Abstract

We introduce a quantitative phase imaging method for homogeneous objects with a bright field transmission microscope by using an amplitude mask and a digitalprocessing algorithm. A known amplitude pattern is imaged on the sample plane containing a thick phase object by placing an amplitude mask in the field diaphragm of the microscope. The phase object distorts the amplitude pattern according to its optical path length (OPL) profile, and the distorted pattern is recorded in a CCD detector. A digitalprocessing algorithm then estimates the object's quantitative OPL profile based on a closed form analytical solution, which is derived using a ray optics model for objects with small OPL gradients.

© 2008 Optical Society of America

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References

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  2. M. Pluta, Advanced Light Microscopy, Vol 2: Specialised Methods (Elsevier, 1989).
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2007

S. V. King, A. R. Libertun, C. Preza, and C. J. Cogswell, "Calibration of a phase-shifting DIC microscope for quantitative phase imaging," Proc. SPIE 6443, 64430M (2007).
[CrossRef]

A. C. Sullivan and R. R. McLeod, "Tomographic reconstruction of weak, replicated index structures embedded in a volume," Opt. Express 15, 14202-14212 (2007).
[CrossRef] [PubMed]

2006

2005

2004

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

2003

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

2002

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]

W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
[CrossRef] [PubMed]

2000

C. Preza, "Rotational-diversity phase estimation from differential-interference-contrast microscopy images," J. Opt. Soc. Am. A 17, 415-424 (2000).
[CrossRef]

M. G. L. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J. Microsc. 198, 82-87 (2000).
[CrossRef] [PubMed]

1999

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1984

1978

1971

R. V. Shack and B. C. Platt, "Production and use of a lenticular Hartmann screen," J. Opt. Soc. Am. 61, 656 (1971).

1900

J. Hartmann, "Bemerkungen uber den Bau und die Justirung von Spektrographen," Z. Instrumentenkd. 20, 47-58 (1900).

Appl. Opt.

J. Microsc.

M. G. L. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," J. Microsc. 198, 82-87 (2000).
[CrossRef] [PubMed]

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]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

R. V. Shack and B. C. Platt, "Production and use of a lenticular Hartmann screen," J. Opt. Soc. Am. 61, 656 (1971).

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. SPIE

S. V. King, A. R. Libertun, C. Preza, and C. J. Cogswell, "Calibration of a phase-shifting DIC microscope for quantitative phase imaging," Proc. SPIE 6443, 64430M (2007).
[CrossRef]

Rep. Prog. Phys.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Science

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Z. Instrumentenkd.

J. Hartmann, "Bemerkungen uber den Bau und die Justirung von Spektrographen," Z. Instrumentenkd. 20, 47-58 (1900).

Other

I. Ghozeil, "Hartmann and other screen tests," in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 367-396.

A. F. Fercher and C. K. Hitzenberger, "Optical coherence tomography," in Progress in Optics (Elsevier, 2002).

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

M. Pluta, Advanced Light Microscopy, Vol 2: Specialised Methods (Elsevier, 1989).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

R. M. Haralick and L. G. Shapiro, Computer and Robot Vision (Addison-Wesley, 1992).

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Figures (11)

Fig. 1
Fig. 1

Beam propagation imaging simulations of a sinusoidal amplitude mask (a) without and (b) with a linear phase object (prism). Transverse cross sections of (c) the sinusoidal mask and images of the sinusoidal mask (d) without and (e) with the linear phase object show a shift in the image of the mask due to the linear phase object.

Fig. 2
Fig. 2

Path followed by a ray propagating perpendicular to the x-axis from a point A through an OPL profile p(x). AO is the actual optical path traveled, while BO is the apparent path as seen from above the surface. θ i is the angle of incidence, θ r is the angle of refraction, and t ( x ) is the distance between the points A and B. n 1 and n 2 are the indices of refraction of the phase object and the surrounding medium, respectively.

Fig. 3
Fig. 3

Simulation of quantitative phase estimation: (a) object with quadratic OPL profile, (b) deformation in x, (c) deformation in y, (d) integration of the x deformation along x, (e) integration of the y deformation along y, (f) calculated square of the OPL profile, (g) calculated quantitative OPL profile, and (h) difference between the horizontal line plots through the center of the OPL profiles of the actual object in (a) and the calculated object in (g) showing a very good match between the actual and the calculated profiles.

Fig. 4
Fig. 4

Quantitative phase imaging experiment: the test phase object is a drop of cured optical cement with refractive index 1.507 surrounded by air. (a) Image of the original amplitude pattern containing periodic 16.5 μ m diameter dots, (b) image of the deformed amplitude pattern after introducing the phase object, and (c) difference between images (a) and (b) (after noise removal) shows the predicted shift in the location of dots. Black regions represent the original dot locations, and the white regions represent the new locations of the dots after introducing the phase object.

Fig. 5
Fig. 5

(a) x and (b) y deformation matrices showing the distances through which the dots move after the phase object is introduced.

Fig. 6
Fig. 6

Numerical integration of (a) x deformation matrix along the x direction and (b) y deformation matrix along the y direction. (a) and (b) represent the square of the OPL profile along the x and y dimensions, respectively.

Fig. 7
Fig. 7

Square of the OPL profile calculated from the numerical integrations of x and y deformation matrices.

Fig. 8
Fig. 8

Quantitative OPL profile.

Fig. 9
Fig. 9

(a) Comparison of the horizontal line plots through the center of the surface thickness profile (OPL profile divided by the object's refractive index) calculated using the new QSIP method with the one measured by a Veeco stylus profilometer. (b) Difference between the two measurements shows a very good match in regions with small phase gradients.

Fig. 10
Fig. 10

Differently angled rays originating from point A, after refraction, appear to have originated from the distortion spot CD. For objects with small phase gradients, the centroid of this distortion spot closely corresponds to the virtual image point B, produced by the central ray R 2.

Fig. 11
Fig. 11

Centroid error (ε) as a function of the cone angle (ϕ) and the phase gradient [ tan ( α ) ] for a 10 μ m thick object with refractive index 1.5 surrounded by air. For small phase gradients, ε is close to 0, which means that the virtual image point produced by the central ray can be well approximated by the centroid of the distortion spot for a wide range of cone angles. TIR is the angle at which total internal reflection occurs for the parameters of this example.

Equations (172)

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2 π
S ( x , y )
exp [ i P ( x , y ) ]
| F 1 ( F { S ( x , y ) exp [ i P ( x , y ) ] } H ( u , v ) ) | 2
H ( u , v )
p ( x )
p ( x )
p ( x ) = n 1 t ( x ) tan ( θ r θ i ) | x = A ,
θ i
θ r
θ r θ i
θ i
θ r θ i = sin 1 [ n 1 n 2   sin ( θ i ) ] θ i ,
n 2
tan ( θ i ) = | d p ( x ) d x | x = A .
p ( x )
p ( x ) = n 1 t ( x ) tan [ sin 1 ( n 1 n 2   sin { tan 1 [ d p ( x ) d x ] } ) tan 1 [ d p ( x ) d x ] ] .
x = 0
p ( x ) = n 1 t ( x ) { 1 ( n 1 ) d p ( x ) d x + [ 1 n 2 1 3 1 6 ( n 1 ) 3 + 1 3 ( 3 n 2 + n 3 2 ) ( n 1 ) 2 ] d p ( x ) d x + higher   order   terms } ,
n = n 1 / n 2
p ( x ) = t ( x ) k = 0 A k [ d p ( x ) d x ] 2 k 1 ,
A k
OPL   p ( x )
t ( x )
( 1 )
k = 0
k = 0
p ( x ) = n 1 n 2 t ( x ) ( n 1 n 2 ) d p ( x ) d x .
p ( x )
p ( x ) d p ( x ) = n 1 n 2 ( n 1 n 2 ) t ( x ) d x .
p 2 ( x ) 2 + C 1 = I ( x ) + C 2 ,
n 1 n 2 ( n 1 n 2 ) t ( x ) d x = I ( x ) + C 2 ,
C 1
C 2
p 2 ( x ) = 2 I ( x ) + C ,
C = 2 ( C 2 C 1 )
n 1
n 2
p ( x )
t ( x )
p ( x )
n 1
n 2
p ( x )
p 2 ( x )
t ( x )
2 I ( x ) = p 2 ( x ) C
p 2 ( x )
2 I ( x ) + C
C = 2 I ( x min ) ,
x min
I ( x )
p ( x )
p ( x ) = [ 2 I ( x ) + C ] 1 / 2 .
p ( x )
p ( x )
p 2 ( x )
P x 2
P y 2
P x 2
P y 2
P x 2
P y 2
( P 2 )
P x 2
P y 2
P x 2
P x 2 ( m , 1 : N ) = P 2 ( m , 1 : N ) + C m ,
C m
P y 2
P y 2 ( 1 : M , n ) = P ( 1 : M , n ) + K n ,
K n
C m + 1 C m = [ P x 2 ( m + 1 , d ) P x 2 ( m , d ) ] [ P 2 ( m + 1 , d ) P 2 ( m , d ) ] ,
P y 2 ( m + 1 , d ) P y 2 ( m , d ) = P 2 ( m + 1 , d ) P 2 ( m , d ) .
C m + 1 C m = [ P x 2 ( m + 1 , d ) P x 2 ( m , d ) ] [ P y 2 ( m + 1 , d ) P y 2 ( m , d ) ] .
C m
C o
C o
C o
P 2
C o
P 2
C o
P 2
P 2
p 2 ( x )
I ( x )
I ( x )
t ( x )
w / M
p 2 ( x )
Δ p 2
Δ p 2 = n 1 n 2 ( n 1 n 2 ) s w M .
p 2 ( x )
Δ p 2
p 2 ( x )
Δ p
Δ p [ n 1 n 2 ( n 1 n 2 ) s w M ] 1 / 2 .
( Δ p max )
p max
Δ p 2
p max 2
p 2 ( x )
p max 2
Δ p 2 = ( p max 2 ) ( p max 2 Δ p 2 )
( Δ p min )
Δ p min = p max ( p max 2 Δ p 2 ) 1 / 2 .
Δ p max
100 ×
7 μ m
458   nm
753   nm
500   nm
Δ p min
97   nm
145 μ m
145 μ m
16.5 μ m
16.5 μ m
10 ×
10 ×
T x
T y
T x ( T y )
x ( y )
T x
T y
P x 2
P y 2
P x 2
P x 2
P y 2
P 2
P 2
tan ( α )
R 1
R 3
R 2
ε = centroid's   distance   from   A distance   of   B   from   A = ( C D 2 + D A ) B A .
C D = C Q A Q P A D P ,
C Q = Q Q   tan ( θ r 3 α ) ,
A Q = S Q S A = Q Q O A tan ( α ) ,
P A = S A S P = O A P P tan ( α ) ,
D P = P P   tan ( θ r 1 α ) ,
θ r 1
θ r 3
R 1
R 3
tan ( ϕ ) = A Q Q Q = P A P P .
Q Q = O A 1 tan ( α ) tan ( ϕ ) ,
P P = O A 1 + tan ( α ) tan ( ϕ ) .
θ r 1 = sin 1 [ n   sin ( α ϕ ) ] ,
θ r 2 = sin 1 [ n   sin ( α ) ] ,
θ r 3 = sin 1 [ n   sin ( α + ϕ ) ] ,
θ r 2
R 2
C D = O A   tan { sin 1 [ n   sin ( α + ϕ ) ] α } 1 tan ( α ) tan ( ϕ ) 2 O A   tan ( ϕ ) 1 tan 2 ( α ) tan 2 ( ϕ ) O A   tan { sin 1 [ n   sin ( α ϕ ) ] α } 1 + tan ( α ) tan ( ϕ ) .
D A = D P + P A = P P   tan ( θ r 1 α ) + O A P P tan ( α ) .
D A = O A ( tan { sin 1 [ n   sin ( α ϕ ) ] α } + tan ( ϕ ) ) 1 + tan ( α ) tan ( ϕ ) .
B A = O A   tan { sin 1 [ n   sin ( α ) ] α } .
ε = O A [ 1 2 ( tan { sin 1 [ n   sin ( α + ϕ ) ] α } 1 tan ( α ) tan ( ϕ ) tan { sin 1 [ n   sin ( α ϕ ) ] α } 1 + tan ( α ) tan ( ϕ ) 2   tan ( ϕ ) 1 tan 2 ( α ) tan 2 ( ϕ ) ) + tan { sin 1 [ n   sin ( α ϕ ) ] α } + tan ( ϕ ) 1 + tan ( α ) tan ( ϕ ) tan { sin 1 [ n   sin ( α ) ] α } ] .
n = 1.5
O A = 10 μ m
ϕ + | α | = sin 1 ( 1 / n )
θ i
θ r
t ( x )
n 1
n 2
16.5 μ m
[ tan ( α ) ]
10 μ m

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