Design of grazing-incidence multilayer supermirrors for hard-x-ray reflectors

Karsten D. Joensen; Peter Voutov; Andrew Szentgyorgyi; John Roll; Paul Gorenstein; Peter Høghøj; Finn E. Christensen

doi:10.1364/AO.34.007935

Applied Optics
Vol. 34,
Issue 34,
pp. 7935-7944
(1995)
•https://doi.org/10.1364/AO.34.007935

Design of grazing-incidence multilayer supermirrors for hard-x-ray reflectors

Karsten D. Joensen, Peter Voutov, Andrew Szentgyorgyi, John Roll, Paul Gorenstein, Peter Høghøj, and Finn E. Christensen

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Karsten D. Joensen, Peter Voutov, Andrew Szentgyorgyi, John Roll, Paul Gorenstein, Peter Høghøj, and Finn E. Christensen, "Design of grazing-incidence multilayer supermirrors for hard-x-ray reflectors," Appl. Opt. 34, 7935-7944 (1995)

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Abstract

Extremely broadband grazing-incidence multilayers for hard-x-ray reflection can be obtained by a gradual change of the layer thicknesses down through the structure. Existing approaches for designing similar neutron optics, called supermirrors, are shown to provide respectable performance when applied to x-ray multilayers. However, none of these approaches consider the effects of imperfect layer interfaces and absorption in the overlying layers. Adaptations of neutron designs that take these effects into account are presented, and a thorough analysis of two specific applications (a single hard-x-ray reflector and a hard-x-ray telescope) shows that an improved performance can be obtained. A multilayer whose bilayer thicknesses are given by a power law expression is found to provide the best solution; however, it is only slightly better than some of the adapted neutron designs.

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Model	Model No.	Structure	Parameters	FOM
Mezei	a1	γ = 0.50		0.3097
	a2	constant γ	γ = 0.477	0.3146
	a3	constant Γ	Γ = 0.473	0.3114
Schelten and Mika	b1	γ = 0.5	ℛ = 1.12, α = 0.91, d(1) = 84	0.3209
no roughness	b2	constant γ	ℛ = 1.41, α = 0.98, d(1) = 90, γ = 0.36	0.3354
	b3	constant Γ	ℛ = 1.32, α = 0.8, d(1) = 99, Γ = 0.39	0.3386
roughness	c1	γ = 0.5	ℛ = 0.83, α = 1.05, d(1) = 96	0.3486
	c2	constant γ	ℛ = 0.98, α = 1.04, d(1) = 109.4, γ = 0.36	0.3686
	c3	constant Γ	ℛ = 0.93, α = 0.99, d(1) = 125, Γ = 0.39	0.3697
Hayter and Mook	d1	γ = 0.5	ℛ = 0.82	0.3197
no roughness	d2	constant γ	ℛ = 0.87, γ = 0.36	0.3386
	d3	constant Γ	ℛ = 0.89, Γ = 0.38	0.3449
roughness	e1	γ = 0.5	ℛ = 0.67	0.3489
	e2	constant γ	ℛ = 0.74, γ = 0.36	0.3731
	e3	constant Γ	ℛ = 0.75, Γ = 0.38	0.3751
Yamada et al.	f1	γ = 0.5	d(1) = 90, Δ = 25.5, a = 5.73	0.3466
	f2	constant γ	d(1) = 110, Δ = 45.6, a = 5.37, γ = 0.36	0.3660
	f3	constant Γ	d(1) = 110, Δ = 41.7, a = 5.29, Γ = 0.40	0.3611
Joensen et al.	g1	γ = 0.5	a = 70, b = −0.918, c = 0.173	0.3589
	g2	constant γ	a = 81, b = −0.921, c = 0.196, γ = 0.34	0.3823
	g3	constant Γ	a = 81, b = −0.978, c = 0.193, Γ = 0.36	0.3840

Model	Model No.	Structure	Parameters	FOM
Mezei	a1	γ = 0.50		1.788
	a2	constant γ	γ = 0.48	1.794
	a3	constant Γ	Γ = 0.52	1.728
Schelten and Mika	b1	γ = 0.5	ℛ = 1.32, α = 0.91, d(1) = 184	2.303
no roughness	b2	constant γ	ℛ = 1.33, α = 0.88, d(1) = 191, γ = 0.41	2.370
	b3	constant Γ	ℛ = 1.57, α = 0.95, d(1) = 273, Γ = 0.42	2.395
roughness	c1	γ = 0.5	ℛ = 1.05, α = 1.01, d(1) = 202	2.311
	c2	constant γ	ℛ = 1.01, α = 1.13, d(1) = 214, γ = 0.41	2.398
	c3	constant Γ	ℛ = 0.95, α = 1.23, d(1) = 254, Γ = 0.42	2.407
Hayter and Mook	d1	γ = 0.5	ℛ = 0.88	2.309
no roughness	d2	constant γ	ℛ = 0.89, γ = 0.43	2.369
	d3	constant Γ	ℛ = 0.91, Γ = 0.46	2.331
roughness	e1	γ = 0.5	ℛ = 0.82	2.281
	e2	constant γ	ℛ = 0.84, γ = 0.42	2.346
	e3	constant Γ	ℛ = 0.85, Γ = 0.45	2.280
Yamada et al.	f1	γ = 0.5	d(1) = 183, Δ = 57, a = 4.36	2.306
	f2	constant γ	d(1) = 198, Δ = 70, a = 4.38, γ = 0.40	2.394
	f3	constant Γ	d(1) = 207, Δ = 74.3, a = 4.237, Γ = 0.44	2.344
Joensen et al.	g1	γ = 0.5	a = 136, b = −0.85, c = 0.238	2.315
	g2	constant γ	a = 145, b = −0.85, c = 0.248, γ = 0.41	2.444
	g3	constant Γ	a = 146, b = −0.96, c = 0.246, Γ = 0.41	2.471

Model	Model No.	Structure	Parameters	FOM
Mezei	a1	γ = 0.50		0.3097
	a2	constant γ	γ = 0.477	0.3146
	a3	constant Γ	Γ = 0.473	0.3114
Schelten and Mika	b1	γ = 0.5	ℛ = 1.12, α = 0.91, d(1) = 84	0.3209
no roughness	b2	constant γ	ℛ = 1.41, α = 0.98, d(1) = 90, γ = 0.36	0.3354
	b3	constant Γ	ℛ = 1.32, α = 0.8, d(1) = 99, Γ = 0.39	0.3386
roughness	c1	γ = 0.5	ℛ = 0.83, α = 1.05, d(1) = 96	0.3486
	c2	constant γ	ℛ = 0.98, α = 1.04, d(1) = 109.4, γ = 0.36	0.3686
	c3	constant Γ	ℛ = 0.93, α = 0.99, d(1) = 125, Γ = 0.39	0.3697
Hayter and Mook	d1	γ = 0.5	ℛ = 0.82	0.3197
no roughness	d2	constant γ	ℛ = 0.87, γ = 0.36	0.3386
	d3	constant Γ	ℛ = 0.89, Γ = 0.38	0.3449
roughness	e1	γ = 0.5	ℛ = 0.67	0.3489
	e2	constant γ	ℛ = 0.74, γ = 0.36	0.3731
	e3	constant Γ	ℛ = 0.75, Γ = 0.38	0.3751
Yamada et al.	f1	γ = 0.5	d(1) = 90, Δ = 25.5, a = 5.73	0.3466
	f2	constant γ	d(1) = 110, Δ = 45.6, a = 5.37, γ = 0.36	0.3660
	f3	constant Γ	d(1) = 110, Δ = 41.7, a = 5.29, Γ = 0.40	0.3611
Joensen et al.	g1	γ = 0.5	a = 70, b = −0.918, c = 0.173	0.3589
	g2	constant γ	a = 81, b = −0.921, c = 0.196, γ = 0.34	0.3823
	g3	constant Γ	a = 81, b = −0.978, c = 0.193, Γ = 0.36	0.3840

Model	Model No.	Structure	Parameters	FOM
Mezei	a1	γ = 0.50		1.788
	a2	constant γ	γ = 0.48	1.794
	a3	constant Γ	Γ = 0.52	1.728
Schelten and Mika	b1	γ = 0.5	ℛ = 1.32, α = 0.91, d(1) = 184	2.303
no roughness	b2	constant γ	ℛ = 1.33, α = 0.88, d(1) = 191, γ = 0.41	2.370
	b3	constant Γ	ℛ = 1.57, α = 0.95, d(1) = 273, Γ = 0.42	2.395
roughness	c1	γ = 0.5	ℛ = 1.05, α = 1.01, d(1) = 202	2.311
	c2	constant γ	ℛ = 1.01, α = 1.13, d(1) = 214, γ = 0.41	2.398
	c3	constant Γ	ℛ = 0.95, α = 1.23, d(1) = 254, Γ = 0.42	2.407
Hayter and Mook	d1	γ = 0.5	ℛ = 0.88	2.309
no roughness	d2	constant γ	ℛ = 0.89, γ = 0.43	2.369
	d3	constant Γ	ℛ = 0.91, Γ = 0.46	2.331
roughness	e1	γ = 0.5	ℛ = 0.82	2.281
	e2	constant γ	ℛ = 0.84, γ = 0.42	2.346
	e3	constant Γ	ℛ = 0.85, Γ = 0.45	2.280
Yamada et al.	f1	γ = 0.5	d(1) = 183, Δ = 57, a = 4.36	2.306
	f2	constant γ	d(1) = 198, Δ = 70, a = 4.38, γ = 0.40	2.394
	f3	constant Γ	d(1) = 207, Δ = 74.3, a = 4.237, Γ = 0.44	2.344
Joensen et al.	g1	γ = 0.5	a = 136, b = −0.85, c = 0.238	2.315
	g2	constant γ	a = 145, b = −0.85, c = 0.248, γ = 0.41	2.444
	g3	constant Γ	a = 146, b = −0.96, c = 0.246, Γ = 0.41	2.471

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Equations (25)

Applied Optics