Title: Phase determination of a radiation wave field
Abstract: A method of quantitative determination of the phase of a radiation wave field including the steps of producing a representative measure of the rate of change of intensity of the radiation wave field over a selected surface extending generally across the wave field; producing a representative measure of intensity of the radiation wave filed over the selected surface; transforming the measure of rate of change of intensity to produce a first integral transform representation and applying to the first integral transform representation a first filter corresponding to the inversion of a first differential operator reflected in the measure of rate of change of intensity to produce a first modified integral transform representation; applying an inverse of the first integral transform to the first modified integral transform representation to produce an untransformed representation; applying a correction based on the measure of intensity over the selected surface to the untransformed representation; transforming the corrected untransformed representation to produce a second integral transform representation and applying to the second integral transform representation a second filter corresponding to the inversion of a second differential operator reflected in the corrected untransformed representation to produce a second modified integral transform representation; and applying an inverse of the second integral transform to the second modified integral transform representation to produce a measure of phase of the radiation wave field across the selected plane.
Patent Number: 6,885,442 Issued on 04/26/2005 to Nugent, et al.
| Inventors:
|
Nugent; Keith (North Fitzroy, AU);
Paganin; David (North Carlton, AU);
Barty; Anton (Brunswick, AU)
|
| Assignee:
|
The University of Melbourne (Victoria, AU)
|
| Appl. No.:
|
830393 |
| Filed:
|
November 1, 1999 |
| PCT Filed:
|
November 1, 1999
|
| PCT NO:
|
PCTAU99/00949
|
| 371 Date:
|
June 7, 2001
|
| 102(e) Date:
|
June 7, 2001
|
| PCT PUB.NO.:
|
WO0026622 |
| PCT PUB. Date:
|
May 11, 2000 |
Foreign Application Priority Data
| Current U.S. Class: |
356/121; 250/201.9 |
| Intern'l Class: |
G01J 001//00 |
| Field of Search: |
356/121-127,451,435,520
250/201.9,331,306
359/845-849
|
References Cited [Referenced By]
U.S. Patent Documents
| 4309602 | Jan., 1982 | Gonsalves et al.
| |
| 4690555 | Sep., 1987 | Ellerbroek.
| |
| 4953188 | Aug., 1990 | Siegel et al.
| |
| 5004918 | Apr., 1991 | Tsuno et al.
| |
| 5298747 | Mar., 1994 | Ichikawa et al.
| |
| 5367375 | Nov., 1994 | Siebert.
| |
| 5633714 | May., 1997 | Nyyssonen.
| |
| 5715291 | Feb., 1998 | Momose et al.
| |
| 5841125 | Nov., 1998 | Livingston.
| |
| 6219142 | Apr., 2001 | Kane.
| |
Other References
Copy of Search Report.
Barty et al., "Quantitative Optical Phase-Amplitude Microscopy", Conference paper,
XI Conference of the Australian Optical Society, (Dec. 1997).
Barty et al., Quantitative Optical Phase Microscopy, (1998), Opt. Lett. (in press).
Barty et al., "Quantitative Optical Phase-Amplitude Microscopy", Conference paper,
Focus on Microscopy, (Apr. 1998).
Gureyev et al., "Partially coherent fields, the transport-of-intensity equation,
and phase uniqueness" J. Opt.Soc.Am. vol. 12 1942-1946.
Gureyev et al., "Phase retrieval with the transport-of- intensity equation. II.
Orthogonal series solution for nonuniform illumination", J. Opt. Soc. Am . vol.
13 1670-1682.
Gureyev et al., "Phase retrieval with the transport-of-intensity equation: matrix
solution with the use of Zernlike polynomials", J. Opt. Soc. Am. vol. 12 1932-1941.
Gureyev et al., "Rapid phase retrieval using the Fast Fourrier Transform" Adaptive
Optics, vol. 23, 1995 Technical Digest Series (Optical Society of America, Washington,
D.C., 1995) pp 77-79.
Gureyev et al., "Rapid quantitative phase imaging using the transport of intensity
equation", (1997), 133 Opt. Comm. 339-346.
N. Streibl, "Phase imaging by the transport of intensity equation", (1984), 49
Opt. Comm. 6.
Nugent et al., "Quantitative phase imaging using hard X-rays", (1996), 77 Phys.
Rev. Lett. 2961-2964.
Paganin et al., "Noninterferometric phase imaging with partially coherent light",
(1998), 80 Phys. Rev. Lett. 2586-2589.
Paganin et al., "Non-interferometric phase imaging with partially coherent radiation",
Conference paper, XI Conference of the Australian Optical Society, (Dec. 1997).
Wilkins et al., "Phase-contrast imaging using polychromatic hard X-rays", (1996),
384 Nature 335.
|
Primary Examiner: Smith; Zandra V.
Assistant Examiner: Nguyen; Sang H.
Attorney, Agent or Firm: Fish & Richardson P.C.
Claims
1. A method of quantitative determination of the phase of a radiation wave field
including the steps of
(a) producing a representative measure of the rate of change of intensity of
said radiation wave field over a selected surface extending generally across the
wave field;
(b) producing a representative measure of intensity of said radiation wave filed
over said selected surface;
(c) transforming said measure of rate of change of intensity to produce a first
integral transform representation and applying to said first integral transform
representation a first filter corresponding to the inversion of a first differential
operator reflected in said measure of rate of change of intensity to produce a
first modified integral transform representation;
(d) applying an inverse of said first integral transform to said first modified
integral transform representation to produce an untransformed representation;
(e) applying a correction based on said measure of intensity over said selected
surface to said untransformed representation;
(f) transforming the corrected untransformed representation to produce a second
integral transform representation and applying to said second integral transform
representation a second filter corresponding to the inversion of a second differential
operator reflected in the corrected untransformed representation to produce a second
modified integral transform representation;
(g) applying an inverse of said second integral transform to said second modified
integral transform representation to produce a measure of phase of said radiation
wave field across said selected plane.
2. A method as claimed in claim 1 wherein said first and second integral transforms
are produced using a Fourier transform.
3. A method as claimed in claim 2 where said Fourier transform is a Fast Fourier transform.
4. A method as claimed in claim 2 wherein said radiation wave field propagates
in a z-direction of a Cartesian co-ordinate system and further including the step
of producing an x component and a y component of phase separately.
5. A method as claimed in claim 4 wherein said first and said second filters
have a component Ω
x for producing the x component of phase and
a component Ω
y for
##EQU18##
##EQU19##
producing the y component of the phase of the form
where k
x, k
y are the Fourier variables conjugate to x and
y;
α is a constant determined by noise in the intensity measurements.
6. A method as claimed in claim 5 including the step of multiplying said representative
measure of rate of change of intensity by the negative of average wave number of
the radiation before said integral transformation.
7. A method as claimed in claim 2 wherein at least one of said first filter and
said second filter include a correction for aberrations in said representative
measures of intensity and rate of change of intensity by including at least one
component dependent on the aberration, coefficients of a system producing the representative measures.
8. A method as claimed in claim 7 wherein said radiation wave field propagates
in a z-direction of a Cartesian co-ordinate system and further including the step
of producing an x component and a y component of phase separately.
9. A method as claimed in claim 8 wherein said first and said second filters
have a component Ω
x for producing the x component of phase and
a component Ω
y for producing the y component of phase both of
the form
##EQU20##
where k
x, k
y are the Fourier variables conjugate to x and
y;
{overscore (λ)} is the average wave length of the radiation;
aberrated(x,y) is the aberrated intensity measured at defocus distance
δz,
A
mn are the aberration coefficients which characterize the imperfect
imaging system.
10. A method as claimed in claim 1 wherein said first and second differential
operators are second order differential operators.
11. A method as claimed in claim 1 wherein said first filter is substantially
the same as said second filter.
12. A method as claimed in claim 1 wherein said first filter includes selectively
suppressing first higher frequencies of the first integral transform representation.
13. A method as claimed in claim 1 wherein at least one of said first and second
filters includes a correction for noise in said representative measure of intensity.
14. A method as claimed in claim 1 including the step of producing said representative
measures of intensity and rate of change of intensity over said selected surface
by producing representative measurements corresponding to intensity over at least
two spaced apart surfaces extending across the wave field.
15. A method as claimed in claim 14 wherein said selected surface is between
two of said spaced apart surfaces.
16. A method as claimed in claim 14 wherein said selected surface is one of said
spaced apart surfaces.
17. A method as claimed in claim 14 including the step of directly detecting
representative measures of intensity over said spaced apart surfaces.
18. A method as claimed in claim 14 including the step of producing said representative
measure of intensity over at least one of said spaced apart surfaces by imaging
that surface.
19. A method as claimed in claim 14 wherein said spaced apart surfaces are substantially parallel.
20. A method as claimed in claim 19 wherein said spaced apart surfaces are substantially planar.
21. A method as claimed in claim 14 wherein said representative measure of rate
of change of intensity is produced by subtraction of representative measurements
of intensity respectively made at locations over said spaced apart surfaces.
22. A method as claimed in claim 1 wherein said representative measures of intensity
and rate of change of intensity are obtained by sampling measurements at selected
locations over said surface.
23. A method as claimed in claim 22 wherein said sampling measurements are made
at locations defining a regular array over said surface.
24. A method as claimed in claim 1 including the step of obtaining said representative
measure of rate of change of intensity by obtaining a first representative measurement
over a measurement surface across the wave field for radiation of a first energy
and obtaining a second representative measurement over said measurement surface
for radiation of a second different energy.
25. An apparatus for quantitative determination of the phase of a radiation wave
field including
(a) means to produce a representative measure of the rate of change of intensity
of said radiation wave field over a selected surface extending generally across
the wave field;
(b) means to produce a representative measure of intensity of said radiation
wave field over said selected surface;
(c) processing means to sequentially
(I) transform said measure of rate of change of intensity to produce a first
integral transform representation;
(II) apply to said first integral transform representation a first filter corresponding
to the inversion of a first differential operator reflected in said measure of
rate of change of intensity to produce a first modified integral transform representation;
(III) apply an inverse of said first integral transform to said first modified
integral transform representation to produce an untransformed representation;
(IV) apply a correction based on said measure of intensity over said selected
surface to said untransformed representation;
(V) transform the corrected untransformed representation to produce a second
integral transform representation;
(VI) apply to said second integral transform representation a second filter corresponding
to the inversion of a second differential operator reflected in the corrected untransformed
representation to produce a second modified integral transform representation;
and
(VII) apply an inverse of said second integral transform to said second modified
integral transform representation to produce a measure of phase of said radiation
wave field across said selected plane.
26. An apparatus as claimed in claim 25 wherein said first and second integral
transforms are produced using a Fourier transform.
27. An apparatus as claimed in claim 26 wherein said Fourier transform is a Fast
Fourier transform.
28. An apparatus as claimed in claim 26 wherein said radiation wave field propagates
in a z-direction of a Cartesian co-ordinate system and processing means produces
an x component and a y component of phase separately.
29. An apparatus as claimed in claim 28 wherein said processing means applies
said first and said second filters have a component Ω
x for producing
the x component of phase and a component Ω
y for producing the
y component of phase of the form
##EQU21##
##EQU22##
where k
x, k
y are the Fourier variables conjugate to x and
y;
α is a constant determined by noise in the intensity measurements.
30. An apparatus as claimed in claim 26 wherein at least one of said first filter
and said second filter include a correction for aberrations in said representative
measures of intensity and rate of change of intensity by including at least one
component dependent on the aberration, coefficients of a system producing the representative measures.
31. An apparatus as claimed in claim 30 wherein said radiation wave field propagates
in a z-direction of a Cartesian co-ordinate system and wherein an x component and
a y component of phase are produced separately.
32. An apparatus as claimed in claim 31 where in said first and said second filters
have a component Ω
x for producing the x component of phase and
a component Ω
y for producing the y component of phase both of
the form
##EQU23##
where k
x, k
y are the Fourier variables conjugate to x and
y;
{overscore (λ)} is the average wave length of the radiation;
I
aberrated (x,y) is the aberrated intensity measured at defocus distance
δz,
A
mn are the aberration coefficients which characterize the imperfect
imaging system.
33. An apparatus as claimed in claim 25 wherein said first and second differential
operators are second order differential operators.
34. An apparatus as claimed in claim 25 wherein said first filter is substantially
the same as said second filter.
35. An apparatus as claimed in claim 25 wherein said first filter includes selectively
suppressing first higher frequencies of the first integral transform representation.
36. An apparatus as claimed in claim 25 wherein at least one of said first and
second filters includes a correction for noise in said representative measure of intensity.
37. An apparatus as claimed in claim 25 including means to produce representative
measurements corresponding to intensity over at least two spaced apart surfaces
extending across the wave field.
38. An apparatus as claimed in claim 37 wherein said selected surface is between
two of said spaced apart surfaces.
39. An apparatus as claimed in claim 37 wherein said selected surface is one
of said spaced apart surfaces.
40. An apparatus as claimed in claim 37 including detector means positioned to
directly detect representative measures of intensity over said spaced apart surfaces.
41. An apparatus as claimed in claim 40 wherein said representative measure of
rate of change of intensity is multiplied by the negative of the average wave number
of the radiation before said integral transformation.
42. An apparatus as claimed in claim 37 including detector means to produce said
representative measure of intensity over at least one of said spaced apart surfaces
and imaging means to image that surface onto the detector.
43. An apparatus as claimed in claim 37 wherein said spaced apart surfaces are
substantially parallel.
44. An apparatus as claimed in claim 37 wherein said spaced apart surfaces are
substantially planar.
45. An apparatus as claimed in claim 37 wherein said means to produce said representative
measure of rate of change of intensity subtracts representative measurements of
intensity respectively made at locations over said spaced apart surfaces.
46. An apparatus as claimed in claim 25 wherein said means to produce a representative
measure of intensity and said means to produce a representative measure of rate
of change of intensity sample at selected locations over said surface.
47. An apparatus as claimed in claim 46 wherein said samples are made at locations
defining a regular array over said surface.
48. An apparatus as claimed in claim 25 wherein said representative measure of
rate of change of intensity is produced by obtaining a first representative measurement
over a measurement surface across the wave field for radiation of a first energy
and obtaining a second representative measurement over said measurement surface
for radiation of a second different energy.
49. A method of imaging an object including the steps of
(a) exposing the object to a radiation wave field from a source;
(b) producing a representative measure of the rate of change of intensity over
a selected surface extending generally across the wave field on the side of the
object remote from incident radiation;
(c) producing a representative measure of intensity of said radiation wave field
over said selected surface;
(d) transforming said measure of rate of change of intensity to produce a first
integral transform representation and applying to said first integral transform
representation and applying to said first integral transform representation a first
filter corresponding to the inversion of a first differential operator reflected
in said measure of rate of change of intensity to produce a first modified integral
transform representation;
(e) applying an inverse of said first integral transform to said first modified
integral transform representation to produce an untransformed representation;
(f) applying a correction based on said measure of intensity over said selected
surface to said untransformed representation;
(g) transforming the corrected untransformed representation to produce a second
integral transform representation and applying to said second integral transform
representation a second filter corresponding to the inversion of a second differential
operator reflected in eh corrected untransformed representation to produce a second
modified integral transform representation;
(h) applying an inverse of said second integral transform to said second modified
integral transform representation to produce a measure of phase of said radiation
wave field across said selected plane.
50. A method as claimed in claim 49 including the step of producing said representative
measures of intensity and rate of change of intensity over said selected surface
by producing representative measurements corresponding to intensity over at least
two spaced apart surfaces extending across the wave field.
51. A method as claimed in claim 50 wherein said selected surface is between
two of said spaced apart surfaces.
52. A method as claimed in claim 50 wherein said selected surface is one of said
spaced apart surfaces.
53. A method as claimed in claim 50 wherein said spaced apart surfaces are substantially parallel.
54. A method as claimed in claim 50 wherein said spaced apart surfaces are substantially planar.
55. A method as claimed in claim 50 wherein said representative measure of rate
of change of intensity is produced by subtraction of representative measurements
of intensity respectively made at locations over said spaced apart surfaces.
56. A method as claimed in claim 49 including the step of producing said measures
of intensity and rate of change of intensity over said selected surface by producing
first representative measurements corresponding to intensity over a first surface
extending across the wave field, changing the distance between said source and
said object, and producing second representative measurements corresponding to
intensity over said first surface for the changed distance between said object
and said source.
57. A method as claimed in claim 56 wherein said selected surfaces is one of
said spaced apart surfaces.
58. A method as claimed in claim 49 including the step of directly detecting
representative measures of intensity over said spaced apart surfaces.
59. A method as claimed in claim 49 wherein said selected surface is spaced apart
from said object in the direction of propagation of said radiation.
60. A method as claimed in claim 49 wherein said source is substantially a point source.
61. A method as claimed in claim 49 wherein said first and second integral transforms
are produced using a Fourier transform.
62. A method as claimed in claim 49 wherein said Fourier transform is a Fast
Fourier transform.
63. An apparatus for imaging an object including:
(a) a source to irradiate the object with a radiation wave field;
(b) means to produce a representative measure of the rate of change of intensity
of said radiation wave field over a selected surface extending generally across
the wave field;
(c) means to produce a representative measure of intensity of said radiation
wave field over said selected surface;
(d) processing means to sequentially;
(i) transform said measure of rate of change of intensity of produce a first
integral transform representation;
(ii) apply to said first integral transform representation a first filter corresponding
to the inversion of a first differential operator reflected in said measure of
rate of change of intensity to produce a first modified integral transform representation;
(iii) apply an inverse of said first integral transform to said first modified
integral transform representation to produce an untransformed representation;
(iv) apply a correction based on said measure of intensity over said selected
surface to said untransformed representation;
(v) transform the corrected untransformed representation to produce a second
integral transform representation;
(vi) apply to said second integral transform representation a second filter corresponding
to the inversion of a second differential operator reflected in the corrected untransformed
representation to produce a second modified integral transform representation;
and
(vii) apply an inverse of said second integral transform to said second modified
integral transform representation to produce a measure of phase of said radiation
wave field across said selected plane.
64. An apparatus as claimed in claim 63 including means to produce representative
measurements corresponding to intensity over at least two spaced apart surfaces
extending across the wave field.
65. An apparatus as claimed in claim 64 wherein said selected surface is between
two of said spaced apart surfaces.
66. An apparatus as claimed in claim 64 wherein said selected surface is one
of said spaced apart surfaces.
67. An apparatus as claimed in claim 64 including detector means positioned to
directly detect representative measures of intensity over said spaced apart surfaces.
68. An apparatus as claimed in claim 64 including detector means to produce said
representative measure of intensity over at least one of said spaced apart surfaces
and imaging means to image that surface onto the detector.
69. An apparatus as claimed in claim 64 wherein said spaced apart surfaces are
substantially parallel.
70. An apparatus as claimed in claim 64 wherein said representative measure of
rate of change of intensity is produced by subtraction of representative measurements
of intensity respectively made at locations over said spaced apart surfaces.
71. An apparatus as claimed in claim 63 including means to produce said measures
of intensity and rate of change of intensity overs said selected surface by producing
first representative measurements corresponding to intensity over a first surface
extending across the wave field; means to change the distance between said source
and said object, and means to produce second representative measurements corresponding
to intensity over said first surface for the changed distance between said object
and said source.
72. An apparatus as claimed in claim 71 wherein said selected surface is one
of said spaced apart surfaces.
73. An apparatus as claimed in claim 67 including means to directly detecting
representative measures of intensity over said spaced apart surfaces.
74. An apparatus as claimed in claim 67 wherein said selected surface is spaced
apart from said object in the direction of propagation of said radiation.
75. An apparatus as claimed in claim 63 wherein said source is substantially
a point source.
76. An apparatus as claimed in claim 63 wherein said first and second integral
transforms are produced using a Fourier transform.
77. An apparatus as claimed in claim 76 wherein said Fourier transform is a Fast
Fourier transform.
78. A method of phase amplitude imaging including the steps of
(a) irradiating an object with a radiation wave field;
(b) focusing radiation from the object through an imaging system to an imaging
surface extending across the wave field;
(c) producing a first representative measure of intensity distribution of radiation
over said imaging surface at a first focus of the imaging system;
(d) introducing a change in focus of the image on said imaging surface through
the imaging system;
(e) producing a second representative of measure intensity distribution over
said imaging surface; and
(f) using said first and second representative measures to produce a representative
measure of intensity and a representative measure of rate of change of intensity
in the direction of radiation propagation over a selected surface extending across
the wave field;
(g) transforming said measure of rate of change of intensity to produce a first
integral transform representation and applying to said first integral transform
representation a first filter corresponding to the inversion of a first differential
operator reflected in said measure of rate of change of intensity to produce a
first modified integral transform representation;
(h) applying an inverse of said first integral transform to said first modified
integral transform representation to produce an untransformed representation;
(i) applying a correction based on said measure of intensity over said selected
surface to said untransformed representation;
(j) transforming the corrected untransformed representation to produce to said
second integral transform representation a second integral transform representation
and applying a second filter corresponding to the inversion of a second differential
operator reflected in the corrected untransformed representation to produce a second
modified integral transform representation;
(k) applying an inverse of said second integral transform to said second modified
integral transform representation to produce a measure of phase of said radiation
wave field across said selected plane.
79. A method as claimed in claim 78 wherein said radiation wave field has a numerical
aperture smaller than the numerical aperture of said imaging system.
80. A method as claimed in claim 78 said first focus of the imaging system produces
an infocus image at the imaging surface and said second focus of the imaging system
produces a slightly defocused image at the imaging surface.
81. A method as claimed in claim 78 wherein said imaging surface is substantially planar.
82. A method as claimed in claim 78 wherein the imaging surface is an intensity detector.
83. A method as claimed in claim 78 wherein said imaging surface is said selected surface.
84. A method as claimed in claim 78 wherein said integral transform is a Fourier transform.
85. A method as claimed in claim 84 wherein said Fourier transform is a Fast
Fourier transform.
86. A method as claimed in claim 85 wherein said radiation wave field propagates
in a z-direction of a cartesian co-ordinate system and further including the step
of producing an x component and a y component of phase separately.
87. A method as claimed in claim 86 wherein said first and said second filters
have a component Ω
x for producing the x component of phase and
a component Ω
y for producing the y component of phase of the form
##EQU24##
##EQU25##
where k
1, k
y are the Fourier variables conjugate to x and
y;
α is a constant determined by noise in the intensity measurements.
88. A method as claimed in claim 87 including the step of multiplying said representative
measure of rate of change of intensity by the negative of the average wave number
of the radiation before said integral trasnformation.
89. A method as claimed in claim 78 wherein said representative measure of rate
of change of intensity is produced by subtraction of said first and second representative
measurements of intensity.
90. A method as claimed in claim 78 wherein said representative measures of intensity
and rate of change of intensity are obtained by sampling measurements at selected
location over said imaging surface.
91. A method as claimed in claim 90 wherein said sampling measurements are made
at locations defining a regular array over said imaging surface.
92. An apparatus for phase amplitude imaging of an object including
a radiation wave field source to irradiate said object;
an imaging system to focus radiation from said object to an imaging surface extending
across the wave field propagating from the object;
means to produce a representative measure of radiation intensity over said imaging
surface;
said imaging system including selectively operable means to adjust said focus
of said radiation to said imaging surface to at least at a first focus and a second
focus;
processing means to:
(i) produce a representative measure of intensity and a representative measure
of rate of change of intensity in the direction of radiation propagation over a
selected surface extending across the wave field from representative measures of
radiation intensity over said image surface at said first focus and said second
focus;
(ii) transform said measure of rate of change of intensity to produce a first
integral transform representation;
(iii) apply to said first integral transform representation a first filter corresponding
to the inversion of a first differential operator reflected in said measure of
rate of change of intensity to produce a first modified integral transform representation;
(iv) apply an inverse of said first integral transform to said first modified
integral transform representation to produce an untransformed representation;
(v) apply a correction based on said measure of intensity over said selected
surface to said untransformed representation;
(vi) transform the corrected untransformed representation to produce a second
integral transform representation;
(vii) apply to said second integral transform representation a second filter
corresponding to the inversion of a second differential operator reflected in the
corrected untransformed representation to produce a second modified integral transform
representation; and
(viii) apply an inverse of said second integral transform to said second modified
integral transform representation to produce a measure of phase of said radiation
wave field across said selected plane.
93. An apparatus as claimed in claim 92 wherein said radiation wave field has
a numerical aperture smaller than the numerical aperture said imaging system.
94. An apparatus as claimed in claim 92 said first focus of the imaging system
produces an infocus image at the imaging surface and said second focus of the imaging
system produces a slightly defocused image at the imaging surface.
95. An apparatus as claimed in claim 92 wherein said imaging surface is substantially planar.
96. An apparatus as claimed in claim 92 wherein the imaging surface is an intensity detector.
97. An apparatus as claimed in claim 94 wherein said imaging surface is said
selected surface.
98. An apparatus as claimed in claim 92 wherein said integral transform is a
Fourier transform.
99. An apparatus as claimed in claim 98 wherein said Fourier transform is a Fast
Fourier transform.
100. An apparatus as claimed in claim 99 wherein said radiation wave field propagates
in a z-direction of a Cartesian co-ordinate system and further including the step
of producing an x component and a y component of phase separately.
101. An apparatus as claimed in claim 100 wherein said first and said second
filters have a component Ω
x for producing the x component of phase
and a component Ω
y for producing the y component of phase of the
form
##EQU26##
##EQU27##
where k
x, k
y are the Fourier variables conjugate to x and
y;
α is a constant determined by noise in the intensity measurements.
102. A method as claimed in claim 101 including the step of multiplying said
representative measure of rate of change of intensity by the negative of the average
wave number of the radiation before said integral transformation.
103. An apparatus as claimed in claim 92 wherein said representative measure
of rate of change of intensity is produced by subtraction of said first and second
representative measurements of intensity.
104. An apparatus as claimed in claim 92 wherein said representative measures
of intensity and rate of change of intensity are obtained by sampling measurements
at selected location over said imaging surface.
105. An apparatus as claimed in claim 104 wherein said sampling measurements
are made at location defining a regular array over said imaging surface.
Description
FIELD OF THE INVENTION
This invention relates to the determination of phase of a radiation wave field.
The invention also relates to a range of applications in which phase information
about a radiation wave field can be used. As used in this specification the term
"radiation wave field" is intended to include all forms of radiation that propagate
in a wave like manner including but not limited to examples such as X-rays, visible
light and electrons.
BACKGROUND OF THE INVENTION
Techniques for the measurement of the phase of a radiation wave field
have many applications in fundamental physics and as a basis for a number of measurement
techniques involving various physical properties. Examples of applications of phase
measurement techniques include the fields of x-ray imaging, electron microscopy,
optical microscopy as well as optical tomography and x-ray phase tomography.
Phase is typically measured using interferometers of various types. The key
feature of interferometry is the ability to quantitatively measure the phase of
a wave field. Whilst interferometry based techniques retain significant importance
it has been recognised that non-interferometric techniques may be used to provide
phase information. A number of non-interferometric approaches involve attempting
to solve a transport of intensity equation for a radiation wave field.
This equation relates the irradiance and phase of a paraxial monochromatic wave
to its longitudinal irradiance derivative and is described in M. R. Teague, "Deterministic
Phase Retrieval: A Green's Function Solution" J. Opt. Soc. Am. 73 1434-1441 (1983).
The article "Phase imaging by the transport of intensity equation" by N. Streibl,
Opt. Comm. 49 6-10 (1984); describes an approach based on the transport of intensity
equation by which phase structure can be rendered visible by the use of defocus
and digital subtraction of intensity data obtained at various defocused distances.
This approach only provides for phase visualisation and does not provide for the
measurement of phase shift. Another approach based on solving the transport of
intensity equation is disclosed in T. E.
Gureyev, K. A. Nugent, D. Paganin and A. Roberts, "Rapid phase retrieval
using a Fast Fourier transform", Adaptive Optics, Volume 23, (1995) Optical Society
of America Technical Digest Series, page 77-79 and T. E. Gureyev and K. A.
Nugent, "Rapid quantitative phase imaging using the transport of intensity
equations", Opt. Comm., 133 339-346 (1997). This approach allows the phase of a
light field to be recovered from two closely spaced intensity measurements when
an illuminating beam has an arbitrary, but everywhere non zero intensity distribution
limited by rectangular aperture. Whilst this approach can be used for non-uniform
intensity distributions the extent of the non uniformity is limited and introduces
significant computational complexity. Consequently this approach is not able to
cope with non uniformities introduced by some sample absorption profiles or in
some intensity illumination distributions. This approach is strictly also only
applicable to coherent wave fields.
The article K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin and Z. Barnea
"Quantitative phase imaging using hard X-rays" (1996) 77 Phys. Rev. Lett. 2961-2964
is also based on a solution to the transport of intensity equation. Again the technique
described cannot be applied to a non-uniform intensity distribution.
Other approaches based on a solution to the transport of intensity equation
limited to a requirement of uniformity are described in T. E. Gureyev, K. A.
Nugent, A. Roberts "Phase retrieval with the transport-of-intensity equation:
matrix solution with the use of Zemike polynomials" J. Opt. Soc. Am. A Vol 12,
1932-1941 (1995) and T. E. Gureyev, A. Roberts and K. A. Nugent "Partially coherent
fields, the transport-of-intensity equation, and phase uniqueness", J.
Opt. Soc. Am. A Vol 12, 1942-1946 (1995).
A technique for recovery of phase in the case of non-uniform illumination is
described
in T. E. Gureyev and K. A. Nugent "Phase retrieval with the transport-of-intensity
equation. II. Orthogonal series solution for nonuniform illumination", J. Opt.
Soc. Am. A Vol 13, 1670-1682 (1996). This approach is based on a method of orthogonal
expansions and can be computationally complex in implementation. In many applications
this complexity makes the technique impractical.
SUMMARY OF THE INVENTION
The present invention provides a non-interferometric method and apparatus for
the measurement of phase. In combination with a direct measurement of intensity
a measurement of phase allows the phase and intensity at any other plane in the
radiation wave field to be determined using known techniques. The invention also
provides the basis for a number of measurement techniques.
In accordance with a first aspect of the invention there is provided a method
of quantitative determination of the phase of a radiation wave field including
the steps of
(a) producing a representative measure of the rate of change of intensity
of said radiation wave field over a selected surface extending generally across
the wave field;
(b) producing a representative measure of intensity of said radiation wave
field over said selected surface;
(c) transforming said measure of rate of change of intensity to produce
a first integral transform representation and applying to said first integral transform
representation a first filter corresponding to the inversion of a first differential
operator reflected in said measure of rate of change of intensity to produce a
first modified integral transform representation;
(d) applying an inverse of said first integral transform to said first modified
integral transform representation to produce an untransformed representation;
(e) applying a correction based on said measure of intensity over said selected
surface to said untransformed representation;
(f) transforming the corrected untransformed representation to produce a
second integral transform representation and applying to said second integral transform
representation a second filter corresponding to the inversion of a second differential
operator reflected in the corrected untransformed representation to produce a second
modified integral transform representation;
(g) applying an inverse of said second integral transform to said second
modified integral transform representation to produce a measure of phase of said
radiation wave field across said selected plane.
In accordance with a second aspect of the invention there is provided an apparatus
for quantitative determination of the phase of a radiation wave field including
(a) means to produce a representative measure of the rate of change of intensity
of said radiation wave field over a selected surface extending generally across
the direction of propagation;
(b) means to produce a representative measure of intensity of said radiation
wave field over said selected surface;
(c) processing means to sequentially
- (I) transform said measure of rate of change of intensity to produce
a first integral transform representation;
- (II) apply to said first integral transform representation a first filter
corresponding to the inversion of a first differential operator reflected in said
measure of rate of change of intensity to produce a first modified integral transform representation;
- (III) apply an inverse of said first integral transform to said first
modified integral transform representation to produce an untransformed representation;
- (IV) apply a correction based on said measure of intensity over said
selected surface to said untransformed representation;
- (V) transform the corrected untransformed representation to produce
a second integral transform representation;
- (VI) apply to said second integral transform representation a second
filter corresponding to the inversion of a second differential operator reflected
in the corrected untransformed representation to produce a second modified integral
transform representation; and
- (VII) apply an inverse of said second integral transform to said second
modified integral transform representation to produce a measure of phase of said
radiation wave field across said selected plane.
The selected surface can take any form that extends across the direction of propagation
of the radiation including planar, part-spherical and part-cylindrical surfaces.
The first and second integral transforms can be of any suitable type and include
approximations employed for computational convenience, speed or efficiency.
The first and second integral transforms are preferably produced using a Fourier
transform. More preferably, the transform is a Fast Fourier transform. The method
and apparatus of this invention provide for determination of phase of a radiation
wave field in a manner that is computationally significantly less complex than
prior art approaches. This results in significantly lower computation times. In
some examples computation times improved by many orders of magnitude have been achieved.
The first and second differential operators are preferably second order differential
operators. In the preferred implementation of the method and apparatus the first
filter is substantially the same as the second filter. It is further preferred
that at least one of the first and second filters includes a correction for noise
in the representative measure of intensity.
In one form of the invention the first filter can comprise selectively suppressing
first higher frequencies of the first integral transform representation. In this
form of the invention the second filter can comprise selectively suppressing second
higher frequencies of said second integral transform representation.
The correction based on the measure of intensity over a selected surface can
be a nil correction where the intensity variations are less than a predetermined
selected amount.
Preferably, the measure of the rate of change of intensity and intensity
distribution over the selected surface are produced from measurements of the intensity
distribution over at least two surfaces extending across the wave field and spaced
apart along the direction of propagation of the radiation. In another form of the
invention the representative measure of rate of change intensity in the direction
of radiation propagation is produced by obtaining a first representative measurement
over a measurement surface extending across the direction of propagation for radiation
of a first energy and obtaining a second representative measurement over said measurement
surface for radiation of a second different energy. In the case of X-ray radiation,
for example, the change in radiation energy can be achieved by changing the X-ray
target or by suitable filtering.
The selected surface for which measurements of intensity and rate of change of
intensity are produced is preferably located between two of the spaced apart surfaces
over which intensity distribution is measured.
In the preferred form of the invention the selected surface and spaced apart
surfaces
are planar. It is further preferred that the planes are generally perpendicular
to the average direction of propagation of the radiation.
The method and apparatus of this invention can be at least partly implemented
using a suitably programmed computer. In particular the processing means preferably
comprises a suitably programmed computer and the steps of the method are preferably
performed using a suitably programmed computer. In such forms of the invention
intensity input information may take the form of digitised images or data containing
information from such images. In other implementations of the invention a dedicated
Fast Fourier transform chip can be employed as at least part of the processing means.
The representative measure of rate of change of intensity is preferably produced
by subtraction of representative measurements respectively made at locations over
the spaced apart surfaces. In the preferred form of the invention the representative
measures of intensity and rate of change of intensity are obtained by sampling
measurements at selected locations over the surface. Preferably the sampling and
measurements are made at locations defining a regular array over the surface. This
can be readily achieved for example by using a CCD (charge coupled device) as the detector.
In the preferred method the direction of propagation of the radiation wave field
is selected to be the z direction of a Cartesian co-ordinate system and x and y
components of phase are produced separately.
In this Cartesian co-ordinate system where the z direction is the direction of
propagation of the radiation, the preferred filters are of the form
##EQU1##
##EQU2##
where
- kx, ky are the Fourier variables conjugate to
x, y and
- α is a constant determined by noise in the intensity measurements
and is equal to zero for a no noise case.
The measure of rate of change of intensity is preferably multiplied by the negative
of the average wave number of the radiation before the integral transformation
into the Fourier domain.
The representative measure of intensity over the spaced apart surfaces can be
obtained by imaging of that surface through an appropriate imaging system.
That is, the intensity information may be imaged to a detector rather than measured
at the surface.
The method of this invention thus provides for the quantitative and decoupled
determination of phase and intensity of a radiation wave field at any surface across
the direction of propagation of the radiation. From this phase and intensity determination
it is possible to calculate the phase and intensity at any other surface along
the direction of propagation. Accordingly, the invention provides the basis for
a number of measurement techniques.
In a further aspect of the invention there is provided a method of imaging an
object including the steps of
- (a) exposing the object to a radiation wave field from a source;
- (b) producing a representative measure of the rate of change of intensity
over a selected surface extending generally across the wave field on the side of
the object remote from incident radiation;
- (c) producing a representative measure of intensity of said radiation
wave field over said selected surface;
- (d) transforming said measure of rate of change of intensity to produce
a first integral transform representation and applying to said first integral transform
representation a first filter corresponding to the inversion of a first differential
operator reflected in said measure of rate of change of intensity to produce a
first modified integral transform representation;
- (e) applying an inverse of said first integral transform to said first
modified integral transform representation to produce an untransformed representation;
- (f) applying a correction based on said measure of intensity over said
selected surface to said untransformed representation;
- (g) transforming the corrected untransformed representation to produce
a second integral transform representation and applying to said second integral
transform representation a second filter corresponding to the inversion of a second
differential operator reflected in the corrected untransformed representation to
produce a second modified integral transform representation;
- (h) applying an inverse of said second integral transform to said second
modified integral transform representation to produce a measure of phase of said
radiation wave field across said selected plane.
In a still further aspect of the invention there is provided an apparatus for
imaging an object including
- (a) a source to irradiate the object with a radiation wave field;
- (b) means to produce a representative measure of the rate of change
of intensity of said radiation wave field over a selected surface extending generally
across the wave field;
- (c) means to produce a representative measure of intensity of said radiation
wave field over said selected surface;
- (d) processing means to sequentially
- (I) transform said measure of rate of change of intensity to produce
a first integral transform representation;
- (II) apply to said first integral transform representation a first
filter corresponding to inversion of a first differential operator reflected in
said measure of rate of change of intensity to produce a first modified integral
transform representation;
- (III) apply an inverse of said first integral transform to said
first modified integral transform representation to produce an untransformed representation;
- (IV) apply a correction based on said measure of intensity over
said selected surface to said untransformed representation;
- (V) transform the corrected untransformed representation to produce
a second integral transform representation;
- (VI) apply to said second integral transform representation a second
filter corresponding to the inversion of a second differential operator reflected
in the corrected untransformed representation to produce a second modified integral
transform representation; and
- (VII) apply an inverse of said second integral transform to said
second modified integral transform representation to produce a measure of phase
of said radiation wave field across said selected plane.
The radiation used to irradiate the object can be a planar wave field or spherical
wave field or an arbitrary wave field. If it is desired to reproduce the phase
in the object plane the phase wave field determined by the above method and apparatus
is back propagated and the wave field used to irradiate, is subtracted.
The method and apparatus of imaging substantially incorporates the determination
of phase as disclosed in relation to the first and second aspects of the invention.
The preferred aspects of the invention described in relation to those aspects above
are also applicable to the method and apparatus of imaging.
It is possible in some applications to use a zero object to image plane distance
corresponding to contact-imaging with zero propagation distance.
If desired the object can be reconstructed in the object plane by back propagating
the intensity and quantitative phase information to numerically reconstruct an
image of the actual object phase and intensity structure.
In other forms of the method more than two image plane intensity distribution
measurements can be made to obtain a better estimate of the rate of change of intensity
or intensity derivative. In this case one or both of the source to object or object
to image plane distances is changed and another intensity distribution measurement
is made. The procedure is repeated until the desired number of measurements is
made. The measurements provide data to which a function can be fitted for the determination
of rate of change of intensity.
The method of imaging an object has particular application to point projection
microscopy using X-rays, visible light or electrons.
In another aspect this invention provides a method of phase amplitude imaging
including the steps of.
- (a) irradiating an object with a radiation wave field;
- (b) focussing radiation from the object through an imaging system to
an imaging surface extending across the wave field propagating from the object;
- (c) producing a first representative measure of intensity distribution
of radiation over said imaging surface at a first focus of the imaging system;
- (d) introducing a change in focus of the image on said imaging surface
through the imaging system;
- (e) producing a second representative measure of intensity distribution
over said imaging surface; and
- (f) using said first and second representative measures to produce a
representative measure of intensity and a representative measure of rate of change
of intensity over a selected surface extending across the wave field;
- (g) transforming said measure of rate of change of intensity to produce
a first integral transform representation and applying to said first integral transform
representation a first filter corresponding to the inversion of a first differential
operator reflected in said measure of rate of change of intensity to produce a
first modified integral transform representation;
- (h) applying an inverse of said first integral transform to said first
modified integral transform representation to produce an untransformed representation;
- (i) applying a correction based on said measure of intensity over said
selected surface to said untransformed representation;
- (j) transforming the corrected untransformed representation to produce
a second integral transform representation and applying to said second integral
transform representation a second filter corresponding to the inversion of a second
differential operator reflected in the corrected untransformed representation to
produce a second modified integral transform representation;
- (k) applying an inverse of said second integral transform to said second
modified integral transform representation to produce a measure of phase of said
radiation wave field across said selected plane.
In yet another aspect of this invention there is provided an apparatus for phase
amplitude imaging of an object including
- a radiation wave field source to irradiate said object;
- an imaging system to focus radiation from said object to an imaging
surface extending across the wave field propagating
