1138schnopper[1]

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