X-ray diffractive optics of curved crystals: focusing properties on a diffraction-limited basis

W. Z. Chang; E. Förster

doi:10.1364/JOSAA.14.001647

Journal of the Optical Society of America A
Vol. 14,
Issue 7,
pp. 1647-1653
(1997)
•https://doi.org/10.1364/JOSAA.14.001647

X-ray diffractive optics of curved crystals: focusing properties on a diffraction-limited basis

W. Z. Chang and E. Förster

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W. Z. Chang and E. Förster, "X-ray diffractive optics of curved crystals: focusing properties on a diffraction-limited basis," J. Opt. Soc. Am. A 14, 1647-1653 (1997)

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Abstract

For a given x-ray focusing geometry with curved crystals, we derive, on a diffraction-limited basis, analytical expressions for the intensity distribution at focus that account for both the Bragg and the Fresnel diffractions in an optical system. Within a monochromatic x-ray wavelength range of 0.709–3.294 Å, a doubly bent quartz crystal with (20-2) reflection is chosen to calculate analytically intensity distributions for point, 5-μm, and 20-μm x-ray sources. It is found that, whereas the fundamental width (for a point source) of the intensity distribution is broader than that deduced from a sinc function analogous to those used in optics, for the case of extended sources the width can be simply expressed as a sum of the fundamental width and the width obtained from demagnification of the source.

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$A$	$[π sin θ_{B} (1 - 3 {cos}^{2} θ_{B}) (1 / {L_{h}}^{2} + 1 / {L_{0}}^{2})]^{2} X^{4} / {8 λ [sin θ_{B} / R_{z}$
	$+ {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$
$B$	${- π sin θ_{B} [cos θ_{B} (x_{0} / L_{0} + x_{h} / L_{h}) + i (Im χ_{0} / sin θ_{B})$ $- sin θ_{B} ({y_{h}}^{2} / {L_{h}}^{2} + {y_{0}}^{2} / {L_{0}}^{2}) / 2]$ $\times (1 - 3 {cos}^{2} θ_{B}) (1 / {L_{h}}^{2} + 1 / {L_{0}}^{2}) X^{2}} /$ ${2 λ [sin θ_{B} / R_{z} + {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$
$C$	$π (x_{h} / L_{h} - x_{0} / L_{0}) cos θ_{B} sin θ_{B} X / λ$
$D$	$[π sin θ_{B} (1 / {L_{h}}^{2} + 1 / {L_{0}}^{2})]^{2} Y^{4} / {8 λ [sin θ_{B} / R_{z} + {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$
$F$	$- π sin θ_{B} (1 / {L_{h}}^{2} + 1 / {L_{0}}^{2}) [cos θ_{B} (x_{0} / L_{0} + x_{h} / L_{h}) + i (Im χ_{0} / sin θ_{B}) - sin θ_{B} ({y_{h}}^{2} / {L_{h}}^{2} + {y_{0}}^{2} / {L_{0}}^{2}) / 2] Y^{2} /$ ${2 λ [sin θ_{B} / R_{z} + {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$
$G$	$2 π (y_{h} / L_{h} + y_{0} / L_{0}) Y / λ$
$H$	$π [cos θ_{B} (x_{0} / L_{0} + x_{h} / L_{h}) + i (Im χ_{0} / sin θ_{B})$ $- sin θ_{B} ({y_{h}}^{2} / {L_{h}}^{2} + {y_{0}}^{2} / {L_{0}}^{2}) / 2]^{2} / {2 λ [sin θ_{B} / R_{z}$ $+ {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$

$λ = 1.541$ Å $(θ_{B} = 27.45 °)$			$λ = 2.749$ Å $(θ_{B} = 55.31 °)$
$R_{x}$ (mm)	200	400	200	400
$R_{y}$ (mm)	42.50	85.00	135.22	270.43
$L_{0}$ (mm)	92.19	184.39	164.45	328.90
∊	3.6%	1.2%	3.7%	0.85%

$L_{0} (mm)$	$Δ x_{h} (Δ x_{0} = 0)$ $(μ m)$	$M$	$L_{h} λ / (sin 2 θ_{B} X)$ (μm)
$λ = 3.294 Å$ $(θ_{B} = 80.62 °)$
197.3	4.00	1.00	0.51
300	2.90	2.04	0.38
600	2.00	5.08	0.30
900	2.00	8.12	0.28
$λ = 2.749 Å$ $(θ_{B} = 55.31 °)$
164.5	0.88	1.00	0.12
300	0.70	2.65	0.083
600	0.55	6.3	0.07
900	0.55	9.95	0.066
$λ = 1.541 Å$ $(θ_{B} = 27.45 °)$
92.2	0.30	1.00	0.043
300	0.18	5.51	0.026
600	0.16	12.02	0.024
900	0.16	18.53	0.023
$λ = 0.709 Å$ $(θ_{B} = 12.24 °)$
42.4	0.15	1.00	0.018
300	0.07	13.15	$\sim 0.01$
600	0.07	27.30	$\sim 0.01$
900	0.07	41.45	$\sim 0.01$

	$λ = 3.294 Å$ $(θ_{B} = 80.62 °)$			$λ = 2.749 Å$ $(θ_{B} = 55.31 °)$			$λ = 1.541 Å$ $(θ_{B} = 27.45 °)$			$λ = 0.709 Å$ $(θ_{B} = 12.24 °)$
$L_{0}$	300	600	900	300	600	900	300	600	900	300	600	900
$Δ x_{h}^{(0)}$ $(Δ x_{0} = 0) (μ m)$	2.90	2.00	2.00	0.70	0.55	0.55	0.18	0.16	0.16	0.07	0.07	0.07
$M$	2.04	5.08	8.12	2.65	6.3	9.95	5.51	12.02	18.53	13.15	27.30	41.45
$Δ x_{0}^{(5)} / M$ $(Δ x_{0} = 5 μ m)$	2.45	0.98	0.62	1.89	0.79	0.50	0.91	0.42	0.27	0.38	0.18	0.12
$Δ x_{h}^{(5)}$ $(Δ x_{0} = 5 μ m)$	4.50	2.98	2.54	2.50	1.60	1.20	1.00	0.60	0.40	0.40	0.20	0.20
$Δ x_{h}^{(0)} + Δ x_{0}^{(5)} / M$	5.35	2.98	2.62	2.59	1.34	1.05	1.09	0.58	0.43	0.45	0.25	0.19
$Δ x_{0}^{(20)} / M$ $(Δ x_{0} = 20 μ m)$	9.80	3.94	2.46	7.55	3.17	2.01	3.63	1.66	1.08	1.52	0.73	0.48
$Δ x_{h}^{(20)}$ $(Δ x_{0} = 20 μ m)$	13.50	5.80	4.60	4.50	3.80	2.50	4.00	1.80	1.20	1.53	0.80	0.52
$Δ x_{h}^{(0)} + Δ x_{0}^{(20)} / M$	12.7	5.94	4.46	8.25	3.72	2.56	3.81	1.82	1.24	1.59	0.80	0.55

$A$	$[π sin θ_{B} (1 - 3 {cos}^{2} θ_{B}) (1 / {L_{h}}^{2} + 1 / {L_{0}}^{2})]^{2} X^{4} / {8 λ [sin θ_{B} / R_{z}$
	$+ {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$
$B$	${- π sin θ_{B} [cos θ_{B} (x_{0} / L_{0} + x_{h} / L_{h}) + i (Im χ_{0} / sin θ_{B})$ $- sin θ_{B} ({y_{h}}^{2} / {L_{h}}^{2} + {y_{0}}^{2} / {L_{0}}^{2}) / 2]$ $\times (1 - 3 {cos}^{2} θ_{B}) (1 / {L_{h}}^{2} + 1 / {L_{0}}^{2}) X^{2}} /$ ${2 λ [sin θ_{B} / R_{z} + {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$
$C$	$π (x_{h} / L_{h} - x_{0} / L_{0}) cos θ_{B} sin θ_{B} X / λ$
$D$	$[π sin θ_{B} (1 / {L_{h}}^{2} + 1 / {L_{0}}^{2})]^{2} Y^{4} / {8 λ [sin θ_{B} / R_{z} + {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$
$F$	$- π sin θ_{B} (1 / {L_{h}}^{2} + 1 / {L_{0}}^{2}) [cos θ_{B} (x_{0} / L_{0} + x_{h} / L_{h}) + i (Im χ_{0} / sin θ_{B}) - sin θ_{B} ({y_{h}}^{2} / {L_{h}}^{2} + {y_{0}}^{2} / {L_{0}}^{2}) / 2] Y^{2} /$ ${2 λ [sin θ_{B} / R_{z} + {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$
$G$	$2 π (y_{h} / L_{h} + y_{0} / L_{0}) Y / λ$
$H$	$π [cos θ_{B} (x_{0} / L_{0} + x_{h} / L_{h}) + i (Im χ_{0} / sin θ_{B})$ $- sin θ_{B} ({y_{h}}^{2} / {L_{h}}^{2} + {y_{0}}^{2} / {L_{0}}^{2}) / 2]^{2} / {2 λ [sin θ_{B} / R_{z}$ $+ {cos}^{2} θ_{B} (1 / L_{h} + 1 / L_{0}) / 2]}$

Abstract

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Figures (4)

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Equations (8)

Journal of the Optical Society of America A