Download Coherent Optical Tomography

Optical Diffraction Tomography
A.J. Devaney
Department of Electrical Engineering
Northeastern University
Boston, MA 02115 USA
E-mail: [email protected]
• Review problems with classical optical microscopy
• Review experimental setup and goal of optical diffraction tomography (ODT)
• Describe two approaches to ODT
• Phase retrieval
• Holographic
• Review results to date
• Outline future goals
Holography, Acoustical, Encyclopedia of Applied Physics, Vol.7 , 511-530, 1992
10/17/97
Optical Diffraction Tomography
1
What’s Wrong With Optical
Microscopy?
Semi-transparent Object
Condenser
Image
Objective Lens
• Illuminating light spatially coherent over small scale:
• Poor image quality for 3D objects
• Need to thin slice
• Cannot image phase only objects:
•Need to stain
•Need to use special phase contrast methods
• Require high quality optics
10/17/97
Optical Diffraction Tomography
2
Experimental Setup
Digital Camera
Test tube
with sample
Diffraction Plane
collimator
d
HE-NE Laser
incident plane wave
transmitted wave
Magnifying Lens
Magnified diffraction pattern
Digital Camera Images Intensity Distribution Over Diffraction Plane
Image is Gabor hologram of diffraction plane field distribution
10/17/97
Optical Diffraction Tomography
3
Inverse Problem
Diffraction Plane
d
Measure transmitted intensity
over diffraction plane
transmitted wave
Inverse Problem: Given intensity of transmitted wave estimate the complex
index of refraction distribution of the object.
Difficulties:
• Phase Problem
• Phase retrieval
• Holography
• Quantitative Inversion
• Diffraction tomography
• Born Model
• Rytov Model
• Limited Data
• Multiple experiments
10/17/97
Optical Diffraction Tomography
4
Scattering Models
V
(r
)



2  k 2  (r)  k 2 [1  n 2 (r)]  (r)
eiks0 r

s0
Born Model 
( s)
B
( s)
1
(r )  
4
1
(r )  
4
d
3
r ' V (r ' ) e
ik [ s r  W ( r )]
Rytov Model   e 0
 W (r ) 
10/17/97

( s)
B
iks 0 r
ik e
(r )
eik |r r '|
 d r ' V (r' )  (r' ) | r  r'|
3
iks0 r '
eik |r r '|
| r  r'|
Diffraction tomography solves inverse
problem within either Born or Rytov
approximation. Requires phase of field.
Optical Diffraction Tomography
5
Why Tomography?
Measurement Plane
 W ( rp ) 
i
d
4 k 
ik |r  r '|
3
r ' V (r ' ) e
iks 0 ( r p  r ')
e p
| r  r '|
  d 3r ' V (r ' ) ( rp  s 0  r ' )
eiks0 r
s0
Integral along straightline ray path: Inversion via CT
Diffraction tomography (DT) is generalization of CT to diffracting wavefields
Inversion methods include:
• Filtered backpropagation
• Generalized ART and SIRT
• Various non-linear and limited view algorithms
10/17/97
Optical Diffraction Tomography
6
Diffraction Tomography
Scattered Field
Filtering
Filtered Scattered Field
Backpropagation
Induced Source
iks  r
V (r )e 0
V (r; s0 )  hs0  V (r)
hs0 (R )  eiks0 R sinc( kR)
Filtered Backpropagation Algorithm
V (r)  V (r; s0 )  H  V (r)
Sum over Views
s0
H (R )   hs0 (R )  sinc( kR) e iks0 R
s0
10/17/97
Optical Diffraction Tomography
s0
7
Quality of Inversion
V (r)  V (r; s0 )  H  V (r)
s0
H (R )   hs0 (R )  sinc( kR) e iks0 R
s0
s0
Point Spread Function approaches delta function as
number of views and wavenumber k approach infinity
Single View
Infinite View
1
1
0.8
0.8
0.6
0.4
0.6
0.2
0.4
0
0.2
-0.2
-0.4
60
0
60
60
40
60
40
40
20
20
0
10/17/97
0
40
20
20
0
0
Optical Diffraction Tomography
8
Phase Retrieval
Camera # 2
Diffraction Plane # 1
Test tube
with sample
Magnifying Lens
Beam Splitter
collimator
HE-NE Laser
incident plane wave
Diffraction Plane # 2
Phase Retrieval
• Gerchberg Saxton iterative procedure
• Approximate algebraic method
Camera # 1
Diffraction tomography (DT) generates quantitative image of
real and imaginary parts of object’s index of refraction distribution
from complex (amplitude and phase) distribution of field
10/17/97
Optical Diffraction Tomography
9
Holography
Diffraction Plane
  eikd   ( s )
d
I  1  I ( s )  eikd ( s )  e ikd ( s )
*
transmitted wave


  eikd I  1  I ( s )   ( s )  e 2ikd ( s )
Filter and backpropagate 
 ( s)
10/17/97
*
A.J. Devaney, Phys. Rev. Letts. 62 (1989)
e2ikd ( s)
conjugate
image
*
Optical Diffraction Tomography
10
Born Inversion Procedures
Measured intensity
distribution(s)
• Phase retrieval
• 
Diffraction Tomographic
Reconstruction Algorithm
Complex index of refraction
distribution of object
d
backpropagated filtered data
Diffraction Plane
Can employ single view theory to deal with thin phase only objects
10/17/97
Optical Diffraction Tomography
11
Quest for a Better Microscope
Coherent Tomographic Microscope
• Use coherent light and one or more Gabor holograms of diffraction plane field
• Employ phase retrieval and DT reconstruction algorithm to reconstruct object
• Employ direct holographic based DT reconstruction algorithm
• Can operate in thin object or thick object mode
Comparison with scanning confocal microscope
• Theoretical better image quality
• No need to stain or use floresence
• Much less expensive
10/17/97
Optical Diffraction Tomography
12