ULTRA HIGH RESOLUTION FOURIER TRANSFORM X-RAY INTERFEROMETER HERBERT W. SCHNOPPER MAD CITY LABS MADISON, WISCONSIN and
SMITHSONIAN ASTROPHYSICAL OBSERVATORY CAMBRIDGE, MA and
JAMES A. MACKAY MAD CITY LABS MADISON WISCONSIN
ATTENUATION AND PHASE SHIFT
d
BRAGGS LAW: n位 = 2d sin胃 where: n = the reflection order 位 = the X-ray wavelength d = either the crystal lattice spacing or the multilayer pair thickness, and 胃 = the incident angle
For a wave propagating in a medium with refractive index n = 1 – δ + iβ
-i(t - r/c)
E(r,t) = E0e
-i(2πδ/λ)r
e
vacuum propagation
Φ-shift
-(2πβ/λ)r
e
absorption
The phase shift, ∆Φ, relative to vacuum that is caused by propagation through a thickness ∆r is:
∆Φ= (2πδ/λ)∆ r
MICHELSON’S TWO-BEAM INTERFEROMETER REFLECTOR
PATH 1
SCANNING STAGE
PATH 2 PARTIALLY REFLECTING BEAM SPLITTER
DETECTOR
A MIRROR STEP EQUAL TO 位/2 PRODUCES A PHASE SHIFT OF 位 AND A MAXIMUM SIGNAL PATH 1
0 deg phase shift PATH 1
A MIRROR STEP EQUAL TO 位/4 PRODUCES A PHASE SHIFT OF 位/2 AND A MINIMUM SIGNAL PATH 1
180 deg phase shift PATH 1
SIN(X)
(sinx)/x
1
0
-30
-20
-10
0
x (radians)
10
20
30
TELESCOPE
FIXED REFLECTOR (MULTILAYERED)
BEAM SPLITTER (MULTILAYERED) TELESCOPE FOCUS d
SCANNING REFLECTOR (MULTILAYERED)
DETECTOR
MICHELSON INTEFRFEROMETER FOR X-RAY SPECTROSCOPY
d INTERMEDIATE SCAN POSITION
TO/FROM BEAMSPLITTER
c
MAXIMUM SCAN POSITION
SCANNING MIRROR
θ CENTRAL RAY
TELESCOPE FOCUS
cmax
SCAN RANGE
dmax
MICHELSON INTERFEROMETER SCANNING ARM
The maximum path difference between these rays (including the reflection) is ∆max = 2(cmax - dmax). For a Michelson interferometer, the resolution element (in wave numbers), ∆kmax = 1/ (2dmax) where dmax is the maximum on-axis displacement of the scanning reflector. We adopt the convention of limiting the maximum phase shift arising from the path difference ∆max to λ/10. This choice implies a maximum phase shift, Φmax= 2π(1/10) rad. For small values of θ, 2
1/(cosθmax) = cmax/dmax ~ 1 + θmax /2, cmax/dmax = 1 + (cmax - dmax )/dmax = 1 + ∆max/(2dmax); therefore, 2
θmax ~ ∆max/dmax = λ/(10 dmax).
Table 1. Spectral resolution (λ/dλ) vs focal length FL (m) 100 200 500 1000 -3 -3 -3 θmax(rad) 5.0 x 10 2.5 x 10 1.0 x 10 5.0 x 10-4 0.286 0.143 0.0573 0.0286 θmax(deg) 3 4 5 4 x 10 1.6 x 10 1 x 10 4 x 105 λ /dλ Table 2. dmax vs focal length (H-like ions) FL (m) dmax ( µm) C VI N VII O VIII 3.4 nm 2.5 nm 1.9 nm 100.0 13.6 10.0 7.6 200.0 54.4 40.0 30.4 500.0 340.0 250.0 190.0 1000.0 1360.0 1000.0 760.0
FOUR BOUNCE FOURIER TRANSFORM X-RAY SPECTROMETER
NANOPOSITIONED SLIDE
REFERENCE DETECTOR
MULTILAYER BEAMSPLITTER
INTERFEROMETER DETECTOR
MULTILAYERED REFLECTOR
FRACTIONAL PATH DIFFERENCE vs TOTAL DISPLACEMENT FOR A MACH-ZEHNDER INTERFEROMETER
∆P/dmax
2
(∆P/dmax) = 2[(1-cosθ/sinθ] 1
0 0
20
40
60
BRAGG ANGLE (θ)
80
100
Table 3. Species to be studied with the 4-bounce Mach-Zhender interferometer. Ion Mg Si S Ar Ca Ti Fe (VII) (XIV) (XVI) (XVIII) (XX) (XXII) (XXVI) Wavelength 0.84 0.62 0.47 0.37 0.30 0.25 0.18 (nm) Bragg 31.6 22.8 17.1 13.4 10.8 9.0 6.5 angle (deg)
BRAGG ANGLE (DEG)
100
O (VIII)
80
N (VII)
C (VI)
MICHELSON INTERFEROMETER
BRAGG ANGLE vs WAVELENGTH FOR COSMICALLY ABUNDANT H-LIKE IONS (multilayer d-spacing = 0.8 nm)
60
40 Mg (VII) Si (XIV)
20 Ti (XXII) Fe (XXVI)
0
0
S (XVI) Ar (XVIII) Ca(XX)
1
MACH-ZHENDER INTERFEROMETER
2
WAVELENGTH (nm)
3
4
MACH-ZEHNDER INTERFEROMETER