## Phase-contrast imaging using a scanning-double-grating configuration

Optics Express, Vol. 16, Issue 8, pp. 5849-5867 (2008)

http://dx.doi.org/10.1364/OE.16.005849

Acrobat PDF (314 KB)

### Abstract

A new double-grating-based phase-contrast imaging technique is described. This technique differs from the conventional double-grating imaging method by the image acquisition strategy. The novelty of the proposed method is in lateral scanning of both gratings simultaneously while an image is collected. The collected image is not contaminated by a Moiré pattern and can be recorded even by using a high-spatial-resolution integrating detector (e.g. X-ray film), thus facilitating improved resolution and/or contrast in the image. A detailed theoretical analysis of image formation in the scanning-double-grating method is carried out within the rigorous wave-optical formalism. The transfer function for the scanning-double- grating imaging system is derived. An approximate geometrical-optics solution for the image intensity distribution is derived from the exact wave-optical formula using the stationary-phase approach. Based on the present formalism, the effects of finite source size on the preferred operating conditions and of polychromaticity on the image contrast and resolution are investigated.

© 2008 Optical Society of America

## 1. Introduction

- X-ray interferometry [1–3] which is sensitive to the phase itself but modulo 2π;
1. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett.

**6**, 155–156 (1965). [CrossRef] - Differential phase-contrast methods, including analyzer-based imaging (ABI) [4–7] and grating-based imaging [8–18], which are sensitive to the phase derivative in a certain direction or to the phase gradient;
4. K. Goetz, M. P. Kalashnikov, Yu. A. Mikhailov, G. V. Sklizkov, S. I. Fedotov, E. Foerster, and P. Zaumseil, “Measurements of the parameters of shell targets for laser thermonuclear fusion using an x-ray schlieren method,” Sov. J. Quantum Electron.

**9**, 607–610 (1979). [CrossRef] - Near-field propagation-based imaging (PBI) [19–21] where the image contrast is proportional to the two-dimensional (2D) Laplacian of the phase.
19. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.

**66**, 5486–5492 (1995). [CrossRef]Some of the salient advantages and disadvantages of the above methods are briefly summarized in Table 1 below.

*x*-axis are fixed for each collected image. The relative offset of the second grating with respect to the first one, along the x-axis, may be changed in order to obtain different types of contrast in the images (e.g. dark-field and differential-contrast images) as well as for collecting multiple images (phase-stepping technique) and subsequent processing of them in order to extract a phase derivative distribution (see, for example, [17

17. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase Tomography by X-ray Talbot Interferometry for Biological Imaging,” Jpn. J. Appl. Phys. **45**, 5254–5262 (2006). [CrossRef]

18. Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-Ray Phase Imaging with Single Phase Grating,” Jpn. J. Appl. Phys.46, L89–L91 (2007). [CrossRef]

*x*-axis fixed for any given scan but adjustable) along the

*x*-axis while collecting an image. The resultant image has potentially better resolution and contrast (if used with a high resolution detector) compared to the standard double-grating imaging modality.

18. Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-Ray Phase Imaging with Single Phase Grating,” Jpn. J. Appl. Phys.46, L89–L91 (2007). [CrossRef]

*x*-axis while keeping their relative motion and orientation fixed to significantly less than a grating period.

## 2. Scanning double-screen image formation

### 2.1. Monochromatic plane incident wave

*OE*1 and

*OE*2 are the first and the second optical elements (e.g. slits or gratings) respectively, separated along the

*z*-axis by a distance

*R*

_{2}. Both optical elements are characterized by their complex transmission functions,

*t*

_{1}(

*x*) and

*t*

_{2}(

*x*), which are here assumed to depend only on the

*x*-coordinate. We restrict our consideration to the case where the first optical element is located immediately after an object, in the exit plane of the object (the object plane, for brevity), and the distance between the second optical element and the detector is negligibly small. Designating

*q*(

*x*,

*y*) as the complex transmission function of the object and assuming that a monochromatic plane wave (i.e.

*R*

_{1}≫

*R*

_{2}) of unit intensity and wavelength

*λ*is incident onto the object, the complex amplitude of the wave immediately after the

*OE*1 is written as follows,

*x*

_{1}is a position of the

*OE*1 along the

*x*-axis. The corresponding wave amplitude in the detector plane is then

*x*

_{1}+Δ

*x*is a position of

*OE*2 along the

*x*-axis (

*OE*2 is off-set a distance Δ

*x*with respect to

*OE*1),

*P*(

_{z}*x*,

*y*)=

*P*(

_{z}*x*)

*P*(

_{z}*y*)≡(

*iλz*)

^{-1}exp[

*iπ*(

*x*

^{2}+

*y*

^{2})/(

*λz*)] is a paraxial approximation for the two-dimensional (2D) free-space propagator at a distance

*z*, and the asterisk,*, between two functions denotes convolution of the functions. The intensity in the detector plane for the fixed positions

*x*

_{1}and

*x*

_{1}+Δ

*x*of the first and the second optical elements is written as follows,

*T*

_{2}(

*x*)≡|

*t*

_{2}(

*x*)|

^{2}is the transmittance function of the second optical element.

*x*, constant) along the

*x*-axis while the image is collected. At this stage we should distinguish between two cases: periodic and non-periodic optical elements. If both the optical elements are periodic (e.g. gratings), with period

*d*, then scanning over an integer number of periods is carried out (actually, scanning over one period is sufficient). In the opposite case of non-periodic optical elements (e.g. slits), scanning of the optical elements has to be carried out across the whole horizontal field of view [-

*A*,

*A*]. Mathematically, such scanning results in integration of the intensity

*I*(

_{det}*x*,

*y*,

*x*

_{1},Δ

*x*) over

*x*

_{1}in the interval

*L*, equal to [0,

*d*] and [-

*A*,

*A*] in the periodic and non-periodic case, respectively,

*y*-component of the free-space propagator to the object wave and

*x*-axis and the function

*G*(

*x*′,

*x*″) is defined as

*x*[this is indicated by superscript (1)] is written as follows:

*T̂*(

_{x}*u*,

*u′*;Δ

*x*) of the imaging system along the

*x*-axis (the Fourier transform of the propagation function of the imaging system) can be presented as

### 2.2. Partially coherent incident illumination

22. T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. **259**, 569–580 (2006). [CrossRef]

*x*,

*y*) and (

*x*′,

*y*″) are the Cartesian coordinates of two arbitrary points in the object plane and

*R*

_{1}is the distance from the source to the object. Taking into account transmission through the object and the first grating, propagation from the first to the second grating and transmission through the second grating, the spectral density in the detector plane, located immediately after the second grating, can be expressed as follows,

*x*

_{1,2}are the positions along the x-axis of the first and second gratings respectively. Equation (13) can be straightforwardly transformed to the equivalent form,

*M*≡(

*R*

_{1}+

*R*

_{2})/

*R*

_{1}is the geometrical magnification of the imaging system and

*R*′≡

*R*

_{1}

*R*

_{2}/(

*R*

_{1}+

*R*

_{2}) is the effective object-to-detector distance.

22. T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. **259**, 569–580 (2006). [CrossRef]

*W*in the incident cross-spectral density has the following form,

_{in}*S*is the spectral density of the incident wave and we allowed for an additional phase term φ

_{in}_{in}in the incident wave (apart from the explicit parabolic term in Eq. (12)). Substituting Eq. (15) into Eq. (14), one obtains

*Q*≡

*S*

^{1/2}

_{in}exp(

*iφ*)

_{in}*q*. Assuming that

*x*

_{2}=

*Mx*

_{1}+

*M*Δ

*x*and integrating Eq. (16) over

*x*

_{1}(therefore the second optical element is shifted

*M*times faster than the first one), one obtains the following expression for the spectral density in the detector plane,

*G*(

*x*,

*x*′) is defined as

*L*was defined in section 2.1. Fourier transforming to Eq. (17), one easily obtains the following general expression,

*T̂*(

_{sys}*u*,

*v*,

*u*′,

*v*′;

*λ*,Δ

*x*), corresponds to partially coherent incident illumination characterized by the spectral degree of coherence

*g*(

_{in}*x*′-

*x*,

*y*′-

*y*,

*λ*) that, according to the generalized Schell model used in this paper, depends only on the distance between two arbitrary points (

*x*,

*y*) and (

*x*′,

*y*′) in the plane of incidence. This “partially coherent” transfer function can be expressed in terms of the “ideal” transfer function,

*T̂*(

_{id}*u*,

*v*,

*u*′,

*v*′,;

*λ*,Δ

*x*) that corresponds to coherent incident illumination with

*g*≡1, as

_{in}*S*(

_{det}*x*,

*y*,

*λ*;Δ

*x*), the corresponding intensity distribution

*I*(

_{det}*x*,

*y*;Δ

*x*) in the detector plane is calculated as follows,

24. Ya. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Cryst. A **62**, 296–308 (2006). [CrossRef]

*Therefore the object wave phase/amplitude reconstruction algorithms*[24

24. Ya. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Cryst. A **62**, 296–308 (2006). [CrossRef]

*developed for analyser-based imaging (using either the weak-object approximation or the geometrical-optics approximation) can be successfully applied to the scanning double-grating imaging, though with a different definition for the transfer function of the imaging system.*

### 2.3. Geometrical optics approximation

*Q*is slowly varying compared to the system transmission function

*T*. Then applying the stationary-phase method [25

_{sys}25. M. V. Fedoryuk, “The stationary phase method and pseudodifferential operators,” Russ. Math. Surveys **26**, 65–115 (1971). [CrossRef]

*S*

_{0}≡|

*Q*|

^{2}=

*S*

_{in}|

*q*|

^{2},φ≡arg(

*Q*) and

*u*

_{0}≡-(2π)

^{-1}∂

_{x}φ(

*x*,

*y*). Note that a similar result has been obtained for the analyzer-based imaging system [24

24. Ya. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Cryst. A **62**, 296–308 (2006). [CrossRef]

*r*(

_{sys}*θ*)≡∫∫

*dUdV*

*ĝ*(

_{in}*U*,

*V*,

*λ*)

*r*(

_{id}*θ*+

*λU*) and

*r*(

_{id}*θ*)≡

*T̂*(0,0,0,0;

_{id}*λ*,

*R*′

*θ*). The newly introduced functions

*r*(

_{sys}*θ*) and

*r*(

_{id}*θ*) are analogous to the rocking curve and the intrinsic reflectivity curve of the analyzer crystal in analyzer-based imaging, respectively [24

**62**, 296–308 (2006). [CrossRef]

*u*

_{0}, depend on the wavelength. In order to obtain a formula for the intensity distribution in the detector plane that could be used for the phase/amplitude reconstruction of the object wave, we introduce the following assumptions. We assume that the spectrum of the incident beam is narrow, so that

*Δλ*/

*λ*<0.1, and is far from the absorption edges of the materials constituting the object and the gratings. Then introducing the refraction angle,

*α*(

*x*,

*y*;

*λ*)≡

*λu*(

_{0}*x*,

*y*;

*λ*), and the absorption function of the object,

*b*(

*x*,

*y*;

*λ*)≡-(1/2)ln(|

*q*(

*x*,

*y*;

*λ*)|

^{2}), both the refraction angle and the absorption function can be well approximated using the following linear decompositions,

*ε*≡

*λ*/

*λ*

_{0}-1<0.1,

*λ*

_{0}is some ‘central’ value of the wavelength (an exact definition for the

*λ*

_{0}is derived below), and

*k*(

*x*,

*y*) is the slope coefficient of the absorption function for a point (

*x*,

*y*) in the object plane (its value depends on the chemical composition of the object’s voxels contributing to the absorption at this point of the object plane).

*r*(

_{sys}*θ*) of the imaging system and that the absorption function of the object does not exceed one, the product 2

*ε*

*α*(

*x*,

*y*;

*λ*

_{0}) is small compared to the period of the rocking curve and the product 2

*ε b*(

*x*,

*y*;

*λ*

_{0})

*k*(

*x*,

*y*) is small compared to one. This allows us to present the right-handside of Eq. (26) and the transmission function of the object, exp[-2

*b*(

*x*,

*y*;

*λ*)], as follows (

*θ*

_{0}≡Δ

*x*/

*R*′),

*S*(

_{in}*x*,

*y*;

*λ*)=

*S*(

_{in,spat}*x*,

*y*)

*S*(

_{in,spec}*λ*)),

*θ*(

*x*,

*y*;

*λ*

_{0})≡

*θ*

_{0}+

*α*(

*x*,

*y*;

*λ*

_{0}). It is convenient to choose the ‘central’ wavelength

*λ*

_{0}=

*λ*

_{0}(

*x*,

*y*) such that the integral over

*λ*in Eq. (30) is zero. Then the image formation in the polychromatic geometrical-optics approximation is described by the following simple formula,

*Δλ*with respect to the wavelength

*λ*for which the Talbot self-imaging condition is satisfied (see Eq. (45) below). If the spectrum distribution

_{T}*S*(

_{in,spec}*λ*) is an even function with respect to some wavelength

*λ*(this is the mean wavelength in the spectrum) then by choosing

_{c}*λ*=

_{T}*λ*, the integral in Eq. (30) is equal to zero for all refraction angles

_{c}*α*if the wavelength

*λ*

_{0}is chosen to be equal to

*λ*. Then applying Eq. (31) to the reconstruction of the refraction angle distribution

_{c}*α*(

*x*,

*y*) and the absorption function

*b*(

*x*,

*y*), both these reconstructed distributions correspond to the wavelength

*λ*. Equation (31) closely resembles the geometrical-optics approximation in analyzer-based imaging [24

_{c}**62**, 296–308 (2006). [CrossRef]

## 3. Scanning double-grating imaging using partially coherent incident illumination

*d*be the period of the first grating and let

*Md*be the corresponding period of the second grating. It is convenient to present both the transmission function

*t*

_{1}(

*x*) of the first grating and the transmittance function

*T*

_{2}(

*x*) of the second grating in the form of Fourier series,

*g*=1, giving

_{in}*x*and Δ

*y*,

*i.e.*

*Ŝ*

_{⊥}(

*u*,

*v*,

*λ*;Δ

*x*) and

*T̂*(

_{x}*u*,

*u*′;

*λ*,Δ

*x*) are defined below,

*Q*′ designates a result of applying only the

_{R}*y*-component of the free-space propagator to the object transmission function (

*y*-axis is parallel to the grating lines),

## 4. Numerical results and discussion

*M*, the effective source size in the object plane is expressed via the source size,

*w*(FWHM), as follows

_{S}*M*≥1.

*d*of the self-image (referred hereafter to the object plane), i.e.

*n*should be sufficiently smaller than one (

*n*=1/8 found to be sufficient).

- source size does not exceed the critical value,
*w*≤_{S}*nd*; - source size is larger than the critical value,
*w*>_{S}*nd*.

### 4.1. Small X-ray source

16. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE **6318**, 63180S (2006). [CrossRef]

*d*

_{1}is a period of the first grating, the integer number

*m*is the Talbot order which should be odd for a phase grating and even for an amplitude grating. The factor

*η*depends on the choice of the first grating. In the case of either a phase grating with the phase modulation φ

_{0}=π/2 or an amplitude grating the corresponding value is

*η*=1. In this case the period of the self-image is equal to the period of the first grating,

*d*=

*d*

_{1}. Another widely used configuration corresponds to the first phase grating having phase modulation φ

_{0}=π. In this case the corresponding value of the factor is

*η*=2 and the period of the self-image is half of the first grating period, i.e.

*d*=

*d*

_{1}/2. In both cases,

*d*=

*d*

_{1}/

*η*and the period of the second (amplitude) grating is chosen to be equal to the period of the corresponding self-image multiplied by the magnification, i.e.

*d*

_{2}=

*M*

*d*

_{1}/

*η*.

*R*′, is expressed via the magnification,

*M*, and the source-to-detector distance,

*R*, as follows,

*d*

_{2}of the amplitude grating is an independent parameter. Then the period of the phase grating is defined as follows,

*d*

_{1}=

*η*

*d*

_{2}/

*M*. Equating

*R*′, given by Eq. (46), to the Talbot distance

*z*, Eq. (45), one can easily obtain the following equation for the appropriate magnification as a function of

_{m}*d*

_{2},

*λ*and

*R*,

*d*/

*R*′. The smaller this ratio the more the GBI system is sensitive to small deflection angles. This angular acceptance can be presented in terms of

*d*

_{2},

*λ*and

*R*as follows,

*R*and

*λ*, the ratio

*d*/

*R*′ has large values for both small and large values of

*d*

_{2}and has a minimum at the optimum value of

*d*

_{2},

*λ*or by increase of the total distance

*R*. Note, however, that the deflection angles due to the object are inversely proportional to the second power of

*λ*which results in decrease in the image contrast for smaller wavelengths, as

*λ*

^{3/2}. On the other hand, only a large increase in the total distance can result in significant improvement of the contrast (for example, in order to double the differential contrast,

*R*should be increased four times). It should be mentioned that according to Eq. (47), the optimum period of the second grating (

*d*

_{2})

_{opt}corresponds to magnification

*M*=2 independently of the x-ray energy and total distance. According to Eq. (44), the maximum allowable source size corresponding to the optimum period (

*d*

_{2})

_{opt}is

*w*

_{S,max}=(

*n*/2)(2

*λR*/

*m*)

^{1/2}. Remarkably, the optimum period (

*d*

_{2})

_{opt}not only minimises the ratio

*d*/

*R*′ but also maximises the period of the self image

*d*, namely

*R*=250mm,

*d*

_{2}=24µm,

*λ*=3Å and

*m*=1. Then according to Eq. (47) the corresponding magnification is

*M*=4.84. At the same time the source size should satisfy Eq. (44); assuming

*n*=1/8 this gives

*w*≤0.62µm. These and other important geometrical parameters are summarized in Table 2 below. Some simulated images of a spherical pure phase object of diameter 0.5mm (maximum phase shift is 12 radians at the wavelength

_{S}*λ*=3Å) which was radially smeared using a Gaussian function with 100µm FWHM are shown in Fig. 2. The contrast in the images was calculated using the following formula,

*C*′≡(

*I*

_{max}-

*I*

_{min})/2, where

*I*

_{max}and

*I*

_{min}are correspondingly the maximum and minimum intensity values in the images of the object (hereafter a unit intensity of the incident wave is assumed).

*λ*=3Å), the magnification

*M*changes almost as the second power of the period

*d*

_{2}of the second (amplitude) grating:

*M*=1.96 in the case of

*d*

_{2}=12µm,

*M*=4.84 in the case of

*d*

_{2}=24µm and

*M*=16.36 in the case of

*d*

_{2}=48µm. The first of the above three values of

*d*

_{2}, namely

*d*

_{2}=12µm is close to the optimum value, Eq. (49), (

*d*

_{2})

_{opt}=12.25µm for the chosen

*R*=250mm and

*λ*=3Å. The maximum source size and contrast decrease when

*d*

_{2}deviates from (

*d*

_{2})

_{opt}.

### 4.2. Large X-ray source

*nd*is written as follows,

*d*

_{2}=

*Md*of the second (amplitude) grating as an independent variable, the maximum magnification that still satisfies Eq. (43) is written as follows,

*R*, and the angular acceptance of the period of the self-image,

*d*/

*R*′, are written in terms of

*d*

_{2},

*λ*and

*M*as follows,

*d*

_{2},

*λ*and

*M*. For example, in terms of magnification, the maximum allowable magnification, Eq. (53), minimizes the total distance and if one assumes that the

*M*

_{max}does not exceed 2, then the maximum increase of the ratio

*d*/

*R*′ due to the magnification is 2 (note that the optimum magnification minimizing this ratio is 1). In the following we shall assume that

*M*=

*M*

_{max}which results in the transformation of Eq. (54) to

*w*, decreasing the second grating period,

_{S}*d*

_{2}, or increasing the x-ray wavelength

*λ*. However, decrease of both

*w*and

_{S}*d*

_{2}results in increase of the ratio

*d*/

*R*′ and, as a result, in decrease of the contrast. Notwithstanding the increase of the ratio

*d*/

*R*′ with the x-ray wavelength, the overall effect of the wavelength is positive for the contrast (differential contrast is proportional to

*λ*). Note, however, that the total absorbed dose increases significantly with increase of

*λ*(the linear absorption coefficient is approximately proportional to

*λ*

^{3}). Thus there are two definite tradeoffs, the first one is between the contrast and the total distance, and the second one is between the contrast and the absorbed dose.

*d*and the distance

*R*

_{2}between the gratings increase only slightly. Also, according to Eq. (55), the total source-to-detector distance,

*R*, is proportional to the source size. At the same time, the angular acceptance,

*d*/

*R*′, slightly decreases with increasing source size; this results only in slight improvement of the contrast in the SDG images. Thus, in order to use grating-based imaging in laboratory conditions (with the total source-to-detector distance limited to several meters) the source size should not exceed ~10µm. If the total source-to-detector distance can be made significantly larger, of the order of 20–100m (as at synchrotrons) then the source size can be of the order of 100µm which is typical for most modern synchrotrons. Note however that the potential improvement in the contrast achievable by going to large distances and large sizes of the source is quite moderate. The greatest advantage of using synchrotron sources is the many orders of magnitude higher intensity in the incident beam. In order to be able to use standard (laboratory) X-ray sources (with the focus size of the order of several hundred microns), an additional amplitude grating

*G*0 should be mounted in front of the source [8,14]. The period of this grating is calculated according to the following formula,

*d*

_{0}=(

*R*

_{1}/

*R*

_{2})

*d*

_{2}. Such a grating produces an array of line sourcelets whose width (the width of a transparent part of the grating period) should be chosen appropriately, according to our considerations above. This would guarantee high performance of the gratings (visibility of fringes in the self-image and contrast in the image of the object formed by each individual sourcelet). Note, however, that the total spatial resolution of the system in this case is limited by the total source size, not the size of an individual sourcelet.

*d*

_{2}results in the period of the self-image decreasing almost linearly with

*d*

_{2}. Also, and more importantly, the distance

*R*

_{2}between the gratings decreases almost as the second power of

*d*

_{2}. This reduction in the distance between the gratings is, however, accompanied by linear increase of the ratio

*d*/

*R*′ and results in linear decrease of the contrast in the images. The source-to-detector distance

*R*also decreases linearly with decrease of

*d*

_{2}. Thus, the amplitude and phase grating with small period are preferable if one needs to minimize the overall size of the SDG imaging system. Note however that this compactness is achieved at the expense of the system effectiveness (contrast in the images).

*R*and

*R*

_{2}. Notwithstanding that the ratio

*d*/

*R*′ decreases two times given the two times decrease of the wavelength, the contrast in the SDG images decreases two times. This effect is shown in Fig. 3 where the simulated images of a spherical object of diameter 0.5mm, radially smeared using a Gaussian function with 100µm FWHM, are shown. The observed reduction of contrast in the image calculated using 0.5Å wavelength as compared to the image calculated using 1Å wavelength is due to the fact that the deflection angles induced by the object decrease four times. This is because the deflection angle is proportional to the second power of the wavelength. Thus, by increasing X-ray energy, on the one hand, one can deliver significantly smaller radiation dose to the sample (the linear absorption coefficient is inversely proportional to the third power of the X-ray energy); on the other hand, performance of the imaging system (contrast in the images) decreases linearly with increasing X-ray energy.

### 4.3. Effect of polychromaticity on the SDG image formation

13. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**, 6295–6304 (2005). [CrossRef]

17. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase Tomography by X-ray Talbot Interferometry for Biological Imaging,” Jpn. J. Appl. Phys. **45**, 5254–5262 (2006). [CrossRef]

- The complex transmission function of an object is wavelength dependent. In the simplest case of the whole spectrum of the source being far from the absorption edges of the materials constituting the object, a phase induced by the object varies linearly with the wavelength and an absorption coefficient varies as the third power of the wavelength.
- The phase induced by the first (phase) grating varies linearly with the X-ray wavelength (assuming that the whole spectrum of the source is far from the absorption edges of the material of the phase grating). Thickness of the lines in the second (amplitude) grating is assumed to be sufficiently large so that the transmittance of the lines in the grating is zero for all the energies in the spectrum of the source.
- The Talbot distances are inversely proportional to the X-ray wavelength. Therefore, if the distance between the gratings is equal to one of the Talbot distances for a particular wavelength and a self-image is observed for that wavelength, the chosen (fixed) distance does not coincide with Talbot distances for other wavelengths.

**17**

17. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase Tomography by X-ray Talbot Interferometry for Biological Imaging,” Jpn. J. Appl. Phys. **45**, 5254–5262 (2006). [CrossRef]

_{0}=π/2,

13. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**, 6295–6304 (2005). [CrossRef]

_{0}=π the maximum allowable polychromaticity, that preserves efficiency of the interferometer, is defined as follows,

*m*=1,3,5,… is the order of the Talbot distance used. We should note however that dispersion in the object (sample) and in the gratings was ignored in this estimation (the dispersion effects in the gratings were assessed in [13

13. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**, 6295–6304 (2005). [CrossRef]

*λ*=3Å was 12 radians. The phase modulation in the first (phase) grating was π/2 at the wavelength 3Å. The period

*d*of the gratings was 8µm. The corresponding Talbot distance was calculated using Eq. (45) with

*m*=1 and

*η*=1,

*z*

_{1}=

*d*

^{2}/(2λ)=0.107m.

*λ*) have been generated (see Fig. 4(a)). The differential (Δ

*x*=

*d*/4) and dark-field (Δ

*x*=

*d*/2) images of the model object have been calculated assuming monochromatic incident radiation as well as using the generated spectra. Some characteristics of the images (including minimum and maximum intensities and the contrast) are summarized in Table 4. The dark-field images are shown in Fig. 5.

*I*in the dark-field images increases considerably with the increase in the degree of polychromaticity.

_{BG}*θ*in Fig. 4(b)). Analysis of Fig. 4(c) shows that the magnitude of the ‘reflectivity’’ derivative at this working point remains almost unchanged. On the other hand, contrast in dark-field images is proportional to the second derivative of the ‘reflectivity’ curve in the working point positioned at the minima of the ‘reflectivity’ curve (this corresponds to

*θ*≈4 arcsec in Fig. 4(b)–(d)). This second derivative is shown in Fig. 4(d). Analysis of this figure shows that the magnitude of the second derivative monotonically decreases.

## 5. Conclusion

## References and links

1. | U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. |

2. | M. Ando and S. Hosoya, “An attempt at x-ray phase-contrast microscopy,” in Proc. 6th Intern. Conf. On X-ray Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinokawa Eds. (Univ. of Tokyo Press, Tokyo, 1972) pp. 63–68. |

3. | A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Methods A |

4. | K. Goetz, M. P. Kalashnikov, Yu. A. Mikhailov, G. V. Sklizkov, S. I. Fedotov, E. Foerster, and P. Zaumseil, “Measurements of the parameters of shell targets for laser thermonuclear fusion using an x-ray schlieren method,” Sov. J. Quantum Electron. |

5. | V. A. Somenkov, A. K. Tkalich, and S. Sh. Shil’shtein, “Refraction contrast in x-ray introscopy,” Sov. Phys. Tech. Phys. |

6. | V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. |

7. | T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature |

8. | J. F. Clauser, “Ultrahigh resolution interferometric X-ray imaging,” US patent No. 5,812,629 (1998). |

9. | C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. |

10. | A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. |

11. | C. David, “Apparatus and method to obtain phase contrast x-ray images,” Europ. patent No. EP 1,447,046-A1; Internat. publ. No. WO 2004/071298-A1; Aust. patent No. AU 2003/275964-A1 (2004). |

12. | T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. |

13. | T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express |

14. | C. David, T. Weitkamp, and F. Pfeiffer, “Interferometer for quantitative phase contrast imaging and tomography with an incoherent polychromatic x-ray source,” Europ. patent No. EP 1,731,099-A1; Internat. publ. No. WO 2006/131235-A1 (2006). |

15. | F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Physics |

16. | T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE |

17. | A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase Tomography by X-ray Talbot Interferometry for Biological Imaging,” Jpn. J. Appl. Phys. |

18. | Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-Ray Phase Imaging with Single Phase Grating,” Jpn. J. Appl. Phys.46, L89–L91 (2007). [CrossRef] |

19. | A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. |

20. | S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature |

21. | P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. |

22. | T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. |

23. | L. Mandel and E. Wolf, |

24. | Ya. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Cryst. A |

25. | M. V. Fedoryuk, “The stationary phase method and pseudodifferential operators,” Russ. Math. Surveys |

**OCIS Codes**

(110.4850) Imaging systems : Optical transfer functions

(110.4980) Imaging systems : Partial coherence in imaging

(110.6760) Imaging systems : Talbot and self-imaging effects

(110.7440) Imaging systems : X-ray imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 24, 2008

Revised Manuscript: February 21, 2008

Manuscript Accepted: February 21, 2008

Published: April 11, 2008

**Citation**

Ya. I. Nesterets and S. W. Wilkins, "Phase-contrast imaging using a scanning-doublegrating configuration," Opt. Express **16**, 5849-5867 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5849

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### References

- U. Bonse and M. Hart, "An x-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965). [CrossRef]
- M. Ando and S. Hosoya, "An attempt at x-ray phase-contrast microscopy," in Proc. 6th Intern. Conf. On X-ray Optics and Microanalysis, G. Shinoda, K. Kohra and T. Ichinokawa Eds. (Univ. of Tokyo Press, Tokyo, 1972) pp. 63-68.
- A. Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods A 352, 622-628 (1995). [CrossRef]
- K. Goetz, M. P. Kalashnikov, Yu. A. Mikhailov, G. V. Sklizkov, S. I. Fedotov, E. Foerster, and P. Zaumseil, "Measurements of the parameters of shell targets for laser thermonuclear fusion using an x-ray schlieren method," Sov. J. Quantum Electron. 9, 607-610 (1979). [CrossRef]
- V. A. Somenkov, A. K. Tkalich, and S. Sh. Shil’shtein, "Refraction contrast in x-ray introscopy," Sov. Phys. Tech. Phys. 36, 1309-1311 (1991).
- V. N. Ingal and E. A. Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D: Appl. Phys. 28, 2314-2317 (1995). [CrossRef]
- T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature 373, 595-598 (1995). [CrossRef]
- J. F. Clauser, "Ultrahigh resolution interferometric X-ray imaging," US patent No. 5,812,629 (1998).
- C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002). [CrossRef]
- A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, "Demonstration of X-Ray Talbot Interferometry," Jpn. J. Appl. Phys. 42, L866-L868 (2003). [CrossRef]
- C. David, "Apparatus and method to obtain phase contrast x-ray images," Europ. patent No. EP 1,447,046-A1; Internat. publ. No. WO 2004/071298-A1; Aust. patent No. AU 2003/275964-A1 (2004).
- T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, "X-ray wavefront analysis and optics characterization with a grating interferometer," Appl. Phys. Lett. 86, 054101 (2005). [CrossRef]
- T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, "X-ray phase imaging with a grating interferometer," Opt. Express 13, 6295-6304 (2005). [CrossRef]
- C. David, T. Weitkamp, and F. Pfeiffer, "Interferometer for quantitative phase contrast imaging and tomography with an incoherent polychromatic x-ray source," Europ. patent No. EP 1,731,099-A1; Internat. publ. No. WO 2006/131235-A1 (2006).
- F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nat. Phys. 2, 258-261 (2006). [CrossRef]
- T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, "Tomography with grating interferometers at low-brilliance sources," Proc. SPIE 6318, 63180S (2006). [CrossRef]
- A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, "Phase Tomography by X-ray Talbot Interferometry for Biological Imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006). [CrossRef]
- Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, "X-Ray Phase Imaging with Single Phase Grating," Jpn. J. Appl. Phys. 46, L89-L91 (2007). [CrossRef]
- A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995). [CrossRef]
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335-338 (1996). [CrossRef]
- P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996). [CrossRef]
- T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569-580 (2006). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
- Ya. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, "On qualitative and quantitative analysis in analyser-based imaging," Acta Cryst. A 62, 296-308 (2006). [CrossRef]
- M. V. Fedoryuk, "The stationary phase method and pseudodifferential operators," Russ. Math. Surveys 26, 65-115 (1971). [CrossRef]

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