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Coherent X-Ray Optics$

David Paganin

Print publication date: 2006

Print ISBN-13: 9780198567288

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567288.001.0001

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(p.397) Appendix B Fresnel scaling theorem

(p.397) Appendix B Fresnel scaling theorem

Coherent X-Ray Optics
Oxford University Press

As shown in Fig. B.1(a), let us consider an in-vacuo point source A of monochromatic scalar X-ray waves, which lies on a nominal optic axis z at a distance R upstream of a sample B. A diffraction pattern of the object is to be recorded over the plane z = ▵, this being parallel to the nominal exit surface z = 0 of the object. The source is considered to be sufficiently distant from the sample for the radiation to be paraxial over its entrance, with the object being sufficiently weakly scattering for the projection approximation to hold (see eqns (2.39), (2.40), and (2.42)). Further, we assume that Fresnel diffraction theory is adequate for calculation of the diffraction pattern over the plane z = ▵, given the wave-field over the exit surface z = 0 of the object (see Section 1.4). The Fresnel diffraction pattern (‘point-projection image’) will have a geometric magnification M, which depends on the source-to-sample distance R, together with the sample-to-detector distance ▵. A limiting case of this scenario is obtained by taking the source-to-sample distance R to infinity, as sketched in Fig. B.1(b). This unit-magnification limiting case corresponds to a Fresnel diffraction pattern of the sample which is obtained for the case of plane-wave illumination.

Subject to the approximations listed above, the Fresnel scaling theorem yields a simple mapping between the Fresnel diffraction pattern of an object which is obtained for the case of illumination by a point source, and a Fresnel diffraction pattern obtained for the limiting case of plane-wave illumination. This theorem states that the point-source-illuminated Fresnel diffraction pattern, obtained over a plane at a propagation distance z = ▵ from the exit surface of the object with a source-to-sample distance of R corresponding to geometric magnification M, is related to a certain plane-wave-illuminated diffraction pattern by the following sequence of steps: (i) take the plane-wave-illuminated Fresnel diffraction pattern over a plane at a propagation distance of z = ▵/M downstream of the exit surface of the sample, before (ii) transversely magnifying it by a factor of M and then (iii) dividing the intensity at each point in the resulting image by M 2. This scaling theorem, which will be derived using Fresnel diffraction theory in the succeeding paragraphs, is of great utility in the computation and analysis of point-projection X-ray images.

With a view to deriving the Fresnel scaling theorem, recall that one may expand the free-space propagator in the convolution formulation (1.45) of the Fresnel diffraction integral, thereby recasting it in the manner given by eqn (1.46): (p.398)

Appendix B Fresnel scaling theorem

Fig. B.1. (a) A monochromatic point source A emits X-rays into vacuum, which subsequently pass through an object B. A Fresnel diffraction pattern is measured over a plane at distance ▵ from the exit surface of the sample. The geometric magnification M of this image is (R + ▵)/R. (b) The same object is illuminated by plane waves, with a diffraction pattern being recorded at a distance of ▵/M from the exit surface of the object. According to the Fresnel scaling theorem, the diffraction patterns of (a) and (b) are identical to one another, up to transverse and multiplicative scale factors, if: (i) θ is sufficiently small for the paraxial approximation to be valid for all rays/streamlines crossing the detection plane; (ii) the object is sufficiently weakly scattering to obey the projection approximation.

ψ ω ( R ) ( x , y , z = Δ 0 ) = i k exp ( i k Δ ) 2 π Δ exp [ i k 2 Δ ( x 2 + y 2 ) ] × ψ ω ( R ) ( x , y , z = 0 ) exp [ i k 2 Δ ( x 2 + y 2 ) ] × exp [ i k Δ ( x x + y y ) ] d x d y .
Here, ψ ω ( R ) ( x , y , z = 0 ) denotes the exit-surface monochromatic scalar X-ray wave-field over the plane z = 0 in Fig. B.1(a), ψ ω ( R ) ( x , y , z = Δ 0 ) denotes the corresponding propagated disturbance over the plane z = ▵ ≥ 0, k = 2π/λ is the usual wave-number corresponding to a wavelength λ and angular frequency ω of the illuminating radiation, (x, y) denotes a Cartesian coordinate system in the plane perpendicular to the optic axis z, and the R superscript on exit-surface and propagated wave-fields indicates the distance between the illuminating on-axis point source and the exit surface of the object.

Under the paraxial and projection approximations, the exit-surface wave-field ψ ω ( ) ( x , y , z = 0 ) for the case of plane-wave illumination is related to the exit-surface wave-field ψ ω ( R ) ( x , y , z = 0 ) for point-source illumination via:

ψ ω ( R ) ( x , y , z = 0 ) = ψ ω ( ) ( x , y , z = 0 ) exp [ i k 2 R ( x 2 + y 2 ) ] .
(p.399) Note that the phase factor, on the right side of the above equation, is the second-order approximation to the phase variation over the exit surface z = 0 due to an on-axis point source at distance R upstream of this plane. Note, also, that in writing the above approximation we have taken R to be sufficiently large that one may neglect the intensity variations introduced into the exit-surface wave-field in moving from plane-wave to point-source illumination.

Having written down the relation (B.2) between the unpropagated exit-surface fields for the case of plane-wave and point-source illumination, we are ready to seek a corresponding relation between the intensities of the propagated fields, under the Fresnel approximation. To this end, insert eqn (B.2) into eqn (B.1) to give:

ψ ω ( R ) ( x , y , z = Δ 0 ) = i k exp ( i k Δ ) 2 π Δ exp [ i k 2 Δ ( x 2 + y 2 ) ] × ψ ω ( ) ( x , y , z = 0 ) × exp [ i k 2 ( x 2 + y 2 ) ( 1 Δ + 1 R ) ] × exp [ i k Δ ( x x + y y ) ] d x d y .
To proceed further, we note from Fig. B.1(a) that the geometric magnification M for point-source illumination is equal to the ratio of the length CD to the length EO. By similar triangles, this is equal to the ratio of the length of AD to that of AO. Thus:
M = R + Δ R ,
so that:
1 Δ + 1 R = 1 Δ ( R + Δ R ) = M Δ .
Substituting the above expression into the squared modulus of eqn (B.3), and letting I ω ( R ) ( x , y , z = Δ ) | ψ ω ( R ) ( x , y , z = Δ ) | 2 denote the intensity of the propagated coherent wave-field, we arrive at:
I ω ( R ) ( x , y , z = Δ 0 ) = k 2 4 π 2 Δ 2 × | ψ ω ( ) ( x , y , z = 0 ) exp [ i k M 2 Δ ( x 2 + y 2 ) i k Δ ( x x + y y ) ] d x d y .
If we take the limit R → ∞ in the above formula, so that M → 1, we see that: (p.400)
I ω ( ) ( x , y , z = Δ 0 ) = k 2 4 π 2 Δ 2 × | ψ ω ( ) ( x , y , z = 0 ) exp [ i k 2 Δ ( x 2 + y 2 ) i k Δ ( x x + y y ) ] d x d y | 2 .

As may be readily checked by direct substitution, the left sides of the above pair of equations are related to one another through the Fresnel scaling theorem:

I ω ( R ) ( x , y , z = Δ 0 ) = M 2 I ω ( ) ( x M , x M , z = Δ M 0 ) .
To supplement the verbal description of this result given in the second paragraph of this appendix, we note that: (i) The factor of M −2 embodies energy conservation; (ii) R may be negative, corresponding to the object being illuminated by a collapsing spherical wave.