LETTERS

d between G

and G

corresponding to the first Talbot distance

²⁰

of d = p

²1

/ 8l, a coherence length of ξ

= ^l l/γ

p

≥ p

is required, where l is the distance between G

and G

. With a typical value of a few micrometres for p

, the spatial coherence length required is of the order of ξ

≥ ¹⁰

⁻⁶

m, similar to the requirements of existing methods (see above). It is important to note, however, that a setup with only two gratings (G

and G

) already requires, in principle, no spatial coherence in the direction parallel to the grating lines, in contrast with propagation-based methods

^8–13

.

As the source mask G

can contain a large number of individual apertures, each creating a suﬃciently coherent virtual line source, standard X-ray generators with source sizes of more than a square millimetre can be used eﬃciently. To ensure that each line source produced by G

contributes constructively to the image-formation process, the geometry of the setup should satisfy the condition (Fig. 1b)

p

= p

× l d .

It is important to note that the total source size w only determines the final imaging resolution, which is given by wd/l . The arrayed source thus decouples spatial resolution from spatial coherence, and allows the use of X-ray illumination with coherence lengths as small as ξ

= ^l l/w ∼ ¹⁰

⁻⁸

^{m in both} directions, if the corresponding spatial resolution wd/l can be tolerated in the experiment. Finally, as a temporal coherence of ξ

≥ ¹⁰

⁻⁹

^{m (} # E/E ≥ 10%) is suﬃcient

²⁰

, we deduce that the method presented here requires the smallest minimum coherence volume ξ

× ξ

for phase-sensitive imaging if compared with existing techniques.

The DPC image-formation process achieved by the two gratings G

and G

is similar to Schlieren imaging

²¹

and diﬀraction-enhanced imaging

^5–7

. It essentially relies on the fact that a phase object placed in the X-ray beam path causes slight refraction of the beam transmitted through the object. The fundamental idea of DPC imaging depends on locally detecting these angular deviations (Fig. 1b). The angle, α is directly proportional to the local gradient of the object’s phase shift, and can be quantified by

²¹

α = ^l 2 π

∂ Φ (x, y)

∂ x ,

where x and y are the cartesian coordinates perpendicular to the optical axis, Φ ( x, y) represents the phase shift of the wavefront, and l is the wavelength of the radiation. For hard X-rays, with l < 0 . 1 nm, the angle is relatively small, typically of the order of a few microradians.

In our case, determination of the angle is achieved by the arrangement formed by G

and G

. Most simply, it can be thought of as a multi-collimator translating the angular deviations into changes of the locally transmitted intensity, which can be detected with a standard imaging detector. For weakly absorbing objects, the detected intensity is a direct measure of the object’s local phase gradient d Φ (x, y)/ d x . The total phase shift of the object can thus be retrieved by a simple one-dimensional integration along x . As described in more detail in refs 19,20, a higher precision of the measurement can be achieved by splitting a single exposure into a set of images taken for diﬀerent positions of the grating G

. This approach also allows the separation of the DPC signal from other contributions, such as a non-negligible absorption of the object, or an already inhomogeneous wavefront phase profile before the object. Note that our method is fully compatible with conventional absorption radiography, because it simultaneously yields separate absorption and phase-contrast images, so that information is available from both.

p₀ p₁

p₂ α

x y z

G₀

G₀ G₁ G₂

G₁ G₂

Object

Imaging detector

Incoherent X-ray source

Au Si Si Au

l d

p₀

p₂ p₁

200 µm 4 µm 2 µm

Figure 1

Talbot–Lau-type hard-X-ray imaging interferometer. a, Principle: the source grating (G

) creates an array of individually coherent, but mutually incoherent sources. A phase object in the beam path causes a slight refraction for each

coherent subset of X-rays, which is proportional to the local differential phase gradient of the object. This small angular deviation results in changes of the locally transmitted intensity through the combination of gratings G

and G

. A standard X-ray imaging detector is used to record the final images. b, Scanning electron micrographs of cross-sections through the gratings. The gratings are made from Si wafers using standard photolithography techniques, and subsequent electroplating to fill the grooves with gold (G

and G

).

Figure 2 shows processed absorption, DPC and reconstructed phase images of a reference sample containing spheres made of polytetrafluoroethylene (PTFE) and polymethylmethacrylate (PMMA). From the cross-section profiles (Fig. 2d), we deduce that 19 ± ⁰ . 6% (7 . 0 ± ⁰ . 6%) of the incoming radiation is absorbed in the centre of the PTFE (PMMA) sphere. By comparison with tabulated literature values

²²

, a mean energy of the eﬀective X-ray spectrum of E

_mean

≈ ²² . 4 ± ¹ . 2 keV (l ≈ ⁰ . 0553 nm) is deduced. To discuss the results for the object’s phase shift (Fig. 2c), we have to consider the real part of the refractive index, which, for X-rays, is typically expressed as n = ¹ − δ . For a homogenous sphere with radius r , the total phase shift through the centre of the sphere is Φ ₌ 4 πrδ/ l.

The experimentally observed maxima of the integrated phase shifts are Φ

PTFE

= ⁵⁴ ± ² π and Φ

PMMA

= ³² ± ² π (Fig. 2f), from which we obtain: δ

_PTFE

= ⁹ . 5 ± ⁰ . 8 × ¹⁰

⁻⁷

^and δ

_PMMA

= ⁵ . 9 ± ⁰ . 6 × ¹⁰

⁻⁷

^, which is in agreement with the values in the literature

²²

. Note that because our method provides a quantitative measure of the absorption and integrated phase shift of the object in each pixel, the results can be used for further quantitative analysis, such as the reconstruction of a three-dimensional map of the real and

naturephysics VOL 2 APRIL 2006 www.nature.com/naturephysics 259

Untitled-3 2 3/22/06, 3:26:31 PM

Nature Publishing Group

Transport of Intensity (TIE)

I i (z)

no object

I(z)

object

• Images corrected for lateral magnification

• Attenuation neglected

• Long propagation distance  (short wavelength)  

➦ two instead of four measurements  k

z

✓ I(z)

I _i (z) 1

◆ ⇡ r ² x

z

x-ray source

(with small spot size) x-ray detector

k @I

@z = r ^x · (Ir ^x )

I ⟹ ⇡ 1

Experimental geometry

• Experimental arrangement at RMD, Inc., Watertown, Mass.

• Three experiments:

• Polystyrene spheres on flat tape (Andor

camera)

• Polystyrene spheres taped onto drinking straw (Andor camera)

• Beetle (Andor camera)

Hamamatsu micro-focus source

∅5µm spot size operated at 20kVp

(λ=0.06nm) Rad-icon

CMOS camera 48µm pixel size 2000×2000 pixels

100″

object on rotation

stage

30″

Andor EM CCD with scintillator by RMD

16µm pixel size

512×512 pixels

X-ray phase tomography: beetle

y (mm)

x (mm)

2 4 6 8 10 12 14

2

4

6

8

10

12

14 30 ^o

y (mm)

x (mm)

2 4 6 8 10 12 14

2

4

6

8

10

12

14 60 ^o

y (mm)

x (mm)

2 4 6 8 10 12 14

2

4

6

8

10

12

14 0 ^o

intensity measurements

y (mm)

x (mm)

2 4 6 8 10 12 14

2

4

6

8

10

12

14 −0.06

−0.05

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02

90 ^o 0.03

1mm

30 ^o

60 ^o

0 ^o

在文檔中 Compressive phase retrieval (頁 59-63)

d between G

and G

corresponding to the first Talbot distance

of d = p

/ 8l, a coherence length of ξ

= l l/γ

p

≥ p

is required, where l is the distance between G

and G

. With a typical value of a few micrometres for p

, the spatial coherence length required is of the order of ξ

≥ 10

m, similar to the requirements of existing methods (see above). It is important to note, however, that a setup with only two gratings (G

and G

) already requires, in principle, no spatial coherence in the direction parallel to the grating lines, in contrast with propagation-based methods

.

As the source mask G

can contain a large number of individual apertures, each creating a suﬃciently coherent virtual line source, standard X-ray generators with source sizes of more than a square millimetre can be used eﬃciently. To ensure that each line source produced by G

contributes constructively to the image-formation process, the geometry of the setup should satisfy the condition (Fig. 1b)

p

= p

× l d .

It is important to note that the total source size w only determines the final imaging resolution, which is given by wd/l . The arrayed source thus decouples spatial resolution from spatial coherence, and allows the use of X-ray illumination with coherence lengths as small as ξ

= l l/w ∼ 10

m in both directions, if the corresponding spatial resolution wd/l can be tolerated in the experiment. Finally, as a temporal coherence of ξ

≥ 10

m ( # E/E ≥ 10%) is suﬃcient

, we deduce that the method presented here requires the smallest minimum coherence volume ξ

× ξ

× ξ

for phase-sensitive imaging if compared with existing techniques.

The DPC image-formation process achieved by the two gratings G

and G

is similar to Schlieren imaging

and diﬀraction-enhanced imaging

α = l 2 π

∂ Φ (x, y)

∂ x ,

where x and y are the cartesian coordinates perpendicular to the optical axis, Φ ( x, y) represents the phase shift of the wavefront, and l is the wavelength of the radiation. For hard X-rays, with l < 0 . 1 nm, the angle is relatively small, typically of the order of a few microradians.

In our case, determination of the angle is achieved by the arrangement formed by G

and G

Talbot–Lau-type hard-X-ray imaging interferometer. a, Principle: the source grating (G

) creates an array of individually coherent, but mutually incoherent sources. A phase object in the beam path causes a slight refraction for each

coherent subset of X-rays, which is proportional to the local differential phase gradient of the object. This small angular deviation results in changes of the locally transmitted intensity through the combination of gratings G

and G

. A standard X-ray imaging detector is used to record the final images. b, Scanning electron micrographs of cross-sections through the gratings. The gratings are made from Si wafers using standard photolithography techniques, and subsequent electroplating to fill the grooves with gold (G

and G

).

, a mean energy of the eﬀective X-ray spectrum of E

The experimentally observed maxima of the integrated phase shifts are Φ

= 54 ± 2 π and Φ

= 32 ± 2 π (Fig. 2f), from which we obtain: δ

= 9 . 5 ± 0 . 8 × 10

and δ

= 5 . 9 ± 0 . 6 × 10

, which is in agreement with the values in the literature

. Note that because our method provides a quantitative measure of the absorption and integrated phase shift of the object in each pixel, the results can be used for further quantitative analysis, such as the reconstruction of a three-dimensional map of the real and

Transport of Intensity (TIE)

I i (z)

no object

I(z)

object

• Images corrected for lateral magnification

• Attenuation neglected

• Long propagation distance (short wavelength)

➦ two instead of four measurements k

z

✓ I(z)

I i (z) 1

◆

⇡ r 2 x

z

x-ray source

(with small spot size) x-ray detector

k @I

@z = r x · (Ir x )

I ⟹ ⇡ 1

Experimental geometry

= ^l l/γ

≥ ¹⁰

= ^l l/w ∼ ¹⁰

^{m in both} directions, if the corresponding spatial resolution wd/l can be tolerated in the experiment. Finally, as a temporal coherence of ξ

≥ ¹⁰

^{m (} # E/E ≥ 10%) is suﬃcient

α = ^l 2 π

= ⁵⁴ ± ² π and Φ

= ³² ± ² π (Fig. 2f), from which we obtain: δ

= ⁹ . 5 ± ⁰ . 8 × ¹⁰

^and δ

= ⁵ . 9 ± ⁰ . 6 × ¹⁰

^, which is in agreement with the values in the literature

• Long propagation distance  (short wavelength)  

➦ two instead of four measurements  k

I _i (z) 1

⇡ r ² x

@z = r ^x · (Ir ^x )

30 ^o

60 ^o

0 ^o