψ I ( x , y ) = Ψ id I ( x , y ) ,
lim Γ H R Γ I ( x , y ) = exp [ i π ( x 2 + y 2 ) λ ( h o + d ) ] ,
lim Γ H R Υ I ( x , y ) = i exp ( i k d ) exp [ i π ( x 2 + y 2 ) λ ( h o + d ) ] Ψ id I ( x , y ) .
Γ i I = { F λ d i [ R * O 0 ] } * .
Υ i I = i exp ( i k d i ) { F λ d i [ R * O 0 ] } * F λ d i [ R * O ] .
S D = 0 , lim Γ H 1 h d = , S D h d = 0 ,
lim h d h = h r , lim h d M = h r d h r = M i .
Γ i I ( x , y ) = lim Γ H 1 Γ I ( x , y ) = exp { i π λ ( h r + d i M i ) [ 1 M i 2 ( x S R x ) 2 + 1 M i 2 ( y S R y ) 2 ] } exp { i π λ ( h 0 + d i M i ) [ 1 M i 2 ( x d i h r S R x ) 2 + 1 M i 2 ( y d i h r S R y ) 2 ] } ,
Υ i I ( x , y ) = lim Γ H 1 Υ I ( x , y ) = i exp [ i k ( d i d i M ) ] 1 M i Υ id I ( x d i h r S R x M i , y d i h r S R x M i ) .
sin θ = S R x 2 + S R y 2 h r .
L shift = d i h r ( S R x 2 + S R y 2 ) 1 2 = d i sin θ .
Γ h H = R O 0 * .
Ψ h H = Γ h H R * O = O 0 * Ψ id H = Υ h H .
Υ h I = i exp ( i k d h ) F λ d h [ Ψ h H ] = i exp ( i k d h ) F λ d h [ O 0 * O ] .
Υ h I ( x , y ) = i exp ( i k d h ) F λ d h { exp [ i π λ h o ( x 2 + y 2 ) ] Ψ id H ( x , y ) }
= exp [ i k ( d h d h M h ) ] 1 M h Υ id I ( x M h , y M h ) .
1 h o = 1 d id + 1 d h , M h = d h d id ,
Γ m H = P W m H = exp [ i 2 π λ ( S R x h r x + S R y h r y ) ] .
Γ m I ( x , y ) = { P W m H ( x , y ) F λ d m [ R * O ( x + d m h r S R x , y + d m h r S R y ) ] } * ,
Υ m I ( x , y ) = i exp ( i k d m ) Γ i I ( x + d m h r S R x , y + d m h r S R y ) F λ d m [ R * O ( x + d m h r S R x , y + d m h r S R y ) ] .
Γ H = P W m H S D , h d , S D h d = S R h r ,
lim h d h = h r , lim h d M = h r d h r = M i .
Γ m I ( x , y ) = Γ I ( x , y ) Γ H = P W m H = exp [ i π λ ( h r + d m M i ) ( x 2 M i 2 + y 2 M i 2 ) ] × exp [ i π λ ( h 0 + d m M i ) ( x 2 M i 2 + y 2 M i 2 ) ] ,
Υ m I ( x , y ) = Υ I ( x , y ) Γ H = P W m H
= exp [ i k ( d m d m M i ) ] 1 M i Υ id I ( x M i , y M i ) .
Υ I ( x , y ) = exp [ i k ( d d M ) ] 1 M Υ id H ( x a M , y b M ) .
M = d d id .
Υ I ( n , m ) = exp [ i k ( d d M ) ] 1 M Υ id H ( ( n a ξ ) λ d N pts T ξ 1 M , ( m b η ) λ d N pts T η 1 M ) .
d = M d id .
Υ I ( n , m ) = exp [ i k ( d d M ) ] 1 M Υ id H ( ( n a ξ ) λ d id N pts T ξ , ( m b η ) λ d id N pts T η ) = Υ id I ( n a ξ , m b η ) .
Υ I ( n , m ) = exp [ i k ( d d M ) ] 1 M Υ id H ( ( n a ξ ) T ξ M , ( m b η ) T η M ) ,

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