The document describes an ultra-high resolution Fourier transform x-ray interferometer using Michelson and Mach-Zehnder interferometers. It discusses how Bragg's law relates x-ray wavelength and angle of incidence to crystal lattice spacing. It also explains how the maximum path difference and scanning reflector displacement determine the instrument's spectral resolution. Tables show how resolution varies with focal length and species that can be studied with a 4-bounce Mach-Zehnder design.

1. ULTRA HIGH RESOLUTION FOURIER TRANSFORM X-RAY INTERFEROMETERHERBERT W. SCHNOPPERMAD CITY LABSMADISON, WISCONSINandSMITHSONIAN ASTROPHYSICAL OBSERVATORYCAMBRIDGE, MAandJAMES A. MACKAYMAD CITY LABSMADISON WISCONSIN

2. ATTENUATION ANDPHASE SHIFTdBRAGGS LAW: nλ = 2d sinθwhere: n = the reflection orderλ = the X-ray wavelengthd = either the crystal lattice spacing or the multilayer pair thickness, andθ = the incident angle

3. For a wave propagating in a medium with refractive index n = 1 – δ + iβ E(r,t) = E0e-i(t - r/c) e-i(2πδ/λ)r e-(2πβ/λ)r vacuum propagation Φ-shift absorption The phase shift, ΔΦ, relative to vacuum that is caused by propagation through a thickness Δr is: ΔΦ= (2πδ/λ)Δ r

7. MICHELSON’S TWO-BEAM INTERFEROMETERSCANNINGSTAGEPATH 2PATH 1DETECTORREFLECTORPARTIALLY REFLECTINGBEAM SPLITTER

8. SIN(X) PATH 1PATH 1PATH 1PATH 10 deg phase shift180 deg phase shiftA MIRROR STEP EQUAL TO F λ/2 PRODUCES A PHASE SHIFT OλAND A MAXIMUM SIGNALA MIRROR STEP EQUAL TO Fλ/4 PRODUCES A PHASE SHIFT O λ/2AND A MINIMUM SIGNAL

9. x (radians) -30-20-100102030 (sinx)/x 01

10. DETECTORFIXED REFLECTOR(MULTILAYERED) SCANNING REFLECTOR(MULTILAYERED) BEAM SPLITTER(MULTILAYERED) TELESCOPEdTELESCOPEFOCUSMICHELSON INTEFRFEROMETER FOR X-RAY SPECTROSCOPY

11. θcSCANNINGMIRRORCENTRALRAYINTERMEDIATESCANPOSITIONMAXIMUMSCANPOSITIONTELESCOPEFOCUSSCAN RANGEddmaxcmax TO/FROMBEAMSPLITTER MICHELSON INTERFEROMETER SCANNING ARM

12. The maximum path difference between these rays (including thereflection) is Δmax = 2(cmax - dmax). For a Michelson interferometer, the resolution element (in wave numbers), Δkmax = 1/ (2dmax) where dmax is the maximum on-axis displacement of the scanning reflector. We adopt the convention of limiting the maximum phase shiftarising from the path difference Δmaxto λ/10. This choice implies amaximum phase shift, Φmax= 2π(1/10) rad. For small values of θ, 1/(cosθmax) = cmax/dmax ~ 1 + θmax2/2, cmax/dmax = 1 + (cmax - dmax )/dmax = 1 + Δmax/(2dmax); therefore, θmax2 ~ Δmax/dmax = λ/(10 dmax).

13. Table 1. Spectral resolution (λ/dλ) vs focal length FL (m) 100 200 500 1000 θmax(rad) 5.0 x 10-32.5 x 10-31.0 x 10-35.0 x 10-4θmax(deg) 0.286 0.143 0.0573 0.0286 λ /dλ 4 x 103 1.6 x 104 1 x 105 4 x 105 Table 2. dmax vs focal length (H-like ions) FL (m) dmax ( μm) C VI 3.4 nm N VII 2.5 nm O VIII 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

14. FOUR BOUNCE FOURIER TRANSFORM X-RAY SPECTROMETERREFERENCEDETECTORINTERFEROMETERDETECTORMULTILAYERBEAMSPLITTERMULTILAYEREDREFLECTORNANOPOSITIONEDSLIDE

15. FRACTIONALPATHDIFFERENCEvsTOTALDISPLACEMENTFORAMACH-ZEHNDERINTERFEROMETER020406080100012(P/d) = 2[(1-cosΔθ/sinθ]max ΔP/dmax BRAGG ANGLE (θ)

16. Table 3. Species to be studied with the 4-bounce Mach-Zhender interferometer. Ion Mg(VII) Si (XIV) S (XVI) Ar (XVIII) Ca (XX) Ti (XXII) Fe (XXVI) Wavelength (nm) 0.840.620.470.37 0.300.25 0.18 Bragg angle (deg) 31.622.817.113.4 10.89.0 6.5

17. BRAGGANGLEvsWAVELENGTHFORCOSMICALLYABUNDANTH-LIKEIONS(multilayerd-spacing=0.8nm) 01234020406080100Fe (XXVI) Ti (XXII)Ca(XX) Ar (XVIII) S (XVI) Si (XIV) Mg (VII) O (VIII)N (VII)C (VI) WAVELENGTH (nm) BRAGG ANGLE (DEG) MICHELSON INTERFEROMETERMACH-ZHENDER INTERFEROMETER

18. MACH-ZEHNDER INTERFEROMETER