QSIP: phase imaging made possible in a bright field microscope

1
QSIP phase imaging made possible in a bright
field microscope
Sri Rama Prasanna Pavani, Ariel Libertun, Sharon
King, and Carol Cogswell Micro Optical Imaging
Systems Laboratory, ECE, University of Colorado
at Boulder http//moisl.colorado.edu
SPIE Photonics west - Bios 1/23/2008
2
Phase imaging What? How?

• Transparent (phase) objects modulate only the
  phase of light
• Convert phase modulations into detectable
  intensity modulations

• Quantitative phase for weak phase objects
• No phase wrapping
• Halo and shading-off
• Only for thin objects

• Quantitative phase after reconstruction
• No phase wrapping
• Polarization sensitive
• Only for thin objects
• Multiple images

• Quantitative phase after reconstruction
• Thick phase objects
• Single image
• Vibration sensitive
• Phase wrapping

• No quantitative phase

3
QSIP

• Amplitude mask in the field diaphragm
• Pattern is imaged on the sample
• Phase object distorts the pattern
• Record the distorted pattern
• Analytical formula calculates phase

Vs
0.2 0.1
(mm)
0.4 0.2
0 0.2 0.4
(mm)
(mm)
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QSIP 1D

• Analytically relate deformation to the optical
  path length
• Consider a 1D phase object p(x)
• Ray R from point A, after refraction, appears as
  if it originated from B
• Deformation t(x) is the distance between A and B

Normal
Tangent
n2
p(x)
n1
A
B
t(x)
S. R. P. Pavani et al, Quantitative
structured-illumination phase microscopy,
Applied Optics 47, 15-24, (2008) S. R. P. Pavani
et al, Structured-illumination quantitative
phase microscopy, in COSI CMB4, (OSA, 2007)
5
QSIP 2D
1D deformations
After 1D integrations
C1 C2 . . CN
Quantitative Phase
2D deformation
K1 K2 KN
S. R. P. Pavani et al, Quantitative
structured-illumination phase microscopy,
Applied Optics 47, 15-24, (2008) S. R. P. Pavani
et al, Quantitative Phase Estimation with a
Bright Field Microscope, FiO FTuP4, (OSA, 2007)
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Simulation
X 100
18 9 0
5 0 -5
Calculated Phase
Quadratic phase
50 25 0
50 25 0
200 100
200 100
After 1D integrations
1D deformations
X 100
18 9 0
5 0 -5
0 100 200
0 100 200
Error
8 4 0 -4 -8
(nm)
Peak error is 5 orders less than peak phase
Error
0 100 200
7
Experimental Results
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Closed-form solution accuracy

• Small gradient approximation

• Error increases as the phase gradient increases

9
Optimization

• Differential equation of optical path length
  p(x)

• Large phase gradient higher order terms are
  important
• No closed form solution when higher order terms
  are included
• Estimation error (?)
• Nelder-mead optimization to minimize ?
• Starting guess is obtained using the closed form
  solution

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Optimization results

• Optimization uses the first 2 orders of the
  Taylor expansion

• Error decreases as the number of iterations in
  increased

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Effect of Numerical Aperture

• For low phase gradients, deformation distance is
  the distance between the centroids of the
  original and deformed dots for all NAs
• For high phase gradients, lower NA objectives are
  recommended.
• Centroid error (e) is the error in using
  centroids for measuring the deformation distance

Centroid error
S. R. P. Pavani et al, Quantitative
structured-illumination phase microscopy,
Applied Optics 47, 15-24, (2008)
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Spatial Resolution

• Size and the spacing between dots
• Dots sampling the object must obey Nyquist
• Resolution enhancement by shifting

d
s
M
M
shift right
shift down
shift diagonally




N
N

• If dot size diffraction limited spot size,
  quantitative phase imaging with the same
  resolution as a bright field image is possible

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Spatial Resolution

• Size and the spacing between dots
• Dots sampling the object must obey Nyquist
• Resolution enhancement by shifting

d
s
M
M
shift right
shift down
shift diagonally




N
N

• If dot size diffraction limited spot size,
  quantitative phase imaging with the same
  resolution as a bright field image is possible

• Full resolution single image phase imaging with
  multi-colored dots

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Phase resolution
Dot shift

• Smallest detectable change in path length
• Minimum deformation

w detector pixel width M magnification

• Trapezoidal numerical integration

s
x
x
Example
lt
w 7µm s 1µm
n1 1.5 n2 1
M 100x NA 0.9
Depth of field 753nm
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Conclusion

• QSIP is a wide field quantitative phase imaging
  mode in a traditional bright field microscope
• Phase is calculated from the deformation of an
  amplitude pattern using an analytical formula for
  low phase gradients and using numerical
  optimization for high phase gradients
• Low phase gradient objects can be used with wide
  range of NAs and high phase gradient objects need
  to be used with lower NAs.
• Conservative calculations with a 100x objective
  predict a phase resolution of 97nm

16
Acknowledgments

• Prof. Rafael Piestun, Electrical Engineering, CU
  Boulder
• Prof. Gregory Beylkin, Applied Mathematics, CU
  Boulder

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References

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Metrology - Cubic phase mask
120 80 40 0
360 180
480 240 0
Deformation
Quantitative OPL profile
140 70 0
Cubic phase mask
360 180
480 240 0
Deformation
Quantitative OPL profile
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Applications and Future work

• Industrial inspection, biological imaging
• Extracting information from axial deformation
• Extending the depth of field of the system
• Fabrication of an amplitude mask with higher
  spatial resolution

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Our method How?
1 Dimensional analysis
(from geometry)
(Snells law,
)
(Taylor expansion)
C 2 (C2 C1)
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Our method How?
M
2 Dimensional analysis
N
Apply 1D solution along x and y to obtain
and
P2
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Phase contrast Halo and shading off