ARTICLE IN PRESS

Nuclear Instruments and Methods in Physics Research A 560 (2006) 598–605 www.elsevier.com/locate/nima

Interferometer for cold

Christian Prunera, Martin Fallya,Ã, Romano A. Ruppa, Roland P. Mayb,Ju¨rgen Vollbrandtc

aFaculty of Physics, Nonlinear Physics group, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria bInstitut Laue-Langevin, BP 156, F-38042 Grenoble Cedex 9, France cGKSS Forschungszentrum, D-21502 Geesthacht, Germany

Received 17 November 2005; received in revised form 12 December 2005; accepted 21 December 2005 Available online 23 January 2006

Abstract

Design and setup of an interferometer for cold and are described. As optical components three holographically produced gratings are arranged in triple Laue geometry. The gratings are adjusted during their recording process in photosensitized polymer slabs once for ever. Employing this device we measured the coherence function of a cold neutron beam by means of interferometry. r 2006 Elsevier B.V. All rights reserved.

PACS: 03.75.Dg; 61.12.q; 42.40.Eq

Keywords: Atom and neutron interferometry; Neutron and scattering; Holographic optical elements; Holographic gratings

1. Introduction [2]. Cold neutron interferometry in the energy range 0:5 meVtEt2 meV, however, was excluded from experi- Scattering and diffraction experiments proved to be mental activities till recently because of the absence of extremely useful for investigation of materials and have proper diffraction elements [3,4]. But just this energy range become standard techniques in condensed-matter science. is of particular interest in materials sciences: for investiga- Depending on the specific task and the desired informa- tions of large-scale structures, e.g., for structure analysis of tion, the interaction potential between the scattering biological objects (viruses, proteins, enzymes), in metal- radiation and the object determines the choice of the lurgy (alloys, magnetic and superconducting materials) or appropriate experimental technique. The most common for polymer research. The continuous development of cold quanta utilized for scattering and diffraction investigations neutron sources and the increasing number of applications are photons, electrons and neutrons. As the properties of at small-angle scattering facilities for cold neutrons (SANS) the scattered or diffracted radiation depend on the illustrate the demand for a device to determine the interaction of the radiation and the scatterer, it is of great coherence properties of cold neutron beams. importance for the correct interpretation of diffraction and Here we report on an interferometer for cold neutrons scattering experiments to determine the coherence proper- utilizing holographically produced volume-phase gratings, ties of the scattering radiation [1]. An appropriate quantity following preliminary studies and a prototype presented by for the characterization of a beam is its coherence volume Schellhorn et al. [4]. Our aim was to construct a more which can be determined by interferometry. During the last sensitive, compact and flexible instrument that allows 30 years numerous interferometric experiments with interferometric measurements at several facilities for cold thermal as well as with ultracold neutrons were performed neutrons, e.g., for determining the coherence character- istics required for the deconvolution of SANS data. That ÃCorresponding author. Tel.: +43 1 4277 51110; fax: +43 1 4277 9511. was not possible with the interferometer prototype. This E-mail address: [email protected] (M. Fally). might be regarded as a first step towards establishing URL: http://www.univie.ac.at/nlp. interferometry with cold neutrons as a standard tool at

0168-9002/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.12.240 ARTICLE IN PRESS C. Pruner et al. / Nuclear Instruments and Methods in Physics Research A 560 (2006) 598–605 599

SANS beamlines, particularly with respect to prospective where hi denotes the time average, the complex applications in material or solid state physics. conjugate, and GðDfÞ the cross-correlation function of Ctrr and Crrt,andCtrt and Crrr. As sketched in Fig. 1, eight beams leave the interferometer, but only the scattered 2. General description of the interferometer fraction of beam Ctr has the ability to interfere with the transmitted part of the beam Crr and vice versa. The other The interferometer consists of three diffraction gratings partial beams Cttt, Cttr, Crtr, and Crtt are unable to aligned equidistantly in transmission (Laue) geometry interfere because of the spatial separation of 0.8 mm ðl0 ¼ (LLL-interferometer) (see Fig. 1). This arrangement of 2nmÞ behind the third grating, but add up to a constant the optical components is topologically analogous to the contribution to the overall counting rate. Mach-Zehnder interferometer, which is well known from light optics. In neutron optics the beamsplitters are 3. Setup of the interferometer represented by the first and third grating G1 and G3, the mirror by the second grating G2. The underlying operation To open up an opportunity for interferometric measure- principle of an interferometer is the preparation and ments at various instruments for cold neutrons imposes superposition of two (partially) coherent neutron beams. special requirements on the scattering facility as well as on This task is accomplished by coherently dividing a the interferometer: sufficiently collimated incoming beam via Bragg reflection from grating G1, and recombining the partial beams by a The scattering facility must provide an adequately second grating G2. At the position of grating G3 these two collimated beam in the cold or ultracold wavelength neutron beams are superimposed under an angle 2YB. The range with appropriate energy resolution according to neutrons in a certain beam are described by wave functions the scattering vector of the neutron optical components Ctr;rr, i.e., wave packets that are a linear superposition of of the interferometer. An instrument which complies plane waves. Due to the crossing angle between the beams these requirements is for instance a SANS facility. Both, a neutron interference-pattern is generated (the subscripts t the collimation as well as the central wavelength l0 of and r indicate transmission or reflection of the beams). The the beam and its distribution Dl=l0 can be adapted to amplitudes of the interfering beams C0 ¼ Ctrr þ Crrt and the specific requirements over a wide range. The beam is CH ¼ Ctrt þ Crrr depend on the relative phase difference collimated by inserting apertures between the cold Df between Ctr and Crr. Varying this phase difference and the sample position, the choice of results in counter oscillating intensity variations of the the wavelength and its distribution is tunable by 0-beam and H-beam. Evaluating the intensities yields changing the rotation velocity and the rotation axis of the wavelength selector (see Fig. 2). I 0 ¼hðCtrr þ CrrtÞðCtrr þ CrrtÞ i The interferometer has to be compact and flexible to ¼ I þ I þ 2R½GðDfÞ trr rrt enable an easy implementation at the standard sample I H ¼hðCtrt þ CrrrÞðCtrt þ CrrrÞ i position of different scattering facilities. Moreover it ¼ I trt þ I rrr 2R½GðDfÞ should be rigid and stable to allow high precision measurements.

Ψ To successfully setup an interferometer for cold neutrons ttt G1 G2 G3 two crucial problems have to be solved: the production of Ψ Ψ Ψ 0= trr+ rrt the optical components and their mutual alignment. As phase flag R-beam diffraction optics we utilize holographically generated Ψ tr Ψ ttr volume-phase gratings from photosensitized slabs of deut- H 2Θ B Ψ erated (poly)methylmethacrylate (PMMA) [5]. The extre- Ψ rtr incident rr neutron beam α S-beam mely precise mutual alignment of the gratings is conducted Ψ Ψ Ψ H= trt+ rrr already during their holographic production, i.e., we perform successive ‘adjustment by illumination’ of the three Ψ L = 300 mm rtt slabs.

Fig. 1. Sketch of the interferometer for cold neutrons (top view). It is based on holographically generated density gratings G1, G2 and G3 in 3.1. Preparation of the polymer slabs slabs of deuterated PMMA. By rotating a phase flag by an angle a a path difference between the two coherent partial beams Ctr and Crr is induced. The preparation of the photosensitive d-PMMA samples After diffraction from grating G3 the reflected and transmitted amplitude starts with purification of liquid d-MMA monomer superimpose to the 0-beam with amplitude C0 ¼ Ctrr þ Crrt and H-beam ðC5O2D8Þ from a polymerization inhibitor (hydrochinon). with amplitude CH ¼ Ctrt þ Crrr, respectively, which are indicated by 0 bold lines. The entire of all transmitted and diffracted partial beams is After purification a thermal initiator (AIBN, a-a -azo- labeled as R-beam ðCR ¼ C0 þ Cttt þ Crtr) and S-beam isobutyronitril) and a photoinitiator (DMPA, o; o-di- ðCS ¼ CH þ Crtt þ CttrÞ, respectively. methoxy-o-phenylacetophenone) are added, each with a ARTICLE IN PRESS 600 C. Pruner et al. / Nuclear Instruments and Methods in Physics Research A 560 (2006) 598–605

plane of interference kS

Θ 2 B

kR

L/2

yaw

roll

pitch

Fig. 3. Holographic two-wave mixing setup for recording the first interferometer grating G1 (top). The wave vector of the signal and reference beam are labeled as kS and kR, respectively. Translation of the interferometer chassis along a distance L=2, correction of translation errors (roll, pitch, yaw) and recording of the second grating G2 (bottom).

Fig. 2. The interferometer for cold neutrons after recording and aligning (top) and in the experimental environment of SANS-2 at GeNF (bottom). plane waves of equal intensity from an Arþ laser, operating at a wavelength of lL ¼ 351 nm, are brought to inter- content of 0:5mg=ml. The latter shows an absorption ference under an angle 2YB at the sample position. The maximum near 350 nm. The degree of substitution of grating spacing L ¼ lL=ð2 sin YBÞ or the spatial frequency hydrogen 1H by deuterium 2H is 99.7%. To fulfill the high H ¼ 2p=L of the interference pattern is defined by the demands on the optical quality of the slabs, the monomer wavelength and the geometry of the setup, where YB solution is injected into cuvettes that were especially denotes the Bragg angle (in air). Illumination of the developed for this purpose. They consist of a stainless photosensitive PMMA samples with a spatially modulated steel frame and are bounded by two high quality quartz intensity pattern results in a local modification of the windows that are spaced with O-rings composed of an fraction of polymerization. The large difference of the elastomer core enclosed in a seamless sheath of tetrafluor- number densities r for the monomer MMA and the oethylene–hexafluoropropylene (FEP). Moderate stress is polymer PMMA ðrMMA=rPMMA 0:8Þ yields a spatial applied onto the glass plates to compensate volume modulation of the coherent scattering length density shrinkage during polymerization. The polymerization of rbðxÞ¼bcrðxÞ for neutrons. In analogy to light optics we the slabs occurs in a two-stage process: It starts with the call this a photoneutronrefractive effect [6]. Here bc denotes thermal prepolymerization of liquid d-MMA. This pre- the average bound coherent scattering length of d-MMA. polymerization of the photosensitive samples lasts for 48 h Illumination of the slabs with the sinusoidal interference at a temperature of 45 C. The fraction of polymerization pattern records a refractive-index grating for neutrons then reaches approximately 80%. The remaining 20% of rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rbðxÞ dissolved monomer are available for the light-induced nNðxÞ¼ 1 l (1) postpolymerization process. p where nNðxÞ denotes the neutronrefractive index and l the 3.2. Recording and adjustment of the gratings neutron wavelength. The spatial modulation of the coherent scattering length density rbðxÞ can be expanded The production process of gratings for neutrons is in a Fourier series performed by exposing photosensitized slabs to a sinusoi- X dal light interference pattern. Therefore, a conventional rbðxÞ¼rb þ bcrj cosðjHx þ jjÞ. (2) holographic two-wave mixing configuration, as shown in j Fig. 3 is employed. Using a beam expansion system, a Depending on the actual material properties higher beamsplitter and two mirrors, two coherent s-polarized harmonics may occur due to non-linearities of the ARTICLE IN PRESS C. Pruner et al. / Nuclear Instruments and Methods in Physics Research A 560 (2006) 598–605 601 photoneutronrefractive mechanism. The phase jj between autocollimator, the accuracy of this technique is in the the interference pattern and the modulation of the order of 0:01 arcsec. For an adequate determination of scattering length density accounts for a possible non- the roll error along the overall length of the interferometer locality of the photoneutronrefractive effect. we adapted a novel polarization optical measuring system Due to the limited coherence of a cold neutron beam the to our demands [9]. It is based on Se´narmont compensation development of a neutron interferometer requires an and consists of a He–Ne laser, a polarizer, a photoelastic extremely precise adjustment of the gratings. In thermal modulator, a quarter-wave plate, and an analyzer, one neutron interferometry generally the complete interferom- after another. Analyzing the intensity of the modulated eter with three or more individual gratings is manufactured linear polarization states having passed the system by out of a monolithic single [7]. Thus the alignment of means of lock-in-technique, allows a high precision the gratings depends only on the quality of the crystal and determination of angles in a plane perpendicular to the its machining. In interferometry with cold and ultracold direction of the propagation of the laser beam, i.e., the neutrons, beam manipulation is typically accomplished by direction of motion of the translation table. The combina- diffraction from artificial gratings. In contrast to single tion of these two techniques with the compensation system crystal interferometers, here, the grating vectors jH have to in principle enables the realization of ‘ideal’ linear be mutually aligned for each grating individually to obtain guidances along unlimited distances (Fig. 4). In our case interference. Estimations of the loss of interference contrast an accuracy of 0:2 arcsec for the pitch and yaw angle, and due to misalignment and inhomogeneity of the gratings are 1 arcsec for the roll angle was achieved. Characteristic given in Ref. [8]. properties of the interferometer are listed in Table 1. The methodology to fulfill the requirements for a sufficiently precise alignment of the gratings is as follows: 3.3. The hologram as diffractive optics for cold neutrons Three photoneutronrefractive d-PMMA slabs are pre- pared and mounted equidistantly on a chassis that is Utilizing holographically generated gratings in d-PMMA placed on a linear translation stage. For the sake of as beam splitting components opens up several novel stability of the setup during recording and operation, the interferometer is designed compactly and rigidly to ensure bending resistance. Moreover, cuvettes and NIF PT PD2 3 AN2 chassis are fabricated from an alloy with nearly M vanishing thermal expansion at operating conditions. ACS M PD The first slab is exposed to the sinusoidal interference M 1 pattern of the holographic two-wave mixing setup, thus AN1 generating a phase grating for cold neutrons (Fig. 3, PT1 TS PT2 top). L SC Then the second slab is translated to the plane of interference for exposure. Prior to illumination the BS M POL-PEM-λ/4 motion of the translation stage is corrected for devia- tions from the ideal translation such as pitch, yaw, and Fig. 4. Light optical setup to control deviations from ideal translation. An roll errors by piezo driven elements (Fig. 3, bottom). autocollimation system (ACS) is used to control yaw and pitch errors. A This process is repeated for the third slab. polarization optical setup is installed to measure the roll angle. The components are labeled as follows: NIF, neutron interferometer; TS, translation stage; SC, Se´narmont compensator consisting of a polarizer, The correction of translation errors occurring as a photoelastic modulator, and quarter-wave plate; BS, beam splitter; M, consequence of the inevitable guide clearance of the mirrors; AN, analyzers; PT, piezo translators, and PD, photodiodes. translation stage is the decisive step during the production of the interferometer. Alignment parameters such as tilt angles, equidistance, etc. can be sufficiently minimized by Table 1 choosing appropriate tolerances in fabrication of the Characteristic values of the interferometer at a central wavelength of l0 ¼ 2 nm and a distribution of Dl=l ¼ 10% interferometer chassis. Inhomogeneities of the gratings 0 can be suppressed by the use of a stable holographic Property Symbol Value assembly. For other degrees of freedom concerning the Overall length L 300 mm mutual alignment of the gratings, such as pitch, yaw, and Grating spacing L 380:3 0:1nm roll errors, it is indispensable to additionally detect and Weight 15 kg compensate deviations from the ideal translation. For this Dimension l b h 40 20 15 cm3 2 2 purpose a combination of an optical measuring and a piezo Enclosed beam area A0 L l=4L 1:2cm driven compensation system was applied. The measure- Beam separation at the third grating x0 Ll=2L 0:8mm ¼ ment of the pitch and yaw angles was carried out with the Diffraction efficiencies at l0 1:5nm ZG1; ZG2; ZG3 58%, 48%, 5% Transmission at l ¼ 1 nm T 41% help of an autocollimator. Depending on the quality of the 0 ARTICLE IN PRESS 602 C. Pruner et al. / Nuclear Instruments and Methods in Physics Research A 560 (2006) 598–605 perspectives in neutron interferometry. To characterize such 60 ± 1st order (nearly lossless) volume-phase holograms, which act as thick 50 neutronrefractive-index gratings, diffraction of light and/or G1 40 G2 neutrons is applied that is embraced by the basic formulae of G3 the dynamical diffraction theory [10,11].Thejth order 30 diffraction efficiency for a monochromatic plane wave in 20 symmetric transmission geometry is given by [%] 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η 10 2 2 2 2 5 I nj sin nj þ xj Z ¼ Zðn ; x Þ¼ 0 ¼ .(3)4 j j j 2 2 3 I 0 þ I jH n þ x j j 2 1 0 The index j indicates the actual diffraction order, I 0;jH are the forward diffracted and jth order diffracted intensities, -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 Θ [deg] respectively. The parameters

pdn jpd 1.0 ± 2nd order n ¼ j and x ¼ ðYðjÞ YÞ (4) j ðjÞ j L B G1 l cos YB 0.8 G2 contain the relevant material information: the thickness d of the grating, which in the ideal case is identical with its 0.6 [%] geometrical thickness and the amplitude of the neutronre- 2 η fractive index modulation nj. A thick grating may be 0.4 described as a grating that shows Bragg regime diffraction (two-wave regime). In this case only the zeroth and the jth 0.2 diffraction order appear at once. Our gratings by far fulfill the criteria for the Bragg regime [12].AsshowninFig. 5 the 0.0 continuous improvement of the preparation technique now -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 permits the production of gratings with diffraction efficien- Θ [deg] ciesp offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ1480% at the Bragg angle. Here the oscillating 2 2 2 Fig. 6. Diffraction efficiencies of the three interferometer gratings G1, G2 sin n þ x term predicted by Eq. (3) is not resolved and G3 for the 1st (top) and 2nd (bottom) diffraction order, recorded because of the divergence and wavelength distribution of the at a neutron wavelength of l0 ¼ 1:5nm ðDl=l0 ¼ 10%Þ. The diffraction neutron beam. Choosing appropriate recording and sample efficiency of the second order was successfully suppressed ðZ2t1%Þ. preparation conditions allows us to adapt the diffraction efficiency for each interferometer grating separately. This opens up the possibility to attain nearly perfect conditions for a neutron interferometer with Z ’ 50% for the first and 80 λ [nm] third and Z ’ 100% for the second grating. In this ideal case 70 2.6 the annoying separation of parasitic beams is obsolete. 2.3 Moreover, the grating spacing L is not fixed a priori but can 60 2.1 be adapted to the specific task within a wide range of 50 1.9 250 nmtLt10 mm by simply modifying the geometry of the two-wave mixing setup. Due to the non-linearity of the

[%] 1.7

1 40 η 1.5 polymerization process in d-PMMA higher harmonics with 30 1.1 spatial frequencies jH might occur ðj 2 NÞ [13].Diffraction experiments performed from higher diffraction orders j41 20 1.0 with grating spacings down to L=jt50 nm indicate that the 10 amplitude of each diffraction order is tuneable individually 0 by proper choice of exposure. Consequently higher harmo- -0.4 -0.3 -0.2 -0.1 0.0 nics can be suppressed or intensified. Thus it follows that Θ [deg] purely sinusoidal gratings are realizable by suppressing higher harmonics. Fig. 6 shows a measurement of the first Fig. 5. Angular dependence of the diffraction efficiency Z1 for the first and second order diffraction efficiency ZðYÞ of the three diffraction order (grating G2) at various neutron wavelengths l0 interferometer gratings, for a neutron wavelength of (wavelength distribution Dl=l0 ¼ 10%, beam divergence 0:08 ). At the exact Bragg angle efficiencies of up to 82% are reached, i.e., the grating l0 ¼ 1:5 nm. The gratings were optimized for maximum acts nearly as a mirror. diffraction efficiencies of the first order. ARTICLE IN PRESS C. Pruner et al. / Nuclear Instruments and Methods in Physics Research A 560 (2006) 598–605 603

4. Measurement of the coherence function 80 xE

The interferometer was put into operation one month 75 70 after its adjustment and setup (in Vienna, Austria) at the R-beam beamlines of the small angle facility D22 65 of the Institute Laue-Langevin (ILL), Grenoble, France and two month later at the small angle neutron scattering 60 instrument SANS-2 at the neutron facility Geesthacht 55 (GeNF), Germany. The measurements were performed at Counts in 1 sec 50 S-beam central wavelengths between 1 nmpl0p2:6 nm and a 45 nominal wavelength distribution of Dl=l0 ¼ 10%. The neutron beam was well collimated to achieve a beam 40 xR,S xc divergence of about 0:01 . The interferometer was placed at 35 the standard sample location on both instruments. Largest -30 -20 -100 10 20 30 possible collimation and detection distances were used. It ∆x [µm] was operated under ambient conditions without any Fig. 7. Interference pattern recorded at a mean neutron wavelength of precautions against thermal fluctuations or vibrations. l0 ¼ 2 nm by rotation of a phase flag (sapphire, rotation angle of 45 , First of all the interferometer was adjusted to exactly increment 1, axis perpendicular to the plane of the incident neutron match the Bragg condition for the three gratings. This task beams). Squares and triangles show the measured intensity of the R-beam was achieved by measuring the overall diffraction efficiency and S-beam, respectively, as a function of the geometrical path difference as a function of the angle between the incident neutron Dx. The solid lines represent a fit of the real part of the coherence function beam and the grating vector H using a rotation table with according to Eq. (7), the envelope function (dashed) their absolute magnitude. an accuracy of 0:001. The axis of rotation is perpendi- cular to the plane spanned by the transmitted and reflected partial beams. For recording an interference pattern, a sapphire phase flag was inserted in both beam paths of the R- and S-beam, read () between the first and second grating G1 and G2 (see 2 Fig. 1). Rotating this phase flag by an angle a generates a 2ðDx xEÞ I R;SðDxÞ¼JR;S AR;S exp phase difference between the neutron beams, represented xc by the wave functions Ctr and Crr. This phase difference cos½rbl0ðDx xR;SÞ. ð7Þ ¼ x Df rbl0D (5) The Gaussian decay of the coherence function (dashed depends on the coherent scattering length density rb of the envelope function) accounts for the limited coherence of sapphire slab, its thickness D, the central neutron the beam and is suggested by regarding the typical neutron wavelength l0, and on the geometry of the setup, with Dx wavelength distribution of a SANS instrument. The as the geometrical path difference cosinusoidal intensity modulation reflects the coherent superposition of the approximately plane wave functions 1 1 Dx ¼ D . (6) Ctrr þ Crrt and Ctrt þ Crrr after the third grating as a cosðYB aÞ cosðYB þ aÞ function of the relative phase. The rotation axis of the phase flag is also perpendicular to Fitting Eq. (7) to the experimental data the following the plane incidence, a being the angle between the surface parameters can be extracted (compare Table 2): normal of the phase flag and the symmetry axis of the partial beams. The maximum visibility of the interference fringes The overall intensity of the beams leaving the inter- vR;S ¼ AR;S=JR;S. AR;S limits the amplitude of the ferometer along the direction of the incident beam interference fringes, JR;S is the constant offset intensity, (R-beam) as well as the diffracted beam (S-beam) is which depends only on the incident flux of the neutron monitored with the help of a two dimensional area sensitive beam and the diffraction efficiencies of the three detector (as a function of the phase flag angle). Fig. 7 gratings G1, G2, and G3. shows an interference pattern of counter oscillating The decrease of the visibility with increasing phase shift intensities for the R-beam and S-beam. The interferogram is directly related to the limited coherence of the neutron was recorded at the instrument D22 at a mean neutron beam. The experimental determination of the coherence wavelength of l0 ¼ 2 nm, the size of the detector was 128 function, including the longitudinal and transverse 128 pixels each with 7:5mm 7:5 mm. coherence properties allows an estimation of the 2 For fitting the interference pattern we accounted for a coherence length lc ¼ xcl0rb=ð2pÞ of the actual neutron cosinusoidal intensity modulation and assumed a Gaussian beam. The coherence length specifies the length, for decay of the coherence function. Thus the intensities of all which the normalized absolute value of the coherence partial beams, leaving the interferometer in the directions function decreases to 1=e. ARTICLE IN PRESS 604 C. Pruner et al. / Nuclear Instruments and Methods in Physics Research A 560 (2006) 598–605

Table 2 Parameters obtained from evaluating data shown in Fig. 7 ðl0 ¼ 2nmÞ

Property Symbol Value

Maximum visibility vR;S ¼ AR;S=JR;S 18.3%, 21.0% 14 2 Coherent scattering length density Al2O3 rb ¼ rbc ð5:49 0:01Þ10 m Coherence length 2 10:85 0:24 nm lc ¼ xcl0rb=ð2pÞ ‘Internal’ phase fE ¼ xErbl0 ð1:047 0:077Þp Phase of the third grating fR;S ¼ðxE xR;SÞrbl0 ð0:593 0:081Þp ðmod 2pÞ Resolution DE 1012 eV

Evaluating the periodicity of the interference fringes the direction of propagation of the neutrons [15]. For yields the interaction strength responsible for the phase typical collimations and wavelength distributions applied shift. In our example the coherent scattering length here, the transverse coherence length is more than two density rb ¼ bcr of sapphire ðAl2O3Þ was determined. orders larger than the longitudinal coherence length. Errors in the adjustment of the gratings give rise to Taking into account the accuracy of the present measure- geometrically caused time of flight differences along the ments, contributions of transverse phase shifts can be several paths of the interferometer. Hence an ‘internal’ neglected in our measurements. phase fE ¼ xErbl0 of the interferometer results which The described experiments were performed under ambient can be obtained directly by comparing the maximum of conditions, i.e., neither thermal stabilization nor isolation the absolute value of the coherence function with the against vibrations were applied. Implementation of the angular position of the phase shifter. The determination latter, optimizing the diffraction efficiencies for each grating, of this ‘internal’ phase presupposes a limited coherence, and increasing the counting rate allows a further enhance- i.e., a decrease of the coherence function of the neutron ment concerning the energy resolution of the interferometer. beam. The accuracy of its determination rises with We reported on the setup and adjustment of a Mach- decreasing coherence length of the beam. Zehnder type interferometer for cold neutrons based on The geometrical path difference between the zero position holographically generated gratings. The interferometer is of the phase flag ða ¼ 0Þ and the maximum (minimum) composed of three equally spaced volume-phase gratings in values of the real part of the coherence function is triple Laue geometry, acting as beam splitters or mirrors determined by the parameter xR;S. The additional for neutrons. The gratings were produced by illuminating information on this internal path difference of the photoneutronrefractive slabs from d-PMMA with intensity interferometer xE allows to evaluate the phase fR;S ¼ fringes of a holographic recording setup. In such materials ðxE xR;SÞrbl0 of the third interferometer grating. the light pattern is transformed into a refractive index The phase uncertainty of the interferometer depends on the profile, i.e., a grating for neutrons. The alignment of the counting statistics as well as on the stability of the gratings occurs during their recording once for ever. To interferometer, for instance with respect to thermal ensure a successful operation of the interferometer for cold influences or vibrations. The interferogram shown in neutrons, the mutual orientation of the grating vectors H Fig. 7 was recorded at a neutron counting rate of 135 s1. was adjusted with an accuracy better than 1 arcsec. To The phase flag was rotated around an angle of 45 with achieve this accuracy a novel method for measuring and an increment of 1. At each position the neutrons were compensating yaw, pitch and roll errors was implemented. counted for 120 s. For this measurement the calculated The interferometer then was tested at various cold neutron statistical error of the phase uncertainty is 1% [14]. beamlines. With the help of this device the coherence Evaluating several interferograms recorded under compar- properties of neutrons were studied. This task was done by able conditions during a longer period permits an estima- continuously increasing the phase difference between two tion of the phase stability and energy resolution of the beam paths of the interferometer through rotation of a phase interferometer. When recording interferograms during 4 h flag. By monitoring the decay of the interference fringes we at a neutron wavelength of 2 nm we observed a maximum performed the first direct measurement of the complete phase drift of 0:088p 0:011p which corresponds to an coherence function for cold neutrons. Evaluation of the energy of DE ¼ 3 1010 eV. The energy resolution of the interference pattern allows the determination of the coherence interferometer itself is about DE 1012 eV. length of the neutron beam, the coherent scattering length density of the phase flag and the absolute phase of the third interferometer grating. The latter was accomplished by 5. Discussion and summary exploiting the limited coherence properties of the neutron beam, i.e., the decrease of the coherence function. The A rotation of the phase flag induces not only a experimental determination of the intensity and the phase of longitudinal but also a transverse phase shift in a direction the scattering radiation permits a direct Fourier transform perpendicular to the rotation axis of the phase flag and to from reciprocal to real space and thus the solution of the ARTICLE IN PRESS C. Pruner et al. / Nuclear Instruments and Methods in Physics Research A 560 (2006) 598–605 605

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