Abstract

We present a procedure that compensates for phase aberrations in digital holographic microscopy by computing a polynomial phase mask directly from the hologram. The phase-mask parameters are computed automatically without knowledge of physical values such as wave vectors, focal lengths, or distances. This method enables one to reconstruct correct and accurate phase distributions, even in the presence of strong and high-order aberrations. Examples of applications are shown for microlens imaging and for compensating for the deformations associated with a tilted thick plate. Finally we show that this method allows compensation for the curvature of the specimen, revealing its surface defects and roughness. Examples of applications are shown for microlenses and metallic sphere imaging.

© 2006 Optical Society of America

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References

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  1. E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
    [CrossRef]
  2. I. Yamaguchi, J. Kato, and H. Matsuzaki, "Measurement of surface shape and deformation by phase-shifting image digital holography," Opt. Eng. 42, 1267-1271 (2003).
    [CrossRef]
  3. U. Schnars, "Direct phase determination in hologram interferometry with use of digitally recorded holograms," J. Opt. Soc. Am. A 11, 2011-2015 (1994).
    [CrossRef]
  4. E. Cuche, P. Marquet, C. Depeursinge, "Simultaneous amplitude and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms," Appl. Opt. 38, 6994-7001 (1999).
    [CrossRef]
  5. E. Cuche and C. Depeursinge, "Method for simultaneous amplitude and quantitative phase contrast imaging by adjusting reconstruction parameters for definition of digital replica of reference wave and aberration parameters correction digitally," Patent WO200020929-A (13 April 2000).
  6. P. Ferraro, S. De 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]
  7. P. Massatsch, E. Cuche, and C. Depeursinge, "Low coherence digital holographic tomography," Novel Opt. Instrum. Biomed. Appl. 5143, 18-21 (2003).
  8. T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, "Polarization imaging by use of digital holography," Appl. Opt. 41, 27-37 (2002).
    [CrossRef] [PubMed]
  9. T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, "Jones vector imaging by use of digital holography: simulation and experimentation," Opt. Commun. 231, 137-147 (2004).
    [CrossRef]
  10. M. Liebling, "On Fresnelets, interference fringes, and digital holography," Ph.D. thesis (Swiss Federal Institute of Technology, Lausanne, Switzerland, 2004).
  11. 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]
  12. 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]
  13. J. Braat, "Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate," Appl. Opt. 36, 8459-8467 (1997).
    [CrossRef]

2004 (1)

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, "Jones vector imaging by use of digital holography: simulation and experimentation," Opt. Commun. 231, 137-147 (2004).
[CrossRef]

2003 (3)

I. Yamaguchi, J. Kato, and H. Matsuzaki, "Measurement of surface shape and deformation by phase-shifting image digital holography," Opt. Eng. 42, 1267-1271 (2003).
[CrossRef]

P. Massatsch, E. Cuche, and C. Depeursinge, "Low coherence digital holographic tomography," Novel Opt. Instrum. Biomed. Appl. 5143, 18-21 (2003).

P. Ferraro, S. De 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]

2002 (1)

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 (2)

1997 (1)

1994 (1)

Beghuin, D.

Bevilacqua, F.

Braat, J.

Colomb, T.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, "Jones vector imaging by use of digital holography: simulation and experimentation," Opt. Commun. 231, 137-147 (2004).
[CrossRef]

T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, "Polarization imaging by use of digital holography," Appl. Opt. 41, 27-37 (2002).
[CrossRef] [PubMed]

Cuche, E.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, "Jones vector imaging by use of digital holography: simulation and experimentation," Opt. Commun. 231, 137-147 (2004).
[CrossRef]

P. Massatsch, E. Cuche, and C. Depeursinge, "Low coherence digital holographic tomography," Novel Opt. Instrum. Biomed. Appl. 5143, 18-21 (2003).

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]

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

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

E. Cuche and C. Depeursinge, "Method for simultaneous amplitude and quantitative phase contrast imaging by adjusting reconstruction parameters for definition of digital replica of reference wave and aberration parameters correction digitally," Patent WO200020929-A (13 April 2000).

Dahlgren, P.

De Nicola, S.

Depeursinge, C.

P. Massatsch, E. Cuche, and C. Depeursinge, "Low coherence digital holographic tomography," Novel Opt. Instrum. Biomed. Appl. 5143, 18-21 (2003).

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]

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

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

E. Cuche and C. Depeursinge, "Method for simultaneous amplitude and quantitative phase contrast imaging by adjusting reconstruction parameters for definition of digital replica of reference wave and aberration parameters correction digitally," Patent WO200020929-A (13 April 2000).

Ferraro, P.

Finizio, A

Kato, J.

I. Yamaguchi, J. Kato, and H. Matsuzaki, "Measurement of surface shape and deformation by phase-shifting image digital holography," Opt. Eng. 42, 1267-1271 (2003).
[CrossRef]

Liebling, M.

M. Liebling, "On Fresnelets, interference fringes, and digital holography," Ph.D. thesis (Swiss Federal Institute of Technology, Lausanne, Switzerland, 2004).

Marquet, P.

Massatsch, P.

P. Massatsch, E. Cuche, and C. Depeursinge, "Low coherence digital holographic tomography," Novel Opt. Instrum. Biomed. Appl. 5143, 18-21 (2003).

Matsuzaki, H.

I. Yamaguchi, J. Kato, and H. Matsuzaki, "Measurement of surface shape and deformation by phase-shifting image digital holography," Opt. Eng. 42, 1267-1271 (2003).
[CrossRef]

Montfort, F.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, "Jones vector imaging by use of digital holography: simulation and experimentation," Opt. Commun. 231, 137-147 (2004).
[CrossRef]

Schnars, U.

Yamaguchi, I.

I. Yamaguchi, J. Kato, and H. Matsuzaki, "Measurement of surface shape and deformation by phase-shifting image digital holography," Opt. Eng. 42, 1267-1271 (2003).
[CrossRef]

Appl. Opt. (5)

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

Novel Opt. Instrum. Biomed. Appl. (1)

P. Massatsch, E. Cuche, and C. Depeursinge, "Low coherence digital holographic tomography," Novel Opt. Instrum. Biomed. Appl. 5143, 18-21 (2003).

Opt. Commun. (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]

T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, "Jones vector imaging by use of digital holography: simulation and experimentation," Opt. Commun. 231, 137-147 (2004).
[CrossRef]

Opt. Eng. (1)

I. Yamaguchi, J. Kato, and H. Matsuzaki, "Measurement of surface shape and deformation by phase-shifting image digital holography," Opt. Eng. 42, 1267-1271 (2003).
[CrossRef]

Opt. Lett. (1)

Other (2)

E. Cuche and C. Depeursinge, "Method for simultaneous amplitude and quantitative phase contrast imaging by adjusting reconstruction parameters for definition of digital replica of reference wave and aberration parameters correction digitally," Patent WO200020929-A (13 April 2000).

M. Liebling, "On Fresnelets, interference fringes, and digital holography," Ph.D. thesis (Swiss Federal Institute of Technology, Lausanne, Switzerland, 2004).

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Figures (10)

Fig. 1
Fig. 1

Schematic of the holographic microscope for reflection imaging: NF, neutral filter; λ / 2 , half-wave plate; PBS, polarizing beam splitter; BS, beam splitter; BE, beam expander; M, mirror; O , object wave; R , reference wave. Inset, detail of the off-axis geometry.

Fig. 2
Fig. 2

Schematic of the holographic microscope for transmission imaging. Inset, detail of the off-axis geometry.

Fig. 3
Fig. 3

Use of a lens or MO to match the sampling capacity of a CCD placed in the plane of the hologram with the spatial spectrum of the object.

Fig. 4
Fig. 4

Digital hologram of a U.S. Air Force test target recorded with a 40× MO in a reflection setup.

Fig. 5
Fig. 5

Automatic adjustment of a second-order phase mask. The left column presents phase images obtained for different values of the PRPs, which are evaluated by fitting a second-order 1D polynomial function on unwrapped phase data extracted along two profiles [horizontal and vertical lines in (a)–(d)]. The right column presents plots of unwrapped phase data, and fitted curves, for the horizontal direction. In the presence of strong aberrations and with zero parameter values (a), the phase unwrapping and fitting procedures fail to define directly the correct PRP values, which are reached after two iterations (c). The correct PRPs can be obtained only with the profile position located in flat regions where the absolute phase contributions of the specimen are constant, as in (d).

Fig. 6
Fig. 6

Phase reconstructions of a wafer of quartz microlenses recorded in transmission with a 10× MO (lens diameter, 230   μm) . Panels (a)–(c) illustrate a multiprofile PRP adjustment procedure. The number of extracted profiles (white lines) increases from (a) two profiles to (c) nine profiles. (d) The perspective view of the 2D unwrap of (c). The black lines on (c) indicate the position of the profile used for the shape-compensation procedure. (e), (f) The perspective views obtained by adjusting the second-order shape-compensation parameters on the centered and lower-right microlenses, respectively.

Fig. 7
Fig. 7

Phase reconstructions for increasing orders of a polynomial phase mask. The computed coefficients are (a) P 0 , 0 ; (b) adding P 1 , 0 , P 0 , 1 , P 2 , 0 ,  and  P 0 , 2 ; (c) adding P 1 , 1 ; (d) adding P α , β with α + β = 3 ; and (e) adding P 4 , 0   and   P 0 , 4 . Left column, compensation for a computed phase distribution; right column, phase distributions reconstructed from a hologram recorded with a strongly aberrated microscope. The accuracy of the aberration compensation is measured by the standard deviation (Std) of the phase distribution evaluated over the entire field of view.

Fig. 8
Fig. 8

Phase reconstructions in perspective view of a U.S. Air Force test target obtained (a) by computing a second-order phase mask with the method presented in Fig. 5, (b) by using the PRPs (cross and noncross terms up to order four) calibrated with a mirror [see Fig. 7(e)], and (c) by adding a first-order (tilt-compensation) procedure to the phase distribution of Fig. 8(b).

Fig. 9
Fig. 9

(a) Phase reconstructions of the vertex of a metallic sphere, (b) 2D unwrapped phase image, (c) digitally flattened phase image revealing surface roughness.

Fig. 10
Fig. 10

Perspective views computed by compensating for the curvature induced by the lens with (a) a third-order SCM and (b) a fourth-order SCM. (c) Phase profiles defined along the directions of the white arrows in (a) and (b).

Equations (69)

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I H ( x , y ) = | R | 2 + | O | 2 + R O * + R * O .
I H ( k , l ) = k Δ x Δ x / 2 k Δ x + Δ x / 2 l Δ y Δ y / 2 l Δ y + Δ y / 2 I H ( x , y ) d x d y ,
Ψ ( ξ , η ) = Φ ( ξ , η ) exp ( i 2 π d / λ ) i λ d exp [ i π λ d ( ξ 2 + η 2 ) ]
× R D ( x , y ) I H ( x , y ) × exp { i π λ d [ ( x ξ ) 2 + ( y η ) 2 ] } d x d y .
R D ( x , y ) = exp [ i 2 π λ ( k x x + k y y ) + i ϕ ( t ) ] ,
Φ ( ξ , η ) = exp [ i π λ D ( ξ 2 + η 2 ) ] ,
F τ [ f ( x , y ) ] ( ξ , η ) = 1 τ 2 f ( x , y ) exp { i π τ 2 [ ( x ξ ) 2 + ( y η ) 2 ] } d x d y ,
F τ [ exp ( i 2 π ν x ) f ( x ) ] ( ξ , η ) = exp ( i 2 π ν ξ ) exp ( i π ν 2 τ 2 ) × F τ [ f ( x ) ] ( ξ υ x τ 2 , η υ y τ 2 ) ,
Ψ ( ξ , η ) = i Φ ( ξ , η ) exp ( i 2 π d / λ ) F τ [ R D I H ] ( ξ , η ) ,
τ = λ d
= i Φ ( ξ , η ) exp ( i 2 π d / λ ) F τ { exp [ i 2 π λ ( k x x + k y y ) + i ϕ ( t ) ] I H } ( ξ , η ) .
Ψ ( ξ , η ) = - i Φ ( ξ , η ) exp ( i 2 π d / λ ) exp [ i 2 π λ ( k x ξ + k y η ) + i ϕ ( t ) ] exp { i π λ d [ ( k x λ ) 2 + ( k y λ ) 2 ] } F τ [ I H ] ( ξ + k x d , η + k y d ) .
ϕ ( k x , k y , t ) = ϕ ( t ) π d ( k x     2 + k y     2 ) / λ
Ψ ( ξ , η ) = - i Φ ( ξ , η ) exp ( i 2 π d / λ ) R ( ξ , η ) × F τ [ I H ] ( ξ , η ) ,
R ( ξ , η ) = exp [ i 2 π λ ( k x ξ + k y η ) + i ϕ ( k x , k y , t ) ] .
Ψ ( ξ , η ) = i exp ( i 2 π d / λ ) Γ ( ξ , η ) F τ [ I H ] ( ξ , η ) ,
Γ ( ξ , η ) = exp [ i π λ ( 2 k x ξ + 2 k y η ξ 2 + η 2 D ) + i ϕ ( k x , k y , t ) ] .
Γ ( ξ , η ) = exp [ i α = 0 H β = 0 V P α , β ξ α η β ] ,
P 0 , 0 = ϕ ,
P 1 , 0 = - 2 π λ k x ,    P 0 , 1 = 2 π λ k y ,
P 2 , 0 = P 0 , 2 = π λ D .
Ψ ( m , n ) = A Γ ( m , n ) exp [ i π λ d ( m 2 Δ ξ 2 + n 2 Δ η 2 ) ] DFT   { I H ( k , l ) exp [ i π λ d ( k 2 Δ x 2 + l 2 Δ y 2 ) ] } m , n ,
Δ ξ   =    Δ η = λ d / ( N Δ x ) .
Γ ( m , n ) = exp [ i α = 0 H β = 0 V P α , β ( m Δ ξ ) α ( n Δ η ) β ] .
Ψ ( m , n ) = A Γ ( m , n ) DFT { I H ( k , l ) exp [ i π λ d ( k 2 Δ x 2 + l 2 Δ y 2 ) ] } m , n
Γ ( m , n ) = exp ( i α = 0 H β = 0 V P α , β m α n β ) .
P 0 , 0 = ϕ ,
P 1 , 0 = - 2 π λ k x Δ ξ , P 0 , 1 = 2 π λ k y Δ η ,
P 2 , 0 = P 0 , 2 = π λ ( 1 D 1 d ) Δ ξ 2 .
Ψ ( m , n ) = Γ ( m , n ) DFT { I H ( k , l ) × exp [ i / d ( k 2 + l 2 ) ] } m , n
=     Γ ( m , n ) Ω ( m , n ) ,
Γ ( m , n ) = exp ( i α = 0 H β = 0 V P α , β m α n β ) .
Y h = a 0 + a 1 x + a 2 x 2 ,
Y v = b 0 + b 1 x + b 2 x 2 ,
Γ ( m , n ) = exp [ i ( a 1 m + b 1 n + a 2     2 m 2 + b 2     2 n 2 ) ] ,
Y h     ( i ) = a 0     ( i ) + a 1     ( i ) x + a 2     ( i ) x 2 ,
Y v     ( i ) = b 0     ( i ) + b 1     ( i ) x + b 2     ( i ) x 2 .
P j , 0           ( i ) = P j , 0           ( i 1 ) + a j     ( i ) ,
P 0 , j           ( i ) = P 0 , j           ( i 1 ) + b j     ( i ) ,
Y ( x , p ) = a 0 ( p ) + a 1 ( p ) x + a 2 ( p ) x 2 .
a j = 1 N h    p = 1 N h a j ( p ) .
Y ( k p , p ) = a 0 ( p ) + a 1 x ( k p , p ) + a 2 x 2 ( k p , p ) ,
X × A = Y ,
A = [ a 0 ( 1 ) a 0 ( p ) a 0 ( N h ) a 1 a 2 ] ,
Y = [ Y ( 1 , 1 ) Y ( n 1 , 1 )    Y ( 1 , p ) Y ( n p , p )    Y ( 1 , N h ) Y ( n N h , N h ) ]  , 
X =
[ 1 0 0 0 x ( 1 , 1 ) x 2 ( 1 , 1 ) 1 0 0 0 x ( n 1 , 1 ) x 2 ( n 1 , 1 ) 0 1 0 0 x ( 1 , 2 ) x 2 ( 1 , 2 ) 0 1 0 0 x ( n 2 , 2 ) x 2 ( n 2 , 2 ) 0 1 0 0 x ( 1 , p ) x 2 ( 1 , p ) 0 1 0 0 x ( n p , p ) x 2 ( n p , p ) 0 0 1 x ( 1 , N h ) x 2 ( 1 , N h ) 0 0 1 x ( n N h , N h ) x 2 ( n N h , N h ) ] .
Y h = a 0 + a 1 x + a 2 x 2 + + a α x α + + a H x H ,
Y v = b 0 + b 1 x + b 2 x 2 + + b β x β + + b V x V ,
A = [ a 0 ( 1 ) a 0 ( p ) a 0 ( N h ) a 1 a 2 a H ] .
ϕ + 1 ( m ) = P 1 , 0 m + P 0 , 1 ( m + τ ) + P 2 , 0 m 2 + P 0 , 2 ( m + τ ) 2 + P 1 , 1 m ( m + τ )
= ( P 2 , 0 + P 0 , 2 + P 1 , 1 ) m 2 + ( P 1 , 0 + P 0 , 1 + 2 P 0 , 2 τ + P 1 , 1 P 0 , 2 ) m + ( P 0 , 1 τ + P 0 , 2 τ 2 )
= a 2     ( + 1 ) m 2 + a 1     ( + 1 ) m + a 0     ( + 1 ) .
ϕ 1 ( m ) = ( P 2 , 0 + P 0 , 2 P 1 , 1 ) m 2 + = a 2     ( 1 ) m 2 + a 1     ( 1 ) m + a 0     ( 1 ) ,
P 1 , 1 = ( a 2     ( + 1 ) a 2     ( 1 ) ) / 2 ,
ϕ + 1 ( m ) = ( P 3 , 0 + P 0 , 3 + P 1 , 2 + P 2 , 1 ) m 3 +
= a 3     ( + 1 ) m 3 + ,
ϕ 1 ( m ) = ( P 3 , 0 P 0 , 3 + P 1 , 2 P 2 , 1 ) m 3 +
= a 3     ( 1 ) m 3 + .
ϕ + 2 ( m ) = ( P 3 , 0 + 8 P 0 , 3 + 4 P 1 , 2 + 2 P 2 , 1 ) m 3 +
= a 3     ( + 2 ) m 3 + ,
ϕ 2 ( m ) = ( P 3 , 0 8 P 0 , 3 + 4 P 1 , 2 2 P 2 , 1 ) m 3 +
= a 3     ( - 2 ) m 3 + .
P 1 , 2 = ( a 3     ( + 2 ) + a 3     ( 2 ) a 3     ( + 1 ) a 3     ( 1 ) ) / 6 ,
P 2 , 1 = 2 3 ( a 3     ( + 1 ) a 3     ( 1 ) - a 3     ( + 2 ) a 3     ( 2 ) 8 ) .
ϕ ( m , n ) = α + β = 1 α + β = 4 C α , β m α n β ,
Ψ SCM ( m , n ) = Γ SCM ( m , n ) Ψ ( m , n ) ,
Γ SCM ( m , n ) = exp ( i α =0 H SCM β = 0 V SCM P α , β            SCM m α n β ) ,
Γ SCM ( m , n ) = exp [ i ( P 0 , 0           SCM + P 1 , 0             SCM m + P 0 , 1           SCM n + P 2 , 0          SCM m 2 + P 0 , 2         SCM n 2 ) ] .

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