## A low-absorption x-ray energy filter for small-scale applications

Optics Express, Vol. 17, Issue 14, pp. 11388-11398 (2009)

http://dx.doi.org/10.1364/OE.17.011388

Acrobat PDF (1260 KB)

### Abstract

We present an experimental and theoretical evaluation of an x-ray energy filter based on the chromatic properties of a prism-array lens (PAL). It is intended for small-scale applications such as medical imaging. The PAL approximates a Fresnel lens and allows for high efficiency compared to filters based on ordinary refractive lenses, however at the cost of a lower energy resolution. Geometrical optics was found to provide a good approximation for the performance of a flawless lens, but a field-propagation model was used for quantitative predictions. The model predicted a 0.29 Δ*E/E* energy resolution and an intensity gain of 6.5 for a silicon PAL at 23.5 keV. Measurements with an x-ray tube showed good agreement with the model in energy resolution and peak energy, but a blurred focal line contributed to a 29% gain reduction. We believe the blurring to be caused mainly by lens imperfections, in particular at the periphery of the lens.

© 2009 Optical Society of America

## 1. Introduction

1. P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, “A prototype of a quasi-monochromatic system for mammography applications,” Phys. Med. Biol. **50**, 225–240 (2005).
[CrossRef]

3. F. Sugiro, D. Li, and C. MacDonald, “Beam collimation with polycapillary x-ray optics for high contrast high resolution monochromatic imaging,” Med. Phys. **31**, 3288–3297 (2004).
[CrossRef]

6. E. Fredenberg, B. Cederström, M. Åslund, C. Ribbing, and M. Danielsson, “A Tunable Energy Filter for Medical X-Ray Imaging,” X-Ray Optics and Instrumentation 2008, Article ID 635024, 8 pages (2008), http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/635024.

7. W. Jark, “A simple monochromator based on an alligator lens,” X-Ray Spectrom. **33**, 455–461 (2004).
[CrossRef]

4. J. Motz and M. Danos, “Image information content and patient exposure,” Med. Phys. **5**, 8–22 (1978).
[CrossRef] [PubMed]

6. E. Fredenberg, B. Cederström, M. Åslund, C. Ribbing, and M. Danielsson, “A Tunable Energy Filter for Medical X-Ray Imaging,” X-Ray Optics and Instrumentation 2008, Article ID 635024, 8 pages (2008), http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/635024.

7. W. Jark, “A simple monochromator based on an alligator lens,” X-Ray Spectrom. **33**, 455–461 (2004).
[CrossRef]

8. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature **384**, 49–51 (1996).
[CrossRef]

*π*removed. If this strategy is applied to the MPL, the extremely high aspect ratios that would otherwise be needed for a large-aperture Fresnel lens for hard x-rays are avoided. In the prism-array lens (PAL) [10

10. B. Cederström, C. Ribbing, and M. Lundqvist, “Generalized prism-array lenses for hard x-rays,” J. Synchrotron Rad. **12**, 340–344 (2005).
[CrossRef]

11. W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, “Focusing X-rays with simple arrays of prism-like structures,” J. Synchrotron Rad. **11**, 248–253 (2004).
[CrossRef]

12. L. D. Caro and W. Jark, “Diffraction theory applied to X-ray imaging with clessidra prism array lenses,” J. Synchrotron Rad. **15**, 176–184 (2008).
[CrossRef]

13. C. Fuhse and T. Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. **45**, 4603–4608 (2006).
[CrossRef] [PubMed]

19. S. Panknin, A. K. Hartmann, and T. Salditt, “X-ray propagation in tapered waveguides: Simulation and optimization,” Opt. Commun. **281**, 2779–2783 (2008).
[CrossRef]

14. Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to X-ray diffraction optics,” Opt. Commun. **118**, 619–636 (1995).
[CrossRef]

15. V. Kohn, I. Snigireva, and A. Snigirev, “Diffraction theory of imaging with X-ray compound refractive lens,” Opt. Commun. **216**, 247–260 (2003).
[CrossRef]

12. L. D. Caro and W. Jark, “Diffraction theory applied to X-ray imaging with clessidra prism array lenses,” J. Synchrotron Rad. **15**, 176–184 (2008).
[CrossRef]

## 2. Geometrical optics approach to the PAL

### 2.1. PAL focusing

*x*is the optical axis,

*y*is the focusing direction, and

*z*is the depth of the lens. The projection of the two prism rows of the MPL approximates a parabola with straight line segments [9]. It is a planar lens and therefore focuses radiation into a line focus. In a PAL, each large prism in the MPL is exchanged for a column of smaller prisms with height

*h*, base

*b*, and prism angle

*θ*[10

10. B. Cederström, C. Ribbing, and M. Lundqvist, “Generalized prism-array lenses for hard x-rays,” J. Synchrotron Rad. **12**, 340–344 (2005).
[CrossRef]

*y*-displacement increases in steps of

*d*along the

*x*-axis, and the focal lengths of the two lenses are the same if

*θ*and

*d*are equal. The projection of the PAL approximates a Fresnel lens, superimposed on a linear profile with slope

*K*=

*γb*/2

*h*for

*γ*>1. A large

*γ*=

*h/d*indicates a good approximation of the ideal Fresnel pattern, however at the cost of a steep linear profile and high absorption.

*b*corresponds to an x-ray phase shift of 2

*πq*, where

*q*is an integer, the blazing condition is fulfilled, and

*kbδ*=2

*πq*for a wave with propagation number

*k*=2

*π/λ*. The deflection of the first diffraction order then equals the refractive deflection of a single prism, and

*F*

_{diff}=

*F*

_{ref}[12

12. L. D. Caro and W. Jark, “Diffraction theory applied to X-ray imaging with clessidra prism array lenses,” J. Synchrotron Rad. **15**, 176–184 (2008).
[CrossRef]

*q*=1 is assumed, and parameters that fulfill the blazing condition are indicated with an asterisk, in particular the design energy of the lens (

*E**).

*F*

_{diff}≠

*F*

_{ref}, and the question arises which one will be dominating. In accordance with [16], the fraction radiation in the mth diffraction order of a blazed saw-tooth grating can be shown to be proportional to [(2

*πm*-

*kbδ*)

^{2}+(

*kbβ*)

^{2}]

^{-1}for given

*k, b, δ*, and

*β*. If

*kbδ*=2

*π*, virtually all of the deflected intensity is found in the first diffraction order because (

*kbβ*)

^{2}is small. In fact, as long as

*π*<

*kbδ<*3π, or

^{2}/3

*E**<

*E*<2

*E** for

*δ*∝

*E*

^{-2}and

*b*=

*b**, the peak intensity in the focal plane is found at

*m*=1. If we define the focus to be at the peak intensity, the focal length of the PAL is equal to

*F*

_{diff}in the stated interval. Note, however, that the intensity in the focus decreases in favor of secondary maxima when the blazing condition is violated, and the focal line is blurred along the optical axis.

### 2.2. PAL energy filtering

*s**

_{i}will transmit

*E**, whereas other energies from a polychromatic incident beam are out of focus and preferentially blocked. This energy filtering scheme is outlined in Fig. 2, and is henceforth referred to as the PAL filter. It is similar to previously presented MPL filters [6

6. E. Fredenberg, B. Cederström, M. Åslund, C. Ribbing, and M. Danielsson, “A Tunable Energy Filter for Medical X-Ray Imaging,” X-Ray Optics and Instrumentation 2008, Article ID 635024, 8 pages (2008), http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/635024.

7. W. Jark, “A simple monochromator based on an alligator lens,” X-Ray Spectrom. **33**, 455–461 (2004).
[CrossRef]

*F*∝

*E*.

*s**

_{i}, with a width

*d*. The intensity gain of the focused radiation compared to the incident beam is [6

_{s}*y*

_{1}refers to the plane of the lens, and the integral is taken over the lens aperture (

*D*

_{p}). The lens transmission has been derived previously [10

10. B. Cederström, C. Ribbing, and M. Lundqvist, “Generalized prism-array lenses for hard x-rays,” J. Synchrotron Rad. **12**, 340–344 (2005).
[CrossRef]

*η*

_{t}(

*y*

_{1})=exp(-

*Kµ*|

*y*

_{1}|), where

*µ*is the linear attenuation coefficient.

*η*is the part of the image (

_{i}*f*

_{i}) from each point on the lens that falls within the slit, i.e.

*y*

_{2}refers to the plane of the slit.

*κ*(

*E*)=[

*s**

_{i}-

*s*

_{i}(

*E*)]/

*s*

_{i}(

*E*) accounts for blurring at

*s**

_{i}, and is zero at

*E*

^{*}.

*d*

_{0}, and

*d*equal to the FWHM of the image (

_{s}*d**

_{i}), i.e.

*d*

_{s}=

*d**

_{i}=

*d*

_{0}

*s**

_{i}/

*s*

_{o}. For an infinitely large lens at

*E*

^{*},

*G**

_{∞}can be increased indefinitely by minimizing

*h*, but clearly there are physical constraints. Dependence on lens material is represented by the factor

*δ*

^{*}/

*µ*

^{*}, which grows with a decreasing atomic number.

*G*

_{∞}(

*E*)=0.5

*G*

^{*}

_{∞}for

*F*

_{0.5}is the focal length at

*G*

_{∞}(

*E*)=0.5

*G**

_{∞}. Assuming high gain and taking

*F*∝

*E*, we find an approximate expression for the energy resolution of a filter with an infinitely large lens,

*E*is the FWHM of the gain peak.

*γ*is favorable from an absorption point of view, but there is a tradeoff in representing the Fresnel pattern correctly.

## 3. Physical optics approach to the PAL: field propagation

### 3.1. The parabolic wave equation

*∂*

^{2}

*ψ/∂x*

^{2}

*+∂*

^{2}

*ψ/∂y*

^{2}+

*k*

^{2}

*n*

^{2}(

*x,y*)

*ψ*=0. For a wave

*ψ*travelling nearly parallel to the

*x*-axis, Δ

*x*≫Δ

*y*, and the paraxial approximation is applicable. Hence,

*ψ*=

*ψ*

_{0}exp[-

*ikn*(Δ

*x*

^{2}+Δ

*y*

^{2})

^{1/2}]≈

*u*(

*x,y*)exp(-

*ikn*

_{0}Δ

*x*) with

*u*varying slowly with

*x*and

*n*

_{0}being a reference refractive index (1 for vacuum) [13

13. C. Fuhse and T. Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. **45**, 4603–4608 (2006).
[CrossRef] [PubMed]

13. C. Fuhse and T. Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. **45**, 4603–4608 (2006).
[CrossRef] [PubMed]

19. S. Panknin, A. K. Hartmann, and T. Salditt, “X-ray propagation in tapered waveguides: Simulation and optimization,” Opt. Commun. **281**, 2779–2783 (2008).
[CrossRef]

14. Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to X-ray diffraction optics,” Opt. Commun. **118**, 619–636 (1995).
[CrossRef]

### 3.2. The Kirchhoff diffraction integral

*n*=

*n*

_{0}), a Fourier transform of Eq. (7) with respect to

*y*yields -2

*ikn*

_{0}

*dU/dx-υ*

^{2}

*=0, with*

_{y}U*U*being the Fourier transform of

*u*, and

*υ*the spatial frequency in the

_{y}*y*-direction. The solution of this differential equation is

*U*

_{0}is the transform of the initial field. The inverse Fourier transform is a convolution,

*u*=(

*u*

_{0}∗

*h*)(

*y*), known as the paraxial Kirchhoff diffraction integral. It can be solved directly in the Fourier domain, according to Eq. (8), to speed up the calculation.

### 3.3. Finite differences

*x*- and

*y*-axes within the computational window be divided into

*P*and

*Q*segments, respectively, so that the electric field is discretized into local fields

*u*with

^{p}_{q}*p*∊ {1,2,…,

*P*} and

*q*∊ {1,2,…,

*Q*}. By using a Taylor expansion, the field and its derivatives at the center of segment

*p*can be expressed as

**45**, 4603–4608 (2006).
[CrossRef] [PubMed]

### 3.4. Field-propagation model

*F*∝

*E*does not hold. Therefore, for more reliable, quantitative predictions than provided by the GM model, we have used the methods described in the last three sections to assemble a field-propagation (FP) model of the PAL filter. Note that the FP model too has restricted validity, mainly because of the paraxial approximation.

*x*. A monochromatic finite x-ray source of energy

*E*was treated as a superposition of point sources, incoherent relative to each other, and the intensity at any point in the setup was found from the field

*u*

_{E}as

*I*

_{E}(

*x,y*)=∑|

*u*

_{E,y0}(

*x,y*)|

^{2}, where

*y*

_{0}is a point in the plane of the source. Accordingly, a polychromatic x-ray source was regarded a superposition of monochromatic sources over a distribution of energies. G was found by integrating

*I*over a slit in the image plane.

21. M. Berger, J. Hubbell, S. Seltzer, J.S., Coursey, and D. Zucker, *XCOM: Photon Cross Section Database*, (National Institute of Standards and Technology, Gaithersburg, MD, 2005), http://physics.nist.gov/xcom.

22. B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” Atomic Data and Nuclear Data Tables **54**, 181–342 (1993).
[CrossRef]

## 4. Measurements and experimental setups

### 4.1. The experimental PAL

**12**, 340–344 (2005).
[CrossRef]

*θ*′=tan

*θ*/2, where

*θ*′ is the prism angle of the modified lens. Silicon was chosen as lens material because of readily available manufacturing methods, but, according to Eq. (4), higher gains can be expected for lighter materials, such as plastics.

*µ*m.

*E**=23 keV and

*γ*=3.75, which yields

*b*=59

*µ*m,

*d*=1.6

*µ*m,

*h*=6

*µ*m,

*θ*=5.8°, and

*K*=18.4. A physical aperture of

*D*

_{p}=194

*µ*m was chosen, resulting in a total of 960 prisms arranged in

*N*=63 columns with 2×16 prisms in the first column, and a lens length of

*L*≈9 mm. The lens was equipped with support structures,

*t*

_{1}=100

*µ*m and

*t*

_{2}=8

*µ*m thick, at the entrance and exit of the lens, and for each prism column, respectively, thus adding a constant term of 2

*t*

_{1}+

*Nt*

_{2}to the linear projection. To facilitate etching, the columns were separated by 14

*µ*m.

### 4.2. PAL focusing, synchrotron setup

*µ*m FWHM in

*y*, and the beam was kept at 23 keV by a double crystal Si(111) monochromator. The lens was mounted on precision stages for lateral translation and tilting at

*s*

_{o}=40 m, yielding

*s**

_{i}≈

*F**, and the beam was collimated in y and z by an adjustable slit. As it can be expected that focusing properties vary across the aperture, the slit could be used to restrict the illumination to a small part of the lens.

*x*-ray CCD camera. We scanned the camera in the

*x*-direction to find an intensity maximum, and

*s*◇

_{i}was defined as the distance from the entrance of the lens to the maximum. All measurements at

*s*◇

_{i}are denoted with a diamond and are ideally equal to the corresponding parameters at

*E**.

*G*◇ was measured over a

*d*=14 µm distance in

_{s}*y*, corresponding to a reasonable slit size in an energy filter. The camera had a pixel size of 0.68

*µ*m, but the line spread function (LSF) was measured to 5.5

*µ*m FWHM from a slanted edge image. This is substantially larger than

*d**

_{i}, and all measurements are subject to blur. To facilitate comparisons, results from the FP model were convolved in the

*y*-direction with the LSF. In the

*z*-direction, the free beam above the lens added an almost constant background to the measurement, and we therefore subtracted from the measured image the mean value in an area far away from the lens focus.

### 4.3. PAL filtering, small-scale bremsstrahlung setup

*µ*m source size in y as specified by the manufacturer. An edge scan confirmed this size and determined the shape to approximately trapezoidal.

*s*

_{o}=585 mm, and an 11

*µ*m 23 keV image of the source can be expected at

*s*

_{i}=256 mm for a thin lens. A 200 µm slit collimated the beam in

*y*upstream of the lens, and two 50

*µ*m slits on either side of the lens restricted the beam in the

*z*-direction.

*µ*m-step tantalum edge scan. To avoid noise amplification at differentiation, the profile was fitted to an error function with an additional slope and displacement; Φ(

*x*)=

*p*

_{1}erf[

*p*

_{2}(

*x*+

*p*

_{3})]+

*p*

_{4}

*x*+

*p*

_{5}. The derivative is a Gaussian with a constant background, which is a good approximation if the ideally trapezoidal image is distorted by lens imperfections.

*x*to find

*s*◇

_{i}, and

*G*(

*E*) was again calculated over a

*d*

_{s}=14

*µ*m slit. The energy resolution is Δ

*E/E*◇, where

*E*◇ is the peak energy.

## 5. Results and discussion

### 5.1. PAL focusing, synchrotron setup

*s*◇

_{i}can be assigned to the thin lens approximation in the GM model. The measured

*s*◇

_{i}is 40 mm longer than predicted, but it is biased by a large measurement uncertainty of ±20 mm.

*s*◇

_{i}is shown in the top part of Fig. 4(a). The focusing efficiency varies in z, and an intensity maximum is located at approximately 60

*µ*m depth in the lens. A cross section of the focal line at this depth is shown in the lower part of Fig. 4(a), together with results from the FP model. The measured focal line intensity is 40% lower than predicted, and

*d*◇

_{i}is a factor 2.1 larger. Put together, these deviations yield a 19% lower

*G*◇. Figure 4(a) also shows abundant background radiation that extends over a larger area than

*D*

_{p}.

*d*

^{◇}

_{i}as a function of an increasing collimator slit. The measurements correspond well to the FP model predictions for the central 25

*µ*m part of the lens, but

*d*

^{◇}

_{i}increases rapidly with the active part of the aperture. The FP model predicts an almost constant

*d*

^{◇}

_{i}, close to the LSF of the CCD camera. As can be seen in the lower part of Fig. 4, the increased

*d*

^{◇}

_{i}at larger apertures is accompanied by a gain that levels off when

*d*

^{◇}

_{i}approaches

*d*

_{s}=14 µm. In fact, the peripheral half of the lens contributes only 10% to the total gain, compared to 30% as predicted by the model. This effect is owing to the particular choice of

*d*

_{s}, but also to a lower focusing efficiency towards the periphery of the lens.

### 5.2. PAL filtering, bremsstrahlung setup

*z*-collimator slits for the measurements were at approximately 60

*µ*m depth in the lens in accordance with the SR measurements. The measured

*s*◇

_{i}was 33 mm longer than predicted by the FP model, but the exact location of the

*x*-ray tube focal spot was only known within ±10 mm, and the scan in

*x*to find

*s*

^{◇}

_{i}was relatively coarse.

*d*

^{◇}

_{i}was 2.1 times wider than predicted, but the comparison was affected by uncertainty and instability of the source size, shape, and lateral position. The two models agreed well in

*s*

^{◇}

_{i}with a correction for the lens length, but deviated slightly in

*d*

^{◇}

_{i}.

*µ*m slit is plotted in Fig. 5(a) as a function of energy together with the unfiltered spectrum. These curves yield

*G*◇(

*E*) by division, with the result shown in Fig. 5(b) along with the model predictions. Compared to the FP model, the measured

*G*◇ deviates -29%,

*E*

_{p}is in close agreement, and the energy resolution deviates less than 4%. The GM model predicts a slightly lower energy resolution and higher gain, mainly because it does not take blurring of the focal spot at

*E*≠

*E** into account. Both models predict

*E*

_{p}=

*E**+0.5, which was not seen in the measurement, but it may be accounted for by the coarse

*x*scan.

*Z*≈6) and without support structures would have a gain

*G**

_{∞}=126, which is 6.2 times higher than a corresponding MPL (Eqs. (4) and (6)).

### 5.3. Discussion of possible lens imperfections

*s*

^{◇}

_{i}>

*s*

^{*}

_{i},

*d*

^{◇}

_{i}>

*d*

^{*}

_{i},

*G*

^{◇}<

*G*

^{*}, and measured background radiation that is not predicted by the model. Some of these metrics are coupled; the reduction in gain is an effect of the wider

*d*

_{i}and some of the transmitted radiation being deflected into background radiation, and parts of the wider

*d*

_{i}is an effect of the larger magnification factor at the longer

*s*

_{i}. Apart from the discussed uncertainties for the respective setups, a large part of the deviations are likely caused by lens imperfections.

*z*and visual inspection of the lens hint on two partly competing manufacturing problems: (1) over etch in the horizontal direction, which is pronounced at the lens surface (low

*z*) because of longer exposure to the etching species, and (2) insufficient etching at large etching depths because of transport problems in the high aspect ratio structure (depletion of etching species and insufficient removal of etching debris). Over etch in the

*x*-direction and/or material deposition in

*y*could result in systematic figure errors and a longer

*F*

_{ref}as

*F*

_{ref}∞

*h/b*.

*F*

_{diff}, on the other hand, is only affected by a change in periodicity. Moderate figure errors therefore cannot move the intensity peak along

*x*, but the peak may be flattened and stretched toward higher

*x*.

*d*

_{i}and the background radiation may also be partly owing to figure errors. Additionally, scattering caused by random phase errors from surface roughness may play a role; even modest roughness could cause large errors because of the steep prism angle. The declining focusing efficiency at the periphery of the lens is reasonable since a peripheral beam passes a larger number of imperfect prisms and is also in accordance with previous studies [10

**12**, 340–344 (2005).
[CrossRef]

## 6. Conclusions

## Acknowledgments

## References and links

1. | P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, “A prototype of a quasi-monochromatic system for mammography applications,” Phys. Med. Biol. |

2. | R. Lawaczeck, V. Arkadiev, F. Diekmann, and M. Krumrey, “Monochromatic x-rays in digital mammography,” Invest. Radiol. |

3. | F. Sugiro, D. Li, and C. MacDonald, “Beam collimation with polycapillary x-ray optics for high contrast high resolution monochromatic imaging,” Med. Phys. |

4. | J. Motz and M. Danos, “Image information content and patient exposure,” Med. Phys. |

5. | G. Pfahler, “A roentgen filter and a universal diaphragm and protecting screen,” Trans. Am. Roentgen Ray Soc. , pp. 217–224 (1906). |

6. | E. Fredenberg, B. Cederström, M. Åslund, C. Ribbing, and M. Danielsson, “A Tunable Energy Filter for Medical X-Ray Imaging,” X-Ray Optics and Instrumentation 2008, Article ID 635024, 8 pages (2008), http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/635024. |

7. | W. Jark, “A simple monochromator based on an alligator lens,” X-Ray Spectrom. |

8. | A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature |

9. | B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, “Focusing hard X-rays with old LP’s,” Nature 404, 951 (2000). |

10. | B. Cederström, C. Ribbing, and M. Lundqvist, “Generalized prism-array lenses for hard x-rays,” J. Synchrotron Rad. |

11. | W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, “Focusing X-rays with simple arrays of prism-like structures,” J. Synchrotron Rad. |

12. | L. D. Caro and W. Jark, “Diffraction theory applied to X-ray imaging with clessidra prism array lenses,” J. Synchrotron Rad. |

13. | C. Fuhse and T. Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. |

14. | Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to X-ray diffraction optics,” Opt. Commun. |

15. | V. Kohn, I. Snigireva, and A. Snigirev, “Diffraction theory of imaging with X-ray compound refractive lens,” Opt. Commun. |

16. | D. Attwood, |

17. | B. Cederström, |

18. | J. W. Goodman, |

19. | S. Panknin, A. K. Hartmann, and T. Salditt, “X-ray propagation in tapered waveguides: Simulation and optimization,” Opt. Commun. |

20. | D. R. Lynch, |

21. | M. Berger, J. Hubbell, S. Seltzer, J.S., Coursey, and D. Zucker, |

22. | B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” Atomic Data and Nuclear Data Tables |

23. | F. Laermer, A. Schilp, K. Funk, and M. Offenberg, “Bosch deep silicon etching: improving uniformity and etch rate for advanced MEMS applications,” in |

**OCIS Codes**

(120.2440) Instrumentation, measurement, and metrology : Filters

(170.7440) Medical optics and biotechnology : X-ray imaging

(340.0340) X-ray optics : X-ray optics

(350.5610) Other areas of optics : Radiation

(050.1965) Diffraction and gratings : Diffractive lenses

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: March 19, 2009

Revised Manuscript: May 15, 2009

Manuscript Accepted: June 10, 2009

Published: June 23, 2009

**Virtual Issues**

Vol. 4, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Erik Fredenberg, Björn Cederström, Peter Nillius, Carolina Ribbing, Staffan Karlsson, and Mats Danielsson, "A low-absorption x-ray energy filter for
small-scale applications," Opt. Express **17**, 11388-11398 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11388

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### References

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