Moiré pattern from a multiple Bragg–Laue interferometer

A moiré pattern is observed by dividing the incident beam into two and formed by multiple Bragg–Laue interference fringes corresponding to the two incident beams. The coherency of X-rays from a bending-magnet beamline is evaluated using the moiré pattern.


Introduction
Interference fringes have been observed in the diffraction from the lateral surface of a plane-parallel crystal (Fukamachi et al., 2004(Fukamachi et al., , 2005 when X-rays are incident on the surface at the incident angle where the anomalous transmission (Borrmann) effect is dominant. Under this condition the refracted beam in the crystal disperses widely from the direction of the diffraction plane to that of the transmitted beam even when the dispersion angle of the incident beam (Á) is less than 1 arcsec. The refracted beam refers to the beam corresponding to the Poynting vector . The diffraction geometry of the incident beam is the Bragg mode and that of the emitted beam is the Laue mode, i.e. the diffraction geometry as a whole is the Bragg-Laue mode. The dispersion angle (Á) of the refracted beam is approximately the same as the Bragg angle ( B ) and the ratio of the dispersion angles Á/ Á is of the order of 10 4 -10 5 (Authier, 2001). The amplification of the dispersion angle is quite large and the crystal works as a lens. Since the refracted beam propagates as a spherical wave, interference between the refracted beam in the Bragg-Laue (BL) mode (its electric displacement field D BL ) and that in the Bragg-Bragg-Laue (BBL) mode (its electric displacement field D BBL ) occurs (Hirano et al., , 2009aFukamachi et al., 2011a), as shown in Fig. 1(a). If the distance from the incident point of the X-rays to the lateral surface is long, multiple diffractions in Bragg mode occur. Then the interference fringes observed in this mode are known as multiple Bragg-Laue (MBL) interference fringes. A MBL interferometer has been developed using MBL interference fringes. It works as a monochromator with high angular resolution, a beam splitter and an analyzer for a two-beam Schematic illustrations of the beam geometry in a plane-parallel single crystal (a) in an unbent crystal, (b) in a bent crystal, and (c) in a crystal for a two-beam MBL interferometer. L is the distance between the incident point of the X-rays and the crystal edge, and H is the crystal thickness. E 0 and E h1i h are the electric fields of the incident beam and the diffracted beam. E 0 h is the electric field of the diffraction beam in the direction of the diffracted beam at the exit point (L; z) from the lateral surface. The refracted beams in the BL and BBL modes are denoted by the Poynting vectors S BL and S BBL , respectively. In (c) the second subscript 1 (2) of the electric field E 0 and the Poynting vectors S BL and S BBL denotes the beam passing through the upper (lower) side of the platinum wire.
In this paper we report that a moiré pattern has been observed by using an MBL interferometer and by inserting a platinum wire into the incident beam on the interferometer to divide the beam into two. The cause of the moiré pattern is analyzed and the coherency of the incident beam is discussed using the moiré pattern.

Experiments
The sample was a plane-parallel Si single crystal. The top and bottom surfaces were polished at Sharan Instruments Corporation by using the non-disturbance polishing technique. The sample size was 50 mm long, 15 mm wide and 0.28 mm thick. The beam geometry around the sample is shown in Fig. 2(a) and a schematic diagram of the measuring optical system is shown in Fig. 2(b). The Si 220 diffraction experiments were carried out using X-rays from synchrotron radiation at bending-magnet beamline BL-15C, Photon Factory, KEK, Tsukuba, Japan. The X-rays were -polarized and monochromated using a Si 111 double-crystal monochromator. The X-ray energy was 11100 AE 0.5 eV. The vertical width of slit 1 was adjusted to be 30 mm or 70 mm. In Fig. 2, P h1i h represents the intensity of the diffracted beam from the inci-dent point, P 0 h represents that from the lateral surface. The beam intensities were recorded on a nuclear plate (ILFORD L4; emulsion thickness 25 mm) and were measured using a scintillation counter (SC). Fig. 2(c) shows the beam geometry around the MBL interferometer.      (a) Magnified MBL interference fringes of the lower part of Fig. 3(a). (b) Magnified MBL interference fringes of the lower part of Fig. 3(b).

Theoretical basis
The electric field E 0 h ðL; zÞ of the diffracted beam from the lateral surface shown in Fig. 1(a) is given by using the electric fields in the BL (E BL ) and BBL (E BBL ) modes as Here L is the distance between the incident point of the X-rays and the crystal edge where the beams are emitted, H is the thickness of the crystal, and z is the depth in the crystal. BL and BBL are the correction factors of the beam width on the lateral surface. D h;BL and D h;BBL are the electric displacement fields of the diffracted beam in the BL and BBL modes, respectively. S BL in Fig. 1(a) is the Poynting vector corresponding to the refracted beam and is related to the electric displacement fields as S BL = s 0 jD 0;BL j 2 + s h jD h;BL j 2 . S BBL is the Poynting vector corresponding to the refracted beam and is related to the electric displacement fields by S BBL = s 0 jD 0;BBL j 2 + s h jD h;BBL j 2 . Here D 0;BL and D 0;BBL are the primary (0th) beam components of the electric displacement fields of the refracted beam in the BL and BBL modes, respectively. The phase angles in the BL and BBL modes are given by where k ð j Þ h;x and k ð j Þ h;z are x and z components of the wavevector of the refracted beam, respectively. The superscript ( j) represents the branch number. The branch j = 1 corresponds to the beam propagating in the transmitted direction such as the refracted beam in the BL mode, and j = 2 corresponds to the beam propagating in the diffracted beam direction such as that in the BBL mode. The diffraction intensity from the lateral surface is given as P 0 h = jE 0 h ðL; zÞj 2 . Figs. 5(a) and 5(b) show the top and side views of the beam geometry when a platinum wire is inserted into the incident beam. L 1 is the distance between the incident point and the crystal edge for the incident beam E 0;1 passing through the upper side of the wire, and L 2 is that for the incident beam E 0;2 passing through the lower side. w 1 and w 2 are the relative widths (w 1 + w 2 = 1) of the beams E 0;1 and E 0;2 . L 1 , L 2 , w 1 and w 2 vary as a function of distance (y) along the crystal edge.
The electric field of the diffraction beam from the lateral surface corresponding to the incident beam E 0;1 is given by Here 01 and 02 are K 0ix and K 0iz are x and z components, respectively, of the wavevector in a vacuum corresponding to the incident beam E 0i (i = 1, 2). x 0 and z 0 are x and z components of the path difference r between the wavefronts of the beams E 0;1 and E 0;2 , as shown in Fig. 5(b). The electric field of the diffraction beam from the lateral surface corresponding to the incident beam E 0;2 is given by If the beams E 0;1 and E 0;2 are coherent with each other, the intensity is given by and if they are incoherent it is given by and 6(c) show the calculated patterns using equations (7) and (8), respectively. In the calculation, L 1 , L 2 , w 1 and w 2 are assumed to vary linearly as a function of y. L 1 varies from 2.08 to 2.21 mm and L 2 from 2.23 to 2.41 mm. w 1 varies from 0.1 to 0.9 and w 2 from 0.9 to 0.1. In the calculated results using equation (7) the third fringe (black contrast) from the bottom left disappears in the middle of the horizontal axis as shown in Fig. 6(b), while in the calculated result using equation (8) the fifth fringe disappears as shown in Fig. 6(c). The agreement between the measured and calculated results using equation (7) shows that the beams E 0;1 and E 0;2 are coherent with each other. Then the optical system in Fig. 1(a) works as an interferometer and can be called an MBL interferometer. The disappearance of the third fringe around the middle of the figure (w 1 = w 2 = 0.5) can be explained by the difference in the phase between E 0 h ðL 1 ; zÞ and E 0 h ðL 2 ; zÞ being for z corresponding to the height of the third fringe and for L 1 being its mean value ($ 2.15 mm).

Moiré pattern
A moiré pattern is observed when the beam passing through slit 1 is divided into two incident beams on the MBL interferometer by inserting a platinum wire into the beam. The cause of the moiré pattern is attributed to be interference between two MBL interference fringes corresponding to the two incident beams E 0;1 and E 0;2 , as shown in Fig. 5. In the present experiment, L 1 < L 2 and the period of MBL interference fringes ( p 1 ) corresponding to the beam E 0;1 is smaller than that ( p 2 ) corresponding to the beam E 0;2 . If we define the wavenumbers corresponding to the periods p 1 and p 2 as 1 = 2=p 1 and 2 = 2=p 2 , respectively, the electric fields of the diffraction beams corresponding to the incident beams E 0;1 and E 0;2 can be expressed as E 0 h ðL 1 ; z 0 Þ = E 01 sinð 1 z 0 =2Þ and E 0 h ðL 2 ; z 0 Þ = E 02 sinð 2 z 0 =2Þ with z 0 = z cos B . The intensity jE 0 h j 2 consisting of the two electric fields E 0 h ðL 1 ; z 0 Þ and E 0 h ðL 2 ; z 0 Þ is given by Here a represents the degree of coherency between the two beams E 0;1 and E 0;2 assuming that the magnitudes of the two electric fields E 0;1 and E 0;2 are equal to each other and are expressed by E 0 . If the two beams are perfectly coherent, a = 1, and if the two beams are incoherent, a = 0. When a = 0, equation (9) shows an ordinary moiré pattern. It is modified by the second term in (9) when a 6 ¼ 0. In Fig. 6(c) the fifth and 13th fringes become weak from left to right and disappear in the middle of the horizontal axis. This modulation is also a moiré pattern, but it should be a rotation moiré pattern caused by inserting the wire.

Coherency
We discuss the temporal (longitudinal) coherent length and spatial coherent length of the MBL interferometer. The wavevectors of the refracted beams in the BL and BBL modes are different and the paths of the two refracted beams are different, which means that the optical system works as a Michelson interferometer. It is possible to estimate the longitudinal coherent length Á l by measuring the region Á in which the interference fringes are observable as shown in Figs. 1(a) and 3(a). The coherent length Á l is related to Á by In the present experiment the resultant value of Á l is approximately 30 mm, which is in good agreement with the value determined by the energy resolution owing to the dispersion angle of the incident beam. The spatial coherent length can be estimated by dividing the incident beam into two parts to measure the observable range of the moiré pattern. If the observed moiré pattern is reproduced by using equation (7) or by putting a = 1 into (9), the two beams E 0;1 and E 0;2 are coherent with each other. If it is reproduced by using (8) or by putting a = 0 into (9), the two beams E 0;1 and E 0;2 are incoherent with each other. The observed moiré pattern in the present experiment agrees with the calculated pattern using (7) and shows that the beams of E 0;1 and E 0;2 are coherent with each other. Since the distance between two beams of E 0;1 and E 0;2 in the present experiment is 43 mm, the spatial coherent length is larger than 43 mm. According to the theoretical consideration the size of spatially coherent region A is given by A = D=S, where is the X-ray wavelength, S is the source dimension of synchrotron radiation, and D is the distance from the source point to the sample crystal. Under the present experimental conditions at beamline BL15C at KEK-PF, S = 55 mm, D = 31 m and = 0.11 nm, then the size of the spatial coherent region becomes 62 mm, which is in good agreement with the estimated value using the moiré pattern.

Bent crystal
We can see a strong dark contrast (1 0 ) of fringes at z 0 = 0 in Fig. 4. This dark contrast is not part of the MBL interference fringes, because the difference in the path lengths of the beams   in the BL and BBL modes is larger than the coherence length Á l in the present experiment. This dark contrast should be caused by the interference between two mirage diffraction beams [S 1 and S 2 in Fig. 1(b)] as the crystal is bent by gravity, and the paths of the refracted beams become hyperbolic forms (Gronkowski & Malgrange, 1984;Authier, 2001) and those refracted beams produce mirage fringes (Bak-Misiuk et al., 1987;Chukhovskii & Petrashen', 1988;Fukamachi et al., 2010) and/or IFMRBs (interference fringes between a mirage diffraction beam and a reflected beam from the bottom surface) (Fukamachi et al., 2011c).
A moiré pattern has been observed in the MBL interference fringes when the two beams are incident on the interferometer crystal. The spatial coherent length of the beam passing through slit 1 has been estimated. It is expected that a moiré pattern owing to lattice distortion can be observed in the MBL interference fringes and should be useful for analyzing the strain around the distortion. If X-rays from an undulator beamline are used instead of X-rays from a bending-magnet beamline, the number of MBL interference fringes will be increased and detailed analysis of a moiré pattern will be possible.