Calculation is based on the results of works:
Nature, 1996, vol.384, p.49;
Applied Optics, 1998, vol.37, No.4, p.653
as well as on some additional (unpublished) analysis.
Let us introduce a notation:
r_{s} - source-to-lens distanse, w_{s} - width of source (source size), E - energy of X-rays, R - radius of holes, d - wall thickness between holes, N - number of holes in array, h - total height of teeth due to roughness of all surfaces of holes. n = 1 - D + iB - refractive index (D - decrement), L = 12.397 / E - wavelength of X-rays, M = 12.567 B / L - absorption coefficient
The parameters are calculated using the formulas:
Focal length - F = R / (2ND),
Focal distance - r_{f} = F / ( 1 - F/r_{s} ),
Aperture due to circular form - A_{c} = 2R
(4LF / R^{ 2} )^{1/4},
Aperture due to arsorption - A_{a} = 2R
(2 / MRN )^{1/2},
Slit effect parameter - p_{se} = A_{c}
/ A_{a},
Aperture is approximated as - A = min(
A_{c}, A_{a} ),
Focus for point source - w_{fp} =
Lr_{f} / A,
Source size projection - w_{sp} =
w_{s}r_{f} / r_{s},
Gain for point sourse - G = ( A /
w_{fp} )( 1 + r_{f} / r_{s} )
exp( -MNd ) X,
where X = 1 if p_{se} < 1
else X = 0.7854,
Gain for real sourse - g = G / C
if C > 1, where C = (
w_{sp} / w_{fp} ) X,
here X = 0.886 if p_{se} < 1
else X = 1.111,
Real gain for rough surfaces - g_{r} = g
exp( -( 0.507 hDE )^{2} ),
Focus for real source - w_{fr} =
w_{fp} + w_{sp}.
The estimations contain maximum error when the slit effect parameter is
close to unity as well as the projection of source size is close to
focus width for a point source.
The decrement of refractive index D is calculated taking into
account only Rayleigh scattering. The index of absorption B
is calculated taking into account Photoelectric absorption and
Compton scattering. The energy dependence of cross section for these
processes for different chemical elements was calculated using Log-Log
polynomial approximation taken from the book:
Handbook of X-Ray Spectrometry, Methods and Techniques,
edited by E. Van Grieken & Andrjei A. Marcowicz, Marcel Dekker, Inc.,
New York, Basel, Hong Kong.